University of Nevada, Reno

The Homotopy Theory of Commutative dg Algebras and Representability Theorems for Lie Algebra Cohomology

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics

by

Aydin Ozbek

Dr. Christopher L. Rogers/Thesis Advisor

May 2019

THE GRADUATE SCHOOL

We recommend that the thesis prepared under our supervision by

AYDIN OZBEK

Entitled

The Homotopy Theory of Commutative dg Algebras and Representability Theorems for Lie Algebra Cohomology

be accepted in partial fulfillment of the requirements for the degree of

MASTERS OF SCIENCE

Chris L. Rogers, Ph.D. , Advisor

Stanislav Jabuka, Ph.D. , Committee Member

Bruce Blackadar, Ph.D. , Comm ittee Member

Scott Hinton, MFA , Graduate School Representative

David W. Zeh, Ph.D., Dean, Graduate School

May-2019

i

Abstract

Building on the seminal works of Quillen [12] and Sullivan [16], Bousfield and Guggenheim [3] developed a "homotopy theory" for commutative differential graded algebras (cdgas) in order to study the rational homotopy theory of topological spaces. This "homotopy theory" is a certain categorical framework, invented by Quillen, that provides a useful model for the non-abelian analogs of the derived categories used in classical homological algebra.

In this masters thesis, we use K. Brown’s generalization [5] of Quillen’s formalism to present a homotopy theory for the category of semi-free, finite-type cdgas over a field k of characteristic 0. In this homotopy theory, the "weak homotopy equivalences" are a refinement of those used by

Bousfield and Guggenheim. As an application, we show that the category of finite-dimensional Lie algebras over k faithfully embeds into our homotopy category of cdgas via the Chevalley-Eilenberg construction. Moreover, we prove that Lie algebra cohomology with coefficients in a trivial module is representable in this homotopy category, in analogy with the classical representability theorem for singular cohomology of CW complexes. Finally, we show that central extensions of Lie algebras are recovered within our homotopical framework as certain principal cofiber sequences. ii

Acknowledgements

The author would like to thank: his advisor, Chris Rogers, for providing invaluable mentorship, his professors for nurturing his desire to learn, and his friends and family for making his world more colorful. iii

Contents

1 Introduction 1 1.1 Homotopy theory as non-abelian homological algebra ...... 1

1.2 Brown categories ...... 2

1.3 Sullivan algebras and rational homotopy theory ...... 3

1.4 Chevalley-Eilenberg algebras of finite dimensional Lie algebras ...... 3

2 Summary of main results 5 2.1 The Brown category of Chevalley-Eilenberg algebras ...... 5

2.2 Representability of Lie algebra cohomology ...... 7

2.3 Classification of principal cofibrations ...... 7

3 Preliminaries 9 3.1 Conventions ...... 9

0 3.2 Projective model structure on Ch≥ ...... 9 3.3 Commutative differential graded algebras ...... 11

3.4 Semi-free cdgas ...... 13

3.5 Lie algebras ...... 17

4 The Brown category of Chevalley- Eilenberg algebras 24 4.1 Axioms for a Brown Category ...... 24

4.2 Chevalley-Eilenberg algebras ...... 26

4.3 Homotopy category of Chevalley-Eilenberg algebras ...... 41

5 Representability Theorems 49 5.1 Suspensions, cogroups, and principal cofiber sequences in Brown categories . . . . 49

5.2 Cohomology ...... 53

5.3 Classification of principal cofiber sequences ...... 57

6 Future Work 65 iv

A Appendix 66 A.1 Acyclic cofibrations in CEAlg admit left inverses ...... 66

A.2 Every weak equivalence in CEAlg is a quasi-isomorphism of cdgas ...... 67

A.3 Obstruction theory for semi-free cdga morphisms ...... 72 1

1 Introduction

The quintessential characteristic of homotopy theory is the replacement a rigid notion of equivalence with a more flexible one. This can be achieved by localizing the relevant category with respect to the class of desired equivalences. The process of localizing categories shares many similarities with the process of localizing of rings. In both processes, we look for the smallest category, resp. ring, containing the original category, resp. ring, such that the desired morphisms, resp. elements, are

1 invertible. Thus, if it exists, the localization, written as C[W − ], of a category C with respect to some class of morphisms W, is defined by the following universal property: Any functor F : C D → 1 that carries the class W to isomorphisms factors uniquely through C[W− ].

1 For arbitrary C and W we can sometimes construct C[W− ] by formally inverting the morphisms

1 in W. That is, we define C[W− ] to be the category with the same objects as C but with morphisms defined as reduced zig zags of morphisms in C

f w f wj X 1 1 i Y −→ · ←− · · · −→ · ←− where w W. The arrows pointing to the left provide the formal inverses to the morphisms in W. k ∈ This is a clunky category to work with, but that is to be expected since C and W are generic and the only information used in its construction is the universal property. But if we know more about the

1 behavior of the class W in relation to C, we can obtain a cleaner description of C[W− ].

1.1 Homotopy theory as non-abelian homological algebra

Let Ch 0(R) be the category of chain complexes of R-modules for some ring R. We can present the ≥ localization of Ch 0(R) with respect to the class of quasi isomorphisms in the form of the derived ≥ category of R-modules, written as D(R). The objects of the derived category are the same as

Ch 0(R), but the morphisms in D(R) between two complexes, M and N, are equivalence classes ≥ of spans. A span from M to N is a pair of chain maps

M M N ← → ! in which the left arrow is a quasi isomorphism. If M is a complex of projective modules, then 2 the set of equivalence classes of spans from M to any complex N, can be presented as the set of chain homotopy classes of chain maps from M to N. This category satisfies the above universal property, so D(R) can be considered the localization. However, the derived category is equipped with additional structure, which allows one to deduce that

n homD(A)(X, Y [n]) ∼= ExtA(X, Y )

Yet, the classical construction of the derived category does not make sense within the context of non-abelian categories. One such solution to this is the theory of model categories, introduced by

Quillen in [13]. A model category is a category M (with small limits and colimits), equipped with three classes of maps called cofibrations, fibrations and weak equivalences subject to a handful of axioms. It is customary to call the localization of M with respect to the class of weak equivalences the homotopy category, which we denote by Ho(M). Within this theory the bifibrant objects in M play an important role, i.e. the objects in M such that the map from the initial object is a cofibration and the map to the terminal object is a fibration. The morphisms in the homotopy category of the full subcategory of bifibrant objects are in one-to-one correspondence with certain equivalence classes of morphisms in M. Moreover, every object in M is weakly equivalent to a bifibrant object. Thus, we can show this category is equivalent to the homotopy category of M. Analgous to the derived category, the homotopy category is more than just the localization. For example, if M is pointed then we can define a notion of a loop and a suspension endofunctor on the homotopy category. In this case, Quillen [13] showed that certain hom-sets in the homotopy category of such model categories correspond to familiar notions of “cohomology groups”

1.2 Brown categories

Since the abstract framework of model categories requires that the underlying category has small limits and colimits, one must use a different approach to study the homotopy theory of a category if it is not closed with respect to limits and colimits. One such structure is the notion of a category of

fibrant objects (CFO), which was introduced by Brown in [5]. A CFO is a category C (with finite products) equipped with a class of fibrations and weak equivalences subject to axioms similar to that of a model category. One major distinction between these two sets of axioms is that every object in 3 a CFO is required to be fibrant, i.e the unique map from every object in C into the terminal object must be a fibration. One can define a similar structure on a category with finite coproducts equipped with cofibrations and weak equivalences subject to the dualized axioms of a CFO. We will refer to categories with this structure as Brown categories. A prototypical Brown category is the category of

finite-type CW complexes where the cofibrations are retracts of inclusions and the weak equivalences are homotopy equivalences.

