Quillen K-Theory: Lecture Notes

Satya Mandal, U. of Kansas

2 April 2017 27 August 2017 6, 10, 13, 20, 25 September 2017 2, 4, 9, 11, 16, 18, 23, 25, 30 October 2017 1, 6, 8, 13, 15, 20, 27, 29 November 2017 December 4, 6 2

Dedicated to my mother! Contents

1 Background from Category Theory 7 1.1 Preliminary Deﬁnitions ...... 7 1.1.1 Classical and Standard Examples ...... 9 1.1.2 Pullback, Pushforward, kernel and cokernel ...... 12 1.1.3 Adjoint Functors and Equivalence of Categories ...... 16 1.2 Additive and Abelian Categories ...... 17 1.2.1 Abelian Categories ...... 18 1.3 Exact Category ...... 20 1.4 Localization and Quotient Categories ...... 23 1.4.1 Quotient of Exact Categories ...... 27 1.5 Frequently Used Lemmas ...... 28 1.5.1 Pullback and Pushforward Lemmas ...... 29 1.5.2 The Snake Lemma ...... 30 1.5.3 Limits and coLimits ...... 36

2 Homotopy Theory 37 2.1 Elements of Topological Spaces ...... 37 2.1.1 Frequently used Topologies ...... 41 2.2 Homotopy ...... 42 2.3 Fibrations ...... 47 2.4 Construction of Fibrations ...... 49 2.5 CW Complexes ...... 53 2.5.1 Product of CW Complexes ...... 57 2.5.2 Frequently Used Results ...... 59

3 4 CONTENTS

3 Simplicial and coSimplicial Sets 61 3.1 Introduction ...... 61 3.2 Simplicial Sets ...... 61 3.3 Geometric Realization ...... 65 3.3.1 The CW Structure on |K| ...... 68 3.4 The Homotopy Groups of Simplicial Sets ...... 73 3.4.1 The Combinatorial Deﬁnition ...... 73 3.4.2 The Deﬁnitions of Homotopy Groups ...... 77 3.5 Frequently Used Lemmms ...... 83 3.5.1 A Lemma on Bisimplicial Sets ...... 84

4 The Quillen-K-Theory 89 4.1 The Classifying spaces of a category ...... 89 4.1.1 Properties of the Classifying Space ...... 91 4.1.2 Directed and Filtering Limit ...... 92 4.1.3 Theorem A: Suﬃcient condition for a functor to be homotopy equivalence ...... 96 4.1.4 Theorem B: The Exact Homotopy Sequence ...... 101 4.2 The K-Groups of Exact Categories ...... 110 4.2.1 Quillen’s Q-Construction ...... 110 4.2.2 The Higher K-groups of an Exact Category ...... 115 4.2.3 Higher K-Groups ...... 120 4.3 Characteristic Exact Sequences and Filtrations ...... 123 4.3.1 Additivity Theorem ...... 125 4.3.2 The Prime Examples ...... 127 4.4 Reduction by Resolution ...... 130 4.4.1 Extension closed and Resolving Subcategories ...... 132 4.5 Dévissage and Localization in Abelian Categories ...... 143 4.5.1 Semisimple Objects ...... 145 4.5.2 Quillen’s Localization Theorem ...... 146 4.6 K-Theory Spaces and Reformulations ...... 149 4.7 Negative K-Theory ...... 149 4.7.1 Spectra and Negative Homotopy Groups ...... 150 CONTENTS 5

4.8 Suspension of an Exact Category ...... 151 4.8.1 Coﬁnality and Idempotent Completion ...... 154 4.8.2 The K-theory Spectra ...... 156

A Not Used 157 A.0.3 Some Jargon from [H] (SKIP)...... 157 A.0.4 On SSet Realization ...... 160 A.0.5 Inclusion of Geometric Realizations ...... 160 6 CONTENTS Chapter 1

Background from Category Theory

Among the sources I used in this chapter are [HS, Sv, Wc1] and most importantly [HS]. Importance of Category Theory, as a learning tool, is underrated. To me, Category Theory is a way to learn how analogy works in mathematics, from one area to others. This is a good way to understand diﬀerent areas of mathematics, just by learning one single area. First, I recall the basic deﬁnitions from [HS].

1.1 Preliminary Deﬁnitions

Deﬁnition 1.1.1. A Category C consists of the following:

1. A class of objects, denoted by Obj(C ).

2. For any two objected A, B in Obj(C ), a set of morphisms, from A to B, denoted by

Mor(A, B), more precisely by MorC (A, B).(Depending on the context, several other notations are used, for Mor(A, B). One of them is Hom(A, B). Morphisms are also called "arrows", "maps" and, other things appropriate to the speciﬁc context.)

3. To each triple A, B, C of objects, a law of composition

Mor(A, B) × Mor(B,C) −→ Mor(A, C) sending (f, g) 7→ gf.

The above data satisfy the following Axioms:

A 1 Mor(A, B) are mutually disjoint.

A 2 For composeable morphisms f, g, h, the associativity holds: h(gf) = (hg)f.

7 8 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

A 3 For each object A, teethe is a morphism 1A : A −→ A such that, for composable f, g:

1Af = f and g1A = g.

Given a category C , the following are some obvious deﬁnitions:

1. A morphism f : X −→ Y in C , is said to be an isomorphism, if there is a morphism g : Y −→ X such that fg = 1Y and gf = 1X . In this case, g would be called the inverse of f.

2. A morphism f : X −→ Y in C is called a monomorphism or injective, if for g1f = g2f =⇒ g1 = g2, for all pairs of morphisms g1, g2 : A −→ X. A morphism f : X −→ Y in C is called a epimorphism or surjective, if for fg1 = fg2 =⇒ g1 = g2, for all pairs of morphisms g1, g2 : Y −→ B.

3. An object L in C would be called an initial object, if Mor(L, X) is a singleton, for all X in Obj(C . An object R in C would be called a ﬁnal object (orterminal object), if Mor(X,F ) is a singleton, for all X in Obj(C . An object 0 in C would be called zero object, if it is both initial and ﬁnal object.

4. Injective objects and projective objects (will skip for now).

5. For a category C , deﬁne the opposite category C opp. Where Obj(E ) = Obj(C opp), and for two objects in C , MorC opp (A, B) = MorC (A, B).

Deﬁnition 1.1.2. A covariant functor F : C −→ D from a category C to another category D is deﬁned by the following data:

1. To each object A ∈ C , it associates an object FA ∈ D.

2. For objects A, B, there is a set theoretic map

F : MorE (A, B) −→ MorD (F A, F B)

3. Further, F (fg) = F (f)F (g) for any two composeable morphisms f, g.

A contravariant functor F : C −→ D is a covariant functor C opp −→ D.

Deﬁnition 1.1.3. Given two covariant functors F,G : C −→ D, a natural transformation

τ : F −→ G, associates, to each object X in C , a morphism τX : F (X) −→ G(X), such that for any morphism f ∈ Mor(X,Y ) the diagram

τX F (X) / G(X)

F (f) G(f) commutes. F (Y ) / G(Y ) τY 1.1. PRELIMINARY DEFINITIONS 9

If τX is an isomorphism, for all objects X, the τ would be called a natural equivalence. Likewise, deﬁne natural transformation and natural equivalence between two contravariant functors.

Example 1.1.4. Let C be a category and A ∈ Obj(C ). Then

X 7→ MorC (A, X) is a covariant functor C −→ Sets. and

X 7→ MorC (X,A) is a contravariant functor C −→ Sets. They are called representable functors. These are, perhaps, the most signiﬁcant examples of functors.

Some special functors:

Deﬁnition 1.1.5. Let F : C −→ D be a functor.

1. F is said to be full, if

∀ X,Y ∈ Obj(C ) the map F : MorC (X,Y ) −→ MorD (FX,FY ) is surjective.

2. F is called faithful, if

∀ X,Y ∈ Obj(C ) the map F : MorC (X,Y ) −→ MorD (FX,FY ) is injective.

3. F is called fully faithful, if

∀ X,Y ∈ Obj(C ) the map F : MorC (X,Y ) −→ MorD (FX,FY ) is bijective.

4. In particular, if F : Obj(C ) −→ Obj(D) is an "inclusion", then C is called a subcategory. So, the concept a full subcategory is to be distinguished from a non-full subcategory.

1.1.1 Classical and Standard Examples

We recall a some of the classical and Standard examples.

Examples 1.1.6. We recall a few examples and establish notations.

1. The category of sets will be denoted by Set. The objects of Set consists of the class of all sets. For two sets X,Y , a morphism f : X −→ Y is a set theoretic map. 10 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

2. The category of Topological spaces will be denoted by Top. The objects of Top consists of the class of all topological spaces. For two topological spaces X,Y , Mor(X,Y ) is the set of all continuous maps f : X −→ Y . Note the empty set φ is the initial object and singleton sets {x} are the terminal objects in Top. In Topology, there are two more important categories:

(a) The category of Pointed Topological spaces will be denoted by Top•. The

objects of Top• consists of the class of all pairs (X, x), where X is a topological space and x ∈ X is a point, to be called the base point. For two objects (X, x), (Y, y), Mor((X, x), (Y, y)) is the set of all continuous maps f : X −→ Y ,

with f(x) = y. Note, Top• has a zero object, namely ({x}, x) , where {x} is any singleton set. (b) The category of Topological Pairs will be denoted by Top − Pairs. The objects of Top − Pairs consists of the class of all pairs (X,U), where X is a topological space and U ⊆ X is a topological subspace. For two objects (X,U), (Y,V ), Mor((X, x), (Y, y)) is the set of all continuous maps f : X −→ Y , with f(U) ⊆ V .

We have commutative diagram of natural functors:

i Top / Top _ • 8 ι j Top − Pairs

where the vertical functor ι is a full subcategory, while i, j may be called the "forgetful functors". Note, i, j would not be full, while they are faithful.

3. The category of Groups will be denoted by Gr. The objects of Gr consists of the class of all groups. For two groups X,Y , Mor(X,Y ) is the set of all group homomorphisms f : X −→ Y .

4. The category of Abelian Groups will be denoted by Ab. The objects of Ab consists of the class of all abelian groups. For two abelian groups X,Y , Mor(X,Y ) is the set of all group homomorphisms f : X −→ Y .

5. The category of Rings, with unity and 0 6= 1 will be denoted by Ring. The objects of Ring consists of the class of all Rings, with unity and 0 6= 1. For two Rings X,Y , Mor(X,Y ) is the set of all ring homomorphisms f : X −→ Y , sending 1 7→ 1. 1.1. PRELIMINARY DEFINITIONS 11

6. The category of commutative Rings, with unity and 0 6= 1 will be denoted by cRing. The objects of cRing consists of the class of all commutative Rings, with unity and 0 6= 1. For two commutative Rings X,Y , Mor(X,Y ) is the set of all ring homomorphisms f : X −→ Y , sending 1 7→ 1.

7. The category of Schemes will be denoted by Sch. The objects of Sch consists of the

class of all Schemes. For two Schemes (X, OX ), (Y, OY ), Mor ((X, OX ), (Y, OY )) is

the set of all morphisms f :(X, OX ) −→ (Y, OY ) of schemes (see [Hr, Sec II.2]).

8. Lat A be a commutative ring.

(a) Then, QCoh(A) or Mod(A) will denote the category of all A-modules. Ob- jects of Mod(A) is the class of all A-modules. Given two A-modules M,N, a morphism f : M −→ N is an A-linear homomorphism. In particular, if A = k is a field, the Mod(k) is the category of all vectors spaces. (b) Coh(A) or fMod(A) will denote the category of all finitely generated A-modules. Objects of fMod(A) is the class of all finitely generated A-modules. Given two finitely generated A-modules M,N, a morphism f : M −→ N is an A-linear homomorphism. In particular, if A = k is a field, the fMod(k) is the category of all vectors spaces, of finite dimension. (c) P(A) will denote the category of all finitely generated projective A-modules. Objects of P(A) is the class of all finitely generated Projective A-modules. Given two finitely generated A-modules P,Q, a morphism f : P −→ Q is an A-linear homomorphism.

9. Lat (X, OX ) be a Scheme (see [Hr, Sec II.5]) .

(a) Then, QCoh(X) will denote the category of all quasi-coherent OX -modules.

Objects of QCoh(X) is the class of all Quasi-Coherent OX -modules. Given two

Quasi-Coherent OX -modules F, G, a morphism f : F −→ G is a morphism of

OX -modules.

(b) Coh(X) will denote the category of all coherent OX -modules. Objects of Coh(X)

is the class of all Coherent OX -modules. Given two Coherent OX -modules F, G,

a morphism f : F −→ G is a morphism of OX -modules.

V ect(X) is the class of all locally free OX -modules. Given two locally free

OX -modules F, G, a morphism f : F −→ G is a morphism of OX -modules. 12 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

Above examples of categories, lead to a number of inclusion functors. Reader would be well advised to think about them. However, we recall the following.

Example 1.1.7. Let A be a commutative ring.

∗ 1. Given a ﬁnitely generated projective A-module P , let P = HomA(P,A). Then,

P 7→ P ∗ deﬁnes a contravariant functor P(A) −→ P(A)

Given a morphism ϕ : P −→ Q, and f ∈ Q∗, ϕ∗(f) is deﬁned by the commutative diagram f Q / A O ? ϕ ϕ∗(f) P

2. Composition of functors are functors. So, P −→ P ∗∗ is a covariant functor P(A) −→ P(A).

∗ 3. For P ∈ Obj(P(A)) and for p ∈ P , deﬁne the evaluation map evp : P −→ A, by ∗∗ evp(f) = f(p). Then, evp ∈ P . So, we have a homomorphism

ev : P −→ P ∗∗. In fact, it is and isomorphism.

∗∗ Let IDP(A) denote the identity functor. Also, let D(P ) = P denote the double

dual functor. Then, ev : IDP(A) −→ D is a natural equivalence.

1.1.2 Pullback, Pushforward, kernel and cokernel

We deﬁne pullback and push forward.

g f Deﬁnition 1.1.8. Let B / XAo be two morphisms in a category C . Consider the diagram: Z ϕ0 ζ

ϕ # Y / A γ0 γ f B g / X We say that (Y, ϕ, γ), in short Y , is pullback of f, g , if

1. the rectangle commutes, i. e fϕ = gγ, and 1.1. PRELIMINARY DEFINITIONS 13

2. Given morphisms ϕ0, γ0 such that fϕ0 = gγ0, there is a unique morphism ζ such the the outer triangles commute.

We would write P ullback(f, g) := (Y, ϕ, γ), to mean that (Y, ϕ, γ) is the pullback of (f, g).

Likewise, the pushout of morphisms f : X −→ A, g : X −→ B in the category C is given by the commutative diagram:

f X / A

g ϕ ϕ0 B γ / Y ζ

γ0 + Z

That means, we say that (Y, ϕ, γ), in short Y , is the pushout of (f, g), if

1. the rectangle commutes, i. e ϕf = γg, and

2. Given morphisms ϕ0, γ0 such that ϕ0f = γ0g, there is a unique morphism ζ such the the outer triangles commute.

We would write P ushF wd(f, g) := (Y, ϕ, γ) or P ushOut(f, g) := (Y, ϕ, γ), to mean that (Y, ϕ, γ) is the pushout of (f, g). Remarks:

1. While the deﬁnition of pullback and pushout makes sense, the existence is not guaranteed. But, if it exists, it is uniques up to isomorphism.

2. It is easy to see that pullback and pushout exists, in the following categories: (1) Gr, (2) Ab, (3) QCoh(A), where A is a commutative ring.

Very similar and retaliated is the concept of direct product and direct coproduct (sum).

Deﬁnition 1.1.9. Let C be a category and A, B ∈ Obj(C ). Consider the diagram:

Z ϕ0 ζ

ϕ # Y / A γ0 γ B 14 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

We say that (Y, ϕ, γ), in short Y , is the direct product of A, B , if Given morphisms ϕ0, γ0 such that fϕ0 = gγ0, there is a unique morphism ζ such the the outer triangles commute.

1. Again, while deﬁnition makes sense, existence of direct product is not guaranteed. But, it it exists, it is unique up to isomorphism. In that case, the direct product is denoted by A × B.

2. If C has a terminal object 0T , then the direct product is, in deed, the pullback of the diagram A

B / 0T

Likewise, the coproduct of morphisms A, B in the category C is given by the commutative diagram: A

ϕ ϕ0 B γ / Y ζ

γ0 + Z That means, we say that (Y, ϕ, γ), in short Y , is the direct coproduct of (A, B), if Given morphisms ϕ0, γ0 such that ϕ0f = γ0g, there is a unique morphism ζ such the the outer triangles commute.

1. Again, while deﬁnition makes sense, existence of direct co-product is not guaranteed. But, it it exists, it is unique up to isomorphism. In that case, the direct co-product is denoted by A ` B.

2. Depending on the context, direct co-product is also called the "direct sum". In that case, it would be denoted by A ⊕ B.

3. If C has an initial object 0I , then the direct product is, in deed, the pushout of the diagram 0I / A

We deﬁne kernels. 1.1. PRELIMINARY DEFINITIONS 15

Deﬁnition 1.1.10. Suppose C is a category with a zero object 0. Given X,Y ∈ Obj(C ), there is a unique "zero element" 0XY ∈ Mor(X,Y ) given by the commutative diagram:

X / 0

3 ∀ f ∈ Mor(W, X) 0XY f = 0WY , and ∀ g ∈ Mor(Y,Z) g0XY = 0XZ . 0 XY Y

Suppose σ : X −→ Y is a morphism.

1. The Kernel of σ : X −→ Y is deﬁned to be a morphism ι : K −→ X such that

(a) σι = 0 and

(b) Further,

∀ τ ∈ Mor(L, X) with στ = 0 ∃ a unique τ0 ∈ Mor(L, K) 3 τ = ιτ0.

0 + L τ / X σ / Y

τ0 ∃! ι σ K / X 3/ Y 0

2. The coKernel of σ is deﬁned to be a morphism ζ : Y −→ C such that

(a) ζσ = 0 and

(b) Further,

∀ ψ ∈ Mor(Y,Z) with ψσ = 0 ∃ a unique η ∈ Mor(C,Z) 3 ηζ = ψ

0 X / Y +/ C σ ζ ∃! η σ ψ X / Y /3 Z 0 16 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

1.1.3 Adjoint Functors and Equivalence of Categories

We deﬁne Adjoint of a functors next.

Deﬁnition 1.1.11. Let F : C −→ D and G : D −→ C be two covariant functors. We say that F is a Left Adjoint of G and/or G is a Right Adjoint of F , if ∀ X ∈ Obj(C ),A ∈ Obj(D), there is a bijection

∼ η := ηXA : MorD (FX,A) −→ MorC (X, GA), which is natural, in the sense that, for any morphisms f : X −→ Y ∈ C and g : A −→ B ∈ D, the following two diagrams commute:

ηXA ηXA MorD (FX,A) ∼ / MorC (X, GA) MorD (FX,A) ∼ / MorC (X, GA) O O ∗ ∗ g∗ G(g)∗ F (f) f ∼ ∼ Mor (FX,B) / Mor (X, GB) Mor (FY,A) / Mor (Y, GA) D ηXB C D ηYA C In other words, the following two functors: ( (X,A) 7→ Mor (FX,A) C opp × D −→ Sets D are naturally equivalent. (X,A) 7→ MorC (X, GA)

Likewise, we deﬁne Left/Right Adjoint of contravariant functors.

Deﬁnition 1.1.12. A functor F : C −→ D would be called an equivalence of categories, if there is a functor G : D −→ C such that FG is naturally equivalent to 1D and GF is naturally equivalent to 1C (as in 1.1.3). If follows, F is an equivalence of categories if and only if the following two conditions hold:

∼ 1. (Essentially Surjective): For any object Z ∈ Obj(D) there is an isomorphism FX −→ Z , for some object X ∈ Obj(C ).

2. (Fully Faithful): For all objects X,Y ∈ Obj(C ), the map

∼ MorC (X,Y ) −→ MorD (FX,FY ) is a bijection. We will recall the following example. Example 1.1.13. Suppose A is a commutative ring. Then, there is an equivalence of categories QCoh(A) −→∼ QCoh(Spec (A))

Proof. See [Hr, Sec II.5]. Note the functor above is a contra variant functor; not covariant. 1.2. ADDITIVE AND ABELIAN CATEGORIES 17 1.2 Additive and Abelian Categories

Deﬁnition 1.2.1. A category C is called an additive category, if (1) C has a zero object 0, (2) For X,Y ∈ Obj(C ) the direct product X × Y exists, (3) For X,Y ∈ Obj(C ), the set Mor(X,Y ) has the structure of an abelian group, such that the compositions is bilinear. That means, for objects X,Y,Z in C and f ∈ Mor(X,Y ), g ∈ Mor(Y,Z) the composition maps ( g : Mor(X,Y ) −→ Mor(X,Z) h 7→ gh ∗ are group homomorphims. f ∗ : Mor(Y,Z) −→ Mor(X,Z) h 7→ hf

Deﬁnition 1.2.2. Suppose C , D are two additive categories. A functor F : C −→ D is called a functor of additive categories, if for all objects X,Y of C , the map

F : MorC (X,Y ) −→ MorD (FX,FY ) is a group homomorphism.

Proposition 1.2.3. Suppose C is an additive category. Then, for X,Y ∈ Obj(C ), the direct sum exists and X × Y = X ⊕ Y .

Proof. See [HS, Prop II.9.1, pp. 76].

Proposition 1.2.4. Suppose C is an additive category and X,Y ∈ Obj(C ). Then, for f, g ∈ Mor(X,Y ), then we have the following commutative diagram:

∆ X / X × X X ⊕ X

hf,gi f+g - Y

Proof. See [HS, Prop II.9.4, pp. 76]. We clarify, the vertical map (f, g) is, by deﬁnition, given by the diagram: 0 / X

f X / X ⊕ X

(f,g) g #, Y 18 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

Lemma 1.2.5. Suppose C is an additive category and σ : X −→ Y is a morphism. Then, a morphism ι : K −→ X deﬁnes the kernel of σ if and only if ∀ A ∈ Obj(C ) the sequence of groups

ι∗ σ∗ 0 / Mor(A, K) / Mor(A, X) / Mor(A, Y ) is exact.

Likewise, a morphism ζ : Y −→ C deﬁnes the coKernel of σ if and only if ∀ Z ∈ Obj(C ) the sequence of groups

ζ∗ σ∗ 0 / Mor(C,Z) / Mor(Y,Z) / Mor(X,Z) is exact.

1.2.1 Abelian Categories

Deﬁnition 1.2.6. Suppose C is an additive category. Further, assume every morphism σ : X −→ Y has a kernel and a coKernel. Deﬁne the image and coImage

Im(σ) := ker (coKer(σ)) coIm(σ) := coKer (ker(σ))

Then, there is a natural map, coIm(σ) −→ Im(σ).

Deﬁnition 1.2.7. (see [Sv]) Suppose A is an additive category. Then, A would be called an Abelian Category, if

1. Every morphism σ : X −→ Y has a kernel and a coKernel.

2. For all morphisms σ : X −→ Y , the canonical morphism coIm(σ) −→ Im(σ) is an isomorphism.

∼ Remark 1.2.8. For an arrow σ : X −→ Y in an abelian category A , coIm(σ) −→ Im(σ) is an isomorphism. Because of this, throughout rest of these notes, we would usually identify coIm(σ) and Im(σ), without further clariﬁcations.

The deﬁnition in [HS, pp. 78], appears a little diﬀerent:

Deﬁnition 1.2.9. (see [Sv]) Suppose A is an additive category. Then, A would be called an Abelian Category, if

1. Every morphism σ : X −→ Y has a kernel and a coKernel. 1.2. ADDITIVE AND ABELIAN CATEGORIES 19

2. Every monomorphism is the kernel of its coKernel. Every epimorphism is the coK- ernel of its kernel.

3. Every morphism is expressible as composition of an epimorphism and a monomorphism.

Proof of Equivalence of Two Deﬁnitions, follows from the following [HS, Prop 9.6, pp. 78]:

Proposition 1.2.10. Suppose A is an abelian category and σ : X −→ Y is a morphism. Then, we have the diagram:  µ = ker(σ) µ η  K / X / I   = co ker(σ) ν  σ   η = co ker(µ) =: coIm(σ) Y where ν = ker() =: Im(σ)    Think I := image(σ)  C  σ = νη

Corollary 1.2.11. Suppose A is an abelian category.

1. A morphism σ : X −→ Y is isomorphism if and only if it is a monomorphism and an epimorphism.

2. Pullback and push forward exists.

Proof. See [HS, Cor 9.8] for the ﬁrst one. Pullback exists, because it is a kernel:

X 9 _ ϕ

# P (ϕ, ψ) / X ⊕ Y / Z O ;

ψ ? % Y Example 1.2.12. Here are the main examples and comments:

1. Suppose A is a commutative ring. Then, Mod(A) = QCoh(A) and fMod(A) = Coh(A) are Abelian categories. (Perhaps, the deﬁnition of the Abelian category was modeled after these two examples.)

2. More generally, let X be a scheme. Then, QCoh(X) and Coh(X) are Abelian categories. 20 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

3. We remark, for a commutative ring A, the category P(A) is not necessarily an Abelian category. P(A) will be an Exact category, which we deﬁne next.

Lemma 1.2.13. Suppose A is an abelian category and f : M −→ N be an arrow. Then, f is an isomorphism if and only if f epi and mono.

Proof. Assume f is an isomorphism and g : N −→ M be the inverse of f. So, fg = 1N and gf = 1M . Suppose fh = 0 for some arrow h : U −→ M. Then, h = 1M h = gfh = g0 = 0. Hence, f is mono. likewise f is epi. Now suppose f is both epi and mono. Since f is mono, ker(f) = 0 and likewise coKer(f) = 0. From this it follows, coIm(f) = M, Im(f) = N and the induced map is f : coIm(f) = M −→ Im(f) = N, which is an isomorphism by deﬁnition of Abelian categories. The proof is complete.

Lemma 1.2.14. Suppose A is an abelian category and f : X −→ Y is a morphism. Let ν : I,→ Y be the image of σ and ν : X I be the corresponding epi morphism. Then, there is a factorization f = µν, where µ is epi and ν is mono. This factorization is unique, up to isomorphism, in the following sense:

I o ? ? µ ν X o ι / Y f ? q p / J

Given any other factorsation f = µ0ν0, where µ0 is epi and ν0 is mono, there is an isomorphism ι : I −→∼ J, such that the diagram commutes.

Proof. Suppose t : ker(f) ,→ X be the kernel of f. Then, pqt = ft = 0. Since p is mono, qt = 0. Since µ : X −→ I is coKernel of t, there is an arrow ι : I −→ J, such that ιµ = q. Also, (pι)µ = pq = f = νµ. Since µ is mono, pι = ν. This establishes that the diagram commutes. Now, ι is epi, because so is q and ι is mono because so is ν. By Lemma 1.2.13, ι is an isomorphism.

1.3 Exact Category

We deﬁne Exact Categories as follows (see [Sm1]). 1.3. EXACT CATEGORY 21

Definition 1.3.1. Consider additive category E , together with a family of sequence of morphisms i p X / Y / / Z to be called conflations. (1.1) We also say i p 0 / X / Y / Z / 0 is exact. (1.2) The left part X / Y of such a conflation would be called an inflation and be denoted as such. Likewise, the right part Y / / Z would be called a deflation and be deooted as such.

Such an additive category E , together with such a family of conﬂations (1.1), would be called an Exact Category, if the following conditions are satisﬁed:

1. In a conﬂation (1.1), the inﬂation i is the kernel of p and p is a coKernel of i.

2. Conﬂations are closed under isomorphisms.

3. inﬂations and deﬂations are closed under composition.

i 4. Any diagram ZXo / Y of morphisms, with i inﬂation, can be completed to a push forward diagram

i X / Y with j inﬂation. Z / W j

5. Likewise, any diagram X / ZYo o p of morphisms, with p deﬂation, can be completed to a pullback diagram

W / Y q p with q deﬂation. X / Z

6. For all objects X,Y ∈ Obj(C ), the following

X / X ⊕ Y / Y is a conﬂation.

Further, an inﬂation is also called an admissible monomorphism and a deﬂation is also called an admissible epimorphism. 22 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

Definition 1.3.2. Suppose E , B are exact categories and F : B −→ E is a functor of additive categories. We say F is an exact functor, if it sends conflations to conflations.

Deﬁnition 1.3.3. Suppose E , B are exact categories such that B ⊆ E be a full subcategory. We say that B is a fully exact subcategory of E , if B is closed under extensions in E , which means:

For a conﬂation (1.1) in E , if X,Z are isomorphic to an object in B, then so is Y .

1. This gives a structure of an exact category to B, by declaring a sequence in B exact, if it is isomorphic to an exact sequence in E .

2. Inclusion B ⊆ E preserves and detects conﬂations.

3. In this case, we also say that B ⊆ E is an extension closed exact subcategory.

Lemma 1.3.4. Every small exact category E can be embedded into an abelian category C , as a fully exact, extension closed subcategory.

Proof. See [Q, pp 92]

Example 1.3.5. Here is a list of example (see [Sm1, Section 2.1.2]).

1. To start with, any abelian category is an exact category. Where a sequence

ι p 0 / K / M / C / 0 (1.3)

is declared a conﬂations (or exact), if ker(p) = ι and co ker(ι) = p.

(a) So, for a ring R, the category R-mod, of all (left) R-modules is abelian (hence an exact) category, where conflations are defined as above (1.3). (b) For a commutative ring A, QCoh(A) = Mod(A) and Coh(A) = fMod(A) are abelian categories, where conflations are defined as above (1.3).

quasi-coherent) OX -modules, is an abelian (hence an exact) category, where conﬂations are deﬁned as above (1.3).

(d) For a noetherian scheme X, the category Coh(X) of coherent OX -modules is an abelian (hence an exact) category, where conﬂations are deﬁned as above (1.3). 1.4. LOCALIZATION AND QUOTIENT CATEGORIES 23

2. For a noetherian scheme X, the category V ect(X) of locally free sheaves form an exact category. If fact, V ect(X) is an exact subcategory of Coh(A).

In particular, for a commutative noetherian ring A, the category P(A) of ﬁnitely generated projective A-modules form an exact category.

3. Suppose E is an exact category. Let Ch(E ) be the category, whose objects are chain complexes:

• • di (A , d ): ··· / Ai−1 / Ai / Ai+1 / ··· 3 dd = 0.

Depending on the authors, subscripts are also use as follows:

di (A•, d•): ··· / Ai+1 / Ai / Ai−1 / ··· 3 dd = 0.

Morphisms in Ch(E ) are chain complex maps, given by commutative diagrams:

di ··· / Ai−1 / Ai / Ai+1 / ··· • • • • • f :(A , d ) −→ (B , ∂ ): f i ··· / Bi−1 / Bi / Bi+1 / ··· ∂i

A sequence

(A•, d•) / (B•, d•) / (C•, d•)

is declared exact, if

∀ i ∈ Z Ai / Bi / Ci is exact.

Then, Ch(E ) is an exact category. Its full subcategory Chb(E ) of bounded chain complexes is also an exact subcategory. A complex (A•, d•) ∈ Chb(E ), if Ai = 0 ∀ i 0, i 0

1.4 Localization and Quotient Categories

The notes of Schlichting [Sm1] seems most helpful for this section. Inverting multiplicative subsets S of a commutative rings A, is routinely covered in commutative algebra classes. However, this process extends to inversion of a class of morphisms in a category, as follows. 24 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

Deﬁnition 1.4.1. Suppose C is a category and S be a class of morphisms in C .(One can think of S as a subcategory, while we intend to invert all the morphisms in S.) The localization of C with respect to S, is a functor ι : C −→ S−1C , where S−1C is a category, with the following universal property.

Given any functor F : C −→ D, such that F (f) is an isomorphism in D,

ι C / S−1C −1 ∃ a unique functor ϕ : S C −→ D 3 ϕ commute. F " D

As usual, existence on localization S−1C is not guaranteed, while if it exists, it is unique up to a unique equivalence.

We work toward giving a suﬃcient condition for existence.

Deﬁnition 1.4.2. Suppose C is a category and S be a class of morphisms in C . We say that S satisﬁes a "calculus of right fractions" if the following three conditions hold:

1. The class S is closed under composition. The identity 1X is in S, for all X ∈ Obj(C ).

2. The ﬁrst diagram below, can be completed to the second commutative diagram:

t X W / X

f with s ∈ S, g f with t ∈ S (1.4) Y Z s / Y s / Z

3. Suppose f, g : X −→ Y be two morphisms in C and s : Y −→ Z, in S, such that sf = sg. Then, there is a morphism t : W −→ X in S such that ft = gt.

If dual of these three conditions hold, then S is said to satisfy a "calculus of left fractions". If S satisﬁes both the conditions, then we say that S satisﬁes "calculus of fractions".

We remark that, barring set theoretic issues, S−1C exists, whenever S satisﬁes calculus of left or right fractions. We give the following version.

Lemma 1.4.3. Suppose C is a category and S is class of morphisms in C . Assume codomains of morphisms in S has a set of isomorphism classes. Then, the localization category S−1C exists. 1.4. LOCALIZATION AND QUOTIENT CATEGORIES 25

Proof. We assume S satisﬁes calculus or right fractions. The objects of S−1C would be the same as those of C . To deﬁne morphisms, for two objects X,Y , in view of Equation 1.4, let

P air(X,Y ) = {(s, f) ∈ MorC (U, X) × MorC (U, Y ): Z ∈ Obj(C ), s ∈ S} Under the hypothesis, we can treat P air(X,Y ) as a set, by restricting U to a set of represen- tatives of isomorphism classes of objects. Refer to the elements (s, f) ∈ P air(X,Y ), by the s f s f t g diagrams XUo / Y . Two such diagrams XUo / Y , XVo / Y in P air(X,Y ), are said to be equivalent, if U O s f t t : W −→ U ~ ∃ either t or s ∈ S 3 XW Y commute. s : W −→ V ` > s t g V One checks that this is an equivalence relation on P air(X,Y ). Let fs−1 denote the equiv- s f alence class of the diagram XUo / Y and let −1 MorS−1C (X,Y ) := {fs :(s, f) ∈ P air(X,Y )}.

−1 −1 s f Let fs ∈ MorS−1C (X,Y ) and gt ∈ MorS−1C (Y,Z) be give by the diagrams XUo / Y , t g YVo / Z . To define compositions, by the conditions of calculus of right fractions, we complete the following commutative diagram: ϕ g W / V / Z τ t with τ ∈ S. Define (gt−1)(fs−1) := (gϕ)(sτ)−1. XUo / Y s f Again, one checks that this composition is well defined. Further, it is clear that the functor ι : C −→ S−1C satisfies the universal property. The proof is complete.

The quotient of additive categories by a subcategory is deﬁned likewise, as follows.

Deﬁnition 1.4.4. Suppose A is an abelian category and B ⊆ A is an abelian subcategory. The quotient of by , is an abelian category A , together with an exact functor ι : −→ A B B A A such that ι(X) = 0 for all X ∈ Obj( ), with the following universal property: B B Given any exact functor F : A −→ D, where D is an abelian category, such that F (X) = 0, for all objects in B,

ι C / A A B ∃ a unique additive functor ϕ : −→ D 3 ϕ commute. B F D 26 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

If it the quotient A exists, then it is unique up to a unique equivalence However, existence B on localization is not guaranteed. Subsequently, we give a suﬃcient condition for existence of the quotient.

Deﬁnition 1.4.5. Let A be an abelian category. A full subcategory B ⊆ A is said to be a Serre Subcategory, if for any conﬂation

K / M / / C in A K, C Obj(B) ⇐⇒ M ∈ Obj(B) (up to isomorphisms).

That means, if and only if B is closed under subobjects, quotient objects and extensions. A Serre subcategory is an abelian subcategory.

Lemma 1.4.6. Let A be an abelian category (small) and B ⊆ A be a Serre subcategory. Then, the quotient category A exists. B

Proof. Suppose S be the set of morphisms f in A , such that ker(f), co ker(f) ∈ B. Then, S is satisﬁes the calculus of fractions:

1. Let f : X −→ Y , g : Y −→ Z be in S. We show that gf ∈ S. We have ker(f) ker(gf) / / ker(g) ∩ Im(f) is a conﬂation.

Since ker(g) ∈ Obj(B), ker(g) ∩ Im(f) ∈ Obj(B). Also, since ker(f) ∈ Obj(B), ker(gf) ∈ Obj(B). By reversing the arrows, co ker(gf) ∈ Obj(B). So, S is closed under composition. 2. Now, we check the second condition of calculus of right fractions (Equation 1.4). Let Y / ZXo f s be arrows in B with s ∈ S. Consider the pull back diagram

g X ×Z Y / X

t s Y / Z f

By Lemma 1.5.1, ker(t) = ker(s) ∈ Obj(B). Now, extend the pullback diagram, to conﬂesions: g X×Z Y 0 / ker(f) / X / co ker(g) / 0

t s s 0 / Y / Z / co ker(f) / 0 ker(f) f where t is induced by t, co ker(t) = co ker(t), and s is induced by s. 1.4. LOCALIZATION AND QUOTIENT CATEGORIES 27

Now, from Snake Lemma 1.5.4,

ker(s) / ker(s) / co ker(t) / co ker(s) is exact.

Since ker(s), co ker(s) are in B, it follows that co ker(t) is in B. So, t ∈ S. So, the second condition, for right fractions, is satisﬁed by S. By reversing the arrows, it follows that the corresponding condition of left fractions is satisﬁed.

3. To check that third condition, let f, g : M −→ N be two maps in A and s : N −→ W be a map is S, such that sf = sg. With F = f − g, we have sF = 0. It follows, there is an arrow F0 : M −→ ker(s0 such that F = F0ι, as in the diagram u ker(s) ker(F ) / M ×N ker(s) / ker(s) ; 8 F0 ι ζ ι F0 M / N This lead to ker(F ) / M / N F t F s s 0 0 # W ' W Claim t is in S. Clearly, ker(t) = 0 ∈ B. Now, co ker(t) = Im(F ) ⊆ ker(s) ∈ B. Hence, co ker(t) is in B. So, t ∈ S. Dual of this property is checked by reversing the arrows.

This establishes that S satisﬁes the calculus of fractions. Hence S−1A exists. Now, one checks, that S−1 =: A satisﬁes the required universal property. A B

1.4.1 Quotient of Exact Categories

More generally, following [Sm2], we deﬁne quotient of exact categories.

Deﬁnition 1.4.7. Suppose E is an exact category and P ⊆ E be an extension closed full subcategory. An exact category Q, together with an exact functor ι : E −→ Q, would be called a quotient of E by P, if ι satisﬁes the following conditions:

1. ι(P ) ' 0 for all P ∈ Obj(P).

2. The functor ι is universal, in the following sense: Suppose D is an exact category ζ : E −→ D is an exact functor, such that ζ(P ) ' 0 ∀ P ∈ Obj(P). Then, ι E / Q ∃ Unique functor ϕ : Q −→ D ϕ commute. ζ D 28 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

Remarks:

1. The existence of such a quotient Q is not guaranteed. However, when it exists, it is unique up to isomorphism (functorial equivalence). In that case, we denote := E . Q P

2. Readers may compare this deﬁnition with [Sm2, 1.14, 1.16].

3. It follows form the proof of Lemma 1.4.6, for a Serre subcategory of an abelian category , the quotient A exists. A B

4. Following Schlichting [Sm2], we give a suﬃcient condition for existence of such a quotient, as follows. This would be of some interest to us in Section 4.7.

Lemma 1.4.8. Suppose E is a small exact category and P ⊆ E be an extension closed full subcategory. A morphism f : M −→ N in E would be called a weak isomorphism, if f = fnfn−1 ··· f1 is a ﬁnite composition of morphisms fi : Mi−1 −→ Mi is E , for i = 1, . . . , n, with M0 = M and Mn = N, such that

1. either fi is a deﬂation, with ker(fi) ∈ Obj(P),

2. or either fi is an inﬂation, with co ker(fi) ∈ Obj(P).

Let Σ denote the class of all weak isomorphisms. Further assume that Σ satisﬁes the calculus of left or right fractions. Then, E exists. Namely, the functor ι : −→ Σ−1 P E E satisfy the universal property of the the quotient E . P

Proof. By Lemma 1.4.3, Σ−1E exists. A sequence in Σ−1E is declared a conﬂasion, if it is isomorphic to a conﬂation in E . This makes Σ−1E an exact category. Now let ζ : E −→ D be an exact functor, such that ζ(P ) ' 0 for all P ∈ Obj(P). If f ∈ Σ, then it follows that ζ(f) is an isomorphism. Hence, it follows that there is a unique functor ϕ :Σ−1E −→ D such that ϕι = ζ. It is also clear that ϕ is exact. The proof is complete.

Remark 1.4.9. Schlichting [Sm2] gives a suﬃcient condition, for the class Σ in (1.4.8) satisﬁes calculus of left or right fractions.

1.5 Frequently Used Lemmas

In this section, we list a frequently used Lemmas in K-Theory. 1.5. FREQUENTLY USED LEMMAS 29

1.5.1 Pullback and Pushforward Lemmas

Following two lemmas on pullback and push forward lemmas are used frequently in K- Theory. The following two frequently used lemmas in K-Theorey.

v h Lemma 1.5.1. Suppose C is a category with zero. Let X / ZYo be two arrow in C . Let p1 P / X

p2 v be the pullback. Y / Z h

1. Suppose (K, η) = ker(p2). Then, (K, p1η) = ker(v).

2. Let ι : K −→ X be the kernel of v. Then there is an induced arrow η : K −→ P , given by the commutative diagram: K η ι

p1 # P / X 0

p2 v Y / Z h

Then, (K, η) = ker(p2). It is expressed better by the diagram:

K _ K _ η ι

p1 P / X

p2 v Y / Z h

One can write down the same, in terms of the kernels of the horizontal maps p1 and h.

Proof. See [HS, II Theorem 6.2]. Reversing the arrows, we have the following.

h v Lemma 1.5.2. Suppose C is a category with zero. Let XZo / Y be two arrow in C . Let h Z / X

v i1 be the pushout. Y / A i2 30 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

1. Suppose j : A −→ C be the coKernel of i2. Then, ji1 : X −→ C is the coKernel of h.

2. Suppose ζ : X −→ D be the cockerel of h. Then, the following commutative diagram deﬁnes a map β : A −→ C: h Z / X

v i1 ζ Y / A i2

β 0 + C

Then, (C, β) is the coKernel of i2. Same is better expressed by the diagram

h ζ Z / X / C

v i1 Y / A / C i2 β

Proof. Reverse the arrow in the proof of (1.5.1).

1.5.2 The Snake Lemma

A key ingredients to K-Theory, is a classical result in Homological Algebra that is known as the Snake Lemma. Before we state and prove Snake Lemma, we prove the following elementary lemma.

Lemma 1.5.3. Suppose A is an abelian category. Consider the following diagram of arrows:

f g K / M / C 3 fg = 0.

Let t : ker(g) ,→ M be the kernel of g and s : Im(f) ,→ M corresponds to the monomorphism, corresponding to Im(f). Then, there is a unique arrow ι : Im(f) −→ ker(g) such that the diagram s Im(f) / M

ι commute. ker(g) / M t In fact, ι is a mono. 1.5. FREQUENTLY USED LEMMAS 31

Proof. Since, t is mono, ι would also be mono, if it exists. Consider the diagram

f g , K u / / Im(f) s / M / C

v t g K / ker(g) / M /1 C f

Since gsu = gf = 0 and u is epi, gs = 0. now, from deﬁnitions of kernel s, there is a morphism ι : Im(f) ,→ ker(g), as required.

Most elementary version of Snake Lemma is stated for Modules over a ring. The following is one of the most general versions.

Lemma 1.5.4. Suppose A is an abelian category. Consider the commutative diagram

i j K / M / L / 0 f g h in A , 0 / A / B / C ι ζ where the rows are exact in A . Then, there is a natural exact sequence

δ ker(f) / ker(g) / ker(h) / coKer(f) / coKer(g) / coKer(h) , where the maps between the kernels, and between the coKernels are induced by the respective morphisms i, j, ι, ζ. The morphism δ is deﬁned in the proof.

Proof. This proof is improvisation to some literature available in the net (due to Julia Goedecke). We extend the above diagram and name the morphisms:

ik jk ker(f) / ker(g) / ker(h) _ _ _ fk gk hk i j K / M / L / 0

f g h 0 / A / B / C ι ζ

fc gc hc coKer(f) / coKer(g) / coKer(h) ιc ζc

Construction of δ: Let (P, u, v) := P ullback(j, hk) and (Q, ϕ, η) := P ushF orward(ιc, fc), 32 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY which we display in the following diagram:

i0 v ker(j) / P _ / / ker( h) : : _ i0 u hk i0 i1 j K / / ker(j) /2 M / L / 0 i g h ζ # 0 / A / B / / coKer(ι) /, C ι ζ0

fc ϕ coKer(f) / Q / / coKer(ι) η θ

By properties of Exact categories (in this case Abelian Categories) or by Lemmas 1.5.1, 1.5.2, η, u are injective and v, ϕ are surjective Also, by (1.5.1) and (1.5.2), we have ker(j) ∼= ker(v) =: i0 and coKer(η) = coKer(ι). We have

0 0 1. ϕgui i0 = ϕgi = ϕιf = ηfcf = η0 = 0. Since i0 is surjective, ϕgui = 0. Since i0 = ker(v), there is a map

v P / ker(h)

ϕgu α : ker(h) −→ Q 3 α commute. Q |

2. Likewise (dually), θ(ϕgu) = ζ0gu = hju = hhkv = 0v = 0. Since η = ker(θ), there is an arrow P β β : P −→ coKer(f) 3 ϕgu commute. z coKer(f) η / Q

3. Hence αv = ϕgu = ηβ. This implies θ(αv) = θ(ηβ) = 0. Since η = ker(θ), there is an arrow P γ γ : P −→ coKer(f) 3 αv z coKer(f) η / Q

4. Now, ηγi0 = αvi0 = 0. Since η is injective, γi0 = 0. Therefore,

v P / ker(h) γ δ ∃ δ : ker(h) −→ coKer(f) 3 αv commute. Ñ u coKer(f) η / Q 1.5. FREQUENTLY USED LEMMAS 33

This completes the deﬁnition of the desired arrow δ. Now, we proceed to prove exactness.

1. Exactness at ker(g): By Lemma 1.5.3, there is an monomorphism Θ: Im(ik) −→ ker(jk), such that sι = t, where s : ker(jk)) ,→ ker(g) is the kernel of jk, and t : Im(ik) ,→ ker(g) is the monomorphism, corresponding to the image of ik. To deﬁne the morphism in the opposite direction, consider the commutative diagram and label the morphisms:

ik jk - ker(f) / ker(jk) /- ker(g) ker(h) _ w s _ _

fk τ gk hk ϕ η K / / ker(j) /2 M 3 L i j In this diagram the morphism τ is deﬁned by the property of kernel ker(j), because jgks = jkgks = 0. Now let (U, λ, µ) := P ullBack(τ, ϕ). Consider the diagram:

λ s U / / ker(jk) / ker(g) _ µ τ gk ϕ η K / / ker(j) /2 M i Note λ is epi. We have

ϕµ = τλ =⇒ iµ = ηϕµ = ητλ = gksλ =⇒ giµ = 0 =⇒ ιfµ = 0 =⇒ fµ = 0,

because ι is mono. By property of kernel fk, we have

µ f U / K / A

fk f ker(f) / K / A

So, gkikψ = ifkψ = iµ = gksλ =⇒ ikψ = sλ. Diagramatically,

µ s U / / ker(jk) / ker(g)

ψ u t ker(f) / / Im(ik) /1 ker(g) ik By Lemma 1.2.14, µ = Im(sµ). Let = ker(sµ), as an arrow. Then, tuψ = 0 and hence uψ = 0. So, there is an arrow Ψ, as in the commutative diagram:

µ s U / / ker(jk) / ker(g) ψ Ψ Inparticular tΨ = s u t ker(f) / / Im(ik) /1 ker(g) ik 34 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

Since sΘ = t and s, t are mono, it follows ΦΘ = 1 and ΘΨ = 1. This completes the proof of exactness at ker(g).

2. Exactness at ker(h): We have the following commutative diagram:

ker(g) jk w

" v & P / ker(h) gk

u hk K / M / L i j f g h A / B / C ι ζ

First, we want to prove δjk = 0. Since η is monic, we will prove ηδjk = 0, or ηδvw = 0. However, ηδvw = ηγw = αvw = ϕguw = ϕggk = 0. Therefore by Lemma 1.5.3, there is an arrow Ψ: Im(jk) ,→ ker(δ) such that tΨ = s, where t : ker(δ) ,→ ker(h) is the kernel of δ and s : Im(jk) ,→ ker(h) is the monomorphism, corresponding to Im(jk). Same is displayed in the commutative diagram:

s Im(jk) / ker(h)

Ψ ker(δ) / ker(h) t

Now we proceed to deﬁne the inverse of Ψ. Consider the diagram

ik jk λ ker(f) / ker(g) / ker(h) / / coKer(jk) _ _ _ _ fk gk hk µ K / M / L / / F i j ν 3

p where (F, µ, ν) := P ushF orw(λ, hk) coKer(gk)

o Image(g)

If follows, µ is mono and ν is epi. It follows νjgk = 0. So, νj factors through 1.5. FREQUENTLY USED LEMMAS 35

coKer(gk). We further extend the above diagram:

ik jk λ ker(f) / ker(g) / ker(h) / / coKer(jk) _ fk gk hk µ K / M / L / / F i j ν 3 _

p f coKerg (gk) ξ _ q ψ A ι / B 2/ G f c ω coKer(f)

where (G, ξ, ψ) := pushF W D(p, q) = pushF W D(νj, g). Since q is mono, so is ξ. By the similar argument above, ψι factors through coKer(f). Now, we have the pushforward diagram, which deﬁnes an arrow τ.

ι A / B

fc ϕ ψ coKer(f) η / Q

τ ω , G Now,

ωδv = τηδv = τηγ = ταv = τϕgu = ψgu = ξνju = ξνhkv = ξµλv Since v is epi, ωδ = ξµλ. In summary,

λ ker(h) / / coKer(jk) _ µ δ F _ ξ coKer(f) ω / G With t = ker(δ), we decompose:

δ t ker(δ) / ker(h) / / coIM(δ) /. coKer(f)

Ω ω ker(h) / / coKer(j ) / F / G λ k µ ξ 36 CHAPTER 1. BACKGROUND FROM CATEGORY THEORY

We have ξµλt = ωδt = 0. So, λt = 0. Since coIm(δ) = coKer(t), there is a unique morphism Ω, making the ﬁrst rectangle commutative. Again, since λt = 0, by property of kernel s, we have the commutative diagram, for some arrow Θ:

δ t . ker(δ) / ker(h) e / / coIM(δ) m / coKer(f)

Θ Ω ω Im(j ) / ker(h) / / coKer(j ) / F / G k s λ k µ ξ

So, we have sΘ = t. Since tΨ = s, and s, t are mono, we have ΘΨ = Id and ΨΘ = Id. This establishes the exactness at ker(h).

3. Exactness at coKer(f): This follows from the dual argument of the proof of exactness at ker(h).

4. Exactness at coKer(g): This follows from the dual argument of the proof of exactness at ker(g).

The proof is complete.

1.5.3 Limits and coLimits

To be Inserted. Chapter 2

Homotopy Theory

Much of the materials in this chapter would be available in standard textbooks on Topology. However, some of the not-so-standard materials were taken from [H].

2.1 Elements of Topological Spaces

We shall begin this chapter with the definition of topological spaces. The basic definitions included in this section are omnipresent in the literature and the net. First and foremost, Topology is study of continuous functions. First, we define Topological spaces.

Deﬁnition 2.1.1. Suppose X is a nonempty set. A topology on X is a collection U of subsets of X, to be called open sets of X, such that

1. the empty set φ and X are open. That is φ, X ∈ U.

2. Union (ﬁnite or inﬁnite) of open sets is open. That is if {Ui : i ∈ I} ⊆ U, then S i∈I Ui ∈ U.

n 3. Finite intersection of open sets is open. That is, if U1,U2,...,Un ∈ U , then ∩i=1Ui ∈ U.

When the set T is understood, we may just say X is a topological space.

Now, assume (X, U) is topological space. Here are few more deﬁnitions:

1. A subset C ⊆ X is called a closed set, if its complement Cc := X \ C is open. Therefore,

37 38 CHAPTER 2. HOMOTOPY THEORY

(a) Intersection (ﬁnite or inﬁnite) of closed sets is closed. (b) Finite union of closed sets is closed.

2. Let x ∈ X be a point. A subset U is called a neighborhood of x, if there is an open subset V , such that x ∈ V ⊆ U. If U itself is open, we say U is an open neighborhood of x.

3. A set of open sets B ⊆ U is called a basis of X, if given any open neighborhood x ∈ U, of a point x, there is a an open set V ∈ B, such that x ∈ V ⊆ U.

So, given any open set U, we have U = ∪i∈I Vi, where Vi ∈ B for all i ∈ I.

4. Given a subset A ⊆ X, then \ A := {C : A ⊆ C and C is closed}

is called the closure of A. So, A is the smallest closed set containing A.

5. A subset D ⊆ X is called dense in X, if D ∩ U 6= φ, for all nonempty open sets U. It follows, D is a dense subset of X, if and only if the closure D = X.

Deﬁnition 2.1.2. Suppose (X, U) and (Y, V) and two topological spaces. A set theoretic map f : X −→ Y is called a continuous function, if f −1(V ) ∈ U, for all V ∈ V.

1. If follows, the collection of all topological spaces, form a category (see Examples 1.1.6). For topological spaces (X, U), (Y, V), Mor((X, U), (Y, V)) is deﬁned to be the set of all continuous functions. This category was denoted by Top.

2. A continuous function f :(X, U) −→ (Y, V) is called homeomorphism, if f : X −→ Y is bijective and f −1 : Y −→ X is continuous. So, homeomorphism f are the "isomorphisms" in the category Top.

Examples 2.1.3. Here are some examples of topological spaces.

1. (Trivial Topology) Let X be a set and U = {φ, X}. Then, (X, U) is a topological space. This is called the trivial topology on X and also the Indiscrete topology. In this case, given any topological space (Y, V), any set theoretic function f : Y −→ X is continuous.

2. (Discrete Topology) Let X be a set and P(X) be the power set of X, (i. e. P(X) is the set of all subsets of X). Then, (X, P(X)) is topological space, to be called the discrete topology on X. We have the following comments: 2.1. ELEMENTS OF TOPOLOGICAL SPACES 39

(a) In the discrete topology on X, all subsets of X are open (and also closed). (b) B = {{x} : x ∈ X} be the set of all singleton subsets of X. Then, B is base for this topology. (c) Suppose (Y, V) is topological space and f : X −→ Y is any set theoretic function. Then, f :(X, P(X)) −→ (Y, V) is continuous.

3. (Induced Topology) Suppose (X, U) is a topological space and Y ⊆ X is a subset. Let V = {Y ∩ U : U ∈ U}. Then, (Y, V) is a topological space.

4. (Usual Topology on Rn) For our purpose, the most important topology is the n n usual topology on R . For any x0 ∈ R and r > 0, deﬁne the open ball, of radius r, as n B(x0, r) = {x ∈ R :k x − x0 k< r}

A subset U ⊆ Rn is deﬁned to be open, in usual topology, if U is union of open balls (possibly inﬁnite union). Let U be the collection of all such open sets, Then (Rn, U) is a topological space.

(a) Suppose X ⊆ Rn be a subset. Then, the usual topology on Rn induces a topology on X (as in (3)). We continue to call it the usual topology on X. The concept of continuity, taught in undergraduate classes, coincides with the continuity deﬁned by Deﬁnition 2.1.2.

(b) Usual topology on Cn is deﬁned in the same way. (c)( Exercise) Prove that the set Q of rational numbers is a dense subset of R.

5. Examples from Algebra:

(a)( Affine Schemes) Suppose A is a commutative ring. Let Spec (A) be the set of all prime ideals of A. For any ideal I ⊂ A, let V (I) = {℘ ∈ Spec (A): I ⊆ ℘}. The Zariski Topology on Spec (A) is defined to be the one, where closed sets are V (I), where I runs through all the ideals of A. i. So, an open sets in Zariski Topology on Spec (A) are Spec (A)\V (I), where I is an ideal. ii. For f ∈ A, define D(f) = {℘ ∈ Spec (A): f ∈ ℘}. Then, B = {D(f): f∈ / A} forms a basis for this topology. 40 CHAPTER 2. HOMOTOPY THEORY

(b)( Projective Schemes) Suppose S = S0⊕S1⊕S2⊕· · · be a graded commutative

ring. Let S+ = S1 ⊕ S2 ⊕ · · · , to be called the irrelevant ideal. Let

Proj (S) = {℘ ∈ Spec (S): ℘ is homogeneous prime ideal,S+ 6⊆ ℘}

For any homogeneous ideal I ⊂ S, let V (I) = {℘ ∈ Proj (S): I ⊆ ℘}. The Zariski Topology on Proj (S) is deﬁned to be the one, where closed sets are V (I), where I runs through all the homogeneous ideals of A. i. So, an open sets in Zariski Topology on Proj (S) are Proj (S) \ V (I), where I is a homogeneous ideal.

ii. For a homogeneous elemtn f ∈ Sn, deﬁne D(f) = {℘ ∈ Proj (S): f ∈ ℘}. Then, B = {D(f): f∈ / A} forms a basis for this topology. (We caution that, as "schemes" Spec (A) and Proj (S) would have more structures, than what we gage above (see [Hr]))

The following are two fundamental concepts in topology.

Deﬁnition 2.1.4. Suppose X is a topological space. We say that is Hausdorﬀ space, if any two distinct points x, y ∈ X, x 6= y, there are two open neighborhoods x ∈ Ux and y ∈ Uy such that Ux ∩ Uy = φ.

Clearly, Rn is a Hausdorﬀ space. Deﬁnition 2.1.5. Suppose X is a topological space. An open cover of X is a collection S of open sets {Ui : i ∈ I}, such that X = i∈I Ui.

1. X is said to be compact, if every open cover {Ui : i ∈ I} of X has a ﬁnite sub cover. Sn This means, there are i1, . . . , in ∈, such that X = j=1 Uij .

2. A subset Y ⊆ X is called the compact subset of X, if Y , with its induced topology, is a compact topological space.

We have two important exercise.

Exercise 2.1.6. Suppose X is a Hausdorﬀ space and K ⊆ X is a compact subset. Then, K is closed. (This fails, if X is not Hausforﬀ.)

Proof. We prove that Y := X \ K is open. Fix y ∈ Y . Then, ∀ x ∈ K, there are open sets Ux,Vx such that x ∈ Ux, y ∈ Vx and Ux ∩ Vx = φ. Since {Ux ∩ K : x ∈ K} is an open cover of K, and K is compact K ⊆ Ux1 ∪ · · · ∪ Uxn for ﬁnitely many x1, . . . xn ∈ K. Write

V = Vx1 ∩ · · · ∩ Vxn . Then, V is open, y ∈ V and K ∩ V = φ. So, y ∈ V ⊆ Y . Therefore, Y is open. 2.1. ELEMENTS OF TOPOLOGICAL SPACES 41

Exercise 2.1.7. Suppose K ⊆ Rn is a subset. Then, K is compact if and only if K is closed and bounded.

Proof. Usually taught in the undergraduate Analysis courses and above.

2.1.1 Frequently used Topologies

We already deﬁned Hausdorﬀ Spaces and Compact spaces. Following is a list of a few more that we may encounter in these notes.

Deﬁnition 2.1.8. Suppose X is a topological space.

1. The Topological space X is called is called locally compact, if X is Hausdorﬀ and if every point x ∈ X has a compact neighborhood.

2. The Topological space X is called a Compactly generated space, if X is Hausdorff and if a subset A ⊆ X is closed ⇐⇒ A ∩ C is closed for every compact subset C of X (see [W, pp. 18]). (The definition in [H, Def. 2.4.21] seems to differ a little. Perhaps, the goal in [H] was provide better functorial sense (see [H, Prop. 2.4.22].)

Following construction is useful, for our purpose.

Construction 2.1.9. Suppose X is Hausdorﬀ space. We construct a new topology, denoted by k(X), as follows: (1) Underlying set in k(X) is X, i.e. it is a topology on X. (2) A subset A ⊆ X is is closed in k(X) ⇐⇒ A ∩ C is closed for every compact subset C of X. We have the following observations:

1. k(X) ,→ X is continuous. So, closed sets in X are closed in k(X).

2. k(X) is compactly generates.

3. Let K denote the category of compactly generated spaces. Then, there is an inclusion functor K ,→ Top. Further,

X 7→ k(X) deﬁnes as functor Hausdorﬀ −→ K .

where Hausdorﬀ denotes the category of Hausdorﬀ spaces. (Compare with [H, Prop. 2.4.22].) 42 CHAPTER 2. HOMOTOPY THEORY

Given two topological spaces, X,Y , we would have to work with the function spaces MorTop(X,Y ). Following, is how MorTop(X,Y ) topologized.

Deﬁnition 2.1.10. Let X,Y be two topological spaces. Then, MorTop(X,Y ) may be called a function space. We deﬁne two topologies on MorTop(X,Y ).

1. First, we deﬁne the Compact-Open Topology. For any two subsets A, B ⊆ X, let

C(A, B) = {f ∈ MorTop(X,Y ): f(A) ⊆ B}.

B = {C(K,U): K is compact subset and U is open subset of X}.

The Compact-Open Topology on MorTop(X,Y ) is the topology generated by B.

Unless stated otherwise, MorTop(X,Y ) together with Compact-Open Topology will

be devoted by MorTop(X,Y ).

(a) That means, Compact-Open Topology on MorTop(X,Y ) is the smallest topol-

ogy on MorTop(X,Y ), containing B. One can see an open set in this topology is union of ﬁnite intersection of members of B.

(b) Let x0 ∈ X. The the evaluation map evx0 : MorTop(X,Y ) −→ Y is continuous. −1 Because, for an open set U of Y , evx0 (U) = C({x0},U). (c) The following Lemma is about Hausdorﬀ property: Lemma 2.1.11. If X,Y is Hausdorﬀ, then the Compact-Open Topology on

MorTop(X,Y ) is also Hausdorﬀ.

Proof. Let f, g ∈ MorTop(X,Y ) and f 6= y. So, there is x0 ∈ X such that y1 :=

f(x0) 6= g(x0) =: y2. Since Y is Hausdorﬀ, there are open neighborhood y1 ∈ V1

and y2 ∈ V2 such that V1 ∩ V2 = φ. It follows C({x0},V1) ∩ C({x0},V2) = φ and

f ∈ C({x0},V1), g ∈ C({x0},V2). The proof is complete.

2. For topological spaces X,Y , F(X,Y ) would denote the compactly generated topo-

logical space k (MorTop(X,Y )).

2.2 Homotopy

The set of all real numbers would be denoted by R. Denote, the unit sphere and the unit disc/Ball, as follows

n n+1 n n S = x ∈ R :k x k= 1 , D = {x ∈ R :k x k≤ 1} . 2.2. HOMOTOPY 43

For n = 0, we let D0 = {0} and S−1 = φ. Also, write I := [0, 1], for the unit interval. So, n−1 n n n ∼ n we have inclusions S ⊆ D ⊆ R . Note D = I are homeomorphic. Deﬁnition 2.2.1. We give variety of deﬁnitions and notations, related to homotopy.

1. Let X,Y be topological spaces and f, g : X −→ Y be two continuous maps. We say f is homotopic to g, if ( H(x, 0) = f(x) ∃ a continuous map H : X × I −→ Y 3 ∀ x ∈ X (2.1) H(x, 1) = g(x). Homotopy is an equivalence relation. We deﬁne the Homotopy Category H omoTop of topological spaces, as follows:

(a) The objects of H omoTop are the topological spaces. (b) For topological spaces X,Y , and continuous maps f : X −→ Y , let f denote the homotopy equivalence class of f. The set of moronism in H omoTop is given by

MorH omoTop(X,Y ) := Mor(X,Y ) := {f : f : X −→ Y is a continuous map}

2. Now we consider Homotopy of maps in the category Top•. Assume X,Y are pointed

spaces, with base points x0 ∈ X and y0 ∈ Y . Assume f, g : X −→ Y preserve

base points. (In other words, f, g :(X, x0) −→ (Y, y0) are two maps in the category

Top•.) Then, we say that f is homotopic to g, if  H(x, 0) = f(x) x ∈ X  ∃ continuous map H : X × I −→ Y 3 H(x, 1) = g(x) x ∈ X (2.2)   H(x0, t) = y0 t ∈ I We say that H is a base point preserving homotopy. This homotopy of base point preserving maps is also an equivalence relation. Deﬁne the Homotopy Category

H omoTop• of pointed topological spaces, in the same way as H omoTop.

3. Now we consider Homotopy in th category TopPairs, of topological pairs (X,X0).

Recall, a map f :(X,X0) −→ (Y,Y0) in TopPairs is a continuous map f : X −→ Y

such that f(X0) ⊆ Y0.

Given two maps f, g :(X,X0) −→ (Y,Y0) of topological pairs, f is deﬁned to homotopic to g, if  H(x, 0) = f(x) x ∈ X  ∃ continuous map H : X × I −→ Y 3 H(x, 1) = g(x) x ∈ X (2.3)   H(x, t) ∈ Y0 x ∈ X0, t ∈ I 44 CHAPTER 2. HOMOTOPY THEORY

In this case also, homotopy of topological pairs is an equivalence relation. Deﬁne the Homotopy Category H omoTopPair of topological pairs, in the same way as H omoTop. As for the original categories, we have a commutative diagram of functors: ¯i H omoT op• / H omoT op 6 ¯ι ¯j H omoT opP air Here the vertical functor represents a full subcategory.

n n 4. Fix a base point ∗ ∈ S , say ∗ = (1, 0,..., 0) ∈ S . Given a pointed space (X, x0), n πn(X, x0) would denote the set of all equivalences classes of maps (S , ∗) −→ (X, x0). That means,

( n πn(X, x0) := MorH omoTop• ((S , ∗), (X, x0)) is the representable functor, n from H omoT op• −→ Set corresponding to the object (S , ∗). (2.4)

th This set πn(X, x0) is referred to as the n -homotopy set (at x0). This study of these

sets πn(X, x0) constitutes the Classical Homotopy Theory.

Remark. The interpretation (2.4), of πn, as a representable functor, provides a clue, n that one can replace (S , ∗), by any sequence (Tn, ∗) of pointed spaces, and develop n a theory considering the sets τn(X, x0) := MorH omoTop• ((T , ∗), (X, x0)), parallel to the Classical Homotopy Theory.

(a) So, π0(X, x0) is the set of all path connected components of X.

(b) Note, π1(X, x0) is called the Fundamental Group of X.

covariant functor from Top• −→ Set, and we have a commutative diagram of funtors: Top• πn

& omoT op / Set H • πn More explicitly, given continuous maps

ϕ :(X, x0) −→ (Y, y0) denote πn(ϕ): πn(X, x0) −→ πn(Y, y0) 2.2. HOMOTOPY 45

(d) Subsequently, we explain that, for n ≥ 1, there are group structures on πn(X, x0).

In fact, the group structure on πn(X, x0) would be abelian groups, for n ≥ 2.

n n n n n 5. Let ∂I ⊆ I denote the boundary of I . Consider the maps (I , ∂I ) −→ (X, x0) of pairs of topological spaces and homotopy.

(a) If follows:

Lemma 2.2.2. The set πn(X, x0) is in bijection with the set of equivalence n n classes of maps (I , ∂I ) −→ (X, x0) of pairs of topological spaces. In other words, there is a bijection

∼ n n πn(X, x0) −→ MorHomoT opP air ((I , ∂I ), (X, x0)) .

n In Outline of the Proof. First, consider the quotient map λ : I ∂In . Let n In ∗ = [∂I ] ∈ ∂In be the base point. Then, n ∼ In i. There is a homeomorphism θ :(S , ∗) −→ ∂In , ∗ . ii. There is a bijection

( n ∼ Mor I , ∗ , (X, x ) −→ Mor ((In, ∂In), (X, x )) Top• ∂In 0 TopPair 0 sending f 7→ fλ

n iii. A continuous map f :(S , ∗) −→ (X, x0) in Top•, gives rise to a map −1 n n fθ λ :(I , ∂I ) −→ (X, x0). This deﬁnes a bijection

n n n HomTop• ((S , ∗), (X, x0)) −→ HomTopPair ((I , ∂I ), (X, x0))

So, we established something stronger. one further check that this bijection respects homotopy.

(b) Using this description, it follows that, for n ≥ 1, the set πn(X, x0) has a group structure, which is abelian if n ≥ 2. n n For f, g :(I , ∂I ) −→ (X, x0) the product is given by the class of ( f(2t , t , . . . , t ) 0 ≤ t ≤ 1 (fg):(In, ∂In) −→ (X, x ) 3 (fg)(t , . . . , t ) = 1 2 n 1 2 0 1 n 1 g(2t1 − 1, t2, . . . , tn) 2 ≤ t1 ≤ 1

One checks, that this induces a well deﬁned product on πn(X, x0), for n ≥ 1. (c) Obviously, the constant map is the identity, which needs a proof, as well. 46 CHAPTER 2. HOMOTOPY THEORY

(d) For n ≥ 2, the groups πn(X, x0) is abelian. This also needs proof, that used the second coordinate.

6. Suppose ϕ, ψ :(X, x0) −→ (Y, y0) are homotopic, in the sense of (2.2), then

πn(ϕ) = πn(ψ): πn(X, x0) −→ πn(Y, y0) ∀ n ≥ 0.

Proof. This is reinterpretation of the fact (2.4), that

n πn(X, x0) := MorH omoTop• ((S , ∗), (X, x0)) is the representable functor.

∼ 7. Suppose X is a path connected space and x0, x1 ∈ X. Then, πn(X, x0) −→ πn(X, x1). To establish this, in two steps:

(a) As always, I = [01]. Let γ : I −→ X be a path, with γ(0) = x0 and γ(1) = x1. n n Also, let f :(I , ∂I ) −→ (X, x0) is a map in TopPair. Deﬁne

n n µ : I −→ [0, 1] by µ(t) = max{| ti − .5 |}, where t = (t1, . . . , tn) ∈ I .

1 n Note µ(t) ≤ 2 ∀ i ∈ I . Now deﬁne ( f(2t − .5) if µ(t) ≤ 1 f ∗ γ :(In, ∂In) −→ (X, x ) by f ∗ γ(t) = 4 1 1 1 γ(4µ(t) − 1) if 4 ≤ µ(t) ≤ 2

n n Note, f ∗ γ :(I , ∂I ) −→ (X, x1) is a map in TopPair. One checks, if f, g are homotopic in TopPair, then f ∗ γ is homotopic to .f ∗ γ. This establishes that γ induces a map

Ψγ : π0(X, x0) −→ π(X, x1).