1.3 Sullivan algebras and rational homotopy theory

Rational homotopy theory is the study of simply connected spaces up to rational weak equivalence.

In other words, it is the study of the category of simply connected spaces localized with respect to the maps, f : X Y , such that f idQ : πn(X) Q πn(Y ) Q is an isomorphism for all n. There → ∗ ⊗ ⊗ → ⊗ is an equivalent algebraic formulation of this category given by the homotopy category associated to the Bousfield-Guggenheim [3] (BG) model structure on commutative differential graded algebras

(cdgas) over Q. In this model structure, the fibrations are degree wise surjections, and the weak equivalences are the quasi isomorphisms. The definition of a cofibration involves the notion of a

Sullivan algebra. A Sullivan algebra S(V ) is a cdga whose underlying commutative graded algebra is isomorphic to the symmetric algebra on a graded vector space V equipped with an increasing

filtration that is compatible with the differential on S(V ) in a certain way. A cofibration in the (BG) model structure is an inclusion A ↩ A S(V ) of cdgas where S(V ) is a Sullivan algebra. Thus → ⊗ the cofibrant objects are exactly the Sullivan algebras.

1.4 Chevalley-Eilenberg algebras of finite dimensional Lie algebras

For any finite dimensional Lie algebra g over a field of char 0 we can define a differential on the graded symmetric algebra of g∗ concentrated in degree 1 by dualizing its bracket. The fact that the bracket satisfies the Jacobi identity implies that the differential squares to zero. This cdga contains information about the Lie algebra cohomology of g in the sense that the cohomology of the cdga is the cohomology of g with coefficients in the ground field. This cdga is called the Chevalley-Eilenberg algebra (CE algebra) of g, and the assignment of a Lie algebra to its CE algebra defines a functor. In fact, this functor faithfully embeds the category of Lie algebras into the the category of cdgas. One 4 might be tempted to believe that the CE algebra of a Lie algebra is a Sullivan algebra, but this is not the case. A CE algebra is a Sullivan algebra iff g is nilpotent. Hence, if we wish to study all Lie algebras via their CE algebras, the BG model structure on cdga may not be the best fit. 5

2 Summary of main results

2.1 The Brown category of Chevalley-Eilenberg algebras

Upon reviewing relevant algebraic definitions and facts in Sec. 3, we introduce (see Def. 4.4) the category, CEAlg, of Chevalley-Eilenberg algebras (CE algebras). A CE algebra is a semi free, finite type commutative differential graded algebra (cdga) (S(V ), δ), generated by a positively graded vector space V . A morphism of CE algebras is simply a cdga morphism. To every CE algebra

1 (S(V ), δ) we can associate a cochain complex (V, δ1) where the differential is defined by pre and post composing δ with the inclusion V ↩ S(V ) and the projection S(V ) V , respectively. Similarly, → → to every morphism of CE algebras F : (S(V ), δ ) (S(W ), δ ) we associate a morphism of V → W cochain complexes F 1 : (V, δ 1) (W, δ 1) where F 1 is obtained by pre and post composing F 1 V 1 → W 1 1 with the inclusion V ↩ S(V ) and the projection S(W ) W . We use this associated cochain map → → in Def. 4.9 to characterize two classes of morphisms in CEAlg. We say that F is a weak equivalence

1 1 iff F1 is a quasi isomorphism of cochain complexes, and we say that F is a cofibration iff F1 is injective in all degrees greater than or equal to 2. In Thm. A.6 of the Appendix, it is shown that weak equivalence of CE algebras induces a quasi isomorphism of cdgas, but the converse is false in general (see Sec. 4.3).

In Prop. 4.11, we show that every strict morphism of CE algebras can be factored into a cofi- bration followed by a weak equivalence. In particular, there exists a such a factorization of the fold map : S(V ) S(V ) S(V ). Our main theorem below and Brown’s Factorization Lemma (see ∇ ⊗ → Lemma 4.3) imply that any morphism of CE algebras can be factored into a cofibration followed by weak equivalence.

We provide an explicit description of the pushout of a strict cofibration along an arbitrary mor- phism of CE algebras in Prop. 4.14. Using this proposition and Lemma 4.16 we show that the pushout of an arbitrary cofibration along any morphism exists in CEAlg. Moreover, we prove the pushout of an (acyclic) cofibration is itself an (acyclic) cofibration.

Although the category CEAlg is a full subcategory of cdga, which is both complete and cocom- plete, it is not closed under products. For this reason CEAlg cannot form a model category. It is however, closed under finite coproducts (see Prop. ??). However, the main novel result of this thesis 6 establishes that this homotopical structure can be modeled as a Brown category:

Theorem. The category CEAlg, of CE algebras over a field of characteristic 0 and cdga morphisms between them, admits the structure of a Brown category of cofibrant objects in which a morphism

F : (S(V ), δ ) (S(W ), δ ) V → W is:

a weak equivalence iff F 1 : (V, δ 1) (W, δ 1) is a quasi isomorphism of cochain com- • 1 V 1 → W 1 plexes

a cofibration iff F 1 : (V, δ 1) (W, δ 1) is injective in all degrees greater than 1 • 1 V 1 → W 1 where F 1 = pr F , δ 1 = pr δ , and δ 1 = pr δ . 1 W |V V 1 V V |V W 1 W W |W

Let us mention a few initial remarks about this result. First since there are quasi isomorphisms of cdgas that aren’t weak equivalences, the homotopy category of CE algebras, Ho(CEAlg), is dis- tinct from the homotopy category induced by the Bousfield Guggenheim model structure on cdgas,

Ho(cdga).

Second, one of the potential applications we have in mind for this alternative homotopical struc- ture is to study (non-abelian) Lie algebra cohomology via representable functors out of Ho(CEAlg), in analogy with the classical E. Brown representability theorem for the singular cohomology of CW complexes [4]. As a first step, in this thesis we focus on recovering the classical Chevalley-Eilenberg cohomology within this homotopy theoretic context. Indeed, we prove that the category of finite dimensional Lie algebras embeds faithfully into Ho(CEAlg). In contrast, we show in Sec. 4.3 that there is no such embedding into Ho(cdga). Moreover since a g-module, for some Lie algebra g, is a vector space M equipped with a Lie algebra homomorphism g End (M) where End (M) → k k is given the commutator bracket, the category of g-modules can be realized as a category under

Ho(CEAlg). These facts suggest that Ho(CEAlg) is a suitable environment in which to study Lie algebras.