In fact, Ψγ is a homomorphism, for n ≥ 1.

(b) In fact, Ψγ is a bijection (or isomorphism for n ≥ 1). 0 0 Let γ (t) = γ(1 − t). Then γ is a path from x1 to x0. One checks that Ψγ and

Ψγ0 are inverse of other.

8. Now, suppose f, g : X −→ Y are homotopic, in Top only, by a homotopy H : X × I −→ Y . Then,

For, x0 ∈ X α(t) = H(x0, t): I −→ Y, is the trajectory of x0 y0 := f(x0) 7→ y1 = g(x0).

Then, conjugation by α deﬁnes an isomorphism

hα : πn(Y, f(x0)) −→ πn(Y, g(x0)) 2.3. FIBRATIONS 47

It follows, that the diagram

πn(f) πn(X, x0) / πn(Y, y0)

o hα commute. π (g) n & πn(Y, y1)

2.3 Fibrations

I recovered three deﬁnitions of Fibrations [H, S, W]. However, they are about homotopy lifting property. We ﬁrst recall the one from [H].

Definition 2.3.1 (See [H]). Let J denote the set of inclusions Dn ,→ Dn × I sending x 7→ (x, 0). A map f : X −→ Y is called a fibration, if f has right lifting property, with respect to J, in the sense of the diagram, n / X In / X D _ ; _ ; ϕ ϕ f Equivalently, f Dn × I / Y In × I / Y meaning, given the outer commutative diagram of continuous maps, there is a continuous map ϕ, making all the triangles commute. Such fibrations are often known as weak fibrations (or Serre fibration). (While we give two more definitions from [S, W], by default "fibration", would mean a weak fibration.)

Corollary 2.3.2. The constant map X −→ {x0} is trivially a ﬁbration. This fact is stated as, every topological space X is ﬁbrant.

Proof. Suppose F0 : D −→ X. Define ϕ : D × I −→ X, by ϕ(x, t) = F0(x). So, the diagram n / X D _ : ϕ f commute. n D × I / {x0} We recall two more definitions, from Spanier [S] and Whitehead [W]. Definition 2.3.3. Let p : E −→ B be a continuous map.

1. (See Spanier [S, pp. 66]): We say p is a topological ﬁbration, if p has homotopy lifting property, with respective to every space X:

X / E _ ; i0 p , which is called homotopy lifting property. X × I / B F 48 CHAPTER 2. HOMOTOPY THEORY

A topological ﬁbration is also called a Hurewicz ﬁbration.

2. (See [W, pp. 29]): Under the same notations, as above, p is called a ﬁbration, which we may call White ﬁbration, the homotopy lifting property holds for X in the category K of Compactly generated topological space (see [H, pp. 18]).

3. Therefore,

f is topological fibration =⇒ f is White fibration =⇒ f is Weak fibration.

The following [H, Lemma 2.4.16] is of our primary interest.

−1 Lemma 2.3.4. Suppose p : X −→ Y is a weak ﬁbration, and x0 ∈ X. Let F = p (p(x0)) and y0 = p(x0). Let ι : F,→ X denote the inclusion map. Then, there is a long exact sequence ι∗ p∗ ··· / πn+1(F, x0) / πn+1(X, x0) / πn+1(Y, y0)

∂n+1 ι∗ p∗ / πn(F, x0) / πn(X, x0) / πn(Y, y0)

/ ········· (2.5)

/ π1(F, x0) / π1(X, x0) / π1(Y, y0)

/ π0(F, x0) / π0(X, x0) / π0(Y, y0)

Here ∂n is group homomorphism, ∀ n ≥ 2. For n = 0,

1. π0(X, x0) is to be understood to be a pointed set, where the path component of x0 is the base point.

2. By kernel, we mean the inverse image of the base point.

3. By exactness, we mean that the image is same as the kernel.

Proof. See [S, Theorem 7.2.10] for a detailed proof. While the maps ι∗ and p∗ are the obvious functorial maps, we deﬁne the connecting homomorphism ∂ = ∂n+1 : πn+1(Y, y0) −→ n+1 n+1 πn(F, x0). Let α = [γ] ∈ πn+1(Y, y0), represented by a map γ :(I , ∂I ) −→ (Y, y0). n Let γ0 : I × 0 −→ X be the constant map x0. Then, we have the commutative diagram:

γ0 In × 0 / X ; p g n I × I γ / Y 2.4. CONSTRUCTION OF FIBRATIONS 49 where g is obtained by homotopy/left lifting property of the fibration p. It follows, n n+1 n n pg(∂I ) ⊆ γ(∂I ) = {y0}. Therefore g(∂I ) ⊆ F . Define g1 : I −→ F by g1(t) = g(t, 1). n n n Then, g1(∂I ) = x0. Therefore, g1 :(I , ∂I ) −→ (F, x0). Define ∂(α) = [g1] ∈ πn(F, x0). One checks that ∂ is well defined and the exactness.

2.4 Construction of Fibrations

This section is some excerpts from [W, Secton I.7]. Not surprisingly, we have some interest in the long exact sequence (2.5), while it does not appear to be easy to detect whether a continuous map p : X −→ Y is a ﬁbration or not. Given a continuous map p : X −→ Y , we would replace it by a ﬁbration X −→ Y where X is homotopic to X .

1. As usual I = [0, 1].

2. For a space Y , let P(Y ) := P(I,Y ) := F(I,Y ) be the space of all paths in Y . See (2.1.10) for topology on the function space F(I,Y ).

3. Suppose p : X −→ B be a continuous map.

4. Let Ip = {(x, u) ∈ X × P(B): p(x) = u(0)} . So, Ip ⊆ X × P(Y ). More explicitly, Ip is the set pairs (x, γ) where x ∈ X and γ is part in Y , starting at p(x).

The following are some results from [W, Secton I.7].

Theorem 2.4.1. For any two spaces , X,Y the projections X × Y −→ X is a (White) ﬁbration.

Corollary 2.4.2. The following are (White) ﬁbration.

 p : P(X) −→ X × X p(u) = (u(0), u(1))  p0 : P(X) −→ X p0(u) = u(0)   p1 : P(X) −→ X p1(u) = u(1)

Theorem 2.4.3. Composition of (White) ﬁbration is a (White) ﬁbration.

Theorem 2.4.4 (Hurewicz). Let p : X −→ B be continuous and B is paracompact. Then, −1 p is a (White) ﬁbration if there is a an open cover B = ∪iVi such that p Vi −→ Vi are ﬁbrations. 50 CHAPTER 2. HOMOTOPY THEORY

Theorem 2.4.5. Let p : X −→ B be a ﬁbration (any kind) and f : B0 −→ B be continuous. Then, the pullback p0 : X0 −→ B0 is a ﬁbration:

X0 / X

p0 B0 / B f In particular, for A ⊆ B, the map p−1A −→ A is a ﬁbration.

Proof. Easy, diagram chase. Corollary 2.4.6. Let f : X −→ Y be a map. Then, If −→ X is a (white) ﬁbration.

Proof. The following f 0 If / P(Y )

0 p0 p0 is a pullback. X / Y f 0 Since p0 is a ﬁbration, so is p0. The proof is complete.

We deﬁne a few functions:  For y ∈ Y ey : I −→ Y the constant map y   ε : Y −→ P(Y ) ε(y) = ey   p0 : P(Y ) −→ Y p0(γ) = γ(0)   p1 : P(Y ) −→ Y p(γ) = γ(1) 0 f 0  f : I −→ P(Y ) f (x, γ) = γ  0 f 0  p0 : I −→ X p0(x, γ) = x  f  λ : X −→ I λ(x) = (x, εf)  f 0  p : I −→ Y p = p1f

Following are helpful commutative diagrams:

f X / Y

ε So, εf(x) = ef(x). εf " P(Y )

X εf λ f 0 If / P(Y ) f 0 $ If / P(Y ) 1 p0 p1 p 0 p p0 0 XY" X / Y f Note, the map f : X −→ Y does not ﬁt in the bottom of the last diagram. 2.4. CONSTRUCTION OF FIBRATIONS 51

Theorem 2.4.7. With the notations as above, we have

0 f 1. The map p = p1f : I −→ Y is a (White) ﬁbration.

2. First„ ( 0 0 p0λ = 1X , λp0 ' 1If 0 pλ = f, fp0 ' p

0 3. So, importantly, p0 and λ are homotopy equivalences.

4. Explicitly,

0 p((x, γ)) = p1f (x, γ) = p1(γ) = γ(1) the other end of γ.

0 Proof. See [W, Thereom 7.30]. First, we prove that p = p1f is a (white) ﬁbration. Consider the commutative diagram:

g Z × 0 / If _ p f Write g(z, 0) = (g1(z), g2(z)(s)) ∈ I ⊆ X × P(Y ) with s ∈ I. Z × I / Y H

Since the path g2(z)(s)) starts at f(g1(z)) and because of the commutativity, we have

g2(z)(0) = f(g1(z)) g2(z)(1) = H(z, 0)

For (z, t) ∈ Z × I, deﬁne a path  g2(z)(s) if t = 0  γ(z, t): I −→ Y γ(z, t)(s) = g2(z)(s/t) if t 6= 0, s ≤ t  (s−t)t  H z, 1−t if 0 < t < s

Then, γ(z, t) is a path from g2(z)(0) = f(gi(z)). Also, γ(z, t)(1) = H(z, t). Now, deﬁne

f G : Z × I −→ I G(z, t) = (g1(z), γ(z, t))

So, H lifts to a homotopy G. This establishes that f 0 is a (white) ﬁbration. (It looks correct. But γ(z, t) looks like a loop: γ(z, t)(0) = γ(z, t)(1)) The equalities in (2) are obvious. The last equivalence follows from third and second: 0 0 0 pλ = f =⇒ pλp0 = fp0 =⇒ p ' fp0. 0 0 So, we need to prove λp0 ' 1X . Note λp0(x, γ) = (x, eγ(0)). Now, F (−, 0) = λp0 With F ((x, γ(s)), t) = (x, γ(st)) 0 F (−, 1) = 1

This completes the proof of (2). Now, (3) follows from (2). 52 CHAPTER 2. HOMOTOPY THEORY

Corollary 2.4.8. Suppose ∗ := y0 ∈ Y is the base point. Then, ﬁber

f −1 T := p (∗) = {(x, γ) ∈ X × P(Y ): γ(1) = ∗ = y0, f(x) = γ(0)}

f So, (x, γ) ∈ T , if γ is a path from f(x) 7→ y0.

Following is the summary:

Corollary 2.4.9. Let f : X −→ Y be a continuous map. Then, we have the diagram

p Tf / If / Y

0 p0 o commute up to homotopy X / Y f and p is a White Fibration and hence a weak fibration. The fiber of p is Tf So, up to homotopy, f fits in a Homotopy Fibration, which is the top line. It follows from Lemma 2.3.4, there is a long exact sequence

f ··· / πn(T , x0) / πn(X, x0) / πn(Y, y0)

∂n f / πn−1(T , x0) / πn−1(X, x0) / πn−1(Y, y0)

/ ········· (2.6)

f / π1(T , x0) / π1(X, x0) / π1(Y, y0)

f / π0(T , x0) / π0(X, x0) / π0(Y, y0)

Notation 2.4.10. For a map f : X −→ Y and y0 ∈ Y , Quillen [Q], uses the notation f F (f, y0) := T , as described above (2.4.8). We would possibly, use Quillen’s notations subsequently.

The Loop Space

Deﬁnition 2.4.11. With X = ∗ the singleton and f : X −→ Y , with f(∗) = y0, in Corollary 2.4.9, we write

f ΩY := T to be called the loop space of Y at y0. 2.5. CW COMPLEXES 53

The loop space ΩY is to be considered as a pointed space, with constant path y0 to be the base point. It follows form the long exact sequence (2.4.9) that

πi (ΩY, ∗) = πi+1 (Y, y0) ∀ i ≥ 0

2.5 CW Complexes

An very important category, associated to Top, is the category of CW complexes. Quillen’s work [Q] is heavily dependent on the topology of CW complexes. This material is available in any standard source, and among them is the book of Hatcher [Ha]. Before we give a deﬁnition of CW complexes, we give some deﬁnitions. As before (see Section 2.2), Dn ⊆ Rn will denote the unit disc, ∂Dn := Sn−1 ⊆ Dn will denote the unit sphere, which is the boundary of Dn.

Deﬁnition 2.5.1. Let (X,A) be a topological pair and f : X −→ Y be a continuous map. ` Deﬁne the topological space X f Y by the push forward diagram A / X

f ιX in Top. Y / X ` Y ιY f ` ` So, X f Y is constructed, set theoretically, from the disjoint union X Y by identifying ` a ∼ f(a) ∀ ∈ A. The topology of X f Y , is the largest one that makes ιX and ιY ` continuous. We say that X f Y is obtained from Y , by attaching X, along A, via f. Of our particular interest would be the case, when (X,A) = (Dn, Sn−1), which encap- sulates the concept of "attaching cells".

Deﬁnition 2.5.2. Let n ≥ 0 and Un = {x ∈ Dn :k x k< 1}, denote the interior of Dn. Suppose X is a topological space. An n-cell of X is a subspace U ⊆ X, which is homeomorphic of Un. A cell in X is an n-cell, for some n ≥ 0. An n-cell U would, sometimes, be denoted by U n, with the superscript, indicating the dimension. We also write dim(U) = n, for an n-cell U.

A CW complex is deﬁned as follow.

Deﬁnition 2.5.3. A CW complex is a topological space X, together with a sequence of subspaces X0 ⊆ X1 ⊆ · · · ⊆ Xn · · · ⊆ X, such that, the following conditions hold:

∞ n 1. X = ∪i=0X . 54 CHAPTER 2. HOMOTOPY THEORY

2. X0 has discrete topology.

n n−1 n 3. Inductively, X is obtained from X , by attaching n-cells eα, via maps ϕα : Sn−1 −→ Xn−1. This means, there is an indexing set In and a family of contin- n−1 n−1 n uous maps {ϕα : S −→ X : α ∈ I }, ` `n n−1 ϕα n−1 S / X α∈I _ n−1 n−1 n {ϕα : S −→ X : α ∈ I } 3 is a push forward in Top. `n n n α∈I D / X By deﬁnition (2.5.1), the topology on Xn is determined by this diagram.

4. A subset A ⊆ X is open (or closed) if and only if A ∩ Xn is open (or closed) in Xn, for all n ≥ 0. (This condition is redundant, if X = Xn for some n.)

Following are comments and Information:

1. The above definition was taken from the book of Hatcher [Ha]. As is pointed out in [Ha], a variant of this definition in available in [W]. Many other definitions averrable in the literature, assumes that X is Hausdorff.

2. We have,

n a m a m X = eα ,X = eα are disjoint union. α∈Im,m≤n α∈Im,m≥0 The subspace Xn is called the n-skeleton of X.

n 3. For α ∈ I , the map ϕα extends as follows:

n−1 α n−1 S _ / X

n ` n n D / β∈In D / X _

Φα * X

n n ∼ The map Φα is called the characteristic map of eα. The restriction (Φα)|Un : U −→ n eα is a homeomorphism.

−1 4. It follows, a subset A ⊆ X is is open (or closed) if and only if Φα (A) is s open (or n S∞ n closed) in D , ∀ α ∈ n=0 I . To prove this, given a subset A ⊆ X, one needs to check, A ∩ Xn is open (or closed), which is done by induction. 2.5. CW COMPLEXES 55

n0 n 5. For each n0 ≥ 0, X is closed in X. To see this, ﬁrst note that, for any α ∈ I , −1 n−1 n−1 n n Φα (X ) = S , is closed in D . Therefore, Xn−1 ,→ Xn is closed in X , which follows from pushout topology on Xn. Therefor Xn0 ∩ Xn is closed in Xn, for all n.

6. Important Remark: More formally, one should say that, for given a topological space X, a CW structure on X is given by the above process. Such a process provides n−1 n−1 n the above information, including the families {ϕα : S −→ X : α ∈ I , n ≥ 0}. Alternately, a CW structure on X can be given by

n (a) A family E := {eα ⊆ X : α ∈ I , n ≥ 0} of cells on X such that

a m n a m X = eα and, let X := eα are disjoint union. α∈Im,m≥0 α∈Im,m≤n

n n n ∼ (b) For α ∈ I , a Characteristic maps Φα : D −→ X, such that (Φα)|Un : U −→ eα n−1 n−1 is a homeomorphism and Φα(S ) ⊆ X .

The push forward condition (3) would be a consequence of the above two conditions. Further, some authors like say (X, E ) is a CW complex and E is called a cell decomposition of X. So, a topological space can have more than one CW structures.

We deﬁne sub complexes of CW complexes.

Deﬁnition 2.5.4. Suppose X is a CW complex, with cell decomposition E := {eα ⊆ X : α ∈ In, n ≥ 0}. A closed subset Y ⊆ X is called a sub complex of X, if Y is union of a n n n subfamily of cells {aα : α ∈ J , n ≥ 0}, where J ⊆ I .

The following is a key lemma on CW complex.

Lemma 2.5.5. Suppose X is CW complex. Let Y ⊆ X be a compact subset. Then, Y is contained in a ﬁnite sub complex of X.

Proof. First, it is shown that Y can only meet ﬁnitely many cells in X. Inductively, for i = 1, 2,..., we can choose x ∈ eni ∩ Y , where α 6= α ∀ i 6= j. Let Z = {x , x ..., }. We i αi i j 1 2 claim that Z is closed. Inductively, assume Z ∩ Xn−1 is closed. It would suﬃce to show n −1 n n ∀ α ∈ I , Φα (X ∩ Z) is closed in D . Now,

−1 n −1 n−1 −1 n Φα (X ∩ Z) = Φα (X ∩ Z) ∪ Φα (eα ∩ Z)

By induction, the first one is a closed set in Sn−1 and the second one is a singleton or null. This establishes that Z is a closed set, and hence compact. Same argument shows that 56 CHAPTER 2. HOMOTOPY THEORY any subset of Z is closed. Hence, Z is a discrete compact set. Therefore, Z is a finite set. This is a contradiction. Therefore, Y meets only finitely many cells of X. In other words, Y is contained in finitely many cells. So, it is shown that Finite union of cells is is contained in a finite sub n complex. So, it is enough to show, each eα is contained in a finite sub complex. 0 0 0 0 Clearly, the zero cells eα ⊆ X is closed in X . Since X0 is closed in X, {eα} is closed in X. So, zero cells are contained in a finite sub complex. Now, use induction. n n n−1 n Consider the characteristic map Φα : D −→ X. Then, Φα(D ) = ϕα(S ) ∪ eα. Now, ϕ ( n−1) ⊆ Xn−1. Therefore ϕ ( n−1) ⊆ ∪ ∪m eni , with n ≤ n − 1. So, by induction, α S α S i=1 αi i n−1 n−1 n ϕα(S ) ⊆ A, where A ⊆ X is a finite sub complex. Now, it is checked that A ∪ eα is n n−1 n−1 −1 n n m closed, because (A∪eα)∩X is closed in X and Φα (A∪eα) = D . For β ∈ ∪m≤nI , n−1 n−1 −1 n −1 n α 6= β, Φβ(S ) ⊆ X . Therefore, Φβ (A ∪ eα) = Φβ (A), which is closed, in X and hence in X. Therefore, A ∪ eα is a sub complex. The proof is complete. We proceed to prove that CW complexes are Hausdorff.

Lemma 2.5.6. Suppose X is a CW complex and x0 ∈ X. Then, {x0} is closed.

n n−1 n−1 n Proof. Suppose x0 ∈ X \ X . Then, {x0} ∩ X = φ, and x0 ∈ eα, for some unique n n n α ∈ I . Now, it follows from push out topology on X that {x0} is closed in X and hence in X. The following is an useful construction of open sets.

Construction 2.5.7. Suppose X is a CW complex, as above.

1. Let U ⊆ Xn0 be an open subset, in Xn0 . Then, there are open sets N(U) ⊂ X (not unique) such that N(U) ∩ Xn0 = U. The proof below would provide more explicit construction of such open sets.

2. Let U, V ⊆ Xn0 be open subsets, in Xn0 , with U ∩ V = φ. Then, there are open sets N(U),N(V ) ⊂ X such that N(U)∩Xn0 = U, N(V )∩Xn0 = V and N(U)∩N(V ) = φ.

n n n Proof. First, we give a easy version. For n ≥ n0, define N (U). Suppose, N (U) ⊆ X one in Xn has been defined. For α ∈ In+1 define

−1 n n Uα = Φα {rx : x ∈ Φα N (U) ⊆ S , 0 < r ≤ 1} ! n+1 n [ n+1 Let N (U) = N (U) ∪ Uα , which in open in X . α∈In+1 [ Let N(U) = N n+1(U), which in open in X.

n≥n0 2.5. CW COMPLEXES 57

n This method can be reﬁned, as follows. For all α ∈ ∪n≥n0+1I , let 0 < α < 1. Redeﬁne

−1 n n Uα = Φα {rx : x ∈ Φα N (U) ⊆ S , α < r ≤ 1}

Then, N(U) deﬁned as above, would be an open set in X. Now, for U, V ⊆ Xn0 , the constructions above shows N(U) ∩ N(V ) = φ. The proof is complete.

Proposition 2.5.8. Suppose X is a CW complex. Then, X is Hausdorﬀ.

m n Proof. Let x 6= y ∈ X. Then, x ∈ eα and , y ∈ eβ. We assume m ≤ n. If n = m, we can find open sets U, V , in Xn, with x ∈ U, y ∈ V and U ∩ V = φ (whether or not α = β. Let N(U), N(V ), be the open sets, as in (2.5.7). Then, N(U),N(V ) satisfy the Hausdorff conditions that x ∈ N(U), y ∈ N(V ) and N(U) ∩ N(V ) = φ. Now suppose m ≤ n − 1. Let U be an open neighborhood of x, in Xm. For k ≤ n − 1, n n construct N (U), as in (2.5.7). Let y0 ∈ U such that Φβ(y0) = y. Let k y0 k< β < 1. Now, define

0 −1 n n n U := Uβ = Φβ {rz : z ∈ Φβ N (U) ⊆ S , β < r ≤ 1} ,V := Vβ = Φβ ({z ∈ D :k z k< β})

n Now, Uβ ∩ Vβ = φ and they are open neighborhood of x, y (respectively) in X . Now, N(U 0),N(V ) satisfy the Hausdorﬀ property.

Proposition 2.5.9. Let X be CW complex. Then X is compactly generated.

Proof. We already proved that X is Hausdorﬀ. Let A ⊆ X be a subset such that A ∩ C is closed, for all compact subsets C of X. For α ∈ In, Φ(Dn) is compact. By hypothesis, 0 n A := A ∩ Φα(D ) is closed. Therefore,

−1 −1 0 Φα (A) = Φα (A ) is closed.

Therefore, A is closed. The proof is complete.

2.5.1 Product of CW Complexes

Let (X, E ), (Y, F ) be two CW complexes. Then, it is desirable that the topological space X ×Y would have a natural CW structure. Unfortunately, this is does not a simple answer. We build up our thinking: n n Denote the indexing set of E and I , n ≥ 0 and let Φα : D −→ X be the characteristic n n maps. Denote the indexing set of F and J , n ≥ 0 and let Ψβ : D −→ Y be the characteristic maps. Also,

n n n n E = {eα : α ∈ ∪n≥0 I } , F = εβ : β ∈ ∪n≥0 J 58 CHAPTER 2. HOMOTOPY THEORY

n ∼ n n−1 ∼ n 1. It may be more intuitive, to identify D = I and S = ∂I . So, a characteristic n n map would be considered as Φα : I −→ X or Ψβ : I −→ Y .

m n m+n 2. For α ∈ I , β ∈ J , there are the natural maps Φα × Ψβ : I −→ X × Y . Note, m+n Φα × Ψβ is a homeomorphism on ∂I .

n i j n 3. Let (I ? J ) = ∪i+j=nI × J , I ? J = ∪n≥0 (I ? J ) and

i j E ? F = eα × εβ :(α, β) ∈ I ? J

The following theorem on product of CW complexes would be useful in Quillen K-Theory.

Theorem 2.5.10. Let (X, E ), (Y, F ) be two CW complexes. Use the notations Φα, Ψβ for the characteristic maps, and other notations as above. Then, E ? F provides a CW structure on k(X × Y ), the compactly generated topology corresponding to X × Y . If X or Y is locally compact, then k(X × Y ) = X × Y .

Proof. See [Ha]. For the purpose of this proof, we denote Z := k(X × Y ). By deﬁnition,

n [ i j Z := eα × εβ (α,β)∈I?J )m,m≤n

Denote a a n n (Φ ? Ψ)n = Φα × Ψβ : I −→ Z n (α,β)∈(I?J ) (α,β)∈(I?J )n n n−1 Note, the restriction of (Φ ? Ψ)n to ∂I maps in to Z . One needs to prove that

` ∂In / Zn−1 (α,β)∈I?J )n is a pushout. ` n n (α,β)∈I?J ) I / Z n (Φ?Ψ)n

This obvious, set theoretically. Rest of the question is topological. The CW topology on X × Y is built, from bottom up, while topology on Z = k(X × Y ) is given. We need to check that these two topologies are same.

1. Suppose A ⊆ X ×Y be closed in the Z = k(X ×Y ). For each (α, β) ∈ (I ?J )n n ≥ 0 n the map Φα × Ψβ : I −→ X × Y is continuous, with the product topology. We have n −1 A ∩ (Φα × Ψβ(I )) is closed in the product topology. So, Φα × Ψβ (A) is closed. So, A is closed in the CW topology.

2. Suppose A ⊆ X ×Y be closed in the CW topology. Suppose C ⊆ X ×Y be a compact set. Then, C ⊆ C1×C2, where C1 ⊆ X, C2 ⊆ Y be projections of C. Then, C1 and C2 are compact. So, both C1 and C2 are, respectively, contained in ﬁnite sub complexes r r X = ∪r1 eni ⊆ X, Y = ∪r2 εmi ⊆ Y . It follows, C ⊆ X ×Y = S 1 S 2 eni ×εmi . 1 i=1 αi 1 j=1 βj 1 1 i=1 j=1 αi βj 2.5. CW COMPLEXES 59

To prove A ∩ C is closed in X × Y , we can replace X by X1 and Y by Y1 and assume X,Y are ﬁnite. It follows

r r r r [1 [2 [1 [2 X × Y = eni × εmi = Φ × Ψ (In) 1 1 αi βj αi βj i=1 j=1 i=1 j=1

r r r r [1 [2 [1 [2 A∩(X ×Y ) = Φ ×Ψ (In)∩(A∩(X ×Y )) = Φ ×Ψ Φ × Ψ−1(A) 1 1 αi βj 1 1 αi βj αi βj i=1 j=1 i=1 j=1

which is closed in X1 × Y1 ⊆ X × Y . Hence A ∩ C = A ∩ (X1 × Y1) ∩ C is closed in X × Y .

So, it is established that two topologies are same. The latter part of the theorem is about product of compactly generated topologies. The proof is complete.

2.5.2 Frequently Used Results

First we deﬁne weak equivalences.

Deﬁnition 2.5.11. Suppose f : X −→ Y is a continuous map. We say f is a weak equivalence, if

∀ x0 ∈ X, ∀ n ≥ 0 πn(f): πn (X, x0) −→ πn (Y, f(x0)) is a bijection.

Weak equivalence is an important concept, because of the following theorem of White- head.

Theorem 2.5.12. Let X,Y be two connected CW complexes. Then, a continuous map f : X −→ Y is a weak equivalence, if and only if, f is a homotopy equivalence.

Proof. See [Ha, Theorem 4.5].

The following property of CW complexes is another tool used in [Q].

Proposition 2.5.13. Suppose X is a CW complex and Xn denotes the n-skeleton and 0 x0 ∈ X . Then,

( n ∼ the map πi (X , x0) −→ πi (X, x0) is an isomorphism ∀ i ≤ n − 1 n the map πn (X , x0) πn (X, x0) is surjective

Proof. See [Ha, Corollary 4.12]. 60 CHAPTER 2. HOMOTOPY THEORY Chapter 3

Simplicial and coSimplicial Sets

3.1 Introduction

These notes on Simplicial Sets is based on notes available on the Internet or from [H].

3.2 Simplicial Sets

Deﬁnition 3.2.1. For integers n ≥ 0, denote the set [n] := {0, 1, 2, . . . , n}. Let ∆ be the category, whose objects are {[n]: n = 0, 1, 2,...} and morphisms are order preserving maps. So,

Mor∆([m], [n]) := ∆([m], [n]) := {f :[m] −→ [n]: f(i) ≤ f(j) ∀ 0 ≤ i ≤ j ≤ m}

This category is sometimes referred to as the Simplicial category.

1. ∆ has two obvious subcategories ∆+, ∆− whose objects are same as that of ∆.

The morphisms of ∆+ are consists of order preserving injective maps.

The morphisms of ∆− are consists of order preserving surjective maps.

2. For integers n ≥ 1, and 0 ≤ i ≤ n, deﬁne special type of morphisms in ∆+ ( k if k < i di :[n − 1] −→ [n]: di(k) = ”skip i”. k + 1 if i ≤ k ≤ n − 1

61 62 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Also, for integers n ≥ 1, and 0 ≤ i ≤ n − 1, deﬁne special type of morphisms in ∆− ( k if k ≤ i si :[n] −→ [n − 1] : si(k) = ”identifies i, i + 1”. k − 1 if i < k ≤ n

3. cosimplicial identities: djdi = didj−1 ∀i < j sjdi = disj−1 ∀i < j = id i = j, j + 1 (3.1) = di−1sj ∀ i > j + 1 sjsi = si−1sj ∀i > j

The following is an useful lemma. Lemma 3.2.2. Let f :[n] −→ [m] in ∆ be a morphism in ∆. Then, f can be written (uniquely) as: f = dik dik−1 ··· di1 sj1 sj2 ··· sjl with

0 ≤ j1 < j2 < ··· < jk < n 0 ≤ i1 < i2 < ··· < ik ≤ m

Proof. See [Mc]

Deﬁnition 3.2.3. Given a category C , a Simplicial Object in C a contravariant functor ∆ −→ C . Likewise, a covariant functor ∆ −→ C is called a coSimplicial Object in C We are mainly interested in simplicial sets, which are contravariant functors K : ∆ −→ Set. Such a simplicial set K, is given by

1. A sequence of sets Kn := K[n]. An element of x ∈ Kn is called an n-simplex and an

element x0 ∈ K0 is called a vertex.

i 2. The dual to d above, we have the face map di : Kn −→ Kn−1,

i 3. The dual to s above, we have the degeneracy map si : Kn−1 −→ Kn,

4. These maps are subject to the simplicial identities

didj = dj−1di ∀i < j

disj = sj−1di ∀i < j = id i = j, j + 1 (3.2)

= sjdi−1 ∀ i > j + 1

sisj = sjsi−1 ∀i > j These identities are obtained by reversing the arrows in (3.1). 3.2. SIMPLICIAL SETS 63

5.A map of simplicial sets τ : K −→ L is a natural transformation.

Such a map is equivalent to a collection of maps τn : Kn −→ Ln commuting with the face and degeneracy maps:

si di Kn oo Kn−1 Kn / Kn−1

τ τ τ τ Ln oo Ln−1 Ln / Ln−1 si di

6. Suppose K a simplicial set K and x ∈ Kn is a simplex. In this case, we say dim x = n

Any image of x under arbitrary iterations of face maps dis is called a face of x. Images of x under iterated degeneracies are called degeneracies of x. Since 0-iteration is included, x is both a face and degeneracy of itself.

A simplex x ∈ Kn is called non degenerate, if it is a degeneracy of itself ONLY.

Given any simplex x ∈ Km, it is the degeneracy of a unique non degenerate simplex

y ∈ Kn, for some n ≤ m. So, any x ∈ K0 is non degenerate, but its image s(x) ∈ K1 is not.

7. The simplicial sets, together with the maps of simplicial sets form a category, to be denoted by SSet.

8. A map L −→ K of simplicial sets would be called a sub-simplical set, if Ln −→ Kn is one to one, for all n ≥ 0. In other words, a sub-simplicial set is a simplicial set L

of K, such that Ln ⊆ Kn and the map L −→ K is a map of simplical sets. Further, K given sub-simplicial set L ⊆ K, the quotient L is deﬁne as follows:

K K = n . L n Ln

Example 3.2.4. First, ∀ (ﬁxed) n, the representable functor

∆(−, [n]) : ∆ −→ Sets given by ∆[n]([k]) = ∆([k], [n]) is a simplicial set. This is denoted by ∆[n] := ∆(−, [n]) ∈ SSet. So,

1. For each integer k ≥ 0, we can denote

∆[n]([k]) = {(0 ≤ n1 ≤ n2 ≤ · · · ≤ nk ≤ n)} 64 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

2. Non degenerate zero simplexes are given by ∆[n]([0]) are given by all the choices of integers 0, 1, . . . , n. A non degenerate k simplex is given by a strictly increasing

sequence 0 ≤ n1 < n2 < ··· < nk ≤ n.

Example 3.2.5. Then, ∀ (ﬁxed) n, the boundary ∂∆[n] is deﬁned to the smallest sub simplex of ∆[n] , whose not degenerate k simplexes are those of ∆[n], except the non ∼ degenerate n-simplex ιn :[n] −→ [n].

Example 3.2.6. ∀0 ≤ r ≤ n there is a simplicial set Λr[n] ∈ SSet, to be called the r-horn of ∆[n], whose nondegenerate k-simplices

= {ι ∈ ∆([k], [n]) : ι is injective, ι 6= 1n, ι 6= dr} where

dr = (0, 1, . . . , r − 1, r + 1, . . . , n):[n − 1] −→ [n] is the face opposite of the vertex r.

n ∆[n] Example 3.2.7. For n ≥ 0, we deﬁne S := ∂∆[n] ∈ SSet. It has two non degenerate n n simplexes ∗ ∈ S0 and σn ∈ Sn .(In deed, this quotient would correspond to the n-sphere, and so would ∂∆[n] and (∆[n], ∂∆[n])).

Reversing the arrows, one can also consider coSimplicial sets, which are the covariant functors K : ∆ −→ Set. Such a coSimplicial Set is given by

1. For each integer n ≥ 0, a set Kn.

2. For each f :[m] −→ [n] in ∆, (i. e. and order preserving map), a map K(f): Km −→ Kn, satisfying obvious laws of compositions.

Perhaps, the most natural example of coSimplicial set (in fact, topological space) may be the following.

Example 3.2.8. Let x0, x1, x2,... be a sequence of district symbols. For integers n ≥ 0, deﬁne ( n n ) X X Σn = λixi : ∀ i 0 ≤ λi ≤ 1 and λi = 1, i=0 i=0 For any map f :[m] −→ [n] in ∆, deﬁne

m ! n   X X X Σ(f):Σm −→ Σn by Σ(f) λixi =  λi xk i=0 k=0 f(i)=k 3.3. GEOMETRIC REALIZATION 65 3.3 Geometric Realization

Given a simplical set K, we would define its geometric realization |K|. In order to do that, we first define the geometric realization of ∆[n] as follows:

Example 3.3.1. We work in the category Top of topological spaces.

n 1. Let |∆[n]| ⊆ R be the convex hull of 0, e1, . . . , en with 0 = (0,..., 0) and e1, . . . , en is the standard basis of Rn. So,

n n X o |∆[n]| := (t1, . . . , tn) ∈ R : 0 ≤ ti ≤ 1, ti ≤ 1

∼ n n+1 X o = (t0, t1, . . . , tn) ∈ R : 0 ≤ ti ≤ 1, ti = 1

2. An order preserving map f :[m] −→ [n] induces a map |∆[m]| −→ |∆[n]| sending

ei 7→ ef(i). For the maps di ∈ ∆([n − 1], [n]), si ∈ ∆([n], [n − 1]), denote the corresponding maps di : |∆[n − 1]| −→ |∆[n]|, si : |∆[n]| −→ |∆[n − 1]|.

3. Hence [n] 7→ |∆[n]| is a cosimplicial topological space, to be denoted by |∆[−]|. We write |∆[−]| ∈ Top∆. In fact, |∆[−]| is precisely the coSimplicial set described in Example 3.2.8

4. The topological space |∆[n]| is to be treated as the Geometric Realization of the simplicial set ∆[n]. This is used to give a geometric realization |K| of any simplicial Set K ∈ SSet. The method given in [H], is a little too formal. So, we will use some what is available in the Net.

Deﬁnition 3.3.2. Let K ∈ SSet. Give each Kn the discrete topology. For f ∈ ∆([m], [n]), we have two maps

K(f): Kn −→ Km and |f| : |∆[m]| −→ |∆[n]|.

1. For (τ, y) ∈ Km × |∆[m]| and (σ, x) ∈ Kn × |∆[n]|, deﬁne

(σ, x) ∼ (τ, y) if ∃ f ∈ ∆([m], [n]) 3 K(f)(σ) = τ, |f|(y) = x.

2. We use all three notations, (σ, x) ∼ (τ, y), (σ, x) ∼f (τ, y) or (σ, x) ∼f (τ, y), in the above case. We would also say the f induces this relation. The relationship ∼ is reﬂexive and transitive, but not necessarily symmetric. 66 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

`∞ 3. Let ' denote the equivalence relation generated by ∼, on n=0 Kn × |∆[n]|. Note, this equivalence relation can be described, using the face and degeneracy maps only.

Deﬁne the geometric realization `∞ Kn × |∆[n]| a |K| := n=0 where denotes the disjoint union. (3.3) '

Possible choices of topology on the geometric realization |K| are as follows:

1. The geometric realization |K| inherits a topology from Kn × |∆[n]|. We would refer to this topology by the "usual topology".

2. The more frequently used, and standard, topology on |K| comes from the CW structure on |K|, described subsequently.

Lemma 3.3.3. This association | − | (3.3.2), deﬁnes a functor

| − | : SSet −→ Top

Conversely, deﬁne the following functor:

Deﬁnition 3.3.4. Deﬁne

S : Top −→ SSet by S(Y )n = MorTop (|∆[n]|,Y )

This will be denoted by Sing in [H, Rem 3.1.7].

Lemma 3.3.5. These two functors

| − | : SSet −→ Top and S : Top −→ SSet are adjoint to each other:

∼ Φ: MorTop (|K|,Y ) −→ MorSSet (K,S(Y )) ∀ K ∈ OB(SSet),Y ∈ OB(Top).

Outline of the Proof. For objects K in SSet and Y in Top, we deﬁne the bijections of the adjoint map: Given f : |K| −→ Y and an integer n ≥ 0, we obtain the diagonal compositions map in Top, as follows:

Kn × |∆[n]| / |K|

f This gives a map Φ(f)n : Kn −→ MorTop(|∆[n]|,Y ) = S(Y )n & Y 3.3. GEOMETRIC REALIZATION 67

Deﬁne

∼ Φ: MorTop (|K|,Y ) −→ MorSSet (K,S(Y )) by Φ(f) = {Φ(f)n : n ≥ 0}.

This way we get a map, subject to checking further details:

MorTop (|K|,Y ) −→ MorSSet (K,S(Y ))

Conversely, given F : K −→ S(Y ), we have

Fn : Kn −→ S(Y )n = MorTop (|∆[n]|,Y )

This deﬁnes a map Fen : Kn × |∆[n]| −→ Y Top.

One checks that Fen factors through a continuous map Ψ(F ): |K| −→ Y . Now, Ψ deﬁnes a maps Ψ: MorSSet (K,S(Y )) −→ MorTop (|K|,Y ) One further checks that Φ and Ψ are inverse of each other and Φ satisﬁes other conditions of Adjoints. The proof is complete. 68 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

3.3.1 The CW Structure on |K|

Given a simplicial set K ∈ SSet, we would provide a CW structure on |K|. First, we deﬁne the skeletons of |K|, as follows.

Deﬁnition 3.3.6. Suppose K ∈ SSet.

n n 1. For integers n ≥ 0, let K ⊆ K be the subsimplicial set generated by the set ∪i=0Ki.

2. We would establish that |Kn| deﬁnes a CW structure on |K|.

We prove the following lemma.

0 0 Lemma 3.3.7. Let K ∈ SSet and n ≥ 1 be an integer. Suppose (σ, x), (σ , x ) ∈ Kn−1 ×

|∆[n − 1]| and (τ, y) ∈ Kn × |∆[n]|, such that

(σ, x) ∼ (τ, y) ∼ (σ0, x0) in |Kn| 3 ” ∼ ” are induced by a face or degeneracy.

Then, either τ ∈ Kn−1 or there is a chain of equivalences, of length two,

0 0 0 0 0 0 (σ, x) ∼ (τ , y ) ∼ (σ , x ) where (τ , y ) ∈ Kn−2 × |∆[n − 2]| and the equivalences are induced by face or degeneracy maps.

Proof. We consider various cases:

1. Suppose both relations are induced by face maps. Let

di i j (σ, x) ∼ (τ, y) d , d ∈ ∆([n − 1], [n]) be faces, and j (σ0, x0) ∼d (τ, y)

In this case, K(di)τ = σ |di|(x) = y K(dj)τ = σ0 |dj|(x0) = y Degeneracies in ∆([n−1], [n−2]) would be denoted by ζi and faces in ∆([n−1], [n−2]) would be denoted by δi. If i = j, then σ = σ0 and x = x0. So, we assume i < j. We have j i i j i i K(d δ ) = K(δ )K(d ) = K(δj−1)K(d ) = K(d δj−1) It follows, K(δj−1)K(di)τ = K(δj−1)(σ) |di|(x) = y K(δi)K(dj)τ = K(δi)(σ0) |dj|(x0) = y

With y = (y0, y1, . . . , yn) we have yi = yj = 0 3.3. GEOMETRIC REALIZATION 69

Let

i j j−1 i x0 = |ζ s |(y) = |ζ s |(y) = (y0, . . . , yi−1, yi+1, . . . , yj−1, yj+1, . . . , yn) j−1 0 and σ0 = K(δ )(σ) = K(δi)(σ )

We claim

j−1 j−1 (σ0, x0) = (K(δ )(σ), x0) ' (σ, x) given by δ :[n − 2] −→ [n − 1]. i 0 0 0 i (σ0, x0) = (K(δ )(σ ), x0) ' (σ , x ) given by δ :[n − 2] −→ [n − 1].

One checks,

i j−1 i j−1 |d |(|δ |(x0)) = |d |(x) = y =⇒ |δ |(x0) = x. j i j 0 i 0 |d |(|δ |(x0)) = |d |(x ) = y =⇒ |δ |(x0) = x .

j δi 0 0 So, the claim is established and (σ, x) ∼ δ (σ0, x0) ∼ (σ , x ). 2. Now suppose, one of the two equivalences is induced by a degeneracy. Assume (σ, x) ∼si (τ, y). In this case, τ = K(sj)(σ) ∈ Kn−1.

This completes the proof.

Proposition 3.3.8. Suppose K ∈ SSet For integers m ≤ n, consider the commutative diagram of geometric realizations:

ιmn |Km| / |Kn| q _ ιn ιm " |K|

For all integers m ≤ n, the maps ιmn, ιn, ιn are injective (as the notations indicate).

n−1 n Proof. It is enough to prove, for integers n ≥ 1, the map ι := ι(n−1)n : |K | −→ |K | is injective. Let ω, ω0 ∈ |Kn−1|, with ι(ω) = ι(ω0). We can write ω = [(σ, x)], ω0 = [(σ0, x0)], 0 0 for some (σ, x), (σ , x ) ∈ Kn−1 × |∆[n − 1]|. There a chain of equivalences

µ1 µ2 µr (σ, x) = (τ0, y0) ∼ (τ1, y1) ∼ · · · ∼ (τr, yr) = (τ, y)

where µi are face or degeneracy maps and (τi, yi) ∈ Kni × |∆[ni]|. We use induction of 0 n−1 n−1 r. For r = 2, it follows from Lemma 3.3.7, that ω = ω in |K |. If τi ∈ K , for some i = 1, . . . , r − 1, then by induction, it follows ω = ω0 in |Kn−1|. So, we assume that ni ≥ n for all i = 1, . . . , r − 1. Let N = max{ni : i = 1, . . . , r − 1. Now, again, we use induction of N. If N = n, by repeated application of Lemma 3.3.7, it follows that ω = ω0 in |Kn−1|. Now, assume N ≥ N − 1. Again, by Lemma 3.3.7, for i = 1, . . . , r − 1, with N−1 0 0 either τi ∈ K or can be replaced by a (τi , yi) ∈ KN−2 × |∆[N − 2]|. Since the "height" N has reduces, we have ω = ω0 in |Kn−1|. The proof is complete. 70 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

The CW Structure on |K| We proceed to describe the CW structure on |K|. Subsequently, ∂ |∆[n]| would denote the boundary of l |∆[n]|. Further, X Interior(|∆[n]|) = {(t0, t1, . . . , tn) : 0 < ti < 1 ∀ i, ti = 1}

0 1. Let, |K | = K0, be endowed with discrete topology. For integers n ≥ 0, write a Kfn = Kk × |∆[k]| k≤n

n Then, the map Kfn −→ |K | is surjective, sending (σ, x) 7→ [(σ, x)].

2. From construction of the Geometric representation (Equation 3.3), for each σn ∈ Kn, there is a map ∼ n ασn : |∆[n]| −→ σn × |∆[n]| −→ |K |

Now, suppose σn is non degenerate. (a) For m ≤ n − 1 and

(σm, x) ∈ Km×|∆[m]|, (σn, y) ∈ σn×|∆[n]|, (σm, x) ∼ (σn, y) =⇒ y ∈ ∂|∆[n]| (b) It follows from the proof of Proposition 3.3.8, the map

∼ n ασn : |∆[n]| −→ σn × |∆[n]| −→ |K | is injective on Interior(|∆[n]|) (c) On the other hand,

∀ x ∈ ∂ |∆[n]| ασn (x) ∈ Image(Ken−1). For example,

ασn (0, t1, t2, . . . , tn) = [(σn, (0, t1, t2, . . . , tn))] = [(d0σn, (t1, . . . , tn))]

n−1 (d) Note, |∂∆[n]| = ∂ |∆[n]|. Also note, ασn (|∂∆[n]|) ⊆ |K |. Denote n−1 ∂(ασn ) := (ασn )|∂|∆[n]| : |∂∆[n]| −→ K

0 3. Let Kn ⊆ Kn be the set of all non degenerate simplexes. Let ` ` n D := σ∈K0 |∆[n]|, and α := σ∈K0 ασ : D −→ |K | ` ` n−1 S := σ∈K0 |∂∆[n]|, and ∂(α) := σ∈K0 ∂ (ασ): S −→ |K | Proposition 3.3.9. Let K be a simplicial set. With the notations, as above, for all integers n ≥ 1, the following ∂(α) n−1 / |K | S _ _ i ι(n−1)n

n D α / |K | is a push forward diagram in Set. 3.3. GEOMETRIC REALIZATION 71

Proof. We use the universal property of push forward. Extend the above to the diagram:

∂(α) n−1 / |K | S _ _ i ι(n−1)n f n D α / |K | ϕ

g ", Z

In the diagram, f, g are given, such that that outer diagram commute. We need to prove that there is a unique map ϕ, such the outer triangles commute. When exists, uniqueness n of ϕ follows, because |K | is union of the images of α and ι(n−1)n. For (σ, x) ∈ Km ×|∆[m]|, with m ≤ n and ω = [(σ, x)], defne

f(ω) if m ≤ n − 1 ϕ(ω) = Here x is considered in the σ−componenent of . g(x) if m = n D

One checks that ϕ is well deﬁned. The proof is complete.

Lemma 3.3.10. Suppose K ∈ SSet. Then, the geometric realization |K| has aCW complex structure. In this CW structure |Kn| would be the n-skeleton. The n-cells are given, restrictions of the map ασ to the interior of |∆[n]|.

Proof. Follows from Proposition 3.3.9.

The following is a basic result on geometric realization.

Proposition 3.3.11. Suppose K is a simplicial set and L ⊆ K is a simplical subset. Then, the map |L| −→ |K| is injective. Further, |L| is a closed subset of |K|

Proof. For a non degenerate n-simplex σ ∈ K, let CellK (σ) ⊆ |K| denote the corresponding n-cell in |K|. In addition to the notation Kn ⊆ K, as above, let Ln denote the simplicial subset of L, generated by simpliﬁes of dimension ≤ n. Note, for integers n ≥ 0 there is commutative diagram of natural maps

κn |Ln| / |Kn|

0 ιn ιn |L| κ / |K|

0 0 Let Kn denote the nondegenrate n-simplices of K and Ln denote the nondegenrate n- 0 i i simplices of L. For σ ∈ Ln, if σ = K(s )τ for τ ∈ Kn−1, then τ = K(d )σ ∈ L would 72 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

0 0 0 ∼ ∼ be a degeneracy in L. Hence, Ln ⊆ Kn. Also, note for σ ∈ Ln, |∆[n]| −→ CellL(σ) −→ CellK (σ) are homeomorphism. Now, we have a a |L| = CellL(σ) ⊆ CellK (σ) = |K| , 0 0 σ∈∪n≥0Ln σ∈∪n≥0Kn where ` denotes disjoint union. To see |L| is a closed subset of |K|, we will prove integers n ≥ 0, |Ln| ⊆ |Kn| is a closed subset. Clearly, |L0| −→ |K0| is closed. Now, consider the commutative diagram of pushout diagrams:

n−1 / |Ln−1| SL _ q _ r κn−1

" $ n−1 n−1 SK / |K | _ _ n ` n n−1 n = 0 = ∂ DK α∈Kn D SK DK where n ` n n−1 n L = 0 = ∂ L D α∈Ln D SL D n / |Ln| DL q αL r κn

" $ n / |Kn| DK αK

n−1 n−1 n −1 n n By induction |L | is closed in |K |, hence in |K |. Also, αK (|L |) = DL is closed in n n n DK . So, |L | is closed in |K |. The proof is complete. 3.4. THE HOMOTOPY GROUPS OF SIMPLICIAL SETS 73 3.4 The Homotopy Groups of Simplicial Sets

In this section, we define the homotopy groups of simplicial sets K ∈ SSet. Easiest way to define homotopy groups πn (K, v0) is to define it as that of the geometric realization |K•|.

Deﬁnition 3.4.1. Suppose K ∈ SSet. Note, for all n ≥ 0, ∆[0]n are singletons and |∆[0]| = ? is a point.

1. Let v0 ∈ K0 be a vertex. Then, v0 determines a point in |K|, which is the equivalence

class of (v0,?) ∈ K0 × |∆[0]|. We denote this point as |v0| ∈ |K|.

2. The map [n] −→ [0] induce the map (iterated degeneracy) K0 −→ Kn. Given a

vertex v0 ∈ K0, it deﬁnes a sub simplicial set V (v0) ⊆ K, where V (v0)n has a single

simplex, namely, the image of v0 under the map K0 −→ Kn.

By abuse of notations, V (v0) is, sometimes denoted by v0 ⊆ K. So, |v0| ⊆ |K| is the "same point", deﬁned above.

Deﬁne, nth-homotopy group

πn (K, v0) := πn (|K| , |v0|) (Compare [H, Prop 3.6.3])

However, there has been attempts to deﬁne these homotopy groups/sets combinatorially, with some success. We extract (in Section 3.4.1) the combinatorial deﬁnition of πn (K, v0) from [H], when K is "Fibrant". Although we do this, it would be of little help for our purposes, in the context of K-Theory, which we would clarify latter.

3.4.1 The Combinatorial Deﬁnition

First, we deﬁne product of two simplicial sets.

Deﬁnition 3.4.2. Let X,Y ∈ SSet. Deﬁne X × Y ∈ SSet by

∀ n ∈ N (X × Y )n := Xn × Yn Given f ∈ ∆([m], [n] deﬁne

(X × Y )(f) := X(f) × Y (f): Xn × Yn −→ Xm × Ym

In particular, the face and degeneracy maps are given by product of the same. Remark. In fact, in the category Top, |X × Y | −→∼ |X| × |Y | is a homeomorphism, usual topology is considered. However, as is remarked after the proof of [H, Lemma 3.1.8], |X×Y | and |X| × |Y | are not necessarily homeomorphic, with compactly generated topology. 74 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Now we deﬁne homotopy of simplicial sets. Denote by I := ∆[1] ∈ SSet. So,

I0 = ∆([0], [1]) = {0, 1},I1 = ∆([1], [1]) = {(0, 0); (0, 1), (1, 1)}

I2 = ∆([2], [1]) = {(0, 0, 0), (0, 0, 1); (0, 1, 1); (1, 1, 1)}···

Now, 0, 1 ∈ I0 induces a sub simplicial sets 0, 1 ⊆ I, also viewed as 0, 1 : ∆[0] −→ I = ∆[1].

Deﬁnition 3.4.3. Suppose X,Y ∈ SSet and f, g : X −→ Y are two maps in SSet. We say that f is homotopic to g, if

∃ F : X × I −→ Y in SSet 3 F (x, 0) = f(x),F (x, 1) = g(x) ∀ x ∈ Xn n ∈ N where F (−, 0) and F (−, 1) are deﬁned as follows:

∼ 1X ×0 ∼ 1X ×1 X / X × ∆[0] / X × I X / X × ∆[0] / X × I

F and F F (−,0) F (−,1) * Y * Y

Further, suppose A ⊆ X and B ⊆ Y be subsimplicial sets. A maps, f, g :(X,A) −→ (Y,B) are said to be homotopic, if there is a homotopy

 F (−, 0) = f  F : X × I −→ Y 3 F (−, 1) = g   F (A × I) ⊆ B

It is of our particular interest, when both A, B represent vertices. In that case, we say f, g are base point preserving maps and H is a base point preserving homotopy.

Deﬁnition 3.4.4. A simplicial set X ∈ SSet, is said to satisfy the Kan extension condition if ∀ n ∈ N, 0 ≤ k ≤ n and any map f :Λk[n] −→ X in SSet, extends to a map ∆[n] −→ X in SSet. Diagramatically, f Λk[n] / X _ = ∃ ϕ ∆[n] Such a simplicial set X is called FIBRANT. More generally, a map γ : X −→ Y is called a ﬁbration, if γ has the right lifting property, as in the diagram: f Λk[n] / X _ = ∃ γ ϕ ∆[n] / Y F 3.4. THE HOMOTOPY GROUPS OF SIMPLICIAL SETS 75

Meaning, if the outer diagram of map commute, there in a map ϕ so that the inner triangles commute. In particular, a simplicial set X is ﬁbrant ⇐⇒ the (unique) map X −→ ∆[0] is a ﬁbration.

Example 3.4.5. Note ∆[1] does not satisfy Kan extension condition.

0 f Λ [2] / ∆[1] ( _ f(0, 1) = (0, 1)

f(0, 2) = (0, 0) ∆[2]

Digress: Non degenerate simplices (which we denote with 0)  (∆[2])0 = {(0), (1), (2)} (  0 (Λ0[2])0 = {(0), (1), (2)} (∆[2])0 = {(0, 1), (1, 2), (0, 2)} 0 1 (Λ0[2])0 = {(0, 1), (0, 2)}  0 1 (∆[2])2 = {(0, 1, 2)}

So, f is well deﬁned because rest is forced by simplicial identities. In deed, it follows,

f(0) = f(d1(0, 1)) = d1(f(0, 1)) = d1(0, 1) = 0, Likewise, f(1) = 1, f(2) = 0.

Now, suppose there is an extension F : ∆[2] −→ ∆[1], of f. So, see this let F (0, 1, 2) = (i, j, k) with 0 ≤ i ≤ j ≤ k ≤ 1. Then,

 F (1, 2) = F (d (0, 1, 2)) = d (i, j, k) = (j, k)  0 0 1 = F (1) = F (d1(1, 2)) = d1(F (1, 2)) = d1(j, k) = j   0 = F (2) = F (d0(1, 2)) = d0(F (1, 2) = d0(j, k) = k

This contradicts that j ≤ k.

Example 3.4.6. Recall the adjoint functors | − | : SSet −→ Top, S : Top −→ SSet. Now, for Y ∈ OB(Top), SY has Kan extension property. Proof. Consider the diagram f Λk[n] / SY _

∆[n] By Adjunction

∼ k ∼ k MorTop(|K|,Y ) = MorSSet(K,SY ) =⇒ MorTop(|Λ [n]|,Y ) = MorSSet(Λ [n],SY ) 76 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Corresponding to f, there is a continuous map f˜ : |Λk[n]| −→ Y . I put it in the diagram

f˜ Λk[n] / Y _ k Also, ∃ retract r : |∆[n]| −→ Λ [n] |∆[n]|

Hence, fr˜ : |∆[n]| −→ Y . By Adjunction, there is F corresponding to fr˜ , completing the commutative diagram f Λk[n] / SY _ <

∃ F ∆[n] The proof is complete.

Under certain conditions, homotopy is an equivalence relation. To prove this the following Lemma would be useful.

Lemma 3.4.7. Let X,Y ∈ SSet and Y satisﬁes the Kan extension condition. Then a map of simplicial sets f : X × Λk[n] −→ Y extends to a map F : X × ∆k[n] −→ Y . Diagramatically: f X × Λk[n] / Y :

F X × ∆[n]

Proof. Skip! Now we prove the equivalence.

Lemma 3.4.8. Let X,Y ∈ SSet and Y satisﬁes the Kan extension condition. Then the homotopy (') is an equivalence relation on the set of all simplical maps X −→ Y . Same is true for base point preserving maps.

Proof. Let f, g, h : X −→ Y be maps of simplicial maps.

1. (Reﬂexivity): Consider the diagram

π X × I / X

f where π is the projection, and F = fπ. F # Y

Then, F (0) = F (1) = f. 3.4. THE HOMOTOPY GROUPS OF SIMPLICIAL SETS 77

2. (Symmetry.) Suppose F : X × I −→ Y be a homotopy, with F (0) = f, F (1) = g. Recall, the non degenerate simplexes:

 0, 1, 2 0, 1  0, 1, 2 I0 = ∆[1]0 = ,→ ∆[2]0 = (0, 1), (0, 2), (1, 2) Λ1[2]0 = (0, 1) (0, 1), (1, 2)  (0, 1, 2)

For i < j, the sub simplicial set generated by the nondegenrate (i, j), will be denoted by [(i, j)]. Suppose F : X × I −→ Y be a homotopy: F (0) = f, F (1) = g. Deﬁne

F 0 = F F 0 : X × Λ1[2] −→ Y X×I F 0(x, [(1, 2)]) = g(x)

This is well deﬁned, because F 0(x, 1) = g = F (1), at the vertex 1, where two legs meet. By Lemma 3.4.8, F 0 extends to G0 : X ×∆[2] −→ Y . Let G : X ×[(0, 2)] −→ Y be the restriction of G0. Then, G(0) = f, G(2) = g.

3. (Transitivity.) Consider homotopies

F : X × [(0, 1)] −→ Y F (0) = f F (1) = g 3 G : X × [(1, 2)] −→ Y G(1) = g G(2) = h

In deed, F,G deﬁnes a simplicial map F 0 : X × Λ1[2] −→ Y . By Lemma 3.4.8, F 0 extends to a simplicial map G0 : X × ∆[2] −→ Y . Now, G0(0) = f and G0(2) = h. So, f ' h.

The proof is complete.

3.4.2 The Deﬁnitions of Homotopy Groups

We give two definitions of homotopy sets πn(−, −). The following is the first definition.

Deﬁnition 3.4.9. For integer n ≥ 0, let 0 ∈ ∆[n]0 be the base point. Let (X, x0) be a pointed simplicial set (i., e x0 is a vertex of X. Then, deﬁne

πn (X, x0) = Set of all homotopy equivalence class of maps f :(∂∆[n], 0) −→ (X, x0)

The deﬁnition from the book of Hovey [H]

Definition 3.4.10. Suppose X ∈ SSet is a fibrant simplicial set. For x, y ∈ X0 define x ∼ y (homotopic), if there is z ∈ X1 such that d1z = x, d0z = y.

Lemma 3.4.11. Suppose X is a ﬁbrant simplicial set. Then homotopy of vertices is an equivalence relation. We denote the set of equivalence classes by π0X. 78 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Proof. We have:

1.( Reﬂexivity): Let x ∈ X0. Recall

i d s0 [0] / [1] X0 / X1

∀ i = 0, 1 s0 =⇒ di =⇒ d0(s0(x)) = x = d1(s0(x)) 1 1 ! [0] X0

2.( Symmetry): Suppose x ∼ y ∈ X0. So, ∃ z ∈ X1 3 d1z = x, d0z = y. Recall,  0, 1, 2 0, 1  0, 1, 2 ∆[1]0 = ∆[2]0 = d := (0, 1), d := (0, 2), d := (1, 2) Λ0[2]0 = (0, 1) 2 1 0 d , d  (0, 1, 2) 1 2

(a) Note s0x, z ∈ X1. (b) Nondegenerate simplexes of Λ0[2] (that count) are d1, d2 ∈ ∆([1], [2]). d1 7→ s x (c) Deﬁne f :Λ0[2] −→ X 0 d2 7→ z

(d) Diagramatically, represent ∆[2] by i2 := 1[2] = (0, 1, 2):

•1 •1 d2i2 d2i2

0 ∆[2] : 0• d0i2 and Λ [2] : 0•

d1i2 d1i2 2 • 2 •

The map f :Λ0[2] −→ X IS

•1 •y d2i2 z

0• 7→ x•

s0x d1i2 2 • •x

(e) Since X is ﬁbrant, f extends to F : ∆[2] −→ X:

•1 •y d2i2 z w = d0F (i2) = F (d0(i2)) = F (1, 2) 0• d0i2 7→ x• w =⇒ d1w = y, d2w = x.

s0x d1i2 2 • •x 3.4. THE HOMOTOPY GROUPS OF SIMPLICIAL SETS 79

3.( Transitivity): Suppose x ∼ y ∼ z. By symmetry, we assume z ∼ y. So, ∃ a, b ∈ X1, 3 d1a = x, d0a = y = d1b, d0b = z: d1 7→ b (a) Deﬁne f :Λ0[2] −→ X . Diagramatically: d2 7→ a

•1 •x d2i2 a

0• 7→ y•

d1i2 b 2 • •z (b) By ﬁbrant property, f extends to: •1 •x d2i2 a

0• d0i2 7→ y• c =⇒ d1(c) = x, d0c = z.

d1i2 b 2 • •z

Lemma 3.4.12. Suppose X is a ﬁbrant simplicial set. Then there is a natural isomorphism ∼ π0X = π0|X|.

Outline of the Proof. For a vertex x ∈ X0, let [v] ∈ π0X denote the class of v. Recall, π0|X| is the set of all path components of |X|. Deﬁne a map ` K × |∆[n]| ϕ : π X −→ π |X| by sending [v] 7→ v × ? ∈ π |X| = n n 0 0 0 ∼ One checks that this map is well deﬁned and and a bijection.

Deﬁnition 3.4.13. With notations as above, for any ﬁbrant simplycial set X and any vertex

v ∈ X0, deﬁne π0(X, v) := (π0X, [v]) as a pointed set.

Higher Homotopy Sets πn(X, v):

Now, we proceed to deﬁne πn(X, v), for all integers n ≥ 0. Before that, we introduce the following deﬁnition and notations. 80 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Deﬁnition 3.4.14. Suppose X,Y are two simplicial sets. Deﬁne a new simplicial set Map(X,Y ) as follows:

1. For integers n ≥ 0, let Map(X,Y )n := MorSSet (X × ∆[n],Y ).

2. For a morphism f :[m] −→ [n] in the category ∆, let Map(f): Map(X,Y )n −→

Map(X,Y )m be the map induced by ∆(f) : ∆[m] −→ ∆[n].

So, Map(X,Y ) has the structure os a simplicial set.

Further, we comment, that given a vertex v ∈ Y0, v deﬁnes a vertex in Map(X,Y ), that means an element in Map(X,Y )0 = MorSSet(X,Y ), which we continue to denote by v, as follows: X / ∆[0]

v This is the constant map. v ! Y Of our subsequent interest would be the vertex in Map(∆[n],Y ), for all integers n ≥ 0, deﬁned by the vertex v in Y : ∆[n] / ∆[0]

v v # Y

Definition 3.4.15. Let X ∈ SSet be fibrant and v ∈ X0 be a vertex. Note that the restriction map Map(∆[n],X) −→ Map(∂∆[n],X) is also a map of simplicial sets. Let F be the fiber (pullback) of this map, over the vertex v:

F / Map(∆[n],X)

v / Map(∂∆[n],X)

By [H, Thm. 3.3.1], the second vertical map is a fibration. Hence F is fibrant. Define

πn(X, v) := π0(F, v)=( π0(F ), [v])

The following is some dissections:

1. We have  Map(∆[n],X)0 = MorSSet(∆[n],X) = Xn,   Map(∂∆[n],X)0 = MorSSet(∂∆[n],X), Map(∆[n],X)1 = MorSSet(∆[n] × ∆[1],X)   Map(∂∆[n],X)1 = MorSSet(∂∆[n] × ∆[1],X) 3.4. THE HOMOTOPY GROUPS OF SIMPLICIAL SETS 81

2. So, F0 consists of all α ∈ Xn, all whose faces are v. Equivalently, all α : ∆[n] −→ X such that α(∂∆[n]) = v.

3. To deﬁne π0(F ), recall for α, β ∈ F0 one deﬁnes α ∼ β if

∃ H ∈ F1 3 d0(H) = α, d1(H) = β. This means, there is a map of simplicial sets

H : ∆[n] × ∆[1] −→ X 3 H(−, 0) = α, H(−, 1) = β, H|∂∆[n]×∆[1] = v. Remark: There should be a direct proof, that the above is an equivalence relation on F0. If and when we had that, we be able to avoid using the result [H, Thm. 3.3.1] that Map(∆[n],X) −→ Map(∂∆[n],X) is a ﬁbration, if X is a ﬁbrant. 4. Here is an important correspondence:

Lemma 3.4.16. Let X ∈ SSet be ﬁbrant and v ∈ X0 be a vertex. Then,

πn(X, v) = πn(|X|, |v|) ∀ n ≥ 0.

Therefore, πn(X, v) is a group for n ≥ 1 and is abelian for n ≥ 2.

Outline of the Proof. A complete proof is given in [H, Prop. 3.6.3]. The case n = 0 was proved before (3.4.12). Fix an integer n ≥ 0. Let α ∈ F0. That means α : ∆[n] −→ X such that α|∂∆[n] = v. It follows, the geometric realization of α is a map |α| :(|∆[n]|, |∂∆[n]|) −→ (|X|, |v|). Deﬁne

ϕ :(X, v) −→ πn(|X|, |v|) := MorTopPair (|∆[n]|, |∂∆[n], (|X|, |v|)) by ϕ([α]) = [|α|] It follows easily, by geometric realization of the homotopies, that ϕ is well deﬁned. Now, one checks that ϕ is bijective.

5. Suppose f : X −→ Y is a map of ﬁbrant X,Y ∈ SSet and v ∈ X0 is a vertex. Then, f induces a map of the homotopy groups πn(X, v) −→ πn(Y, f(v)), making these groups functorial:   π0 : SSet −→ Set π1 : SSet −→ Gr  πn : SSet −→ Ab n ≥ 2

The Connecting Homomorphisms

Definition 3.4.17. Let p : X −→ Y be a fibration of fibrant simplical sets X,Y and v ∈ X0 be a vertex. Let [α] ∈ πn (Y, p(v)). We have lift of α as follows:

n v dnγ Λ [n] / X ∆[n − 1] / X = ; γ p =⇒ d p n γ ∆[n] / Y α ∆[n] α / Y 82 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

By deﬁnition pdnγ = dnpγ = dnα = p(v). Let F be the ﬁber of p over p(v):

∆[n − 1] dnγ

% F /& X

p ∆[0] / Y p(v)

n So, dnγ lies in F . Further, didnγ = v, by looking at Λ [n].

So, dnγ ∈ πn(F, v). This deﬁnes a map

∂ : πn (Y, p(v)) −→ πn−1 (F, v) .

Lemma 3.4.18. Let p : X −→ Y be a fibration of fibrant simplicial sets X,Y and v be a vertex of X. Let i : F −→ X be the fiber of p over p(v). Then, there is a sequence of pointed sets ∂ ··· / πn (F, v) / πn (X, v) / πn (Y, p(v))

∂ / πn−1 (F, v) / πn−1 (X, v) / πn−1 (Y, p(v))

·········

∂ / π1 (F, v) / π1 (X, v) / π1 (Y, p(v))

∂ / π0 (F, v) / π0 (X, v) / π0 (Y, p(v)) This sequence is exact, (in the sense that the the kernel (the premiere of the base point) is equal to the image of at each degree.

Remark 3.4.19. We tried to give a flavor that Homotopy Theory of Simiplicial Sets can be developed without any reference to Topological Homotopy Theory (as in [H, Chapter 3]). However, we refrained from going in to deeper details of the homotopy theory of Simplicial Sets. For our purpose, Definition 3.4.1, should suffice, that for a simplicial set X (fibrant or not) and a vertex v of X:

πn(X, v) := πn(|X|, v) ∀ n ≥ 0

Quillen [Q], did not seem to have used the Homotopy Theory of Simplicial Sets. As he indicates [Q, pp. 5-6], such theory was not in existence, at that time. Even in [H, Chapter 3.5. FREQUENTLY USED LEMMMS 83

3], it was deﬁned form Fibrant Simplicial Sets. In our context of Quillen K-Theory, we may not ﬁnd this combinatorial approach, because the simplicial set we consider (i. e. nerve of a category) may not be Fibrant.

3.5 Frequently Used Lemmms

Lemma 3.5.1. Let X ∈ SSet and ` ∈ X1 be an 1-simplex. Then, there is a path γ(`):

I −→ |X|, such that γ(0) = d0(`) and γ(1) = d1(`).

Proof. Recall

|∆[0]| = ? and |∆[1]| = {(t0, 1 − t0) : 0 ≤ t0 ≤ 1}

Deﬁne β : I −→ X1 × |∆[1]|, by β(t) = (`, (1 − t), t). Let γ(`) be the composition:

β I / X1 × |∆[1]|

γ(`) $ |X|

By notations, d0(`) := (d0(`),?) ∼ (`, (0, 1)) = γ(0). Likewise, d1(`) = γ(1). The proof is complete. The following is about push forward and geometric realization of simplical sets.