Finally we note that there is a similar homotopical structure on a class of infinite dimensional topological semi free cdgas, due to Lazarev [11], that shares many of the same properties of Ho(CEAlg). 7

The advantage of the finite type algebras is that they have well-defined spatial realizations as cer- tain simplicial manifolds [9], in analogy with the fact that finite-dimensional Lie algebras can be integrated into Lie groups.

2.2 Representability of Lie algebra cohomology

In the second part of this thesis, we begin in section 5 by introducing the notion of suspension in a pointed Brown category C. Analogous to the topological case, the suspension ΣX of an object X in a Brown category is a cogroup object in the homotopy category. Thus, for all X C, the functor ∈ [ΣX, ] factors through the category of groups, where [ , ] denotes the set of morphisms in the − − − homotopy category.

For n 1 and M a finite dimensional vector space, we denote by K(M, n) the symmetric ≥ algebra on M ∗ concentrated in degree n with trivial differential. We show, in Prop. 5.12, that ΣK(M, n + 1) is weakly equivalent to K(M, n). Thus, [K(M, n), (S(V ), δ)] is a group for all

CE algebras (S(V ), δ). In particular we prove the following:

Theorem. Let C(g) be the Chevalley-Eilenberg algebra of a finite dimensional Lie algebra g. If we consider M as a trivial g-module, then for all n 1 there is a natural isomorphism of abelian ≥ groups

n H (g, M) ∼= [K(M, n), C(g)]

This result is analogous to E. Brown’s representability theorem for singular cohomology [4], which states that Hn(X, A) is naturally isomorphic as a group to [X, K(A, n)] , for any pointed ∗ CW complex X and Eilenberg-Maclane space K(A, n). Just as our group structure arises from the suspension functor, the group structure on [X, K(A, n)] is given by the fact that ΩK(A, n + 1) ≃ K(A, n).

2.3 Classification of principal cofibrations

Finally, motivated by the definition of a principal fiber sequence given in [1], we define, in Sec.

5.1, the notion of a principal cofiber sequence in a pointed Brown category. In a manner similar to classifying principal G-bundles over a CW complex, we show that the set of equivalence classes 8 of principal cofiber sequences with fixed classifying space K(M, n + 1), base C(g), and cofiber

K(M, n) are in one to one correspondence with [K(M, n + 1), C(g)] for all n 1. In addition, in ≥ the n = 1 case, we show that equivalence classes of these principal cofiber sequences are in one to one correspondence with equivalence classes of central extensions of the Lie algebra g by M. Thus the classic correspondence between the second Lie algebra cohomology group with coefficients in a trivial module and central extensions fits into the following commutative diagram of of bijections between sets:

Prop. 5.19 Central extensions of g by M / H2(g, M) { } ∼ ∼=

Prop. 5.20 = ∼ ∼= Thm. 5.14

Principal cofiber sequences with classifying space Prop. 5.16 [K(M, 2), C(g)] K(M, 2), base C(g), and cofiber K(M, 1) ∼ = " #$ ∼

Hence the well established bijection between central extensions and H2 of a Lie algebra is recovered by the homotopy theory of CEAlg. Moreover, the above diagram for the n = 1 case suggests that, for the n 2 case, we interpret the sequence of bijections ≥

Hn(g, M) = [K(M, n + 1), C(g)] = Principal cofiber sequences with classifying space ∼ ∼ K(M, n + 1) base C(g), and cofiber K(M, n) ∼ " #$ as a realization of higher degree Lie algebra cohomology classes as “higher degree” extensions (i.e. cofiber sequences) of g by more general CE algebras. 9

3 Preliminaries

3.1 Conventions

Throughout this thesis, we will adopt the following conventions and notations:

k denotes a field of characteristic zero •

We define x := n if x V n for some graded vector space V • | | ∈

V [k] denotes the graded vector space with V [k]n := V n k for a graded vector space V • −

Non-graded vector spaces will be considered as graded vector spaces considered in degree 0 •

The category of commutative dg algebras and dg algebra morphisms is denoted by cdga. •

The category of semi-free, finite-type commutative dg algebras and dg algebra morphisms is • denoted by CEAlg.

0 3.2 Projective model structure on Ch≥

For a more thorough introduction to homological algebra, we recommend Weibel’s book [19], and for more on the model category structure on complexes, we recommend Goerss and Schemmerhorn’s expository article [8].

n Recall that a cochain complex (D, d) over k is a N-graded k-vector space D = n 0 D ≥ equipped with a degree 1 map of graded vector spaces d: D D, called the differentia%l, with the → property that d2 = 0. The condition that the differenitial squares to 0 implies that im d ker d. ⊆ Hence we define the cohomology of a cochain complex (D, d) to be the N-graded k-vector space n H∗(D) = n 0 H (D) where ≥ % (ker d: Dn Dn+1) Hn(D) := → (im d: Dn 1 Dn) − → A morphism of cochain complexes f : (D, d ) (E, d ) is a degree 0 map of graded vector D → E spaces f : D E with the property that d f = fd . A morphism of complexes f is said to be → E D n n injective (resp. surjective) in degree n iff f n : D E is injective (resp. surjective). Every |D → 10 morphism of complexes f induces a degree 0 map of graded vector spaces H (f): H (D) ∗ ∗ → H∗(E). If H∗(f) is an isomorphism we say that f is a quasi isomorphism. We denote the category 0 of cochain complexes and cochain morphisms by Ch≥ .

Example 3.1.

1. If (D, dD) and (E, dE) are complexes then their direct sum (D E, d ) is defined as D E := ⊕ ⊕ ⊕ n n n 0 D E with differential d (x, y) := (dD(x), dE(y)). With the obvious projections ≥ ⊕ ⊕ 0 a%nd inclusions, (D E, d ) is the categorical product and coproduct in Ch≥ , respectively. ⊕ ⊕

2. If (D, dD) and (E, dE) are complexes then their tensor product (D E, d ) is defined as the ⊗ ⊗ graded vector space whose degree n component is

D E n := Di Ej ⊗ ⊗ i+j=n & ' (

x The differential is defined as d (x y) := dD(x) y + ( 1)| |x dE(y). ⊗ ⊗ ⊗ − ⊗

3. If (D, dD) and (E, dE) are complexes then the mapping complex (Map(D, E), ∂) is graded n n vector space Map(D, E) := n 0 Map (D, E), where Map (D, E) is the k-vector space ≥ of degree n maps from D to E%, equipped with the differential given by

f ∂(f) = d f ( 1)| |fd E − − D

There are three classes of cochain maps that are of particular interest to us. The class of quasi isomorphisms, the class of injections in degrees n 1, and the class of surjections in all degrees. ≥ 0 With these distinguished classes, the category Ch≥ admits the structure of a model category. How- ever, for our purposes it will be more convenient to work with the following reformulation of this model category.

1 Definition 3.2. Let Ch≥ denote the category of positively graded cochain complexes (D, dD) over

0 k, i.e. D = 0. Then a morphism f : (D, dD) (E, dE) is called →

A weak equivalence iff f is a quasi isomorphism •

A cofibration iff f is injective in all degrees n 2 • ≥ 11

A fibration iff f is surjective in all degrees •

1 1 We denote the category Ch≥ equipped with these three classes by Chp≥roj.