Lemma 3.5.2. Suppose

fK A / K

fL ιK is a push forward diagram in SSet or in SimTop. L / K ` L ιL A

Then, the diagram of the geometric realizations:

|fK | |A| / |K|

|fL| |ιK | is a push forward diagram in Top or in CGHaus. ` |L| / |K A L| |ιL| 84 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Proof. (Double Check) Clearly, the diagram of geometric realization commutes. Consider the commutative diagram diagram of continuous maps.

|fK | |A| / |K|

|fL| |ιK | ` α |L| / |K A L| |ιL|

β , X Deﬁne, ` a (Kn Ln) × |∆[n]| α(([σ, x)]) if σ ∈ K ϕ : K L = An −→ X by ϕ(([σ, x)]) n ∼ β(([σ, x)]) if σ ∈ Ln A

One checks that ϕ is well deﬁned and continuous. It is clear that ϕ |ιK | = α and ϕ |ιL| = β. Uniqueness of such a maps ϕ is obvious.

3.5.1 A Lemma on Bisimplicial Sets

In this section, we discuss a lemma on Bisimplicial Sets. Before we proceed, we introduce a nation.

Notation 3.5.3. The category of Simplicial Topological spaces would be denoted by SimTop. So, objects of SimTop, are functors

∆o −→ Top.

Given two objects, K,L : ∆o −→ Top,

Mor(K,L) = Set of all natural transformations τ : K −→ L

In particular, for such a τ : K −→ L, the maps τn : Kn −→ Ln are continuous maps. Remark: First, not that there is a forgetful functor SimTop −→ SSet. On the other hand, any set X with its discrete topology can be considered as a topological space. In this way, there is a natural functor SSet −→ SimTop.

Now, we deﬁne Bisimplicial Sets.

Deﬁnition 3.5.4. A Bisimplicial Set T is a functor

o o o o T : ∆ × ∆ −→ Set. For ([m], [n]) ∈ Obj(∆ × ∆ ) write Tmn = T([m], [n]). 3.5. FREQUENTLY USED LEMMMS 85

Similarly, a functor

o o T : ∆ × ∆ −→ Top. is called a BiSimplicial Space

Subsequently, in this section, we treat BiSimplical Sets as Bisimplicial Topological Spaces.

Given such a Bisimplicial space T, we deﬁne, essentially three Simplicial Spaces, as follows:

o 1. First, the diagonal DT : ∆ −→ SimSet, is deﬁned by (DT)n := (DT)([n]) := Tnn. Then, DT ∈ SimTop. We call it the diagonal of T.

o 2. For a fixed integer m ≥ 0, define Rm• : ∆ −→ SimSet, by Rm•([n]) = Tmn. So, o RTm• ∈ SimSet. Likewise, for a fixed integer n ≥ 0, define LT•n : ∆ −→ Set, by LT•n([m]) = Tmn. So, LT•n ∈ SimSet.

3. Now, deﬁne, two functors ( o (LT)m := (LT)[m] := |Rm•| LT, RT : ∆ −→ Top by (RT)n := (LT)[n] := |L•n|

Therefore, LT, RT ∈ SimTop are simplicial topological spaces. We will compare the geometric realizations of DT, LT, RT ∈ SimTop, in the next lemma (3.5.5)

The following is in [Q, pp. 86].

Lemma 3.5.5. Let T : ∆o × ∆o −→ SimSet be a bisimplicial space and DT, LT, RT be as above (3.5.4). Then, there are natural homeomorphisms:

|LT|o ∼ |DT| ∼ / |RT|

Proof. See [Q]. We establish only the ﬁrst homeomorphism. We have  ` |D | = m≥0 Tmm×|∆[m]|  T ∼ ` ` n≥0 Tmn×|∆[n]| ` m≥0 ∼ ×|∆[m]|  m≥0|Rm•|×|∆[m]|  |LT| = ∼ = ∼

From this, if follows that there is a map

ϕ : |DT| −→ |LT| where ϕ([(τ, x)]) = [((τ, x)], x)] ∀ (τ, x) ∈ Tmm × |∆[m]| 86 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Now, one proves that ϕ is a homeomorphism. This is done in two stages. For notational conveniences, for integers r, s ≥ 0, we introduce the notation

Ω[r, s] := ∆[r] × ∆[s] : ∆o × ∆o −→ Set. So, Ω[r, s]([p], [q]) = ∆([p], [r]) × ∆([q], [s]) ∀ p, q ≥ 0.

Clearly, Ω[r, s] is bisimplicial set. Further, for ﬁxed topological space S and integers r, s ≥ 0, T be deﬁned by

Tmn = Ω[r, s] × S = ∆([m], [r]) × ∆([n], [s]) × S In our familiar notations:

Ω[r,s] ∆o × ∆0 / Set / Top

×S T ( Top

We ﬁst establish the Lemma for such a bisimplicial set T. We have ` ∆([m], [r]) × ∆([m], [s]) × S × |∆[m]| |D | = m≥0 −→∼ S × |∆[r]| × |∆[s]| T ∼ Similarly, for a ﬁxe integer m ≥ 0, we have

` ∆([m], [r]) × ∆([n], [s]) × S × |∆[n]| |R | = n≥0 −→∼ ∆([m], [r]) × S × |∆[s]| Tm• ∼ Therefore, ` ` |RTm•| × |∆[m]| ∆([m], [r]) × S × |∆[s]| × |∆[m]| ∼ |L | = m≥0 = m≥0 −→ S×|∆[r]|×|∆[s]| T ∼ ∼ This establishes that ϕ is a homeomorphism, in this case. Now, establish the same for general Bisimplicial spaces T.

1. For integers r, s ≥ 0, there is a map (natural transformation) of bisimplicial sets,

εrs : Ω([r, s])×Trs = ∆[r]×∆[s]×Trs −→ T deﬁned by εrs(f, g, τ) = T (f, g)(τ) ∈ Tmn,

where (f, g, τ) ∈ ∆([m], [r]) × ∆([n], [s]) × Trs. We note the following:

(a) εrs is a map of bisimplicial spaces.

(b) εrs is determined by the equation εrs(ιr, ιs, τ) = τ, where ιr denotes the non degenerate simplex of ∆[r].

2. Combining, we get a surjective map a a ε := εrs : Ω[r, s] × Trs −→ T r≥0,s≥0 r≥0,s≥0 So, ε is surjective.

0 0 3. Given (f, g) ∈ ∆([r], [r ]) × ∆([s], [s ]) there are maps Tr0s0 −→ Trs and a map ∆[r] × ∆[s] −→ ∆[r0] × ∆[s0]. These two maps, induce maps of bisimplical spaces:

(f,g,1) 0 0 0 0 Ω[r, s] × Tr0s0 ∆[r] × ∆[s] × Tr0s0 / ∆[r ] × ∆[s ] × Tr0s0 Ω[r , s ] × Tr0s0

∆[r] × ∆[s] × Tr0s0 / ∆[r] × ∆[s] × Trs Ω[r, s] × Trs (1,1,(f,g)) In fact, the diagram

(f,g,1) 0 0 Ω[r, s] × Tr0s0 / Ω[r , s ] × Tr0s0

(1,1,(f,g)) εr0s0 commutes. Ω[r, s] × T / T rs εrs

4. By taking coproduct, we have a commutative diagram ` ` (f,g,1)` 0 0 S Ω[r, s] × Tr0s0 / (r0,s0) Ω[r , s ] × Tr0s0 `(1,1,(f,g)) ε (3.4) ` (r,s) Ω[r, s] × Trs ε / T wehre S = {(f, g, r, r0, s, s0): r, r0, s, s0 ≥ 0, (f, g) ∈ ∆([r], [r0]) × ∆([s], [s0])}. One checks, that this is a push forward diagram of bisimplicial spaces. To check this one needs to check that the for all integers, p, q ≥ 0, the (p, q)-component of the above diagram is a push forward in Top:

` ` (f,g,1) ` 0 0 (f,g)∈∆([r],[r0])×∆([s],[s0]) ∆([p], [r]) × ∆([q], [s]) × Tr0s0 / (r0,s0) ∆([p], [r ]) × ∆([q], [s ]) × Tr0s0

`(1,1,(f,g)) ε0 ` ∆([p], [r]) × ∆([q], [s]) × T / T (r,s) rs ε” pq where ε0, ε” denote the restrictions of ε. In fact, both ε0, ε” are surjective maps. It follows form construction of push forward (in Set), that above rectangle is a push forward diagram. Therefore, it is established the the diagram (3.4) is push forward diagram of bisimplicial sets. Hence, if follows the D of the diagram (3.4) are also push forward diagrams. Further it follows from Lemma 3.5.2 that and the L of the above diagram (3.4) are also push forward diagrams. 88 CHAPTER 3. SIMPLICIAL AND COSIMPLICIAL SETS

Now, consider the commutative diagram

` ` 0 0 D ( S Ω[r, s] × Tr0s0 ) / (r0,s0) D (Ω[r , s ] × Tr0s0 ) ϕ ϕ ∼ ∼ * + ` ` 0 0 L ( S Ω[r, s] × Tr0s0 ) / (r0,s0) L (Ω[r , s ] × Tr0s0 )

` D (r,s) Ω[r, s] × Trs / DT ϕ ϕ ∼ ∼ ) ` * L (r,s) Ω[r, s] × Trs / LT

We need to prove that the lower ring hand map ϕ is a bijection. Since other three are bijection, by properties of push out, so is the fourth one. (In the notes, broken arrows have been uses to indicate existence of an arrow. However, here the natural maps ϕ was already deﬁned above. Chapter 4

The Quillen-K-Theory

This chapter is essentially an exposition of the paper of Quillen [Q]. Other such expositions include [Sv, Sm1, Wc2]. Before we proceed, we introduce the following:

Notations 4.0.6. Recall, a category C is called a small category, if Obj(C ) is a set.

1. The category of small categories and functors will be denoted by Cat. ∼ We would often, work with categories C , such that there is an equivalence F : Cs −→

C where Cs is in Cat. Such a category, C is said to have a set of isomorphism classes of objects.

2. The category of CW complexes would be denoted by CW. The category of pointed

CW complexes would be denoted by CW•. Also, the category of pairs CW complexes would be denoted by CWPair.

4.1 The Classifying spaces of a category

This section mostly corresponds to [Q, §1].

Deﬁnition 4.1.1. Suppose C is a small category. To any such category, we associate a simplicial set NC ∈ SSet, as follows:

1. We let, the set of vertices NC0, be the set of all objects in C . The set of n-cells

NCn be the set, sequence of composable arrows

f1 f2 f3 fn σ : X0 / X1 / X2 / ··· / Xn−1 / Xn (4.1)

89 90 CHAPTER 4. THE QUILLEN-K-THEORY

2. The face maps di : NCn −→ NCn−1 is given by sending, the n-call σ in (4.1), by

dropping Xi, and for i 6= 0, n,

fi−1 fi fifi−1 replacing Xi−1 / Xi / Xi+1 by Xi−1 / Xi+1

3. The degeneracy maps si : NCn −→ NCn+1 is given by sending, the n-call σ in (4.1),

fi−1 fi fi−1 fi replacing Xi−1 / Xi / Xi+1 by Xi−1 / Xi Xi / Xi+1

The simplicial set NC ∈ SSet is called the nerve of C .

Deﬁnition 4.1.2. Suppose C is a category. The classifying space BC is deﬁned to be the geometric realization |NC | ∈ Top. So, BC := |NC |. So, objects X ∈ Obj(C ) corresponds to a vertex (a point) in BC . Remark: There is no reason, that the nerve NC is a category is a Fibrant simplicial set.

Following obvious lemma would be very useful.

Lemma 4.1.3. Suppose C is a category and f : X −→ Y is a morphism. Then there is a a path γ := γ(f) : [0, 1] −→ BC such that γ(f)(0) = X and γ(f)(1) = Y .

Proof. Follows from Lemma 3.5.1.

Following are two useful examples of classifying spaces.

Example 4.1.4. Let I denote the category of the partial ordered set {0, 1}. Note, the nerve N (I) = ∆[1]. So, the classifying space BI = [0, 1], the interval.

Example 4.1.5. Suppose G is a group. Let G denote the category with one object ? and Mor(?, ?) = G. Then, ( G if i = 1 πi (G,?) = 0 if i 6= 1

∼ Here the isomorshism G −→ πi (G,?) sends g ∈ G to the loop corresponding to g : ? −→ ? (see Lemma 3.5.1).

Proof. Skip 4.1. THE CLASSIFYING SPACES OF A CATEGORY 91

4.1.1 Properties of the Classifying Space

Proposition 4.1.6. Let C , D be two small categories and F : C −→ D be a functor. Then, F induces a continuous map B(F ): BC −→ BD. This associations ( C 7→ BC deﬁnes a functor B : Cat −→ Top. F 7→ B(F )

Proof. Follows from construction.

Proposition 4.1.7. Let C be a small category. Then, there is a (cellular) homeomorphism ∼ BC −→ BC o, where C o is the dual category. Proof. Follows from construction.

Proposition 4.1.8. Let C , D be two small categories.

∼ 1. It follows easily that N (C × D) −→ N (C ) × N (D) is a bijection.

2. So, there is a continuous bijection B(C × D) −→ BC × BD. This map is a homeomorphism, if the right side is given by compactly generated topology. (Also, if BC or BD is a ﬁnite complex.) In particular, if BC or BD is compact.

3. In particular B(C × I) ≡ BC × I, where I denotes the category (0 < 1) and I = [0, 1]. ∼ (In fact, the nerve N I = ∆[1].) Proposition 4.1.9 (Prop. 2). Let C , D be two small categories and F,G : C −→ D be two functors. Suppose θ : F −→ G is a natural transformation. Then, θ induces a homotopy

H : BC × I −→ BD 3 H(0) = B(F ),H(1) = B(G).

Proof. For a morphism f ∈ MorC (X,Y ) we have the commutative diagram

F (f) F (X) / F (Y )

θX θY G(X) / G(Y ) G(f) Deﬁne a functor   H(X, 0) = F (X), H(X, 1) = G(X) ∀ X ∈ Obj(C )  H(f, ι) = F (f): F (X) −→ F (Y ) ∀ f ∈ Mor (X,Y ), ι ∈ Mor (0, 0) H : × −→ 3 C I C I D H(f, ι) = G(f): G(X) −→ G(Y ) ∀ f ∈ Mor (X,Y ), ι ∈ Mor (1, 1)  C I  H(f, ι) = θY F (f) = G(f)θX ∀ f ∈ MorC (X,Y ), ι ∈ MorI(0, 1) 92 CHAPTER 4. THE QUILLEN-K-THEORY

By (4.1.6, 4.1.8) there is a homotopy

H := H : BC × [0, 1] ≡ B(C × I) −→ BD.

If follows, H(−, 0) = BF and H(−, 1) = BG. The proof is complete.

Corollary 4.1.10. Let F : C −→ D, G : D −→ C be two functors of small categories and F be left adjoint to G. Then, B(F ): BC −→ BD is a homotopy equivalence.

Proof. There is a natural transformation 1C −→ GF as follows:

 η  Mor (F A, F A) / Mor (A, GF A)  D ∼ C ηAX  MorD (F A, X) ∼ / MorC (A, GX) =⇒  ∼  MorD (F GX, X) η / MorC (GX, GX)

Let θ(A) = η(1FA): A −→ GF A. Then, θ : 1C −→ GF is a natural transformation. Therefore 1BC ' B(G)B(F ) are homotopic. Likewise, B(F )B(G) ' 1BC are homotopic. So, B(F ) is a homotopy equivalence.

Corollary 4.1.11. Suppose C is a category with an initial object or a ﬁnal object. Then, BC is contractible.

Proof. Assume C has an initial object 0. Let 0 denote the full subcategory of C consisting of the zero object. (This is the category with on object, one morphism.) Let F : 0 −→ C be the inclusion functor. Let G : C −→ 0 be the functor G(X) = 0 ∀ X ∈ Obj(C ) and G(f) = 10 ∀ f ∈ MorC (X,Y ). Then,

∼ MorC (0,X) / Mor0(0, 0) with A = 0 ∈ Obj(0),X ∈ ObjC

MorC (F A, X) Mor0(A, GX)

This shows that F is a left adjoint functor. Hence ∗ −→ BC is a homotopy equivalence, by (4.1.10).

4.1.2 Directed and Filtering Limit

Recall the following deﬁnition.

Deﬁnition 4.1.12. A partially ordered set (I, ≤) is called a directed set, if ∀ i, j ∈ I there is a k ∈ I such that i ≤ k, j ≤ k. As small category I is called a ﬁltering category, if 4.1. THE CLASSIFYING SPACES OF A CATEGORY 93

1. ∀ i, j ∈ Obj(I) there are two arrows:

i f

k (4.2) @ g j

2. Given two arrows λ, β : i −→ k, there is an arrow γ : k −→ κ such that γλ = γβ. Diagramatically, λ i / k β γ γλ=γβ κ

It follows, a directed set is a ﬁltering set.

Definition 4.1.13. Suppose I is a filtering small category. A functor A : I −→ Set, also written as {Ai : i ∈ I} is called a filtering family of sets.

Then, a functor C : I −→ Cat, also written as {Ci : i ∈ I} is called a ﬁltering family of small categories.

Proposition 4.1.14. Suppose I is a ﬁltering category and {Ci : i ∈ I} is ﬁltering family of small categories, indexed by I.

1. Let C = lim→ Ci.(One should call them coLimits.)

That means, C is a category, with functors Ci −→ C ∀ i ∈ I such that:

ϕ # ∀ i ≤ j / ?C < D

the ﬁrst triangle commute. Further, given the commutative diagram of the outer triangle, there is a unique functor, ϕ such that all the triangles commute.

Since we are working with small categories, this is essentially a limit in in the category of sets, as follows. In particular, such limits exist. 94 CHAPTER 4. THE QUILLEN-K-THEORY

2. It follows, F i∈I Obj(Ci) Obj(C ) = lim Obj(Ci) = . → I ∼

Denote ιi : Obj(Ci) −→ Obj(C ).

MorC (X,Y ) = lim MorCi (Xi,Yi) → I,ιi(Xi)=X,ιi(Yi)=Y

3. It follows that, the nerve N (C ) = lim→ I N (Ci).

4. Let Xi ∈ Obj(Ci) be a family, such that X = ιi(Xi). Then, there are maps

πn(Ci,Xi) −→ πn(C ,X).

Proposition 4.1.15 (Prop 3). With notations, as in (4.1.14), we have

lim πn(Ci,Xi) = πn(C ,X) ∀ n ≥ 0. → I

Proof. We have NC = lim→ I NCi. The rest of the proof follows from the corresponding result of simplicial sets (4.1.16), below.

Proposition 4.1.16. Let I be a ﬁltering category and {Ki : i ∈ I} be a ﬁltering family i i of simplicial sets. Let K = lim→ I K and v = lim→ I vi ∈ K, where vi ∈ K0 be vertices ∀ i ∈ I. Then, i lim πn(|K| , vi) = πn(|K|, v) ∀ n ≥ 0. → I

i Outline of the Proof. Note, (|K|, v) = lim→(|K |, vi), in the Category CW•. The rest of the proof follows, from the corresponding results on CW complexes, as follows (4.1.17). .

Proposition 4.1.17. Let I be a ﬁltering category and {Xi : i ∈ I} be a ﬁltering family in the category of CW complexes (in particular, the maps Xi −→ Xj are in the category i i CW). . Let X = lim→ X and v = lim→ I vi ∈ K, where vi ∈ K0 be vertices ∀ i ∈ I. Then, i lim πn(X vi) = πn(X, v) ∀ n ≥ 0. → I 4.1. THE CLASSIFYING SPACES OF A CATEGORY 95

Outline of the Proof. By usual arguments, we have the following commutative diagrams of the "limit" maps:

i i i i (X , v ) πn (X( , v ) ι π (ι ) ι i n i ij which induce maps ηi % ( (Xj, vj) / (X, v) i i ιj lim→I πn (X , v ) ι / πn (X, v)

Here, the ﬁrst triangle is in the category of pointed CW complexes, and the second one is in Set. Using results on CW-structures, which we did not cover, the proof is completed as follows:

n n i i πn(X, v) = MorTop (( ,?), (X, v)) = MorTop ( ,?), lim(X , v ) • S • S →I

∼ n i i i = lim MorTop ( ,?), (X , v ) = lim πn(X , v). →I • S →I This argument depends on the facts that {Xi, i ∈ I} is a ﬁltered family in CW and that Sn is compact. The proof is complete.

Corollary 4.1.18. Suppose ∀i ≤ j, the map BCi −→ BCj is a homotopy equivalence. Then, BCi −→ BC is a homotopy equivalence.

Proof. Obvious!

Corollary 4.1.19. Suppose I is a ﬁltering category. Then, I is contractible.

Proof. We need to prove that BI is contractible. In fact, I = lim→ I/i (meaning, BI/i is contractible.. Since, I/i has a ﬁnal object, BI/i is contractible. By (4.1.18), BI is contractible.

Deﬁnition 4.1.20. From now on, terminologies and concepts from Topology, would be borrowed freely to Category Theory. A category C would said to have a "topological property", if the classifying space BC has the same property. For example,

1. A functor F : C −→ D would be called a homotopy equivalence, if BC −→ BD is a homotopy equivalence.

2. A category C would be called contractible, if BC is contractible.

3. Two functor F,G : C −→ D would be called a homotopic, if B(F ) and B(G) are homotopic. 96 CHAPTER 4. THE QUILLEN-K-THEORY

4.1.3 Theorem A: Suﬃcient condition for a functor to be homotopy equivalence

Deﬁnition 4.1.21. Suppose F : C −→ D is a functor and Y ∈ Obj(D).

1. Deﬁne the category Y \ F , as follows:

Obj(Y \ F ) = {(A, u): A ∈ Obj(C ), u : Y −→ FA ∈ MorD }. For (A, u), (B, v) ∈ Obj(Y \ F ) a morphism is a morphism

FA = u

w : A −→ B ∈ MorC 3 Y F w commutes.

v ! FB In particular, if f is identity then Y \ Id is a category under Y :

= {(A, u): u : Y −→ A}

2. Likewise, for an object Y ∈ Obj(D), the category F/Y is deﬁned as follows:

Obj(F/Y ) = {(A, u): u : FA −→ Y ∈ MorD }. For (A, u), (B, v) ∈ Obj(F/Y ) a morphism is a morphism

FA u ! w : A −→ B ∈ Mor 3 F w Y commutes. C = v FB

The following is the celebrated Theorem A in the paper of Quillen [Q].

Theorem 4.1.22 (Theorem A). Let F : C −→ D be a functor, such that Y \ F is contractible, ∀ Y ∈ Obj(D). Then, F is a homotopy equivalence. Dually (see Prop 4.1.7), if F/Y is contractible, ∀ Y ∈ Obj(D). Then, F is a homotopy equivalence.

Proof. Let S(F ) be the category, described as follows:

1. The objects are

Obj(S(F )) = {(X, Y, v): X ∈ Obj(C ),Y ∈ Obj(D), v ∈ MorD (Y,FX)} 4.1. THE CLASSIFYING SPACES OF A CATEGORY 97

0 0 0 0 0 0 2. For (X, Y, v), (X ,Y , v ) ∈ Obj(S(F )), Deﬁned MorS(F ) ((X, Y, v), (X ,Y , v )) =

 0 w   Y / Y    0 0 0 v (u, w) ∈ MorC (X,X ) × MorD (Y ,Y ): v commute    FX0 o FX   F u 

There are functors:

p1 S(F ) / C p1(X, Y, v) = X p1(u, w) = u p2 3 p2(X, Y, v) = Y p2(u, w) = w D o

Let T (F ) be the bisimplicial set, meaning the functor

∆o × ∆o −→ Sets deﬁned by   fp v0  Yp / Yp−1 / ··· / Y0 / FX0 in D      T (F )pq := T (F ) p, q =    X / X / ··· / X / X in   0 u1 1 u2 q−1 uq q C 

For morphisms (f, g) in ∆o × ∆o, deﬁne T (F )(f, g) accordingly.

Treat the simplicial set N (C ) (the nerve), as a bisimplicial set, by (p, q) 7→ N (C )q. Now consider the map of bisimplicial sets:

T (F ) −→ N (C ) projections T (F )pq −→ N (C )pq = N (C )q (4.3)

Given an ordered pair in T (F )pp, as described above, let

vk = F (uk) ··· F (u1)v0f1 ··· fk : Yk −→ FXk. So, (Xk,Yk, vk) ∈ S(F ), and

(X0,Y0, v0) / (X1,Y1, v1) / ··· / (Xp,Yp, vp) ∈ (S(F )) . (u1,f1) N p

This shows that, the nerve N (S(F )) is the diagonal p 7→ T (F )pp (i.e. in bijection). The geometric realization of the projective p1, gives a map:

B(p1): BS(F ) = |T (F )pp| −→ BC

Now, for a ﬁxed q a o |p 7→ T (F )pq| = B(D/F X0) X•∈N (C )q 98 CHAPTER 4. THE QUILLEN-K-THEORY

Since D/F X0 has a ﬁnal object, namely, 1 : FX0 −→ FX0, B(D/F X0) ' ? is contractible. So, there is a homotopy equivalence

a o a |p 7→ T (F )pq| = B(D/F X0) −→ ? = NCq X•∈N (C )q X•∈N (C )q is a homotopy equivalence, for all q.

Write Tq = |p 7→ T (F )pq| , So, T = {Tq} is a simplicial topological space. It follows from above, ∀ q ≥ 0, the map

Tq −→ N (C )q is a homopoty equivalence, with discrete topology on N (C )q. It follows from theorem of May and Tornehave (I did not check) or from [Q, Lemma] that

|T | = |(q 7→ |p 7→ T (F )pq|)| ' BC is a homotopy equivalence. By Lemma 3.5.5 we have the commutative diagram

B(p1) S(F ) |T (F ) | / B pp 6 BC o |T | |(q 7→ |p 7→ T (F )pq|)|

Here the vertical map is a homeomorphism, by Lemma 3.5.5 and the diagonal map is a homotopy equivalence. Therefore, the horizontal map B(p1) is a homotopy equivalence. o o Now we consider the functor p2 : S(F ) −→ D . As before, consider {N (D )pq = o N (D )p}, as a bisimplicial set and we have a map of bisimplicial sets. As in (4.3), consider the projection map to the Y -coordinate:

o o T (F ) −→ N (D ) sending T (F )pq 7→ N (D )p (4.4)

As before, one can check that T (F )pp = N (S(F )). So, the classifying map B(p2) is given by B(p2) o BS(F ) |T (F )qq| / BD

o In fact, for any ﬁxed Y• ∈ N (D ), we have

{(x•, y•) ∈ T (F )pq : y• = Y•}' (Y0 \ F ) By hypothesis, there is a homotopy equivalence

B(Y0 \ F ) −→ pt

So, for ﬁxed p, we have |q 7→ T (F )pq| =

a a 0 B(Y0 \ F ) −→ pt = N (D )p

Yp / Yp−1 / ··· / Y0 Yp / Yp−1 / ··· / Y0 4.1. THE CLASSIFYING SPACES OF A CATEGORY 99 which is homotopy equivalence, ∀ p.

B(p2) S(F ) |T (F ) | / o B qq 6 BD o |(p 7→ |q 7→ T (F )pq|)|

The vertical map is a homeomorphism and the diagonal map is a homotopy equivalence. So, B(p2) is a homotopy equivalence. Now, we have a commutative diagram

p2 p1 D o o S(F ) / C

f F o o S(id) / D π2 π1 D

This induces the corresponding map of the classifying spaces:

B(p2) B(p1) B (D o) o B (S(F )) / B (C )

B(f) B(F ) B (D o) o B (S(id)) / B (D) B(π2) B(π1)

All the horizontal maps are homotopy equivalence. Hence B(f) is a homotopy equivalence. And, hence B(F ) homotopy equivalence. 100 CHAPTER 4. THE QUILLEN-K-THEORY

A Corollary to Theorem A

Deﬁnition 4.1.23. Let F : C −→ D be a functor. For Y ∈ Obj(D), let

F −1(Y ) = {X ∈ Obj(C ): FX = Y } ,→ C denote the subcategory

−1 Arrows of F (Y ) are those that map to 1Y .

1. First, we deﬁne preﬁbred category, via a functor F .

(a) We say that F makes C a preﬁbred category, if the functor

−1 ι : F (Y ) −→ Y \ F given by X 7→ (X, 1X )

has a right adjoint.

(b) Given a preﬁbred category C , given by F : C −→ D, denote the right adjoint by ζ : Y \ F −→ F −1(Y ), and write ζ(X, v) = v∗X

−1 0 θ F (Y ) / Y \ F ( θ(X) = (X, v) ζ where v∗ ∗ % ζ(X, v) =: v X F −1(Y )

This functor v∗ is determined up to canonical isomorphism (what does it mean?). This s called the Base Change.

(d) (Not Needed) A preﬁbred category C , given by F : C −→ D, is called a ﬁbred u v category, if for all pair of compassable arrows Y / Y 0 / Y ” the canonical morphism of functors u∗v∗ −→ (vu)∗ is an isomorphism. For X ∈ F −1(Y ), I do not see a natural map u∗v∗(X) −→ (vu)∗(X). I believe it comes from properties of adjoint functors.

2. Dually, we deﬁne precoﬁbred category, via a functor F .

(a) We say, F makes C a precoﬁbred category, if the functor

−1 F (Y ) −→ F/Y has a left adjoint β (X, v : F (X) −→ Y ) 7→ v∗X 4.1. THE CLASSIFYING SPACES OF A CATEGORY 101

(b) (Not Needed) Now, suppose v : Y −→ Y 0 is a morphism in D. Then, we have the commutative diagram of functors:

−1 Θ 0 F (Y ) / F/Y ( Θ(X) = (X, v) β v∗ % β(X, v) =: v∗(X) F −1(Y 0)

We say, v∗ is cobase change.

∼ (c) (Not Needed) We say, C is coﬀered category, if (vu)∗ −→ v∗u∗.

Corollary 4.1.24. Suppose F : C −→ D is either preﬁbred or precoﬁbred, such that F −1(Y ) is contractible, for all objects Y in D. Then, F is a homotopy equivalence.

Proof. We prove the ﬁrst case. By Corollary 4.1.10, F −1(Y ) −→ Y \ F is a homotopy equivalence. Since F −1(Y ) is contractible, so is Y \ F . Now, the corollary follows from Theorem A (4.1.22).

4.1.4 Theorem B: The Exact Homotopy Sequence

In this section, we apply the long exact sequences of homotopy groups/set in topology (2.6), to Classifying spaces of categories and functors, to obtain a long exact sequence of homotopy groups/sets of categories. That would be the objective of Theorem B (4.1.32), which is proved by an application Theorem A (4.1.22).

Some Further Background From Topology: Before we proceed, we bring in some deﬁnitions from topology.

Deﬁnition 4.1.25. Let g : E −→ B be a map of topological spaces and b ∈ B. The, Homotopy Fiber F(g, b) was deﬁned in (2.4.8, 2.4.10). In the cartesian product notation, the Homotopy Fiber can be described, as follows:

I I I F(g, b) = E ×B B ×B {b} = {(z, γ, b) ∈ E × B × {b} : γ ∈ B , γ(1) = b, γ(0) = g(z)}

This leads to a long exact sequence of homotopy groups/sets (2.6).

More generally, we recall the deﬁnition of homotopy cartesian product. 102 CHAPTER 4. THE QUILLEN-K-THEORY

Deﬁnition 4.1.26. Consider the maps h, g of topological spaces

h0 E E0 / E

g and extend it to g0 g (4.5) 0 B / B B0 / B h h a commutative diagram of topological spaces. Write

0 0 0 0 E(g, h) := B ×B F(I,B)×B E := {(b , γ, e) ∈ B × F(I,B) × E : γ(1) = h(b ), γ(0) = g(e)}

There is a map

( ϕ : E0 −→ E(g, h) ϕ(z) = g0(z), hg0(z), h0(z) where hg0(z) denotes the constant path

The rectangle in (4.5) is called a Homotopy cartesian square, if the map ϕ is a homotopy equivalence.

In the framework of the Homotopy Category H omoTop, Homotopy cartesian is char- acterized as follows.

Proposition 4.1.27. Let

h0 E0 / E

g0 g be a commutative diagram of topological spaces. B0 / B h

This diagram a Homotopy cartesian square if and only if (E0, g0, h0) is the Pullback of (g, h) in the Homotopy Category H omoTop.(This means, the space E(g, h) described above satisﬁes the property of pullback, in H omoTop.)

Outline of the Proof. We need to prove, given a diagram

E0 h0 ϕ h0 E0 / E # 0 0 0 & g0 g h g ∼ g h ∃ ϕ : E −→ E(g, h) 3 E(g, h) / E g0 B0 / B g h B0 / B h commute in H omoTop and ϕ is unique in H omoTop. 4.1. THE CLASSIFYING SPACES OF A CATEGORY 103

Suppose H : E0 × [0, 1] −→ B is a homotopy from gh0 to hg0. Deﬁne

ϕ(z) = (g0(z),H(z, −), h0(z)). Note H(z, 0) = gh0(z),H(z, 1) = hg0(z).

So, ϕ(z) ∈ E(g, h). The outer triangles commute (exactly). One needs to further prove, if ψ : E0 −→ E(g, h) is another map, so that the outer triangles commute, up to homotopy, then ϕ ' ψ. The proof is complete.

Corollary 4.1.28. Let g : E −→ B be a continuous map and b ∈ B. Then, the following commutative diagram:

F (g, b) / E

t=1 g is a pullback diagram in H omoTop. {b} / B

Deﬁnition 4.1.29. Suppose g : E −→ B be a continuous map and b ∈ B. Suppuse

ι F / E g is a pullback diagram in H omoTop. {b} / B

ι g Then, we say that F / E // B is a homotopy ﬁbration.

1. From uniqueness of pullback, F ' F (g, b) are homotopicaly equivalent.

2. Therefore, for x0 ∈ F there is long exact sequence

··· / πn (F, x0) / πn (E, ι(x0)) / πn (B, ι(b)) / πn−1 (F, x0) / ···

The following is an useful lemma.

Lemma 4.1.30. If B is contractible, and g : E −→ B is a map, then the map

I π : F(g, b) = E ×B B × {b} −→ E are homotopy equivalences.

Outline of the Proof. We construct the homotopy inverse as follows:

0 H : B × [0, 1] −→ B be 3 H(−, 1) = 1B,H(−, 0) = Cb Let h : B −→ BI be deﬁned by h(x) = H(x, −). Now deﬁne

ϕ : E −→ F(g, b) ϕ(z) = (z, hg(z), b) . So πϕ = 1E. 104 CHAPTER 4. THE QUILLEN-K-THEORY

Also, need to prove

ϕπ(z, γ, b) = ϕ(z) = (z, hg(z), b) ∼ 1F(g,b) which we skip!

Back to Homotopy of Functors:

Now we try to adapt some of the above to categories and functors.

1. Let F : C −→ D be a functor. Let Y ∈ Obj(D). Let

j : Y \ F −→ C 3 j(X, v) = X where v : Y −→ FX

So, the commutative triangle

j B(j) Y \ F / C B (Y \ F ) / B (C )

F =⇒ B(F ) F j B(F j) " % D B (D)

Let CY : Y \ F −→ D be the constant functor Y. This induces, the constant map, of the classifying spaces:

cY : B (Y \ F ) −→ B (D)

2. There is a natural transformation

CY −→ F j by v : CY (X, v) = Y −→ F j(X, v) = FX

So, by Proposition 4.1.9, there is a homotopy

H : B (Y \ F ) × [0, 1] −→ B (D) ∗ = Y 7→ B (F j)

So, we obtain a canonical map

B (Y \ F ) −→ F(B(F j),Y ) sending z 7→ H(z, −) (4.6)

a path starting from Y to B(F j)(z). 3. Is this map (4.6) homotopy equivalence? We answer this question subsequently.

4. Here is a deﬁnition borrowed from topology. 4.1. THE CLASSIFYING SPACES OF A CATEGORY 105

Deﬁnition 4.1.31. A commutative diagram, of categories

C / D BC / BD is called Homotopy Cartesian, if is so. E 0 / D0 BE 0 / BD0

0 In this case, if E is contractible, we say C // D / D0 is a homotopy ﬁbration. Consequently, a long exact sequence of homotopy groups/set would follow.

Now we are ready to state Theorem B.

Theorem 4.1.32 (Theorem B). Let F : C −→ D be a functor such that

0 0 0 ∀ Y −→ Y ∈ MorD(Y,Y ) the induced functor Y \ F −→ Y \ F is a homotopy equivalence. Then, ∀ Y ∈ Obj(D), the commutative diagram

j Y \ F / C  j(X, v) = X  F 0 F F 0(X, v) = (F X, v) is homotopy cartesian. (4.7)  0 0 0 Y \D / D  j (Y , v) = Y j0

The following would be consequences of the above:

1. Note, ∀ Y ∈ Obj(D), the category Y \ D has an initial object (Y, 1Y ). So, Y \D is contractible. Therefore, the upper right part of the cartesian square (4.7) would be a homotopy ﬁbration.

2. Consequently, there ∀ Y ∈ Obj(D) and X ∈ Obj(C) with FX = Y ˜ (so, X := (X, 1Y ) ∈ Obj(Y \ F )), there is a long exact sequence:

˜ ··· / πi+1 (D,Y ) / πi Y \ F, X / πi (C,X) / πi (D,Y ) / ···

3. As always, this theorem admits a dual formulation, replacing the categories Y \ F by categories F/Y etc., in the diagram (4.7).

A Corollary to Theorem B, inn analogy to Corollary 4.1.24: 106 CHAPTER 4. THE QUILLEN-K-THEORY

Corollary 4.1.33. Suppose F : C −→ D is a preﬁbred functor. Assume, for every arrow 0 −1 −1 0 u : Y −→ Y in D, the base change functor u∗ : F (Y ) −→ F (Y ) is a homotopy equivalence. Then, ∀ Y ∈ Obj(D) the diagram

F −1(Y ) / C

pt / D is homotopy cartesian. (We say that the upper right sequence is a homotopy ﬁber.) Consequently, for any X ∈ Obj(F −1(Y )), there is a long exact sequence:

−1 ··· / πi+1 (D,Y ) / πi (F (Y ),X) / πi (C,X) / πi (D,Y ) / ···

Further, same holds, if we replace the word "preﬁbred", by "precoﬁbred".

Proof. To apply Theorem B (4.1.32), we need to establish that Y 0 \ F −→ Y \ F is a homotopy equivalence. We have the diagram of preﬁbred spaces:

F −1(Y 0) / Y \ F

% F −1(Y )

The vertical functor is a homotopy equivalence, by the deﬁnition of preﬁbred spaces. The diagonal map is a homotopy equivalence, by hypothesis of the corollary. Therefore, the horizontal functor is a homotopy equivalence. Now, we have another commutative diagram:

F −1(Y 0) / Y \ F 9

Y 0 \ F

The vertical map is also a homotopy equivalence, by taking u = 1Y 0 (by preﬁbred property). So, the diagonal map is homotopy equivalence. Now, the corollary follows by an application of Theorem B (4.1.32).

We proceed to give proof of Theorem B (4.1.32): First, we have the following deﬁnitions.

Deﬁnition 4.1.34. Let f : E −→ B be a map of topological spaces. Such a map is called a quasi-ﬁbration, if

∀ b ∈ B, the natural map f −1(b) −→ F(f, b) is a weak equivalence (see Defn.2.5.11). 4.1. THE CLASSIFYING SPACES OF A CATEGORY 107

Recall the natural map ι, given by the diagram

f −1(b) ι $ F(f, b) &/ E

f ! b / B

It follows from Whitehead’s Theorem 2.5.12, if E, B, f −1(b) are CW complexes, g is quasi-ﬁbration if and only if f −1(b) −→ F(f, b) is a homotopy equivalence. Hence, in such cases f −1(b) / E

f is homotopy cartesian. b / B For our purpose, this would be the main point of introducing quasi-ﬁbrations. 108 CHAPTER 4. THE QUILLEN-K-THEORY

Lemma 4.1.35. Suppose I is a small category and I −→ Top is a functor, with i 7→ Xi.

1. Let N I be the nerve of I

2. Consider the simplicial topological space a p 7→ Xp := Xσ(0). Let XI := |{Xp}| be its geometric realization.

σ∈N Ip

We have a map of simplicial spaces γ : {Xp} −→ N I, given by the associations:

p / Xp

γp ! N Ip

3. The map γ of simplicial sets, gives a map g := |γ| : XI −→ BI, of topological spaces.

Assume, ∀ i −→ j ∈ MorI , the induced arrow Xi −→ Xj is a homotopy equivalence. Then, g is a quasi-ﬁbration.

−1 Proof. The proof is given by proving the same to the restriction of g, to g Fp −→ Fp, where Fp is the p-skeletons of BI (see Section 3.3.1). That would suﬃce, by Lemma 1.5 of [Dold-Lashof]. We have a commutative diagram of push out (co-cartesian) diagrams:

` −1 σ∈ I σ × |∂∆[p]| × Xσ(0) / g Fp−1 N p0 _ _ g

` * # σ × |∂∆[p]| / Fp−1 σ∈NIp0 _ _

` −1 σ × |∆[p]| × Xσ(0) / g Fp σ∈NIp0 g

` * # σ × |∆[p]| / Fp σ∈NIp0

Here, the front of the cube is the pushout diagram deﬁning Fp and backside is the pushout p for the p-skeleton XI . The rest of the proof given in [Q] consists of multiple cross references to the paper of [Dold-Lashof]. We would avoid getting in to deeper details of the same. 4.1. THE CLASSIFYING SPACES OF A CATEGORY 109

Proof to Theorem B (4.1.32): We recall some of the notations from Theorem A (4.1.22). Let S(F ) be the category, described below:

1. The objects are

Obj(S(F )) = {(X, Y, v): X ∈ Obj(C ),Y ∈ Obj(D), v ∈ MorD (Y,FX)}

0 0 0 0 0 0 2. For (X, Y, v), (X ,Y , v ) ∈ Obj(S(F )), Deﬁne MorS(F ) ((X, Y, v), (X ,Y , v )) =

 0 w   Y / Y   0 0  (u, w) ∈ MorC (X,X ) × MorD (Y ,Y ): v0 v commute  F u   FX0 o FX  There are functors:

p1 S(F ) / C p1(X, Y, v) = X p1(u, w) = u p2 3 p2(X, Y, v) = Y p2(u, w) = w D o

As in the proof of Theorem A (4.1.22), p1 is a homotopy equivalence. o Also, from the proof of Theorem A (4.1.22), the map B(p2): BS(F ) −→ B(D ) is the realization of the map

a a 0 B(Y0 \ F ) −→ pt = N (D )p

Yp / Yp−1 / ··· / Y0 Yp / Yp−1 / ··· / Y0 By hypothesis (4.1.32), Lemma 4.1.35 is applicable, with I := D o, to the functor Y 7→ o B(Y \ F ) from D −→ Top. By Lemma 4.1.35, we see that Bp2 is a quasi ﬁbration. Since all the spaces concerned are CW complexes, by Whitehead’s Theorem 2.5.12:

Y \ F / S(F )

o ∀ Y ∈ Obj(D ) p2 is a homotopy cartesian. pt / o Y D Now, we have the following diagram: ∼ Y \ F / S(F ) / C

F 0 F Y \ D / S(1D ) ∼ / D

o o pt / o Y D All the three rectangles are actually cartesian. The combination of two rectangles on the left is a homotopy cartesian. So, the upper left rectangle is homotopy cartesian. Also, two horizontal maps on the right hand rectangle are homotopy equivalence. Therefore combined two horizontal rectangles is homotopy cartesian. 110 CHAPTER 4. THE QUILLEN-K-THEORY 4.2 The K-Groups of Exact Categories

This is an exposition of [Q, Section 2] form Quilen’s paper. Given an exact category E , Quillen [Q], associates a new category QE , as follows.

4.2.1 Quillen’s Q-Construction

Deﬁnition 4.2.1. Suppose E is a small exact category. Deﬁne a new category QE as follows.

1. The objects of QE are same as that of E .

2. Give two objects X,Y ∈ QE , a morphism X −→ Y in QE is deﬁned as follows: Consider diagrams, as follows:

 0 p  K / Z / / X  p i XZo o / Y 3 ∃ conﬂations in E . (4.8)   Z / Y / / C  i

In other words, p is a deﬂation and i is an inﬂation. It is sometimes convenient to display this diagram as: Z / Y i p X

Two such diagrams (as in (4.8)),  p i  XZo o / Y  are said to be isomorphic, if   XZo o 0 / Y  p0 i0

Z p p i

∼ 0 ~ ~ ∃ isomorphism τ : Z −→ Z 3 XYo τ commutes. ` ` >

0 0 p . i Z0 A morphism X −→ Y in QE is an isomorphism class of diagrams, as in (4.8). A diagram, as in (4.8), will be denoted by (Z, p, i). 4.2. THE K-GROUPS OF EXACT CATEGORIES 111

3.( Compositions): Let X −→ Y and Y −→ Z be two morphisms in QE , repre- p i q j sented by XWo o / Y , YVo o / Z , the composition is represented by (V, pq0, ji0), as in the diagram:

i0 j U / V / Z

q0 q 0 0 W / Y where (U, i , q ) is the pullback. (4.9) i p X

We remark, composition is well deﬁned and associative. It is also imperative that we assume that the isomorphism classes of such diagrams (4.9), form a set.

We list the following:

1. Recall that an inﬂation A,→ X in E is called an admissible monomorphism and a deﬂation X B in E is called an admissible epimorphism. 2. Given an object X in E , by a sub object K of X, we mean an isomorphism classes of admissible monomorphism K,→ X, as described in the following commutative diagram K / X K / X

o o σ , which is equivalent to o K0 / X K0 / X Likewise, by an admissible quotient C of X, we mean an isomorphism classes of admissible epimorphisms X C, as described in the following commutative diagram

X / / C X / / C

o σ o , which is equivalent to o X / / C0 X / / C0

There is a bijection between sub objects of X and quotient objects of X.

3. The set of all sub objects [K] of X forms a partially ordered set. We say [K0] ≤ [K] (or K0 ≤ K) if there is an admissible monomorphism ι (unique):

0 K _ / X ∃ ι K / X

If K0 ≤ K, we say (K0,K) is an admissible layer of sub objects of X. In this case K/K0 is admissible. 112 CHAPTER 4. THE QUILLEN-K-THEORY

4. It follows, a morphism X −→ Y is QE is a pair ((K,Z), θ) of admissible layers of Y , and an isomrphism θ : X : X −→∼ Z/K. Diagramtically:

K _

ι Z / Y where K = ker(p).

p X

5. To see how composition works with respect to this description of morphisms in QE , via admissiible layers, refer to the diagram (4.9):

W ∼ (a) The map X −→ Y is represented by (ker(p),W ), ker(p) −→ X ,

V ∼ (b) the map Y −→ Z is represented by (ker(q),V ), ker(q) −→ Y .

0 U ∼ (c) The composition X −→ Z is represented by (ker(pq ),U), ker(pq0) −→ X Diagrammatically, extending (4.9):

ker( q) ker( q) Ll _ _

y 0 0 i j ker(pq ) / U / V / Z KV := ker(q) q0 q Write 0 (4.10) KU := ker(pq ) W / Y i p X

6. in QE , morphisms 0 −→ Y are is one to one correspondence with admissible sub objects U,→ Y . In terms of layers they are ((U, U), ∗).

7. Give an admissible monomorphism ι : X,→ Y in E , the corresponding map in QE is denoted by ι! : X −→ Y . In [Q], such a morphism in QE was referred to as "injective"(which is not to be confused with "injective maps" in the usual sense).

8. Likewise, given an admissible epimorphism ζ : Y X in E , it gives rise to a morphism ζ! : X −→ Y in QE . In [Q], such a morphism in QE was referred to as "surjective"(which is not to be confused with "surjective maps" in the usual sense).

! 9. By deﬁnition, any morphism u : X −→ Y in QE can be written as u = ι!p . This factorization is unique up to isomorphism (by deﬁnition):

p ι X o o U / / Y

o XVo o / / Y q ζ 4.2. THE K-GROUPS OF EXACT CATEGORIES 113

10. Given a bicartesian square (i.e. pullback and pushout)

ι U / Y

ζ η ! ! =⇒ u = ι!ζ = η β! (4.11) X / Z β

! This follows directly from the deﬁnition of η β!. The latter factorization is also unique (which follows from the deﬁnitions of compositions and equality of maps.).

11.

Lemma 4.2.2. If a map is both injective and surjective, then it is an isomorphism. ! Proof. Suppose u = Θ! = Ψ . Then, we have the commutative diagram:

1X Θ X o o X / Y

o Φ XYo o / Y Ψ 1Y

−1 This shows that Θ = Ψ . Now, we prove that if Θ is an isomorphism in E , the Θ! is an isomorphism in QE and its inverse is Θ−1)!:

1X Θ X o o X / Y

Θ XYo o / Y Θ−1 1Y The proof is complete.

12.

Lemma 4.2.3. Every morphism in QE is a monomorphism. Proof. Since every map in QE is composition of an "injective" followed by a "surjective" map, prove they are mono in QE . Suppose η : Y,→ W is a monomorphism and β : Y W is a epimorphism in QE . Consider two maps in QE : p ι u : XZo o / Y

q ζ v : XZo o 0 / Y

Now suppose η!u = η!v. Then, we have the combative diagram: p ι η η!u : X o o Z / Y / W

o ϕ q 0 ζ η η!v : XZo o / Y / W 114 CHAPTER 4. THE QUILLEN-K-THEORY

where ϕ is an isomorphism. Since, η is a mono, ζϕ = ι and hence u = v. Likewise, we check that β!u = β!v implies u = v. The proof is complete.

13. For an object Z in E , the category QE \Z is equivalent to the ordered set of admissible layers (KU ,U) of Z. It follows from Equation 4.10 that the ordering is given by

(KU ,U) ≤ (KV ,V ) if KV ≤ KU ≤ U ≤ V

Diagramatically,

14. The Universal Property of QE : First, we have the following observations:

(a) There is No functor E −→ QE because, given a morphism f : X −→ Y in E , there is no natural map in QE associated to f. (b) However, given a admissible ι : X,→ Y monomorphism, in E , we can associate ι! in QE . Given two composable admissible ι, ζ monomorphism, in E , we have (ζι)! = ζ!ι! in QE . Likewise, given a admissible p : X Y epimorphism, in E , we can associate p! in QE . Given two composable admissible p, q epimorphism, in E , we have (pq)! = q!p! in QE . ! Further (IdX )! = (IdX ) = IdX . (c) Given the bicartesian square (4.11), where all the maps are admissible (epi or ! ! mono), then ι!ζ = η β!.

Now, let C be a category. Consider the diagram of associations (not functors):

E / QE   X 7→ hX ∀ X ∈ Obj(E ) 3 ι 7→ ι! : hX −→ hY ∀ admissible mono ι : X,→ Y h !  p 7→ p! : hY −→ hX ∀ admissible epi p : X Y C While h is a not a functor, the above deﬁnes two functors:

(a) E− −→ C where E− is the category whose objects are same as those of E , and morphisms are the inﬂations of E ,

(b) E+ −→ C where E+ is the category whose objects are same as those of E , and morphisms are the deﬂations of E . Further assume that conditions 14b, 14c hold. Then, there is a unique functor Θ, as follows:

E / QE ι 7→ ι! Θ ”commute” and compatible with h p 7→ p! ! C

Finally, we have the obvious lemmas: 4.2. THE K-GROUPS OF EXACT CATEGORIES 115

Lemma 4.2.4. For any exact category E , we have

o QE = Q(E ) induced by the contra variant functor E −→ E o.

4.2.2 The Higher K-groups of an Exact Category

First, we deﬁne the Grothendieck group of an exact category.

Deﬁnition 4.2.5. Suppose E is a small exact category. Let Z(E ) be the free abelian group generated by the objects of E . Let R(E ) be the subgroup of Z(E ) generated by the set

Y − X − Z ∈ Z(E ): X / Y / / Z is a conﬂation.

Deﬁne the Grothendieck group Z(E ) K0(E ) := R(E )

We Remark:

1. The conﬂation 0 / 0 / / 0 implies 0 = [0] ∈ K0(E ).

∼ π 2. Suppose π : X −→ Y is an isomorphism. Then, 0 / X / / Y is a conﬂation.

Hence [X] = [Y ] ∈ K0(E ).

3. The map ι : E −→ K0(E ) has the following universal property: Suppose G is an abelian group and β : E −→ G is a set theoretic map, such that

ζ(Y ) = ζ(X)ζ(Z) ∀ conﬂations X / Y / / Z

Then, there is a group homomorphism

ι Obj(E ) / K0(E )

ψ : K0(E ) −→ G 3 ψ commutes. β % G

We say that K0(E ) has the above universal property, in the category Ab of abelian

groups. We will see next, that K0(E ) has the same universal property, in the category Gr of groups. 116 CHAPTER 4. THE QUILLEN-K-THEORY

4. The above deﬁnition K0(E ) makes sense, whenever E has a set of isomorphism classes of objects, by replacing E by an equivalent small exact subcategory E 0 ⊆ E . In this case (and otherwise), let E be the set of all isomorphism classes of objects in E and Z(E) be the free abelian groups generated by E. Let R(E) ⊂ Z(E) be deﬁned as above. Then, Z(E) K0(E ) = . R(E)

Lemma 4.2.6. Let E be a small exact category. Then, ι : Obj(E ) −→ K0(E ) has the following universal property:

Given any (possibly, non-commutative) group G and a set theoretic map ζ : Obj(E ) −→ G such that ζ(Y ) = ζ(X)ζ(Z) ∀ conﬂations X / Y / / Z there is a group homomorphism

ι Obj(E ) / K0(E )

ψ : K0(E ) −→ G 3 ψ commutes. ζ % G

In particular, Z(E ) ∼ F (E ) K0(E ) := = R(E ) N (E ) where F (E ) is a the free (non-abelian group generated by Obj(E ) and N (E ) is the normal subgroup generated by the set

−1 W = Y (XZ) : X / Y / / X is a conﬂation .

Proof. Write ( ) := F (E ) .(Recall, elements of ( ) are "words".) Note that the map G E N (E ) F E η : Obj(E ) −→ G (E ) has the universal property in the category Gr of groups. Using the split exact conﬂations, it follows η(X)η(Y ) = η(X ⊕ Y ) = η(Y ⊕ X) = η(Y )η(X). Form this, it follows, G (E ) is abelian. Therefore, both G (E ) and K0(E ) have same universal property in the category Ab of abelian groups. Hence G (E ) ≡ K0(E ).

The following proposition establishes the key connection between classical K-theory to homotopy groups of BQE .

Proposition 4.2.7. Let E be a small Exact category. The classifying space BQE is considered as a pointed topological space, the base point being the 0-object of E . Then, 4.2. THE K-GROUPS OF EXACT CATEGORIES 117

1. The classifying space BQE is (path) connected. So, π0 (BQE ) = 0.

∼ 2. There is an isomorphism K0(E ) −→ π1 (BQE , 0).

Proof. The following proof is drawn from [Sm1], which is more elaborate than the one in [Q]. Suppose X is an object of E or QE . The diagram 0o o 0 / X represents a map 0 −→ X in . 0X QE

This gives a path from 0 to X. Since all the vertices of BQE are path connected to 0, QE is path connected. Hence π0 (BQE ) = 0.(We comment that MorQ(E )(0,X) is not necessarily a singleton.) With notations, as Lemma 4.2.6, F (E ) K0(E ) ≡ N (E )

Deﬁne a group homomorphism ϕ0 : F (E ) −→ π1 (BQE ), as follows. For an arrow → in QE , the corresponding path in the classifying space B(QE ) will be denoted by γ(→). Suppose X ∈ Obj(E ). Consider two morphisms 0 −→ X in QE :

0X (0, 0, 0X ) : 0o o 0 / X

1X (X, 0, 1X ) : 0 o o X / X

By Lemma 4.1.3, these two morphisms deﬁnes two paths γ0 := γ((0, 0, 0X ), γ1 := γ((X, 0, 1X ) −1 from 0 to X in BQE . So, `X := γ0 γ1 is a loop. Deﬁne

ϕ0 : F (E ) −→ π1 (BQE ) by ϕ(X) = [`X ] Now, consider a conﬂation

i p −1 X / Y / / Z which corresponds to Y (XZ) ∈ N (E ).

−1 + Now, `X = γ(0, 0, 0X ) γ(X, 0, 1X ): 0 3 X is homotopic to

γ(0,0,0X ) γ(X,1 ,i) −1 + X γ ((X, 1X , i)(0, 0, 0X )) γ ((X, 1X , i)(X, 0, 1X )) : 0 3 X / Y γ(X,0,1X )

The compositions (X, 1X , i)(0, 0, 0X ) is given by

0X i 0 / X / Y

0 1X 0 / X which is (0, 0, 0Y ) 0X 0 0 118 CHAPTER 4. THE QUILLEN-K-THEORY

Likewise, the compositions (X, 1X , i)(0, 0, 1X ) is given by

1X i X / X / Y

1X 1X X / X which is (X, 0, i) 1X 0 0

−1 −1 So, [`X ] = [γ(0, 0, 0Y ) γ(X, 0, i)]. Likewise, [`Z ] = [γ(X, 0, i) γ(Y, 0, 1Y )]. So, [`X ][`Z ] = [`Y ]. Therefore, ϕ0 factors through a well deﬁned homomorphism ϕ : K0(E ) −→ π1 (BQE , 0).

Now, we prove that ϕ is surjective. Let ` ∈ π1 (BQE , 0). By Proposition 2.5.13, the map 1 1 π1 (BQE , 0) π1 (BQE , 0) is surjective, where BQE denotes the one skeleton. Therefore ` is represented by a product of paths γnγn−1 ··· γ1, such that

1. γi is a path from Xi−1 to Xi, with Xi ∈ Obj(E ), and X0 = Xn = 0. 2. Further,

(a) either γi = γ(fi), in which case fi ∈ MorQE (Xi−1,Xi), −1 (b) Or, γi = γ(fi) , in which case fi ∈ MorQE (Xi,Xi−1).

3. In fact, for any nonzero object X in E , MorQE (X, 0) = φ. So, we can assume f1 : 0 −→ X1, and fn : 0 −→ Xn−1.

Let gi = γ(Xi, 0, 1Xi ). Up to homotopy, we have

−1 −1 −1 γnγn−1 ··· γi ··· γ1 = (γngn−1) ··· gi γigi−1 ··· g2 γ2g1 g1 γ1

−1 We will prove that, each gi γigi−1 is in the image of ϕ. There are two cases,

1. γi = γ(fi), where fi : Xi−1 −→ Xi. So, fi is given by

Xi−1 o o p Ui ι / Xi

We have, gi−1 = γ(Xi−1, 0, 1Xi−1 ) is given by the arrow

0 o o Xi−1 Xi−1

So, γigi−1 is given by the arrow

0 o o Ui ι / Xi

Therefore,

−1 −1 −1 gi γigi−1 = γ(Xi, 0, 1Xi ) γ(Ui, 0, ι) ∼ γ(Xi, 0, 1Xi ) γ ((Ui, 1Ui , ι)o(Ui, 0, 1Ui )) 4.2. THE K-GROUPS OF EXACT CATEGORIES 119

−1 ∼ γ(Xi, 0, 1Xi ) γ(Ui, 1Ui , ι)γ(Ui, 0, 1Ui ) −1 −1 ∼ γ(Xi, 0, 1Xi ) γ(Ui, 1Ui , ι)γ(0, 0, 0Ui )γ(0, 0, 0Ui ) γ(Ui, 0, 1Ui ) −1 ∼ γ(Xi, 0, 1Xi ) γ ((Ui, 1Ui , ι)o(0, 0, 0Ui )) γ(Ùi ) −1 −1 ∼ γ(Xi, 0, 1Xi ) γ(Xi, 0, 0Xi )γ(Ùi ) ∼ γ(`Xi ) γ(Ùi ) So, −1 −1 gi γigi−1 = ϕ(Xi) ϕ(Ui)

−1 2. γi = γ(fi) , where fi : Xi −→ Xi−1. This case is dealt similarly, as above.

This establishes that ϕ is surjective.

To prove that ϕ is an injective, we deﬁne a map π1 (BQE , 0) −→ K0(E ). As in Example 4.1.5, let K0(E ) denote the category with one object ? and Mor(?, ?) = K0(E ). Deﬁne a functor Φ: QE −→ K0(E ) ∀ X ∈ Obj(QE ) Φ(X) = ? and, ι W / Y

for f := p ∈ MorQE (X,Y ) Φ(f) = [ker(p)]. X We check that Φ is a functor. For another morphism

ι V / Z

g := q ∈ MorQE (Y,Z) composition gf is displayed in Diagram 4.10 Y

We need to prove that [ker(p)] + [ker(q)] = [ker(pq0)]. This follows, because

ker(q) / ker(qp0) / / ker(p) is a conﬂation.

Therefore, Φ is a functor. Hence, it induces a map η := π1 (BΦ) : π1 (BQE , 0) −→ π1 K0(E ),? = K0(E )

Now, for [X] ∈ K0(E ),

−1 η(ϕ([X])) = η(γ(0, 0,X) γ(X, 0, 1X )) = −[0] + [X] = [X].

Therefore, ϕ is a bijection. The proof is complete. 120 CHAPTER 4. THE QUILLEN-K-THEORY

4.2.3 Higher K-Groups

Deﬁnition 4.2.8. Suppose E is a small exact category and 0 denote the/a zero object.

∀ i ∈ N deﬁne Ki(E ) = πi+1 (BQE , 0)

We have the following remarks:

1. The deﬁnition is independent of the choice of the zero object, because BQE is path connected.

2. The deﬁnition extend to any exact category E that has a set of isomorphisms classes of objects.

Lemma 4.2.9. Any exact functor F : E −→ D, of exact categories induces a functor QE −→ QD. This induces homomorphisms of K groups

Fi := Ki(F ): Ki(E ) −→ Ki(D) ∀ i ≥ 0

1. So, for each (ﬁxed) i ≥ 0, the association

i 7→ Ki(E )

is a functor, from the "category" of exact categories and exact functors to the category of abelian groups.

∼ ∼ 2. If F : C −→ D is an equivalence of categories, then Fi : Ki(E ) −→ Ki(D) are isomorphisms, ∀ i ≥ 0.

3. For any exact category E , by Lemma 4.2.4, we have

o o ∀ i ≥ 0 Ki(E ) = πi+1(BQ(E )) = πi+1(B(QE )) = Ki(E ).

4. Suppose E , D are two exact categories. Then, E × D is also an exact category. It follows Q(E × D) = Q(E ) × Q(D). Further, it follows B(Q(E × D)) = B(Q(E )) × B(Q(D)). Therefore,

∀ i ≥ 0 Ki(E × D) ' Ki(E ) ⊕ Ki(D).

5. Suppose E is an exact category. Then the direct sum functor E × E −→ E , sending (M,N) 7→ M ⊕ N is an exact functor. This induces a group homomorphism

∀ i ≥ 0 ⊕∗ : Ki(E ) ⊕ Ki(E ) −→ Ki(E ) 4.2. THE K-GROUPS OF EXACT CATEGORIES 121

The map coincides with the usual direct sum map because of the commutative diagram:

Ki(E ) 1 i1 # Ki(E ) ⊕ Ki(E ) / Ki(E ) O ⊕∗ ; i2 1 Ki(E )

where i1 is induced by the functor M 7→ (M, 0) and i2 is induced by the functor M 7→ (0,M).

6. Suppose I is a small ﬁltering category and j 7→ Ej functor from I to the category of small exact categories.

Let lim→ Ej be the inductive limit ([Q, Prop 3, pp 84]), which is an exact category.

Then, Q(lim→ Ej) = limj (Q(Ej)). Hence,

∀ i ≥ 0 Ki lim Ej = lim Ki(Ej). → → Example 4.2.10. Let A be a ring with unity and P(A) be category of all projective (left) A-modules. We write,

∀ i ≥ 0 Ki(A) := Ki (P(A))

1. Given a ring homomorphism A −→ B induces a functor P(A) −→ P(B), sending P 7→ B ⊗ P . This induces well deﬁned group homomorphism

∀ i ≥ 0 Ki (A) −→ Ki(B).

2. This makes Ki(A) a covariant functor from the category of rings with unit, to the category of abelian groups.

3. It follows,

∀ i ≥ 0 Ki(A × B) = Ki(A) ⊕ Ki(B)

4. Suppose I is a small ﬁltering category and j 7→ Aj functor from I the category of rings with unit. It follows, from (6): ∀ i ≥ 0 Ki lim Aj = lim Ki(Aj). → → Example 4.2.11. Let A be a noetherian commutative ring with unit. 122 CHAPTER 4. THE QUILLEN-K-THEORY

1. The category QCoh(A) be the category of A-modueles, is an exact category. So, this gives rise to a K-theory.

2. The category Coh(A) be the category of ﬁnitely generated A-modules, is an exact category. So, this gives rise to a K-theory. This is usually referred to as G-theory of

A and the K-groups are denoted by Gi(A).

3. Then, we have Example 4.2.10. The category P(A) be the category of ﬁnitely generated projective A-modules over X, is an exact category. So, this gives rise to a K-theory. This is usually referred to as K-theory of X and the the K-groups are

denoted by Ki(A).

Comparison of the K-Theory of these categories constitutes a big chunk of the literature.

Example 4.2.12. More generally, let X be (noetherian) scheme.

1. The category QCoh(X) be the category of quasi coherent modules over X, is an exact category. So, this gives rise to a K-theory.

2. The category Coh(X) be the category of quasi coherent modules over X, is an exact category. So, this gives rise to a K-theory. This is usually referred to as G-theory of

X and the the K-groups are denoted by Gi(X).

3. The category P(X) be the category of locally free modules over X, is an exact category. So, this gives rise to a K-theory. This is usually referred to as K-theory of

X and the K-groups are denoted by Ki(X).

Comparison of the K-Theory of these categories constitutes a big chunk of the literature.

Lemma 4.2.13. Let A be a commutative noetherian ring. The classical deﬁnition of c K1(A), which we denote, temporarily, by K1(A), is as follows:

c GL∞(A) K1(A) = EL∞(A)

Readers are referred to [GM] for further details. We have,

c GL∞(A) K1(A) = ' K1(P(A)) EL∞(A)

Proof. Skip. See [Q, pp. 96], for the statement. 4.3. CHARACTERISTIC EXACT SEQUENCES AND FILTRATIONS 123

Lemma 4.2.14. Let A be a commutative noetherian ring. A classical deﬁnition of K2(A), c which we denote, temporarily, by K2(A), is obtained through the so called "Steinberg Groups", given in [Mi]. We have,

c K2(A) ' K2(P(A))

Proof. Skip. See [Q, pp. 96], for the statement.

4.3 Characteristic Exact Sequences and Filtrations

Let E be an exact category and let ε(E ) be the category of the exact sequences in E . More precisely, Objects X,Y of ε(E ) are the exact sequences (i.e. conﬂations) in E , as the diagram and a morphism f : X −→ Y is a commutative diagram in E : X : 0 / K / M / / C / 0

f ι β ϕ (4.12) Y : 0 / K0 / M 0 / / C0 / 0

1. Give X ∈ Obj(ε(E )) as in (4.12), write s(X) := K, t(X) = M and q(X) = C. Then, s, t, q : ε(E ) −→ E deﬁnes three functors. s(X), t(X), q(X) would be called "sub", "total" and "quotient" objects of X.

2. In fact, ε(E ) has a structure of an exact category, where a sequence X / Y / Z is declared exact, if the corresponding vertical sequences (as in diagram (4.12)) are exact.

3. It follows, s, t, q : ε(E ) −→ E are exact functors.

Theorem 4.3.1. Let E be an exact category. Then, the functor (s, q): Q(ε(E )) −→ Q(E ) × Q(E ) is a homotopy equivalence.

Proof. By Theorem 4.1.22, it would be enough to show that (s, q)/(K,C) is contractible, for any pair of objects (K,C) of Q(E ). Fix such a pair of objects (K,C) and let C := (s, q)/(K,C). So,

C := (s, q)/(K,C) = {(X, u, v): u ∈ MorQE (sX, K), v ∈ MorQE (qX, C)} More explicitly,  p i  u : sXo o U / K  (4.13)  q j  v : qXo o V / C 124 CHAPTER 4. THE QUILLEN-K-THEORY

1. Let E 0 be the full subcategory of C , consisting of objects (X, u, v) such that u is surjective, meaning u looks like

u : sXo o K K

2. Let E ” be the full subcategory of C , consisting of objects (X, u, v) such that u is surjective and v is injective, meaning u, v looks like

u : sXo o K K

v : qX qX / C

Lemma 4.3.2. The inclusion functors C ” −→ C 0 and C 0 −→ C have left adjoints. Consequently, C −→ C ” is a homotopy equivalence (by (4.1.10)).

Proof. First, we look at the functor C 0 −→ C . Given X = (X, u, v) in C , we will establish that there is an universal arrow ι : X −→ X , with X ∈ C 0. This means, given Y ∈ C 0 and ζ : X −→ Y in C , as in the diagram, there is a unique morphism η, as in the commutative diagram: ι X / X

η ∃ unique ζ Y As in Equation 4.13,

p i sXo o U / K / / W

p i p 0 0 ! 0 u : sXo o U / K =⇒ j =⇒ u = (j ) (i )! sX / M / / W i0

! where W is the co-kernel of i and M is the pushout. Then, u = j i!. Deﬁne X , as follows:

X sX _ / tX _ / / qX i0 X := M / T / / qX

0 where T is the pushout. Therefore s(X) = M, t(X) = T , q(X ) = qX. So, j : K s(X), induce the "surjective" map (j0)! : s(X) −→ K in QE . Further, v : q(X) −→ C. Deﬁne X := (X, j!, v). Then, X is in C 0. One checks, X −→ X is universal. It follows, from universality, for X in C and Y in C 0,

∼ MorC (X , Y) −→ MorC 0 (X , Y)

This means, X 7→ X is a Left Adjoint of the inclusion C 0 −→ C . 4.3. CHARACTERISTIC EXACT SEQUENCES AND FILTRATIONS 125

Now, we prove that C ” −→ C 0 has a left adjoint. Let X = (X, u, v) be an object in C 0, where u : sX −→ K is surjective and v : qX −→ C is a morphism in QE , which is given by j i v : qXo o U / C We essentially dualize the above argument. Consider the diagram:

X : 0 / sX / T / U / 0

j X : 0 / sX / tX / qX / 0 where T is the pull back. Write X = (X, u, i!). One checks that X −→ X is universal. This again establishes that C ” −→ C 0 has a left adjoint, namely , X 7→ X . The proof is complete. Proof of Theorem 4.3.1: By Lemma 4.3.2, it would suﬃce to show that C ” has an initial ! object. Let jK : K 0 and iC : 0 −→ C be the unique maps. Let X0 = (0, jK , (iC )!). One checks that X0 is an initial object in C ”. The proof is complete.

4.3.1 Additivity Theorem

Theorem 4.3.3. Let E , D be two exact categories. Let G, F, H : E −→ D be three exact functors, such that

0 / G / F / H / 0 be exact. Then,

∀ i ≥ 0 F∗ = G∗ + H∗ : Ki(E ) −→ Ki(D)

Proof. As before, let ε(D) denote the exact category of short exact sequences in D. Let s, t, q : ε(D) −→ D be the sub, total and quotient functors. Then,

0 / s / t / q / 0 is exact.