1 Theorem 3.3. The category Chp≥roj admits the structure of a model category.

We refer to this structure the projective model structure on the category of positively graded 1 cochain complexes. The interested reader can find a variation of the proof that Chp≥roj is a model category in [8, Thm. 1.5].

3.3 Commutative differential graded algebras

For those readers who desire a more in depth exposition of graded algebra, we recommend Hess’ article [10] and Felix, Halperin, and Thomas’ book [6, Sec. I.3].

n A commutative graded algebra (A, µ, η) over k is a N-graded k-vector space A = n 0 A ≥ equipped with two degree 0 maps of graded vector spaces %

η : k A → called the unit, and µ: A A A, x y xy ⊗ → ⊗ ,→ called multiplication, such that the following are satisfied:

1. µ is associative, i.e. (xy)z = x(yz)

2. µ is graded commutative, meaning xy = ( 1) x y yx − | || |

3. µ is unital, that is TFDC

η idA idA η k A ⊗ A A ⊗ A k ⊗ ⊗ ⊗ µ

A

Notation 3.4. Let 1A = η(1). 12

Condition 3 implies that there exists a 1 A0 such that 1 x = x = x1 . A morphism of A ∈ A A commutative graded algebras F : (A, µ , η ) (B, µ , η ) is a degree 0 map of graded vector A A → B B spaces such that F η = η and F µ = µ (F F ). A B A B ⊗

Definition 3.5. A left A-module over a commutative graded algebra (A, µ, η) is a graded vector space M equipped with a degree 0 linear map

A M M, a m am ⊗ → ⊗ ,→ such that (aa′)m = a(a′m) for all a, a′ A and m M. A right A-module is a graded vector ∈ ∈ space M equipped with a degree 0 linear map M A M that satisfies the analogous associative ⊗ → property. We refer to a graded vector space that is both a left and right module as a module.

Example 3.6. Let F : (A, µA, ηA) (B, µB, ηB) be a morphism of commutative graded algebras. → Then B is an A-module given by the assignments a b F (a)b and b a bF (a) for all a A ⊗ ,→ ⊗ ,→ ∈ and b B. ∈

Definition 3.7. A commutative differential graded algebra (cdga) (A, δ, µ, η) over k consists of a cochain complex (A, δ) over k equipped with two cochain maps η : (k, 0) (A, δ) and → µ: (A, δ) (A, δ) (A, δ), such that conditions 1-3 above are satisfied. A morphism of cdgas ⊗ → F : (A, δ , µ , η ) (B, δ , µ , η ) is a cochain map F : (A, δ ) (B, δ ) such that F η = A A A → B B B A → B A η and F µ = µ (F F ). We denote the category of cdgas and cdga morphisms by cdga. B A B ⊗

Notation 3.8. We will write a cdga as (A, δ) leaving the multiplication and unit implicit, and simi- larly, we will write a commutative graded algebra as A leaving the multiplication and unit implicit.

Example 3.9. Let (A, δA) and (B, δB) be cdgas. The tensor product of these cdgas (A B, δ ) is ⊗ ⊗ defined as the tensor product of cochain complexes with multiplication given by

b a (a b)(a′ b′) := ( 1)| || ′|aa′ bb′ ⊗ ⊗ − ⊗ and unit given by η (1) := 1A 1B. This construction gives the categorical coproduct in cdga. ⊗ ⊗ Indeed, given two morphisms of cdgas (we suppress the differentials) ψ : A Q and ψ : B Q A → B → we can define a morphism of cdgas ψ : A B Q by ψ(a b) = ψ (a)ψ (b), and inclusions of ⊗ → ⊗ A B ) ) 13 cdgas ι : A A B and ι : B A B by ι (a) = a 1 and ι (b) = 1 b. It is easy to A → ⊗ B → ⊗ A ⊗ B B A ⊗ check that the necessary maps commute and that ψ is unique.

Remark 3.10. Note that a cdga is a commutative )graded algebra equipped with a differential δ that satisfies the following for all x, y A ∈

x δ(xy) = δ(x)y + ( 1)| |xδ(y) −

The above relation is called the (graded) Leibniz rule. Hence the differential on a cdga is an example of a derivation.

Definition 3.11. A derivation of degree n is a degree n linear map F : A M, where A is a → commutative graded algebra and M is an A-module, such that

a n F (aa′) = F (a)a′ + ( 1)| | aF (a′) − for all a, a A. ′ ∈

3.4 Semi-free cdgas

Definition 3.12. The tensor algebra, T (V ), of a N-graded k-vector space V is defined to be T (V ) := n 0 n n 0 T (V ) where T (V ) := k and for n 1, T (V ) := V V is the n-fold tensor product. ≥ ≥ ⊗· · ·⊗ M%ultiplication in T (V ) is given by the assignment

(x x )(y y ) x x y y 1 ⊗ · · · ⊗ i 1 ⊗ · · · ⊗ j ,→ 1 ⊗ · · · ⊗ i ⊗ 1 ⊗ · · · ⊗ j and the unit is given by

id 0 k T (V ) ↩ T (V ) −→ → The multiplication in T (V ) is associative and unital, but is not graded commutative.

Definition 3.13. The symmetric algebra S(V ) of a N-graded k-vector space V is defined as S(V ) := T (V )/I where I is the two sided ideal generated by x y ( 1) x y y x x, y V . We write { ⊗ − − | || | ⊗ | ∈ } x x to denote the equivalence class represented by x x in S(V ). 1 ∨ · · · ∨ n 1 ⊗ · · · ⊗ n n n n n n Equivalently, if I = I T (V ), then S(V ) = n 0 S (V ) where S (V ) := T (V )/I . Since ∩ ≥ I0 = 0 = I1, S0(V ) = k and S1(V ) = V . N%ote that Sn(V ) corresponds to words of length n 14 and not degree n elements. In general, x x = x + + x . S(V ) naturally has the | 1 ∨ · · · ∨ n| | 1| · · · | n| structure of a commutative graded algebra. The multiplication and unit are inherited from the tensor algebra T (V ). That is, multiplication is given by the assignment

(x x )(y y ) x x y y 1 ∨ · · · ∨ i 1 ∨ · · · ∨ j ,→ 1 ∨ · · · ∨ i ∨ 1 ∨ · · · ∨ j and the unit is given by

id 0 k S (V ) ↩ S(V ) −→ → Proposition 3.14. Let V be a graded vector space. Then S(V ) is the free commutative graded algebra on V . That is, for any degree 0 linear map f : V A, where A is a commutative graded → algebra, there exists a unique morphism of commutative graded algebras F : S(V ) A such that → F = f. |V Proof. Let n 1 and let x x Sn(V ). Then we define ≥ 1 ∨ · · · ∨ n ∈

F (x x ) = F (x ) F (x ) 1 ∨ · · · ∨ n 1 · · · n and F (1k) = 1A.

There exists a similar method of extending degree n graded linear maps to derivations.

Proposition 3.15. Let V be a graded vector space and let M be a S(V )-module. Then for any degree n linear map f : V M, there exists a degree n derivation F : S(V ) M such that F = f. → → |V Proof. Let n 1 and let x x Sn(V ). Then we define ≥ 1 ∨ · · · ∨ n ∈ n n x1 xi 1 F (x1 xn) = ( 1)| || ∨···∨ − |x1 xi 1δ(xi)xi+1 xn ∨ · · · ∨ − · · · − · · · *i=1 and F (1k) = 0.