First, we prove that t∗ = s∗ + q∗ : Ki(ε(D)) −→ Ki(D). Let f : D × D −→ ε(D) be the functor sending (K,C) to the split exact sequence:

0 / K / K ⊕ C / C / 0

Note (s, q)f = Id. By Theorem 4.3.1, (s, q) is a homotopy equivalence Q(ε(D)) −→ Q(D) × Q(D). Hence, f is also a homotopy equivalence Q(D) × Q(D) −→ Q(ε(D)). 126 CHAPTER 4. THE QUILLEN-K-THEORY

We have f QD × QD / Qε(D)

f tf = ⊕(s, q)f : Qε(D) t

(s,q) × QD QD ⊕ / QD So,

t∗f∗ = ⊕∗(s∗, q∗)f∗ = (s∗ + q∗)f∗

Since f∗ is isomorphism, we have t∗ = s∗ + t∗. To prove the theorem, consider the exact functor ϕ : E −→ ε(D) ending X 7→ 0 / G(X) / F (X) / H(X) / 0

Now, for integers i ≥ 0, we have consider the maps Ki (E ) −→ Ki (D). We have

F∗(X) = (tϕ)∗(X) = t∗(ϕ∗(X)) = s∗(ϕ∗(X))+q∗(ϕ∗(X)) = (sϕ)∗(X)+(qϕ)∗(X) = G∗(X)+H∗(X).

The proof is complete.

Corollary 4.3.4. Let Φi : C −→ D be exact functors, for i = 0, 1, . . . , n. Suppose

0 / Φ0 / Φ1 / Φ2 / ··· Φn / 0 is an exact sequence. Then,

n X p (−1) (Φp)∗ = 0 : Ki(E ) −→ Ki(D) p=0

Proof. Follows from Theorem 4.3.3, by inducetion.

Filtrations of Functors

Let E , D be two exact categories (small).

1. Let C = F un(E , D) be the category al all exact functors, whose objects are exact functors and a morphism Φ: F −→ G is a natural transformation. 4.3. CHARACTERISTIC EXACT SEQUENCES AND FILTRATIONS 127

2. In fact, C = F un(E , D) is an exact category, where a sequence

0 / Φ / Ψ / Θ / 0

is declared short exact, if

∀ X ∈ Obj(E ) 0 / Φ(X) / Ψ(X) / Θ(X) / 0 is exact.

3. A sequence of natural transformations (morphisms), in C ,

0 = Φ0 / Φ1 / Φ2 / ··· Φn = Φ

is deﬁned to be, an admissible ﬁltration, if

∀ X ∈ Obj(E ), ∀ i = 0, . . . , n−1 Φi(X) / Φi+1(X) is an admissible monomorphism.

4. Given an admissible ﬁltration, as above, the quotient morphisms Φi+1 are deﬁned, Φi determined up to canonical isomorphism. There does not seem to be any reason why such a quotient would be exact. However, if Φi+1 is exact ∀ i = 0, 1, . . . , n − 1, then Φi Φp is exact, for all 0 ≤ q ≤ p ≤ n. Φq

Corollary 4.3.5. Let Φ: C −→ D be an exact functor. Suppose Φ admits an admissible ﬁltration 0 = Φ0 / Φ1 / Φ2 / ··· Φn = Φ such that Φp+1 is exact ∀ p = 0, 1, . . . , n − 1. Then, Φp n X Φp Φ∗ = : Ki(E ) −→ Ki(D) Φ p=1 p−1 ∗

Proof. Follows from Theorem 4.3.3.

4.3.2 The Prime Examples

Example 4.3.6. Let A be commutative noetherian ring and P(A) be the category ﬁnitely generated projective A-modules. Then, we write Ki(A) := Ki(P(A)). This example was discussed in (4.2.10).

1. Given a projective A-modules P , the Q 7→ P ⊗ Q deﬁnes an exact functor P ⊗ : P(A) −→ P(A). This will induce homomorphisms:

[P ⊗ −]: Ki(A) −→ Ki(A) 128 CHAPTER 4. THE QUILLEN-K-THEORY

2. In fact, short exact sequences

0 / P−1 / P / P1 / 0

induce exact sequences of exact functors:

0 / P−1 ⊗ − / P ⊗ − / P1 ⊗ − / 0

So, it follows, from (4.3.3),

[P ⊗ −]∗ = [P−1 ⊗ −]∗ + [P1 ⊗ −]: Ki(A) −→ Ki(A) ∀ i ≥ 0

Let X be a ringed space. Let P(X) be the category of locally free sheaves (i.e. category of locally free sheaves of OX -modules). Then, all of the above works for P(X).

Before we discuss the next example, we give the following graded versions of Nakayama’s Lemma.

Lemma 4.3.7. Let A = A0 ⊕ A1 ⊕ A2 ⊕ · · · be a graded commutative ring and M = L M be a ﬁnitely generated graded A-module. Let A = L A and M = A M. n∈Z n + i≥1 i + Then, M = 0.

Proof. Let m1, m2, . . . , mk be a set of homogeneous generators of M. We have       m1 a11 a12 ··· a1k m1  m2   a21 a22 ··· a2k   m2    =      ···   ············   ···  mk ak1 ak2 ··· akk mk where aij ∈ A are homogeneous elements of positive degree. Let A = (aij) be the k × k matrix, as above. So,     m1 0  m2   0  (I − A)   =    ···   ···  mk 0

Multiplying by the Adj(I − A), we have det(I − A)mi = 0 ∀ i = 1, . . . , k. It follows, det(A) = 1 + f1 + ··· + fn, where fj ∈ Aj, for j = 1, . . . , n. Therefore

∀ i = 1, . . . k (1 + f1 + ··· + fn)mi = 0; hence mi = 0.

So, M = 0. The proof is complete.

The following is a graded analogue of the corresponding ungraded result. 4.3. CHARACTERISTIC EXACT SEQUENCES AND FILTRATIONS 129

Lemma 4.3.8. Let A = A ⊕A ⊕A ⊕· · · be a graded commutative ring and P = L P , 0 1 2 n∈Z n Q = L Q be a ﬁnitely generated graded projective A-modules. Let A = L A . Let n∈Z n + i≥1 i P ∼ Q ϕ0 : −→ be an isomprphism of A0-modules. Then, ϕ0 lifts to and isomorphism A+P A+Q ϕ : P −→∼ Q of graded A-moduels.

Proof. Similar to the proof of the corresponding ungraded result, using (4.3.7).

Example 4.3.9. Let A = A0 ⊕ A1 ⊕ A2 ⊕ · · · be a graded commutative ring and Pgr(A) be the category of ﬁnitely generated graded projective modules P = L P . n∈Z n

−1 1. Then, for all integers i ≥ 0, the groups Ki (Pgr(A)) is a Z[t, t ]-module, where multiplication by t is induced by the functor

Pgr(A) −→ Pgr(A) given by P 7→ P (−1), where P (−1)n = Pn−1

Proposition 4.3.10. For all integers i ≥ 0, there is a Z[t, t−1]-module isomorphism

−1 ∼ Z[t, t ] ⊗Z| Ki(A0) −→ Ki (Pgr(A)) 1 ⊗ x 7→ (A ⊗A0 −)∗x

Proof. Note the functor

P(A0) −→ Pgr(A) given by Q 7→ A ⊗A0 Q is an exact functor. So, it induces a the following homomorphisms:

Ki(A0) / Ki (Pgr(A)) 5

−1 Z[t, t ] ⊗Z Ki(A0)

Given P in Pgr(A) and integer k, let ! M Fk(P ) = A Pn ⊆ P n≤k and

Pq = {P ∈ Obj (Pgr(A)) : F−q−1(P ) = 0,Fq(P ) = P } be the full subcategory.

We have a functor P T : Pgr(A) −→ Pgr(A0),T (P ) = A0 ⊗A P = , where A+ = A1 ⊕ A2 ⊕ · · · A+P 130 CHAPTER 4. THE QUILLEN-K-THEORY

It follows from Lemma 4.3.8, that P is non-canonically isomorphic to a A ⊗A0 T (P ) = A(−n) ⊗A0 T (P )n (4.14) n Now, it follows from (4.14):

1. Fk : Pgr(A) −→ Pgr(A) is an exact functor. This is because a Fk(P ) = A(−n) ⊗A0 T (P )n n≤k

2. For each q, there is a ﬁltration of the functors 1Pq :

0 = F−q−1 ⊆ F−q ⊆ · · · ⊆ Fq = 1Pq

(here Fi means the restriction of Fi.

Fk(P ) ∼ 3. = A(−k) ⊗A T (P )n. Fk−1(P ) 0

Deﬁne the map

` n n Φ: −q≤n≤q t ⊗ Ki(A0) −→ Ki(Pq) t ⊗ x 7→ (A(−n) ⊗A0 −)∗x ` n n Ψ: Ki(Pq) −→ −q≤n≤q t ⊗ Ki(A0) x 7→ ⊕−q≤n≤qt ⊗ (Tn)∗(x) By Corollary 4.3.5, one can see Φ and Ψ are inverse of each other. Since Pgr(A) is union of Pq, the proposition follows. The proof is complete. .

4.4 Reduction by Resolution

From the point of view of the Classical K-Theory, the following example motivates this section.

Example 4.4.1. Let A be a noetherian commutative ring. Let P(A) be the category of finitely generated projective A-modules and let H(A) be the category of finitely generated ∼ A-modules, with finite projective dimension. Then, K0(P(A)) = K0(H(A)). (Use Definition 4.2.5 only.)

Proof. For definitions of the classical K0, refer to Definition 4.2.5 (4) . As usual, the ∼ inclusion functor ι : P(A) −→ H(A) induces an homomorphism K0(ι): K0(P(A)) = K0(H(A)). Now, we define the inverse of this map K0(ι). Let H denote the isomorphism classes objects in H(A). For an object in M in H(A), let M denote its isomorphism class in H and [M] denote its class in K0(H(A)) (and likewise for P(A)). For such an M, let

0 / Pn / Pn−1 / ··· / P1 / P0 / M / 0 4.4. REDUCTION BY RESOLUTION 131

be a P-resolution of M. We say that P• is a P(A)-resolution of M. Now, deﬁne

n X i ϕ0 : H −→ K0(P(A)) by ϕ(M) = (−1) [Pi]. i=0

We need to show that ϕ0 is well deﬁned. Let Q• be another P(A)-resolution of M (by augmenting zeros at the tails, we can assume that both P• and Q• have same length n). It would be suﬃcient to prove that there is an exact sequence

··· / P1 ⊕ Q2 / P0 ⊕ Q1 / Q0 / 0

To see this, note that by properties of projective modules, there is a map of chain complexes f• : P• −→ Q•, as in the following commutative diagram:

0 / Pn / Pn−1 / ··· / P1 / P0 / M / 0

fn fn−1 f1 f0 0 / Qn / Qn−1 / ··· / Q1 / Q0 / M / 0

Let C(f•) denote the cone of f• (see [W, pp. 18-19]). So, C(f•)n = Qn ⊕ Pn−1 and the sequence 0 / Q• / C(f•) / P•[−1] / 0 This gives rise to a long exact sequence of homologies

··· / H1(P•) / H1(Q•) / H1(C(f•)) / H0(P•) / H0(Q•) / H0(C(f•)) / 0

Since H0(P•) = H0(Q•) = M, and Hi(P•) = Hi(0•) = 0, ∀ i ≥ 1, it follows Hi(C(f•)) = 0 ∀ i. This establishes the above and hence ϕ0 is an well deﬁned set theocratic map. So, ϕ0 extends to a homomorphism, as follows

ϕ0 H / K0(P(A)) _ 8

ϕ 0 Z(H)

Now, K ( (A)) = Z(H) . We show that ϕ respects the relations (H). To see this, 0 H R(H) 0 R suppose 0 / K / M / C / 0 is exact in H(A).

Given ﬁnite P(A)-resolutions P• of K and Q• of C (say of same length n), inductively, one can build a resolution

0 / Qn ⊕ Pn / Qn−1 ⊕ Pn−1 / ··· / Q1 ⊕ P1 / Q0 ⊕ P0 / M / 0

Therefore

ϕ0(M) = ϕ0(K) + ϕ0(C) 132 CHAPTER 4. THE QUILLEN-K-THEORY

Hence, ϕ0 factors through a homomorphism:

ϕ0 H / K0(P(A)) _ 8 O ϕ ϕ 0 Z(H) / K0(H(A))

It is easy to see that K0(ι) and ϕ are inverse of each other. The proof is complete.

4.4.1 Extension closed and Resolving Subcategories

Deﬁnition 4.4.2. Suppose E is an exact category (with a set of isomorphism classes). For subsequent reference, let

0 / K / M / C / 0 be an exact sequence in E . (4.15)

Recall, short exact sequences (4.15) is also known as "extensions" (see [HS, Chapter III]), or "conﬂations". A full subcategory P of E is called an extension closed subcategory of E ,

1. if P has a zero object,

2. and, if for any exact sequence (4.15), whenever K,C are isomorphic to some objects in P, then so is M.

(see Deﬁnition 1.3.3). We have the following:

1. If P is an extension closed subcategory of E , then P is also an exact category, where a sequence in P is declared exact, if and only if it is exact in E . Consequently, P ,→ E is an exact functor.

2. Further, if P is an extension closed subcategory of E , then Q(P) is a subcategory of Q(E ), while not necessarily a full subcategory.

The Resolution Theorem

Theorem 4.4.3 (Theorem 3). Suppose P is a full subcategory of an exact category E , which is extension closed. Assume

1. Given a short exact sequence (4.15) in E , if M ∈ Obj(P), then K ∈ Obj(P). 4.4. REDUCTION BY RESOLUTION 133

2. For any C ∈ Obj(E ), there is a short exact sequence (4.15), with M ∈ Obj(P).

(Equivalently, (1) P is closed under subobjects and (2) any C ∈ Obj(E ) is subquotient of an object in P.) ∼ Then, the inclusion QP ,→ QE is a homotopy equivalence. Consequently, Ki(P) −→

Ki(E ) ∀ i ≥ 0.

Proof. Let C be the full subcategory of Q(E ), whose objects are those of P. Recall, Q(P) is not necessarily a full subcategory of Q(E ). However, we have the factorization

g Q(P) / C

f $ Q(E )

We prove that both f, g are homotopy equivalence. First, we show g is a homotopy equivalence. We do this, we use Theorem 4.1.22, and show that ∀ P ∈ Obj(P), the category g \ P is contractible. Fix P ∈ Obj(P). Objects in g \ P and given by (C, u), where C ∈ Obj(P) and p u : C −→ P is a morphism in QE , which is given by CMo o / P . In terms of admissible layers, it is given by the following diagram: Let J be the ordered set (category) of E -admissible layers (K,M) in P , such that M/K ∈ Obj(P). Diagramatically:

p 0 / K / M _ / C / 0 horizontal line is a conflation in E , M/K ∼= C ∈ Obj(P), P and M,→ P is an inflation. To describe the morphisms in J , let (K,M), (K0,M 0) in J . We have M/K, M 0/K0 ∈ Obj(P). Define

M 0/M O

K0 / M 0 _ O (K,M) ≤ (K0,M 0) if with K/K0, M/M 0 ∈ Obj(P) ? K / M

K/K0

That means, the vertical sequences are conﬂations in E , to start with. 134 CHAPTER 4. THE QUILLEN-K-THEORY

So, (J , ≤) is an ordered set (category). Deﬁne a functor

Φ: J −→ g/P by Φ((K,M)) = M/Ko o M / P

Also:   M/Ko o M / P  O      0   Φ((K,M) ≤ (K,M )) =  M/K0   _        M 0/K0 o o M 0 / P

This diagram of three maps in QE commute and the vertical map is in QP. We have:

1. Clearly, Φ is essentially surjective.

2. Consider

Φ: Mor((K,M), (K,M 0)) = {≤} −→ Mor(Φ(K,M), Φ(K0,M 0))

Since, the domain is a singleton, Φ is injective. We show that Mor(Φ(K,M), Φ(K0,M 0)) is a singleton. Consider a map in g/P :

M/Ko o M / P O

C _

M 0/K0 o o M 0 / P

The composition of the vertical with the bottom map is given by:

W / M 0 / P M/K o o M / P

The commutativity =⇒ o τ 0 0 M/Ko o C / M /K M/Ko o W / P

Identifying M −→∼ W , we have the commutative diagram

M / M 0 / P

f g M/Ko o C / M 0/K0

It follows from properties of pullback, that ker(f) = ker(g) = K0 (see [HS, II.Thm 6.2]). So, C ∼= M/K0. Therefore Mor(Φ(K,M), Φ(K0,M 0)) is a singleton. 4.4. REDUCTION BY RESOLUTION 135

This establishes that J −→ g/P is an equivalence of categories. So, we will now prove that J is contractible. So, in J , we have

F : J −→ J by F (K,M) = (0,M) (K,M) ≤ (0,M) ≥ (0, 0). Deﬁne C : J −→ J by C(K,M) = (0, 0)

We deﬁne two natural transformations: τ1 : IDJ −→ F by (K,M) ≤ (0,M) τ2 : C −→ F by (0, 0) ≤ (0,M)

By Proposition 4.1.9, there are homotopy: IDBQJ 7→ BF and Constant 7→ BF . So, IDBQJ is homotopic to the constant map. Hence J is contractible. Therefore g/P is contactable. By Theorem 4.1.22, g is a homotopy equivalence.

Now, we prove that f : C −→ QE is a homotopy equivalence. To do this, we will prove that, for M ∈ Obj(QE ) = Obj(E ), M/f is contractible. Recall, the objects (P, u) of M/f are all maps u : M −→ f(P ) = P in QE , (where P ∈ Obj(P)) which looks like MQo o / P j i in E

Write F = M/f and F 0 = {(P, u) ∈ Obj(F ): u surjective} be the full subcategory. We proceed to prove that the inclusion functor F 0 −→ F is a homotopy equivalence. Here are some observations:

1. For an object (P, u) in F 0, P is in P. 2. It follows, that a morphism θ : X −→ Y in F 0 is surjective:

X : M o o Q Q j i O θ j X : MPo o P j i

! MQo o / P 3. Given X = (P, u) ∈ Obj(F ), we have u = i!j an in j i .

! 0 By hypothesis (1), Q ∈ Obj(P). So, X := (Q, j ) ∈ F . Now, ι := i! : X −→ X in QE is a map in F, as given by the diagram:

j X M o o Q Q

ι i! XMQo o / P j i 136 CHAPTER 4. THE QUILLEN-K-THEORY

We claim ι : X −→ X is an universal arrow from an object of F 0 to X. This means, given and arrow α : X 0 −→ X, with X 0 ∈ F 0, there is an unique map β : X 0 −→ X , such that the diagram X 0 α β commute. X / X To see this let X 0 = (Q0, j0) and let α : X 0 −→ X be given by the commutative diagram in QE :

j0 X 0 M o o Q0 Q0 O 0 O j j” i0 j” M o o W / P ∼ 0 α o τ W _ =⇒ =⇒ τ : W −→ Q, i = iτ i0 MQo o / P j i XMQo o / P j i

0 0 0 We can assume W = Q and i = i . It follows, Q is a subquotient j”: Q Q and

X 0 α (j”)! commutes. This establishes existence of β. X ι / X

The uniqueness of the vertical map follows from the fact that, decomposition of α, as composition of injective and surjective maps, is unique: Suppose

X 0 M o o Q0 Q0 O X 0 α β β Write W _ =⇒ W = Q X ι / X X MQo o Q and j0 X 0 M o o Q0 Q0 O ζ

β Q

X MQo o Q j 4.4. REDUCTION BY RESOLUTION 137

Also, j0 X 0 M o o Q0 Q0 O ζ

β Q

X M o o Q Q j _ ι i XMQo o / P j So, β is given by ζ!. Now, ιβ = α means that the vertical map coincide with the vertical map in the diagram of α, which is given by

ζ i Q0 o o Q / P

o ρ =⇒ ρ = Id j” Q0 o o Q / P i Therefore the (j”)! = ζ!. This completes the proof of uniqueness. So, it is established that X −→ X as an universal arrow, from an object of F 0. We denote the functor F : F −→ F 0, i. e. F (X) = X . In fact, universality, establishes that F is a functor. This also implies that F is a right adjoint of the inclusion functor F 0 −→ F: 0 ∼ ∀ U ∈ Obj(F ),X ∈ Obj(F) there is a bijection MorF 0 (U,F (X)) −→ MorF (U,X) By Corollary 4.1.10, it follows that the inclusion functor F 0 −→ F is a homotopy equivalence. So, it would be enough to prove that F 0 is contractible. We will prove that (F 0)op is 0 op contractible. Objects in (F ) are deﬂations P M, with P ∈ Obj(P) and morphisms P / / M > > are commutative diagrams . Fix a deﬂation q0 : P0 M, with P0 ∈ Obj(P), Q 0 op given by hypothesis (2). Given another object q : P M in (F ) , we have the pull back

p1 P ×M P0 / / P0

p2 q0 P q / / M

Since ker(p2) = ker(q0), and since it is in Obj(P) by hypothesis (1), it follows from the left vertical line, that P ×M P0 ∈ Obj(P). We deﬁne functors:  0 op 0 op Id :(F ) −→ (F ) Id(P M) = (P M)  0 op 0 op F :(F ) −→ (F ) F (P M) = (P ×M P0 M) 0 op 0 op  C :(F ) −→ (F ) C(P M) = (P0 M) 138 CHAPTER 4. THE QUILLEN-K-THEORY

So, p2 : F −→ Id and q0p1 : F −→ C are natural transformations. So, by Proposition 4.1.9, 0 there are homotopies: BF 7→ IDBF 0 and BF 7→ Constant. Therefore,F is contractible. The proof is complete.

The hypothesis of the following (4.4.4) is slightly weaker than being a resolving subcategory.

Corollary 4.4.4. Suppose E is an exact category and P is an extension closed full subcategory. For subsequent reference, let

0 / K / M / C / 0 be a conﬂation in E . (4.16)

Assume

1. For any conﬂation (4.16) in E , if C,M ∈ Obj(P), then K ∈ Obj(P).

2. Given a deﬂation j : M P in E , with P ∈ Obj(P),

P 0 f 0 0 0 g ∃ deﬂetions g : P P, f : P M 3 P ∈ Obj(P) and commute. ~ ~ M / / P j

(With P = 0, any object M in E is surjective image of an object P 0 ∈ Obj(P).)

For integers n ≥ 0, let Pn denote the full subcategory of objects M E , that has a P- resolution of length ≤ n. (Here the concept of "resolutions" needs to be understood, in terms of deﬂations and inﬂations.) Then,

1. Pn is an extension closed full subcategory of E . Therefore, Pn is an exact category. S 2. Write P∞ = Pn. Then, P∞ is also an exact subcategory of E .

3. Finally, ∼ ∼ Ki (P) −→ Ki (Pn) −→ Ki (P∞) ∀ i ≥ 0, n ≥ 0.

Proof. The proof follows by an application of Theorem 3 (4.4.3), to the inclusion Pn ⊆ Pn+1, with additional help from the following Lemma 4.4.5. Note, since P ⊆ Pn, any C ∈ Obj(Pn+1) is a subquotient of an object in Pn. Further, Pn is closed under subobjects, in Pn+1, by Assertion 2 of Lemma 4.4.5. So, Theorem 3 (4.4.3) applies to the inclusion Pn ⊆ Pn+1. The proof is complete. 4.4. REDUCTION BY RESOLUTION 139

Lemma 4.4.5. Suppose P and E and for integers n ≥ 0, Pn be as in (4.4.4). Suppose

i j 0 / K / M / C / 0 be exact in E . (4.17)

Under the hypothesis of (4.4.4), and for integers n ≥ 0, we have

1. If K,C ∈ Obj(Pn+1), then M ∈ Obj(Pn+1).

(Therefore, Pn is extension closed in E ).

2. If M ∈ Obj(Pn) and C ∈ Obj(Pn+1), then K ∈ Obj(Pn).

3. If M,C ∈ Obj(Pn+1), then K ∈ Obj(Pn+1).

Proof. (The proof is imitation of the standard arguments in commutative algebra, regarding projective dimension of modules M, over commutative noetherian rings A. For example, see [Mat, Lemma 2, pp. 128].) We prove the case n = 0, 1 and then use induction.

First we prove (1), for n = 0, which is to prove that P1 is extension closed in E . Let Consider the sequence (4.17). Assume K,C ∈ P1. Consider the diagrams: 0 0 0 0

Q1 P1 0 / Q1 / N / P1 / 0

Q0 P0 =⇒ 0 / Q0 / Q0 ⊕ P0 // P0 / 0 η ∂ d ∂ δ=(i∂+η) d ~ K / M / C 0 / K / M / C / 0 i j

0 0 0 0 In the left hand diagram, two vertical lines provide P-resolutions of K and C. The map η is obtained by hypothesis (2) of (4.4.4), with jη = d. The second commutative diagram extends the ﬁrst one, where δ is induced by ∂ and d. Recall, by (1.3.4) E can be embedded as a fully exact, extension closed subcategory of an abelian category C . Let N = ker(δ) in C . By Snake Lemma 1.5.4,

Q1 / N / P1 / coKer(∂) / coKer(δ) / coKer(d) is exact. Since coKer(∂) = coKer(d) = 0, it follows coKer(δ) = 0. Further,

Q1 / N / P1 / 0 is exact in C . Also, 0 / Q1 / N is exact in C . Therefore, 0 / Q1 / N / P1 / 0 is exact in C . 140 CHAPTER 4. THE QUILLEN-K-THEORY

Since P ,→ E ,→ C are extension closed, N ∈ Obj(P). Now, the middle vertical line is a conﬂation is C . Since the E ,→ C is fully exact, the middle vertical line is conﬂation in E . Therefore, M ∈ Obj(P1). This completes the proof that P1 is an extension closed subcategory of E .

Now, we prove (2), for n = 0. So, M ∈ Obj(P0) and C ∈ Obj(P1). Consider the diagram: P1 P1

g 0 / K / P1 ×C M / P0 / / 0

f 0 / K / M / C / 0 i j where the last vertical line is a resolution of C, with P0,P1 ∈ Obj(P) and P1 ×C M is the pullback of (j, f). If follows from (1.5.1) that the middle vertical line and as well the middle horizontal line are conﬂesions, in E . By extension closed property of P it follows that P1 ×C M ∈ Obj(P). Now, by hypothesis (1) of Corollary 4.4.4, K ∈ Obj(P). This establishes (2) for n = 0.

Now, we prove (3), for n = 0. So, let M,C ∈ Obj(P1). Consider the pullback diagram:

P P / / 0 1 _ 1 _ _

jd P0 ×M K / P0 / / C

d K / M / / C i j It follows from Snake Lemma 1.5.4, that the middle horizontal line is a conﬂesion. Since C ∈ Obj(P1) and P0 ∈ Obj(P0), by Case n = 0 of (2), K ∈ Obj(P1). So, the lemma is established in the case n = 0. Now, the proof is completed by induction.

Resolving Subcategories

Deﬁnition 4.4.6. Suppose E is an exact category (with a set of isomorphism classes). For subsequent reference, let

0 / K / M / C / 0 be an exact sequence in E . (4.18)

Now suppose P is an extension closed subcategory of E ,(hence an exact subcategory). Then, P is said to be a resolving subcategory of E , if the following two conditions are satisﬁed: 4.4. REDUCTION BY RESOLUTION 141

1. Given an exact sequence (4.18) in E , if M,C are in P, then so is K.

2. For all objects M in E , there is an exact sequence:

0 / Pn / Pn−1 / ··· / P1 / P0 / M / 0

with Pi in P. In this case, we say P• is a P-resolution of M, using deﬂations.

Example 4.4.7. Suppose A is noetherian commutative ring.

1. Note, P(A) is an exact subcategory of fMod(A) = Coh(A), it is not a resolving subcategory of fMod(A). However, if A is a regular ring, then P(A) is a resolving subcategory of fMod(A) = Coh(A).

2. Let H(A) be the full subcategory of fMod(A) whose objects are the A-modules M in fMod(A), with ﬁnite projective dimension. Then, H(A) is an exact subcategory of fMod(A). Further, P(A) is a resolving subcategory of H(A).

Theorem 4.4.8. Let E be an exact category and P ⊆ E be a resolving subcategory. Then, ∼ for all integers, i ≥ 0, we have Ki(P) −→ Ki(E ) is an isomorphism.

Proof. We apply Corollary 4.4.4. The first condition of (4.4.4) follows from definition of resolving subcategory. Now suppose j : M P in E is a deflation in E , with P ∈ Obj(P). Again by definition of resolving category, there is a deflation ι : P0 M, with P0 ∈ Obj(P). Therefore, the diagram of deflations

jι ι commute. ~ ~ M / / P j

So, the second condition of (4.4.4) is satisﬁed. By the second condition in the deﬁnition of resolving category, P∞ = E . So, the proof is complete by Corollary 4.4.4.

The following is extension of the classical Example 4.4.1.

Corollary 4.4.9 (Corollary 2). Let A be a commutative noetherian ring and P(A) ⊆ H(A) ∼ be as in Example 4.4.7. Then, for all integers i ≥ 0, Ki(P(A)) −→ Ki(H(A)) is an ∼ isomorphism. In particular, if A is a regular ring, then Ki(P(A)) −→ Ki(Coh(A)).

Proof. Follows from Corollary 4.4.8. We comment that if dim A = d then H(A) = (P(A))d (see [Mat]). 142 CHAPTER 4. THE QUILLEN-K-THEORY

Corollary 4.4.10 (Corollary 3). Let T = {Ti : i ≥ 1} be an exact connected sequence of functors from an exact category E to an abelian category A , meaning

1. Ti : E −→ A are functors (not necessarily exact),

2. For each conﬂation K / M / / C in E (4.19)

and i ≥ 2, there is a morphisms δi : Ti(C) −→ Ti(K) such that following

δ2 ··· / T2(K) / T2(M) / T2(C) / T1(K) / T1(M) / T1(C)

is exact.

Let

P = {M ∈ Obj(E ): Ti(M) = 0 ∀ i ≥ 1} be the full subcategroy E .

Then, P is an extension closed subcategory of E . Further assume that

1. for all M ∈ Obj(E ) there is a deﬂation P M for some P ∈ Obj(P).

2. for all M ∈ Obj(E ), Tn(M) = 0 ∀ n 0.

∼ Then, Ki(P) −→ Ki(E ) ∀ i ≥ 0.

Proof. It follows immediately, that P is extension closed and hence Ki(P) are deﬁned. For integers n ≥ 0, deﬁne the full subcategories

Pn = {M ∈ Obj(E ): Ti(M) = 0 ∀ i ≥ n + 1}

It follows, that Pn ⊆ Pn+1 is an extension closed subcategory, for all n ≥ 0. It follows, given a conﬂation (4.19) in Pn+1, if M ∈ Obj(Pn) and K ∈ Obj(Pn). Both the conditions ∼ of Theorem 3 (4.4.3) are satisﬁed. Hence, for all integers i ≥ 0, Ki(Pn) −→ Ki(Pn+1) are isomorphisms.

By hypothesis, E = P∞ = ∪n≥0Pn. Therefore, Ki(E ) = Ki(P0) for all i ≥ 0. The proof is complete.

Remark 4.4.11. Compare the deﬁnition of "connected sequence of fuctors" with that of δ-functors in [Hr, Section III.1, pp. 205]. 4.5. DÉVISSAGE AND LOCALIZATION IN ABELIAN CATEGORIES 143 4.5 Dévissage and Localization in Abelian Categories

We would be working on the following set up.

Deﬁnition 4.5.1. Let A be an abelian category, with an set isomorphism classes of objects. Let B ⊆ A be a subcategory, such that

1. B ⊆ A is full subcategory,

2. B is closed under subobjects, quotient objects and ﬁnite product in A .

Note, in this case,

1. B is an abelian category and the inclusion B ,→ A is an exact functor.

2. Further, QB ,→ QA is a full subcategory.

The following is the well known Dévissage Theorem.

Theorem 4.5.2 (Dévissage). Let B ,→ A be as in the set up (4.5.1). Further assume, every object M ∈ Obj(A ) has a ﬁltration:

Mj 0 = M0 ,→ M1 ,→ M2 ,→ · · · ,→ Mn =: M 3 ∀ j = 1, . . . , n − 1 ∈ Obj(B). Mj−1

∼ Then, QB ,→ QA is a homotopy equivalence and hence Ki(B) −→ Ki(A ) is an isomorphism, for all integers i ≥ 0.

Proof. Let f : QB ,→ QA be the inclusion functor. Fix M ∈ Obj(A ). By Theorem A (4.1.22), it suﬃces to show that f/M is contractible. An object of f/M is given by a pair

(N, u), where N ∈ Obj(B) and u ∈ MorQA (N,M). Such a morphism u is given by an A -admissible layer (M0,M1) in M, as in the diagram:

M 0_

ι M1 / M with M0 = ker(p). p N

Let J (M) be a category (ordered) of all such A -admissible layer (M0,M1) in M, with M1/M0 ∈ Obj(B). Further, (as in the proof of Theorem 4.4.3)

0 0 0 0 (M0,M1) ≤ (M0,M1) in M0 ≤ M0 ≤ M1 ≤ M1 144 CHAPTER 4. THE QUILLEN-K-THEORY

We clarify, diagrammatically:

0 0 M0 M0

0 with M 0 = ker(p),M = ker(q). M1 / M1 / M 0 0 q π p ~ NWo o / N 0

0 ∼ M1 M0 Note, W = 0 ⊆ 0 . Hence W ∈ Obj(B). M0 M0 Since J (0) is trivial and M has a ﬁltration, with quotient in B, it is enough to prove that that inclusion ι : J (M 0) ,→ J (M) is a homotopy equivalence, with M 0 ⊆ M and M M 0 ∈ Obj(B. We deﬁne two functors:

0 0 0 r : J (M) −→ J (M )(M0,M1) 7→ (M0 ∩ M ,M1 ∩ M ) 0 s : J (M) −→ J (M)(M0,M1) 7→ (M0 ∩ M ,M1)

These are well deﬁned because

0 M1 ∩ M M1 M1 M1 0 ⊆ 0 ⊆ × 0 ∈ Obj(B). M0 ∩ M M0 ∩ M M0 M Consider the following:

1. rι = IdJ (M 0). 2. There are natural transformations:

0 0 0 ιr −→ s (M0 ∩ M ,M1 ∩ M ) ≤ (M0 ∩ M ,M1) 0 IdJ (M) −→ s (M0,M1) ≤ (M0 ∩ M ,M1)

Now, by Proposition 2 (4.1.9), r is a homotopy inverse of ι. The proof is complete.

The Nilpotent ideals Here is an important corollary of our primary interest.

Lemma 4.5.3. Suppose A is a noetherian ring (commutative) and I is a nilpotent ideal. A ∼ Then, for all integers i ≥ 0, we have Ki fMod I −→ Ki(fMod(A)).

A Proof. Apply the Dévissage Theorem (4.5.2), to the inclusion fMod I ,→ fMod(A). Note In = 0 for some integer n ≥ 0. So, for any object M in fMod(A), there is a ﬁltration:

0 ,→ In−1M,→ · · · IM,→ M.

The proof is complete. 4.5. DÉVISSAGE AND LOCALIZATION IN ABELIAN CATEGORIES 145

4.5.1 Semisimple Objects

Deﬁnition 4.5.4. Let A be an abelian category.