Let V be a k-vector space and n 2. Recall that the nth-exterior power, Λn(V ), of V is defined ≥ to be the quotient of the n-fold tensor product, V V , by the subspace generated by ⊗ · · · ⊗

x1 xn sgn(σ)xσ(1) xσ(n) σ Sn { ⊗ · · · ⊗ − ⊗ · · · ⊗ | ∈ } 15

We denote the equivalence class represented by x x in Λn(V ) as x x . If V is 1 ⊗ · · · ⊗ n 1 ∧ · · · ∧ n finite dimensional and with basis e , , e , then ΛnV has dimension m with basis given by { 1 · · · m} n & ' e e i < < i (1) { i1 ∧ · · · ∧ in | 1 · · · n}

The exterior power can also be characterized by the following universal property. Recall that a function f : V V W is said to be alternating if f(x1, , xn) = 0 when xi = xj for × · · · × → · · · i = j. Then every multilinear alternating function f : V V W induces a unique k-linear ∕ × · · · × → map f : ΛnV W . Suppose that f : ΛjV k and g : ΛkV k. We define f g : Λj+kV k → → → ∧ → to be the k-linear map given by

(f g)(x x ) = sgn(σ)f(x x )g(x x ) (2) ∧ 1 ∧ · · · ∧ j+k σ(1) ∧ · · · ∧ σ(j) σ(j+1) ∧ · · · ∧ σ(j+k) σ Shuf(j,k) ∈ * where,

Shuf(j, k) = σ Sj+k σ(1) < < σ(j) and σ(j + 1) < < σ(j + k) { ∈ | · · · · · · }

As shown in Warner’s book [18], if V has basis e , , e and θ1, , θn is the corresponding { 1 · · · n} { · · · } dual basis for V ∗, then (θ1 θn)(e e ) = 1 ∧ · · · ∧ 1 ∧ · · · ∧ n

n The exterior algebra of V is the commutative graded algebra Λ(V ) := n 0 Λ (V ) where ≥ Λ0(V ) := k and Λ1(V ) := V . The multiplication is given by the assignment %

(x x )(y y ) x x y y 1 ∧ · · · ∧ i 1 ∧ · · · ∧ j ,→ 1 ∧ · · · ∧ i ∧ 1 ∧ · · · ∧ j and the unit is given by

id 0 k Λ (V ) ↩ Λ(V ) −→ →

Proposition 3.16. Let V be a k-vector space, and let V [1] denote the graded vector space consisting of V concentrated in degree 1. Then there is an isomorphism of graded algebras

S(V [1]) = Λ(V )

Proof. Note that since V [1] is concentrated in degree 1, word length of an element in S(V [1]) cor- 16 responds to its degree, and the two sided ideal I ⊳ T (V [1]) in Def. 3.13 is generated by x y + { ⊗ y x x, y V . Since Sn is generated by (12), (23), , (n 1 n) , we have that ⊗ | ∈ } { · · · − }

n T (V [1]) I = x1 xn sgn(σ)xσ(1) xσ(n) σ Sn ∩ { ⊗ · · · ⊗ − ⊗ · · · ⊗ | ∈ }

Moreover the multiplication and the unit in both commutative graded algebras are given by concate- nation of words and inclusion, respectively.

Notation 3.17. For the remainder of this thesis we will assume that all graded vector spaces V are positively graded, i.e. V n = 0 for all n 0, unless stated otherwise. ≤

Definition 3.18. A semi-free cdga over k is a cdga (S(V ), δ) whose underlying commutative graded algebra is isomorphic to S(V ) with the canonical multiplication and unit for some positively graded

n k-vector space V = n 1 V . ≥ % Let F : S(V ) S(W ) be a graded linear map. We can decompose F by precomposing and post → composing with various combinations of graded vector space inclusions and projections as follows

Sj(V ) ↩ Sn(V ) F Sn(W ) Sk(W ) → −→ ↠ n 0 n 0 (≥ (≥ We denote this composition by F k : Sj(V ) Sk(W ) j →

k By the universal property of the product and coproduct the collection Fj j 1,k 1 determines F . { } ≥ ≥ Suppose that S(V ) and S(W ) are equipped with their canonical multiplication and unit. Then if F is a morphism of commutative graded algebras, it is the unique extension of F : V S(W ). |V → n n Since F V = n 1 F1 , it is completely determined by the collection F1 n 1. Note that this sum | ≥ { } ≥ n n is not infinite +since F1 (x) = 0 for all n x + 1. We call the collection F1 n 1 the structure ≥ | | { } ≥ n maps of F and refer to n as the arity of F1 . Moreover since F is respects multiplication, we have

k F k(x x ) = F k1 (x ) F j (x ) (3) j 1 ∨ · · · ∨ j 1 1 ∨ · · · ∨ 1 j k1+ +k =k ·*·· j F is called strict morphism iff F n = 0 for all n 2. 1 ≥ 17

Composition Let F : S(V ) S(W ) and G: S(W ) S(U) be morphisms of commutative graded algebras. → → n Then GF : S(V ) S(U) is the unique extension of GF V = n 0(GF )1 where → | ≥ n + n n k (GF )1 = Gk F1 (4) *k=1 Differentials Suppose that (S(V ), δ) is a cdga. Then since δ is a derivation, it is the unique extension of

n δ V = n 1 δ1 . Hence it is determined by the collection of degree 1 graded vector space maps | ≥ n n δ1 n +1. Again we call the collection δ1 n 1 the structure maps of δ and refer to n as the arity { } ≥ { } ≥ of δn. Observe that δ δ is a derivation since 1 ◦

x δ δ(x y) =δ δ(x) y + ( 1)| |x δ(y) ◦ ∨ ∨ − ∨ , x +1 - x 2 x =δ δ(x) y + ( 1)| | δ(x) δ(y) + ( 1)| |δ(x) δ(y) + ( 1) | |x δ δ(y) ◦ ∨ − ∨ − ∨ − ∨ ◦ =δ δ(x) y + x δ δ(y) ◦ ∨ ∨ ◦ x δ δ =δ δ(x) y + ( 1)| || ◦ |x δ δ(y) ◦ ∨ − ∨ ◦

Thus δ δ is determined by its restriction to V . Therefore, δ δ = 0 iff (δ δ)n = n δnδk = 0 ◦ ◦ ◦ 1 k=1 k 1 for all n 1. + ≥ Additionally suppose that F : (S(V ), δ ) (S(W ), δ ) is a morphism of commutative graded V → W algebras. Then F δV and δW F are both derivations, where S(W ) is given the structure of an S(V )- module as in Ex. 3.6 , and are thus determined by their restriction to V . Therefore F is a cdga morphism iff n n n n k n k n (F δV )1 = Fk δV 1 = δW k F1 = (δW F )1 *k=1 *k=1 for all n 1. ≥

3.5 Lie algebras

For more on Lie algebras and Lie algebra cohomology, we recommend Weibel’s book [19].

Definition 3.19. A Lie algebra over k is a k-vector space g equipped with a bilinear alternating 18 map [ , ]: 2 g g, called the bracket, such that the bracket satisfies the Jacobi identity, i.e. − − → . [x, [y, z]] = [[x, y], z] + [y, [x, z]]

A Lie algebra homomorphism is a k-linear map f : g h such that f([x, y]) = [f(x), f(y)]. →

Observe that since the bracket is alternating, [x, y] = [y, x] for all x, y g. − ∈

Example 3.20.

1. Given any associative k-algebra A we can define a bracket on A as follows,

[x, y] := xy yx for all x, y A − ∈

This bracket is called the commutator bracket.

2. Let V be a k-vector space. We can make V into a Lie algebra by defining the bracket for any x, y V to be [x, y] = 0. Such a Lie algebra is called abelian. ∈ g-modules

Definition 3.21. A g-module is a k-vector space M and a Lie algebra homomorphism φ: g → End(M) where End(M) is equipped with the commutator bracket. For g g and x M we write ∈ ∈ φ(g)(x) as gx.

The fact that φ is a Lie algebra homomorphism implies that

[g, g′]x = g(g′x) g′(gx) − for all g, g g , x M. ′ ∈ ∈

Example 3.22.

1. The Jacobi identity makes the assignment g [g, ] into a Lie algebra homomorphism g ,→ − → End(g). Hence g is a g-module. 19

2. Let V be a vector space. Then the trivial map map 0: g End(V ) defines a Lie algebra → homomorphism and hence gives V the structure of a g-module. We call such a module as a

trivial g-module.

The Chevalley-Eilenberg functor

A Lie algebra g is said to be finite dimensional if its underlying vector space is finite dimensional. Given a finite dimensional Lie algebra g it is easily verified using the basis constructed in (1) that 2 2 ( g)∗ = g∗, where ( )∗ denotes the k-linear dual. Thus the bracket on g defines a linear map ∼ − [., ] : g . Λ2g . − − ∗ ∗ → ∗

Definition 3.23. The Chevalley-Eilenberg algebra of a finite dimensional Lie algebra g is the semi- free cdga given by (S(g∗[1]), δ), where g∗[1] is the graded vector space consisting of g∗ concentrated in degree 1, whose differential defined as

[ , ]∗ n = 2 n − − δg1 = / 0 100 otherwise

0 We denote this cdga by (C(g), δg). 2

3 2 A straightforward calculation shows that the Jacobi identity implies that δg2δg1 = 0, and hence δg is a differential.

Let f : g h be a Lie algebra homomorphism. From this we define a strict cdga morphism → C(f): (C(h), δ ) (C(g), δ ) where C(f)1 = f : h g . Since f is a Lie algebra homomor- h → g 1 ∗ ∗ → ∗ phism TFDC Λ2f Λ2g Λ2h

[ , ] [ , ] − − − − f g h

2 1 2 2 Thus δg1C(f)1 = C(f)2δh1.

Definition 3.24. The Chevalley-Eilenberg functor C : LieAlg cdga is the contravariant functor → 20 defined by the assignment g (C(g), δ ) on objects and the assignment ,→ g

f : g h C(f): (C(h), δ ) (C(g), δ ) → ,→ h → g , - , - where C(f) is the strict cdga morphism given by the k-linear dual of f.

Proposition 3.25. The Chevalley-Eilenberg functor C : LieAlg cdga is full and faithful. →

Proof. If C(f) = C(g) for some Lie algebra homomorphisms f, g : g h, then C(f)n = C(g)n → 1 1 for all n 1. Thus, f = C(f)1 = C(g)1 = g which implies that f = g. ≥ ∗ 1 1 ∗ Now let F : (C(h), δ ) (C(g), δ ) be a morphism in cdga. For degree reasons F must be h → g strict. Moreover, since F commutes with the differentials, we have

F 2 = Λ2F 1 2 2 1 2 Λ h∗ Λ g∗

[ , ] [ , ] − − ∗ − − ∗ 1 F1 h∗ g∗

Therefore, F 1∗ : g h defines a Lie algebra homomorphism such that C(F 1∗) = F . 1 → 1

Remark 3.26. Let V be a finite dimensional vector space and suppose that (S(V [1]), δ) is a cdga. For

2 degree reasons the structure maps of the differential must be zero except in arity 2. Since (Λ V )∗ ∼= Λ2V one can define a bilinear alternating map δ2∗ : Λ2V V . The fact that δ is a differential ∗ 1 ∗ → ∗ 2 implies that δ1∗ satisfies the Jacobi identity. Thus, the image of the Chevalley-Eilenberg functor is all of the semi-free cdgas that are the symmetric algebras on a finite dimensional vector spaces concentrated in degree 1.

Lie algebra cohomology

Definition 3.27. The Lie algebra cohomology groups H∗(g, M) of a Lie algebra g with coefficients in the g-module M are the cohomology groups of the complex homk(Λ∗g, M), d , where the differential , - d : hom (Λng, M) hom (Λn+1g, M) n k → k 21 is defined as

d0f(x) := xf(1k) and for all n 1 ≥ n+1 d f(x x ) = ( 1)i+1x f(x xˆ x ) n 1 ∧ · · · ∧ n+1 − − i 1 ∧ · · · ∧ i ∧ · · · ∧ n+1 *i=1 ( 1)i+jf([x , x ] xˆ xˆ x ) − − i j ∧ · · · ∧ i ∧ · · · ∧ j ∧ · · · ∧ n+1 *i

Remark 3.28. We will show that differential δg on C(g) agrees with the differential d on hom(Λ∗g, k) where k is a trivial g-module. Let n 2. Since the structure maps of δg are zero except in arity 2, ≥ the differential δ : Λng Λn+1g is g ∗ → ∗

n i 1 2 δ (f f ) = ( 1) − f δ (f ) f g 1 ∧ · · · ∧ n − 1 ∧ · · · ∧ g1 i ∧ · · · ∧ n *i=1 Because δ2(f ) = 2 this can be written as | 1 i |

n i 1 2 = ( 1) − δ (f ) f f f − g1 i ∧ 1 ∧ · · · i · · · ∧ n *i=1 Recall the definition of f g (Λj+kV ) given in (2), where3f (ΛjV ) and g (ΛkV ) . Hence, ∧ ∈ ∗ ∈ ∗ ∈ ∗ if we consider (f f f ) (Λn 1g) , 1 ∧ · · · i · · · ∧ n ∈ − ∗ 3 δ (f f )(x x ) = g 1 ∧ · · · ∧ n 1 ∧ · · · ∧ n+1 n i 1 = ( 1) − sgn(σ)f ([x , x ])(f f f )(x x ) − i σ(1) σ(2) 1 ∧ · · · i · · · ∧ n σ(3) ∧ · · · ∧ σ(n+1) i=1 σ Shuf(2,n 1) * ∈ * − 3 The shuffles, σ Shuf(2, n 1), are given by a pair of integers 1 j < k n + 1 such that ∈ − ≤ ≤