1. An object M in A is called a simple object if M has precisely two subjects, namely, 0 6= M.

2. An object M in A is called a semisimple object if there is a surjective maps ⊕i∈I Si M, where {Si :∈ I} is a set of simple objects.

3. We say and object M has finite length, if M has a finite filtration

Mj 0 = M0 ,→ M1 ,→ M2 ,→ · · · ,→ Mn =: M 3 ∀ j = 1, . . . , n − 1 is simple. Mj−1

(In this case, define length `(M) := n is well defined. It needs a proof that it is well defined.)

Lemma 4.5.5. Suppose A is an abelian category and M be an object. Then, the following are equivalent:

1. M is semi simple.

∼ 2. M = ⊕i∈I Si, where {Si :∈ I} is a set of simple objects. (So, if `(M) = Card(I),)

Proof. (2) =⇒ (1) is obvious. To prove ((1) =⇒ 2), let f : ⊕i∈I Si M, be a surjective mal, where {Si :∈ I} is a set of simple objects. Let

:= {I ⊆ I : f is injective} I 0 |∪i∈I0 Si

By Zorn’s lemma, I has a maximal element J ⊆ I. Write N = ∪i∈J Si. Claim f0 := f|N : ∼ N −→ M is an isomorphism. If not, there is i1 ∈ I \ J such that f(Si1 ) 6⊆ g(N). Let f := f : S −→ M. We have, 1 |Si1 i1

1. f0 : N −→ M is injective and f1 : S1 −→ M is injective.

2. Further, f : N ⊕ S = f ⊕ f . |(N⊕Si1 ) i1 0 1 3. Since N ⊕ S ∼= N × S , it follows that f is injective. This is a contradiction i1 i1 |(N⊕Si1 ) to the maximality of J.

Therefore, f0 is an isomorphism. The proof is complete. 146 CHAPTER 4. THE QUILLEN-K-THEORY

Corollary 4.5.6. Let A be an abelian category (with a set of isomorphism classes of objects), such that every object has ﬁnite length. Then,

a a ∀ i ≥ 0 Ki(A) = Ki(Dj) recall = ⊕ j∈J where {Xj : j ∈ J} is a set of representative for isomorphism classes of simple objects of o A and Dj = End(Xj) . (Note, Dj is a devision ring.)

Proof. Let B be the full subcategory of semi simple objects in A . Then, by Dévissage ∼ Theorem (4.5.2), Ki(B) −→ Ki(A ) is an isomorphism, ∀ i ≥ 0.

Let Bj be the full subcategory of B, only one simple object Xj (up to isomorphism). Qm It follows B is limit of finite products k=1 Bjk . Since Ki commute with finite product and filtered limits, if follows

∼ M ∀ i ≥ 0 Ki(B) −→ Ki(Bj) is an isomorphism. j∈J

Now, the proof follows from the following lemma (4.5.7).

Lemma 4.5.7. Suppose B is an abelian category such that (1) each object has ﬁnite length, and (2) B has only a single simple object X, up to isomorphism. Let D := Mor(X,X)o and P(D) denote the category of ﬁnite dimensional left D-vector spaces. The functor

F : B −→ P(D) 3 F (M) = MorB(X,M) is an equivalence of categories.

Proof. It follows from Lemma 4.5.5 that, each object M ∼= Xn for some integer n ≥ 0.

4.5.2 Quillen’s Localization Theorem

Theorem 4.5.8 (Localization Theorem). Let A be an abelian category and B ⊆ A be a Serre subcategory (1.4.5). Let A be the quotient abelian category. Let ι : ,→ be the B B A inclusion functor and q : −→ A the canonical functor. Then, there is an exact sequence A B

q∗ ι∗ q∗ ··· / K A / K ( ) / K ( ) / K A / 0 1 B 0 B 0 A 0 B

Proof. Fix zero 0 in and let 0 denotes its image in A . From the deﬁntion of Serre A B subcategories (1.4.5) and the construction of the quotient A , it follows, is the full B B 4.5. DÉVISSAGE AND LOCALIZATION IN ABELIAN CATEGORIES 147 subcategory of objects M in A such that qM ' 0.Consider the composition:

Qι QB / QA

Qq up to natural equivalence. 0 " A Q B

Consider 0 \ Qq. Since MorA (0,M) and MorA (M, 0) are singletons, objects in . Since Mor A (0,M) and Mor A (M, 0) are singletons, objects in 0 \ Qq are given by, equivalence B B classes of, A A 0 0 i M in Q , i ∈ . o o / B B If M ∈ Obj( ) then i is an isomorphism in A . We consider two functors B B ∼ F : QB −→ 0 \ Qq F (M) = (M, 0 −→ qM) G : 0 \ Qq −→ QA G(N, u) = N Further, there we have the commutative diagram:

F QB / 0 \ Qq

G ι Q # QA The following would settle of proof of this theorem:

1. We would apply Theorem B (4.1.32) to Qq. To apply Theorem B (4.1.32) to q, we would prove that for all u : V 0 −→ V in A , Q Q B the map u∗ : V \ Qq −→ V 0 \ Qq is a homotopy equivalence. Recall, u∗ sends (W, v) 7→ (W, vu). Since every morhism in A factors in to "surjective" and "injective", and also by Q B reversing the arrows, we can assume u is injective. It is claimed i!(iV 0 )! = (iV )!, which I do not see. So, it is enough to prove this for injective maps (i ) for V in A . V ! B 2. Also, we prove that F : QB −→ 0 \ Qq is a homotopy equivalence. Let FV be the full subcategory of V \ Qq, consisting of pairs (M, u) such that ∼ u : V −→ qM is an isomorphism. In particualar, F0 ,→ 0 \ Qq. It is clear F0 is equivalent to B. Hence, this point follows form the following Lemma 4.5.9.

Lemma 4.5.9. The inclusion functor FV ,→ V \ Qq is a homotopy equivalence.

Proof. For the purposes of this course, we skip the proof. In stead, we do other things during the remaining four lectures. 148 CHAPTER 4. THE QUILLEN-K-THEORY

Remark 4.5.10. We summarize application of the above to K-Theory of schemes. Given a scheme X and a closed subscheme Z ⊆ X, the the localization theorem (4.5.8) applies to the inclusion Coh(Z) ⊆ Coh(X), as follows:

1. Let CohZ (X) ⊆ Coh(X) be the full subcategory of coherent OX -modules, F such

that Supp(F) ⊆ Z. Obviously, i : CohZ (X) ⊆ Coh(X) is an exact functor. There- fore, i induce a map of the K-theories of these two exact categories.

2. Let U := X \ Z. Then, the restriction functor j : Coh(X) −→ Coh(U) is an exact functor. Therefore, j induce a map of the K-theories of these two exact categories.

3. It follows, Coh(X) −→∼ Coh(U) is an equivalence of categories. CohZ (X)

4. By localization theorem (4.5.8) (although I did not state it inthis way):

BQCohZ (X) / BQCoh(X) / BQCoh(U)

is a homotopy ﬁbration of topological spaces (in fact CW-complexes). Further, by ∼ Dévissage Theorem (4.5.2), BQCohZ (X) −→ BQCoh(Z) is a homotopy equivalence. Putting these two fcts together,

BQCoh(Z) / BQCoh(X) / BQCoh(U)

is a homotopy ﬁbration of topological spaces (in fact CW-complexes).

5. This homotopy ﬁbration leads to long exact sequence of K-groups. With notations

Gi(X) := Ki(Coh(X)) := πi+1(BQCoh(X))

there is a long exact sequence

∂1 ··· / G1(Z) / G1(X) / G1(U) / G0(Z) / G0(X) / G0(U) / 0

6. Only when X is regular (noetherian and separated) (hence so is U), we can say ∼ ∼ Ki(X) −→ Gi(X) are , and as well Ki(U) −→ Gi(U). However, unless Z is also ∼ regular, we cannot say that Ki(Z) −→ Gi(Z).

7. Finally, there is no satisfactory analogue of the Localization Theorem (4.5.8) that applies to V ect(X) 4.6. K-THEORY SPACES AND REFORMULATIONS 149 4.6 K-Theory Spaces and Reformulations

As was seen above, the K-groups Ki(E ) of an exact category was determined by the classifying spaces BQE . From this perspective, it may make more sense to deﬁne K- Theory as a topological space, as follows.

Deﬁnition 4.6.1. Let E be an exact category. The K-theory space of |SE is deﬁnes to be the pointed topological space

K(E ) := Ω (BQE ) the loop space of BQE , at 0 with the base point the constant loop (see Deﬁnition 2.4.11). Therefore, the K-groups are given by

Ki(E ) = πi+1 (BQE , 0) = πi (K(E ), 0) ∀ i ≥ 0.

So, most of the above can be reformulated in terms of the K-Theory spaces. The Resolution Theorem is reformulated, as follows.

Theorem 4.6.2. Let E be an exact category and P ⊆ E be a resolving subcategory. Then, the map K(P) −→ K(E ) is a homotopy equivalence.

Theorem 4.6.3 (Dévissage). Under the hypotheses of Theorem 4.5.2, the map

K(B) −→ K(B) is a homotopy equivalence.

Theorem 4.6.4 (Localization). Under the hypotheses of the Localization Theorem (4.5.8)

K( ) / K( ) / K A B A B is a homotopy ﬁbration.

4.7 Negative K-Theory

With my primary interest on P(A), where A is a commutative noetherian ring, I cannot pinpoint what was the need for negative K-Theory. However, failure of applicability of both Dévissage Theorem (4.5.2) and Localization Theorem (4.5.8) , to K-Theory of projective modules, leaves a taste of incompleteness. At this time, I would presume that that would be a justiﬁcation for Negative K-Theory. We follow the papers of Schlichting ([Sm1, Sm2]). Before we introduce negative K-theory, we develop some background. 150 CHAPTER 4. THE QUILLEN-K-THEORY

4.7.1 Spectra and Negative Homotopy Groups

We begin with the related topological background on (negative) homotopy groups of spetra. Deﬁnition 4.7.1. A Spectrum of pointed topological spaces is a sequence of topological spaces

E0,E1,E2,..., together with maps σk : Ei −→ ΩEk+1 to be called bonding or structure maps. Such a Spectrum, would be denoted by (E, σ).

Fix an integer n ∈ Z. To define πn(E), fix integers k ≥ l ≥ −n. So, iteratively, we have maps k−l k−l+1 k−l Ω σk k−l+1 Ω σk+1 k−l+2 Ω Ek / Ω Ek+1 / Ω Ek+2 / ··· Define k−l k−l+1 k−l Ω σk k−l+1 Ω σk+1 k−l+2 πn(E) = co lim πn+l Ω Ek / πn+l Ω Ek+1 / πn+l Ω Ek+2 / ···

1. One checks that this deﬁnition does not depend of the choices on l, k, as long as k ≥ l ≥ −n.

2. If n ≥ 0 then, we can take k = l = 0. In this case, we have

1 σ0 1 Ω σ1 2 πn(E) = co lim πn (E0) / πn (Ω E1) / πn (Ω E2) / ··· We can also take k = l = 1, in which case

1 σ1 1 Ω σ2 2 πn(E) = co lim πn+1 (E1) / πn+1 (Ω E2) / πn+1 (Ω E3) / ··· We can also take k = 2, l = 1, in which case

1 2 1 Ω σk 2 Ω σ3 k−l+2 πn(E) = co lim πn+1 (Ω Ek) / πn+1 (Ω E3) / πn+l Ω Ek+2 / ··· We can pick l = −n and k ≥ 0. In this case, k+n k+n+1 k+n Ω σk k+n+1 Ω σk+1 k+n+2 πn(E) = co lim π0 Ω Ek / π0 Ω Ek+1 / π0 Ω Ek+2 / ···

3. If n ≤ −1 then we need to pick k ≤ l ≤ −n. We can pick k = l = −n. In this case,

1 σ−n 1 Ω σ−n+1 2 πn(E) = co lim π0 (E−n) / π0 (Ω E−n+1) / π0 (Ω E−n+2) / ···

A spectrum would be called Ω-spectrum, if σk : Ei −→ ΩEk+1 are homotopy equivalence, for all k ≥ 0. In this case, all the above maps of homotopy groups/sets are isomorphisms. Therefore,

1. If n ≥ 0 then πn(E) = πn(E0).

2. If n ≤ −1, then πn(E) = π0(E−n). 4.8. SUSPENSION OF AN EXACT CATEGORY 151 4.8 Suspension of an Exact Category

Given an exact categroy E , we would ﬁrst deﬁne an exact category, SE to be called the suspension of E .

Definition 4.8.1. Let E be an exact category. The filter category F `(E ) of E is an exact category, defined as follows:

1. The objects of F `(E ) and sequences

i0 i1 i2 A0 / A1 / A2 / A3 / ··· , where ir are inﬂesion. (4.20)

Such an object would, sometimes be denoted by A•.

2. Given another object

j0 j1 j2 B• : B0 / B1 / B2 / B3 / ···

in F `(E ), a morphism f• : A• −→ B• is a sequence of morphism fn : An −→ Bn such that the obvious diagram

i0 i1 i2 A0 / A1 / A2 / A3 / ···

f0 f1 f2 f3 B0 / B1 / B2 / B3 / ··· j0 j1 j2

commute.

3. A pair of composible morphisms:

f• g• K• / M• / C• in F `(E )

is declared a conﬂasion in F `(E ) , if

fn gn Kn / Mn / Cn are conﬂasions in E ∀ n ≥ 0.

This deﬁnes the structure of an exact category on F `(E ). (See [K, Section 5] for more about the ﬁltered category.

Now, we deﬁne the countable envelop of E .

Deﬁnition 4.8.2. Let E be an exact category. The countable envelop cF (E ) of E is an exact category, deﬁned as follows: 152 CHAPTER 4. THE QUILLEN-K-THEORY

1. The objects of cF (E ) are the same as that of the ﬁltering category F `(E ), as in Equation 4.20.

2. Given two objects A•,B• in cF (E ), the set of morphisms is deﬁned to be

MorcF (E )(A•,B•) := lim co lim MorE (Ai,Bj) i j

One can visualize this set, as the set of limits of diagrams:

MorE (Ai,Bj)

x & MorE (Ai−1,Bj) MorE (Ai,Bj+1)

* t MorE (Ai−1Bj+1)

A morphism in MorcF (E )(A•,B•) would be denoted by {fij} where fij ∈ MorE (Ai,Bj).

3. There is a natural functor Φ: F `(E ) −→ cF (E ). A pair of composable morphisms

fij gij in c ( ) K• / M• / C• F E

is declared a conﬂasion, if it is isomorphic to the image of a conﬂation in F `(E ).

Further, there is a fully exact functor

ι : −→ `( ) sending X 7→ X / X / X / X / ··· 0 E F E 1X 1X 1X

Composing, we get a fully exact functor ι : E −→ cF (E ) as follows

ι0 E / F `(E )

ι " cF (E )

Note, by deﬁnition of cF (E ), the vertical functor is fully exact. (A slightly diﬀerent argument was given in [Sm2].)

Lemma 4.8.3. Use the notations as in (4.8.2). Consider the "inclusion" functor ι : E −→ cF (E ). Then, the construction (1.4.8) applies to this inclusion, and hence the quotient cF (SE) of exact categories exists. E 4.8. SUSPENSION OF AN EXACT CATEGORY 153

We say that the sequence of exact functors:

c (SE) / c ( ) / F is exact. E F E E Consequently, the sequence, of K-theory spaces

c (SE) K ( ) / K (c ( )) / K F is homotopy ﬁbration. E F E E

Proof. To be inserted.

Now, we are ready to define suspension of E . Definition 4.8.4. Let E be an exact category. Define the suspension of S(E ) of E by cF (SE) S(E ) := E It follows from (4.8.3), the sequence, of K-theory spaces

K (E ) / K (cF (E )) / K (S(E )) is homotopy ﬁbration.

Lemma 4.8.5. Use the notations as in (4.8.2). Then, K(cF (E )) is contractible. Consequently, ΩK (S(E )) ' K (E ) is a homotopy equivalence.

Proof. (To be completed. We skip the proof that K(cF (E )) is contractible.) We have K (E ) / K (cF (E )) / K (S(E )) is homotopy fibration. Hence, K (E ) / ? / K (S(E )) is homotopy fibration. Hence, ΩK (S(E )) / K (E ) / ? is homotopy fibration.

Lemma 4.8.6. Suppose E is an exact category and Y Y Y 0 i 1 j 2 be two inﬂations.

Then, Y1 −→ Y2 is an inflation. Y0 Y0 Proof. Consider the diagram j Y1 / Y2 Y0 Y0 p 0 / Y2 Y1 Routine chasing the definitions of kernel and cokernels, it follows that j is the kernel of p. Since the bottom horizontal line is an inflation, so it the top. 154 CHAPTER 4. THE QUILLEN-K-THEORY

4.8.1 Coﬁnality and Idempotent Completion

Deﬁnition 4.8.7. Let A be an additive category and B ⊆ A be a full subcategory.

1. We say that B ⊆ A is cofinal if every object in A is direct factor (i.e summand) of an object in B. We also say that B is cofinal in A . For a commutative ring A, the category F (A) of finitely generated free modules is cofinal in P(A).

2. If A is an exact category, a fully exact subcategory B ⊆ A is said to be cofinal, if it is cofinal in the above sense and if B is fully exact subcategory, meaning (1) B is extension closed and (2) it preserves and detects conflations.

3. An additive category A is called idempotent complete, if for all idempotent arrows p = p2 : A −→ A there is an isomorphism ϕ : A −→∼ X ⊕ Y such that the diagram

p A / A

ϕ o o ϕ X ⊕ Y / X ⊕ Y commute.   1X 0    0 0 

4. Given an additive category A , there is a largest additive category Af, such that A ⊆ Afis coﬁnal. In fact, Afwould be idempotent complete. Such an Afwould be called the idempotent completion of A . Existence of such a completion is proved below (4.8.8)

Lemma 4.8.8. Let A be an additive category. Then, there is a largest additive category Af, such that A ⊆ Afis coﬁnal. In fact, Afwould be idempotent complete. Further, if A ie exact, then Afwould be exact and the inclusion is coﬁnal, in the sense of exact categories.

Proof. Let the objects of Afbe the pairs (A, p) where A ∈ Obj(A ) and p = p2 : A −→ A is an idempotent arrow in A . A morphism (A, p) −→ (B, q) is an arrow f : A −→ B in A , such that the diagram p A / A f f f commute. B q / B 4.8. SUSPENSION OF AN EXACT CATEGORY 155

The composition in Afis inherited from the compositions in A . The identities in Afare deﬁned to be 1(A,p) := p. The group structure on Mor ((A, p), (B, q)) is also inherited from that in Mor (A, B). Ae A The direct product (A, p) × (B, q) = (A × B, p × q). So, Afis also an additive category. Therefore, (A, p) × (B, q) = (A × B, p × q) = (A, p) ⊕ (B, q). An arrow f :(A, p) −→ (A, p) in Afis an idempotent if and only if f is an idempotent in A (as well fp = f = pf). Suppose f :(A, p) −→ (A, p) is an idempotent in Af. Then, the arrows

f f p − f :(A, f) ⊕ (A, p − f) −→ (A, p), :(A, p):−→ (A, f) ⊕ (A, p − f) p − f are inverses of each other. This follows from direct matrix multiplications:

f f f 0 f p − f = p = 1 , f p − f = p − f (A,p) p − f 0 p − f

Now, the diagram,

f (A, p) / (A, p)  f   f    o o   p − f p − f commutes. (A, f) ⊕ (A, p − f) / (A, f) ⊕ (A, p − f)  f 0   0 0  the map f 0 1 0 = (A,f) 0 0 0 0

This establishes that Afis an idempotent complete additive category. Now, the canonical "inclusion" functor is defined by ι : A −→ Afby sending X 7→ ∼ (X, 1X ). Given any object (X, p) ∈ Obj(Af), we can write (X, 1) = (X, p) ⊕ (X, 1 − p). So, A ⊆ Afis cofinal. To establish that Afis the largest cofinal category to A , let ζ : A −→ B be a cofinal inclusion of additive categories. For Y ∈ Obj(B), Y ⊕ X ∼= A for some X ∈ Obj(B), A ∈ Obj(A ). Define p as follows:

A / Y

p Deﬁne ζ(Y ) = (A, p) A

One checks that this is well deﬁned (always up to isomorphism). 156 CHAPTER 4. THE QUILLEN-K-THEORY

If A is an exact category, declare a sequence

f g f g (K, w) / (A, p) / (C, q) exact in Afif K / A / C is exact in A

This gives a structure of an exact category on Afand ι : A −→ Afis fully exact (i. e. extension closed and preserves and detects conﬂations). The proof is complete.

We state the following theorem without proof, for now.

Theorem 4.8.9. Let A be any exact category and B ⊆ A be a coﬁnal fully exact subcat- ∼ egory. Then, the maps Ki(B −→ Ki(A ) are isomorphism for all i ≥ 1 and injective for i = 0. In particular, this holds for the inclusion A ,→ Af. Proof. See [Gd].

4.8.2 The K-theory Spectra

The negative K-Theory is deﬁned as the homotopy groups of the K-theory Spectra, which we discuss in this section. Before that we have the following lemma.

Lemma 4.8.10. Let E be an exact category and Ee be its idempotent completion. Then, ∼ the map S(E ) −→ S(Ee) is an equivalence of categories. Consequently,

∼ ∀ n ≥ 0 Sn+1(E ) −→ S(S]n(E ) is an equivalence of categories.

Proof. The ﬁrst part follows from the construction and universal properties(1.4.8) of the quotient categories. Hence, the latter formula is valid for n = 0. Again, by n = 0 case, ∼ ∼ Sn+1(E ) −→ S (Sn(E )) −→ S(S]n(E ) are equivalence of categories. .

Lemma 4.8.11. Let E be an exact category, Assume E is idempotent complete. Then,

n+1 n ∀ n ≥ 0 ΩK S (E ) ' K (S (E )) is a homotopy equivalence.

Proof. It follows inductively, from (4.8.5).

Deﬁnition 4.8.12. Let E be an exact category. Deﬁne the K-theory Spectra of E by

n 2 3 4 o K(E ) := K Ee , K S (E ) , K S (E ) , K S (E ) , ··· Also,

∀ n ∈ Z deﬁne Kn(E ) := πn(K(E )) Appendix A

Not Used

A.0.3 Some Jargon from [H] (SKIP)

The following is [H, Prop. 3.2.5].

Proposition A.0.13. Suppose C is a category with all small colimits. Then, the category C ∆ is equivalent to the category of adjunction SSet −→ C . We denote

A• ∈ C ∆ 7→ (A• ⊗ −, C (A•, −), ϕ): SSet −→ C

Proof. I will only write down the construction.

1. Given ∆ / SSet

Φ := (F, U, ϕ): SSet −→ C consider F A•(Φ) " C So, A•(Φ) ∈ C ∆. 2. Given A• ∈ C ∆. We need to ﬁrst, give a functor SSet −→ C . (a) Given K ∈ SSet, ∆K ∈ SSet, where

(∆K)n = MorSSet(∆[n],K) Also, there is a functor ∆K −→ ∆ sending and n simplex (∆[n] −→ K) 7→ [n] There is a restriction functor C ∆ −→ C ∆K

157 158 APPENDIX A. NOT USED

(b) There is the colimit functor C ∆K −→ C (c) Combining C ∆ / C ∆K

⊗ # C For A• ∈ SSet, this diagonal associates K −→ A• ⊗ K.

Corollary A.0.14. There is a pointed version of Proposition A.0.13. 159

Now we apply (A.0.13). The following is [H, Rem. 3.1.7].

Remark A.0.15. There are two remarks:

1. A simplicial analogue is possible.

2. By Proposition A.0.13, a functor C −→ C ∆ gives rise to a functor

C −→ ADJ(SSet, C )

This gives rise to a bifunctor

− ⊗ − : C × SSet −→ C

In deed, for A ∈ C , the functor A ⊗ − : SSet −→ C will have a right adjoint.

3. An important application is when C = SSet, when we consider the functor

SSet −→ SSet∆ given by A 7→ A × ∆[n] ∈ SSet∆

The associate bifunctor is just

SSet × SSet −→ SSet (A, L) 7→ A × L

Its adjoint (i. e. Adjoint of (A × −): SSet −→ SSet) is:

Map(A, L) where Map(A, L)n := MorSSet (A × ∆[n],L)

4. Another example comes from Top.

(a) As usual, deﬁne

( n ) ( n n ) X X X |∆[n]| := (t1, . . . , tn) : 0 ≤ ti ≤ 1, ti ≤ 1 = ttei : 0 ≤ ti ≤ 1, ti = 1 i=1 i=0 i=1

n+1 where e0, e1, . . . , en is the standard basis of R . So, |∆[−]| ∈ Top∆ is a cosimplical topological space. (b) By Proposition A.0.13, we get an Adjunction

(|−| , Sing, ϕ): SSet −→ Top.

(c) Not surprisingly, |−| is called the geometric realization and Sing is called the singular functor. While I do not have a proof, I am sure, |−| coincides with the description given above (Deﬁnition 3.3.2) and Sing coincides with the functor S (Deﬁnition 3.3.4). 160 APPENDIX A. NOT USED

A.0.4 On SSet Realization

Prove the following lemma, which may not be true:

Lemma A.0.16. For, σn−1 ∈ Kn−1 and σn ∈ Kn, with σn non degenerate, we have ( ∃ a face di ∈ ∆([n − 1], [n]), x0 ∈ |∆[n − 1]| 3 |di|(x0) = y (σ , x) ' (σ , y) in |K| ⇐⇒ n−1 n i 0 n−1 (σn−1, x) ' (K(d )(σn), x ) in |K |

i 0 Proof. The implication (⇐=) is obvious, because under the hypotheses (σn−1, x) ' (K(d )(σn), x ) ' (σn, y). To prove the implication (=⇒), assume (σn−1, x) ' (σn, y) in |K|. It follows,

1. There are integers n0 = n − 1, n1, . . . , nk−1, nk = n, such that, for j = 1, . . . , k, nj = nj−1 ± 1.

2. Further, for j = 1, . . . , k − 1, there are (τj, xj) ∈ Knj × |∆[nj]|. We write (τ0, x0) = (σn−1, x) and (τn, y) = (σn, xk). 3. Further, for j = 1, 2, . . . , k, we have

i (a) if nj = nj−1 + 1, there is a face map d :[nj−1] −→ [nj] such that

i i K(d )(τj) = τj−1, and |d |(xj−1) = xj

i (b) if nj = nj−1 − 1, there is a degeneracy map s :[nj−1] −→ [nj] such that

i i K(s )(τj) = τj−1, and |s |(xj−1) = xj

A.0.5 Inclusion of Geometric Realizations

This Proof Remains Incomplete. However, we have an alternative proof, given in Proposi- tion 3.3.11 The following is a basic result on geometric realization.

Proposition A.0.17. Suppose K is a simplicial set and L ⊆ K is a simplical subset. Then, the map |L| −→ |K| is injective.

0 0 Proof. For (σ, x) ∈ Lp × |∆[p]|, (σ , x ) ∈ Lq × |∆[q]| and (τ, y) ∈ Kn × |∆[n]|. Assume

(σ, x) ' (τ, y) ' (σ0, x0) in |K| to prove (σ, x) ' (σ0, x0) in |L|

The simplest case this problem is the following:

(σ, x) ' (τ, y) ' (σ0, x0) in |K| 3 ” ' ” are induced by a face or degeneracy.

In this case, we prove (σ, x) ' (σ0, x0) in |L|. 161

1. Suppose, one of the two (σ, x) ' (τ, y) ' (σ0, x0) equivalences, say the ﬁrst one, is induced by a map f ∈ ∆([n], [m1]). It follows,

0 0 τ = K(f)σ. Therefore, τ ∈ Ln and σ, x) ' (σ , x ) in |L|.

2.( Recall, f ∈ ∆([m], [n]) is surjective if and only if it si product of degeneracies.) Suppose one of the two (σ, x) ' (τ, y) ' (σ0, x0) equivalences is induced by a product Qr ji Qr ji of degeneracies s = i=1 si , ∈ ∆([m1], [n]) or s = i=1 si , ∈ ∆([m2], [n]). Assume the ﬁrst one is induced by s ∈ ∆([m1], [n]). Now, s has a right inverse d. We have,

K(s)τ = σ, |s|(x) = y. =⇒ τ = K(d)K(s)τ = K(d)σ ∈ Ln

So, these relations are in |L| and hence (σ, x) ' (σ0x0) in |L|.

3. Suppose (σ, x) ' (τ, y) ' (σ0, x0) are induced by two face maps di, dj ∈ ∆([n−1], [n]). In this case, p = q = n − 1.

di i j (σ, x) ∼ (τ, y) d , d ∈ ∆([n − 1], [n]) be faces, and j (σ0, x0) ∼d (τ, y)

We have, K(di)τ = σ |di|(x) = y K(dj)τ = σ0 |dj|(x0) = y Degeneracies in ∆([n−1], [n−2]) would be denoted by ζi and faces in ∆([n−2], [n−1]) would be denoted by δi. If i = j, then σ = σ0 and x = x0. So, we assume i < j. We have j i i j i i K(d δ ) = K(δ )K(d ) = K(δj−1)K(d ) = K(d δj−1) It follows, K(δj−1)K(di)τ = K(δj−1)(σ) |di|(x) = y K(δi)K(dj)τ = K(δi)(σ0) |dj|(x0) = y

With y = (y0, y1, . . . , yn) we have yi = yj = 0 Let

i j j−1 i x0 = |ζ s |(y) = |ζ s |(y) = (y0, . . . , yi−1, yi+1, . . . , yj−1, yj+1, . . . , yn) j−1 0 and σ0 = K(δ )(σ) = K(δi)(σ ) ∈ Ln−2 We claim

j−1 j−1 (σ0, x0) = (K(δ )(σ), x0) ' (σ, x) given by δ :[n − 2] −→ [n − 1]. i 0 0 0 i (σ0, x0) = (K(δ )(σ ), x0) ' (σ , x ) given by δ :[n − 2] −→ [n − 1]. One checks,

i j−1 i j−1 |d |(|δ |(x0)) = |d |(x) = y =⇒ |δ |(x0) = x. j i j 0 i 0 |d |(|δ |(x0)) = |d |(x ) = y =⇒ |δ |(x0) = x .

So, the claim is established and (σ, x) ' (σ0, x0) in |L|. 162 APPENDIX A. NOT USED

4. Now suppose f ∈ ∆([m1], [n]) and g ∈ ∆([m2], [n]) be product of face maps (which means injective maps) that induce these two equivalence

f 0 0 g (σ, x) ∼ = (τ, y), (σ , x ) ∼ = (τ, y), Write n = m1 + p = m2 + q

K(f)τ = σ |f| (x) = y K(g)τ = σ0 |g| (x0) = y

ip i2 i1 f = d ··· d d 0 ≤ i1 < i2 < ··· < ip jq j2 j1 g = d ··· d d 0 ≤ j1 < j2 < ··· < jp

0 0 This completes the proof that (σn−1, x) ' (σn−1, x ), in the simplest case stated above. (This Inductive Step remains incomplete. Alternately, if we can do the inductive step in (3.3.8), then we can use that to give an alternative proof one (3.3.11), using the CW structure (3.3.10) on |L| and |K|.) 0 0 For the general case, suppose for (σ, x) ∈ Lp × |∆[p]|, (σ , x ) ∈ Lq × |∆[q]|, there is chain of equivalences:

η1 η2 η2 0 0 (σ, x) = (τ0, y0) ∼ (τ1, y1) ∼ · · · ∼ (σr−1, yr−1) ∼ (τr, yr) = (σ , x )

where each equivalence is given by a face or degeneracy ηi, with (τi, yi) ∈ Kni × |∆[ni]|. We will prove that (σ, x) ' (σ0x0) in |L|. If each of the arrow are left to right or right to left, then τi ∈ Lni ∀ i, and we the through.

1. Except in case (out of four) η1 :[n0] −→ [n1] is a face map, it follows τ1 ∈ Ln1 (note for any degeneracy s K(s) has a left inverse), and induction would apply. So, assume i1 d1 := η1 :[n0] −→ [n1] = [n0 + 1] is a face map.

2. Now, we consider various cases of η2.

j2 (a) Suppose η2 = s2 :[n1] −→ [n2] is a degeneracy. Using the commutativity of 0 0 j2 i1 i j s d1 = d s , we can reduce the length of the chain and we would be done. Here i1 j2 τ0 = K(d1 )τ1 τ1 = K(s2 )τ2 Bibliography

[BS] Balmer, Paul; Schlichting, Marco Idempotent completion of triangulated categories. J. Algebra 236 (2001), no. 2, 819-834.

[Gd] Grayson, Daniel R. Localization for ﬂat modules in algebraic K-theory. J. Algebra 61 (1979), no. 2, 463-496.

[GM] Gupta, S. K.; Murthy, M. P. Suslin’s work on linear groups over polynomial rings and Serre problem. ISI Lecture Notes, 8. Macmillan Co. of India, Ltd., New Delhi, 1980.

[GJ] Goerss, Paul G.; Jardine, John F. Simplicial homotopy theory. Reprint of the 1999. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2009.

[Ha] Hatcher, Allen Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp.

[HS] Hilton, P. J.; Stammbach, U. A course in homological algebra. Second edition. Grad- uate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp.

[H] Hovey, Mark Model categories. Mathematical Surveys and Monographs, 63. American Mathematical Society, Providence, RI, 1999. xii+209 pp.

[Hr] Hartshorne, Robin Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp.

[Hu] Husemoller, Dale Fibre bundles. Third edition. Graduate Texts in Mathematics, 20. Springer-Verlag, New York, 1994. xx+353 pp. (First published in 1966)

[K] Keller, Bernhard Chain complexes and stable categories. Manuscripta Math. 67 (1990), no. 4, 379-417.

[Mc] MacLane, Saunders Categories for the working mathematician. Graduate Texts in Mathematics, Vol. 5. Springer-Verlag, New York-Berlin, 1971. ix+262 pp.

[Mat] Matsumura, Hideyuki Commutative algebra. Second edition. Mathematics Lecture Note Series, 56. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980.

163 164 BIBLIOGRAPHY

[Mi] Milnor, John Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. xiii+184 pp.

[Q] Quillen, Daniel Higher algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85?147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973.

[Sm1] Schlichting, Marco Higher algebraic K-theory. Topics in algebraic and topological K-theory, 167-241, Lecture Notes in Math., 2008, Springer, Berlin, 2011.

[Sm2] Schlichting, Marco Delooping the K-theory of exact categories. Topology 43 (2004), no. 5, 1089-1103.

[S] Spanier, Edwin H. Algebraic topology. Corrected reprint. Springer-Verlag, New York- Berlin, 1981. xvi+528 pp. (Original published in 1966)

[Sv] Srinivas, V. Algebraic K-theory. Second edition. Progress in Mathematics, 90. Birkhäuser Boston, Inc., Boston, MA, 1996. xviii+341 pp.

[St] Steenrod, Norman The topology of ﬁbre bundles. Reprint of the 1957 edition. Prince- ton Landmarks in Mathematics. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1999. viii+229 pp.

[Wc1] Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38.Cambridge University Press, Cambridge, 1994. xiv+450 pp.

[Wc2] Weibel, Charles A. The K-book. An introduction to algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. xii+618 pp.

[W] Whitehead, George W. Elements of homotopy theory. Graduate Texts in Mathemat- ics, 61. Springer-Verlag, New York-Berlin, 1978. xxi+744 pp.