1 2 3 n + 1 · · · · · · · · · 4 j k 1 ˆj kˆ n + 1 5 · · · · · · · · · and the sign of such a permutation is ( 1)j+k+1. Hence the sum above can be written as − n i 1 j+k+1 ( 1) − ( 1) f ([x , x ])(f f f )(x xˆ xˆ x ) − − i j k 1 ∧ · · · i · · · ∧ n 1 ∧ · · · j · · · k · · · ∧ n+1 *i=1 *j

n We will now consider (f1 fn) hom(Λ g, k) with k a trivial g-module, and apply the ∧ · · · ∧ ∈ differential defined in Def. 3.27 to it.

d (f f )(x x ) = n 1 ∧ · · · ∧ n 1 ∧ · · · ∧ n+1 = ( 1)j+k(f f )([x , x ] x xˆ xˆ x ) (5) − − 1 ∧ · · · ∧ n j k ∧ 1 ∧ · · · j · · · k · · · ∧ n+1 *j

(yjk yjk) := ([x , x ] x xˆ xˆ x ) 1 ∧ · · · ∧ n j k ∧ 1 · · · j · · · k · · · ∧ n+1

Thus (5) can be written as

= ( 1)j+ksgn(σ)f (yjk ) f (yjk ) − − 1 σ(1) · · · n σ(n) j

Note that Shuf(1, , 1) = Sn. We will partition Sn into n subsets characterized by what they send · · · to 1. n = ( 1)j+ksgn(σ)f (yjk)f (yjk ) f!(yjk) f (yjk ) (6) − − i 1 1 σ(1) · · · i 1 · · · n σ(n) j

Let σ Sn such that σ(i) = 1. Then, if we identify Sn 1 with the set of bijections between ∈ − 1, , i, n and 2, , n , { · · · · · · } { · · · } 3 σ = (12)(23) (i 1i)σ′ · · · −

i 1 for some σ′ Sn 1. Hence, sgn(σ) = ( 1) − sgn(σ′). Thus (6) can be written as ∈ − − n j+k i 1 jk jk !jk jk = ( 1) ( 1) − sgn(σ′)fi(y1 )f1(y ) fi(y1 ) fn(y ) − − − σ′(1) · · · · · · σ′(n) j

Hence

H∗(g, k) = H∗(C(g), δ) 24

4 The Brown category of Chevalley- Eilenberg algebras

In this section we introduce the category, CEAlg, of Chevalley-Eilenberg algebras along with two classes of morphisms called cofibrations and weak equivalences. We show that this category along with theses two classes satisfy the axioms of a Brown category of cofibrant objects. In [5], Brown in- troduces the framework of a category of fibrant objects for a category with finite products. However,

CEAlg is not closed under finite products. It is closed under finite coproducts, and for this reason, we dualize the framework established by Brown to a category of cofibrant objects, which we will refer to as a Brown category.

4.1 Axioms for a Brown Category

Definition 4.1. Let C be a category with finite coproducts endowed with two classes of morphisms called weak equivalences and cofibrations. A morphism that is both a weak equivalence and a cofibration is called an acyclic cofibration. Then C is a Brown Category of cofibrant objects for a homotopy theory iff:

1. All isomorphisms in C are acyclic cofibrations

2. The class of cofibrations is closed under composition and the class of weak equivalences sat-

isfies the two out of three axiom. i.e. given any two composable morphisms f, g such that any

two of f, g, or gf are weak equivalences then so is the third

3. The pushout of an (acyclic) cofibration along any morphism in C exists, and (acyclic) cofibra-

tions are preserved under pushout. That is, if f is an (acyclic) cofibration and the following

pushout square exists, g X Z

f j

Y Y X Z 7 and j is also an (acyclic) cofibration.

4. For every object X C there exists a cylinder object. i.e. there exists a (not necessarily ∈ 25

functorial) factorization (i ,i ) X X 0 1 Cyl(X) s X −→ −→ 8 of the fold map : X X X into a cofibration followed by a weak equivalence. We ∇ → denote the composition7of inclusion into the first coordinate with (i0, i1) by i0, and similarly

denote the inclusion into the second coordinate followed by (i0, i1) as i1.

5. Every object is cofibrant. That is, the unique morphism from the initial object into any object is a cofibration.

Example 4.2. If M is a model category, then its full subcategory of cofibrant objects is a Brown 1 1 category. This implies that Chp≥roj is a brown category since every object in Chp≥roj is cofibrant. Recall that the cofibrant objects in the Bousfield Guggenheim model structure on cdga are exactly the Sullivan algebras. Hence the category of Sullivan algebras is a Brown category.

The following lemma is a useful fact about morphisms in a Brown Category.

Lemma 4.3. Let C be a Brown category and f be any morphism in C. Then f can be factored into f = rj, where j is a cofibration and r is a weak equivalence.

(i0,i1) s Proof. Let f : X Y be a morphism in C and let X X Cyl(X) X be a cylinder → −→ −→ object for X. We denote the pushout of (i , i ) along 7f id: X X Y X by Z, which 0 1 → exists because (i0, i1) is a cofibration. Thus there exists a c7ommutati7ve of the for7m

f id X X Y X !

(i0, i1)7 7 (id, f) Cyl(X) Z ! r ∃

Y fs

Let j be the composition X ↩ Y X Z. Then j is a cofibration, r is a weak equivalence, and → → f = rj. See [5] for details. 7 26

4.2 Chevalley-Eilenberg algebras

Definition 4.4. A semi-free cdga (S(V ), δ) is said to be finite-type if V n is finite dimensional for all n. We call a semi-free, finite-type cdga a Chevalley-Eilenberg algebra (CE algebra). We denote the category of CE algebras and cdga morphisms between them by CEAlg.

Example 4.5.

0 1. Let (D, d) Ch≥ be a finite-type cochain complex (not necessarily positively graded). We ∈ can define a differential δ on S(D[1]) by δ1 = d and δn = 0 for n 2. Then (S(D[1]), δ) 1 n+1 1 ≥ is a CE algebra.

2. Let g be a finite dimensional Lie algebra. Then its Chevalley-Eilenberg algebra (C(g), δg) (see Def. 3.23) is a CE algebra.

Proposition 4.6. The category CEAlg is closed under finite coproducts.

Proof. Let (S(V ), δ) and (S(V ′), δ′) be two CE algebras. In Ex. 3.9 we show that their coproduct in cdga is given by (S(V ) S(V ′), δ ) with inclusion maps ⊗ ⊗

ι: (S(V ), δ) (S(V ) S(V ′), δ ) (S(V ′), δ′): ι′ −→ ⊗ ⊗ ←− defined as ι(x) := x 1 and ι (y) := 1 y. One can check that the linear map ⊗ ′ ⊗

V V ′ S(V ) S(V ′), (x, y) x 1 + 1 y ⊕ → ⊗ ,→ ⊗ ⊗ lifts to an isomorphism S(V V ) ∼= S(V ) S(V ) in the category of graded commutative ⊕ ′ −→ ⊗ ′ algebras. Moreover, since V and V are finite-type, V V is finite-type. Therefore, (S(V ) ′ ⊕ ′ ⊗ S(V ′), δ ) is a CE algebra. ⊗

Remark 4.7. It is not difficult to show that the isomorphism constructed above S(V V ′) = ⊕ ∼ S(V ) S(V ) is natural. Throughout this paper we will frequently use this isomorphism to blur the ⊗ ′ distinction between (x, y) and x 1 + 1 y. ⊗ ⊗ 27

Let (S(V ), δ ) be a CE algebra. Then since δ squares to zero, δ 1 δ 1 = 0. Thus (V, δ 1) V V V 1 ◦ V 1 V 1 is a cochain complex. Let F : (S(V ), δ ) (S(W ), δ ) be a morphism in CEAlg. Then the V → W 1 1 1 1 fact that F commutes with the differentials implies that F1 δV 1 = δW 1F1 . Hence the assignments 1 1 (S(V ), δV ) (V, δV ) and F F induce a well defined functor. We call this functor the functor ,→ 1 ,→ 1 of indecomposables and we denote it by

1 T ∗ : CEAlg Ch≥ →

1 1 Proposition 4.8. A cdga morphism F : (S(V ), δV ) (S(W ), δW ) is an isomorphism iff F : (V, δV ) → 1 1 → 1 (W, δW 1) is an isomorphism of cochain complexes.

Proof. If F : (S(V ), δ ) (S(W ), δ ) is an isomorphism then there exists a G: (S(W ), δ ) V → W W → 1 1 1 1 (S(V ), δV ) such that GF = idS(V ) and F G = idS(W ). Thus F1 G1 = idV and G1F1 = idW , and 1 hence F1 is invertible. Now suppose that there exists and an inverse (F 1) 1 : (W, δ 1) (V, δ 1) of F 1. We will 1 − W 1 → V 1 1 construct a left inverse G: (S(W ), δ ) (S(V ), δ ) for F as follows. Let G1 = (F 1) 1, then W → V 1 1 − δ 1G1 = G1δ 1. Let n 2 and suppose that for all k < n that Gk : W Sk(V ) is defined and V 1 1 1 W 1 ≥ 1 → that k k k k k i δV i G1 = Gi δW 1 (7) *i=1 *i=1 Then we define n Gn := GnF iG1 1 − i 1 1 *i=2 n n n i n 1 1 1 1 1 Thus (GF )1 = i=2 Gi F1 + G1 F1 = 0. Using (7) and the fact that δV 1G1 = G1δW 1 implies n n n n n n δV nGn = GnδW+n, it is not hard to show that (δW G)1 = (GδV )1 . Therefore if G is the unique lift n of G1 n 1 to a cdga morphism, then GF = idS(V ) { } ≥ Similarly we can build a right inverse H : (S(W ), δ ) (S(V ), δ ) for F defined as the W → V n unique lift of H1 n 1 where { } ≥

1 1 (F )− n = 1 n 1 H1 = / 0 n 1 n n i 10 − H F H n > 1 − i=1 n i 1 0 + Then G = GF H = H is a two sided i20nverse for F . 28

The prior proposition shows that the functor T ∗ completely characterizes isomorphisms in CEAlg, and this property motivates the following definition.

Definition 4.9. We say that a morphism F in CEAlg is a weak equivalence (resp. cofibration) iff

1 1 T ∗(F ) is a weak equivalence (resp. cofibration) in Chp≥roj. That is, F is a weak equivalence iff F1 is 1 a quasi isomorphism of cochain complexes, and F is a cofibration iff F1 is injective in all degrees n 2. ≥

4.2.1 Factorization of CE algebra morphisms

1 We first dualize the construction of Goerss and Schemmerhorn in [8] to factor a morphism in Chp≥roj into a cofibration followed by a weak equivalence. We then lift this construction to factor strict morphisms of CE algebras using techniques of Rogers [14]. Since the fold map is strict, this will establish the existence of cylinder objects.

1 Factoring morphisms in Chp≥roj Let f : (D, d ) (E, d ) be a map of positively graded cochain complexes. Define a complex D → E 1 (I(D), d ) Ch≥ as I(D) ∈ proj 0 D2 n = 1 I(D)n := /{ } ⊕ 0 10Di Di+1 n 2 ⊕ ≥ with differential d (x, y) := (y, 0). Ob0serve that the degree 1 map of graded vector spaces I(D) 2 − h: I(D) I(D) defined as h(x, y) = (0, x) serves as a contraction homotopy on (I(D), d ) → I(D) in the sense that id = d h + hd . Thus (I(D), d ) is acyclic. Let γ : (D, d ) I(D) I(D) I(D) I(D) D → 1 (I(D), d ) Ch≥ be defined as I(D) ∈ proj

(0, dD(x)) x = 1 γ(x) = / | | (8) 0 10(x, dD(x)) x 2 | | ≥ 0 Then f can be factored as 20 (f,γ) pr D E I(D) E E −→ ⊕ −→ 29

1 (f, γ) is a cofibration in Chp≥roj since it is injective in all degrees greater than 1, and prE is a weak 1 equivalence in Chp≥roj since I(D) is acyclic.

Factoring strict morphisms in CEAlg We lift the above factorization to the category CEAlg. First, we record an elementary lemma from homological algebra.

Lemma 4.10. Let (D, dD) and (E, dE) be cochain complexes of vector spaces. Let Map(D, E), ∂ be the mapping complex from Example 3.1. Then if E is acyclic, Map(D, E) is ac&yclic as well. '

Proof. If E is acyclic, then since our base ring is a field, there exists a contracting chain homotopy h: E E satisfying id = d h + hd . One then observes that the degree 1 map → E E E −

h : Map(D, E) Map(D, E), h (f) := h f ∗ → ∗ ◦ is a contracting chain homotopy on the mapping complex.

Proposition 4.11. Let F : (S(V ), δ) (S(V ′), δ′) be a strict morphism between CE algebras. → Then F can be factored in CEAlg as

J P (S(V ), δ) (S(V ), δ) (S(V ′), δ′) −→ −→ ) ) Where J is a cofibration and P is a weak equivalence.

Proof. Let the differential on S(I(V )) be the strict differential induced by dI(V ) and define V := V I(V ) so that S(V ) = S(V ) S(I(V )). Let δ be the differential on S(V ) induced by the ′ ⊕ ∼ ′ ⊗ ) 1 tensor product. Define P : S(V ) S(V ′) be the strict map with P = pr . A simple check shows ) → ) 1 V ′ ) that this defines a morphism of CE algebras, which is in fact a weak equivalence because pr is a ) V ′ quasi isomorphism of cochain complexes. We define J 1 : V V I(V ) as J 1 := (F 1, γ) where 1 → ′ ⊕ 1 1 γ is defined in equation (8). Then observe that P 1J 1(x) = F 1(x) = F (x) for all x V . It remains 1 1 1 ∈ to construct the higher arity structure maps of J satisfying P nJ n = 0 for all n 2. We proceed by n 1 ≥ induction on n. 30

2 1 2 Consider V and S (V ) as cochain complexes with differentials δ1 and δ2 respectively. Define degree 1 map of graded vector spaces q : V S2(V ) by ) 2 → ) ) q : = J 2δ2 δ2J 1 2 2 1 − 1 1 ) Now observe that P 1J 1 = F 1 implies that P nJ n = F n for all n 1 by equations (3) and (4). 1 1 1 n n n ≥ Thus, since F and P are strict morphisms of CE algebras, we have

P 2q = P 2J 2δ2 P 2δ2J 1 2 2 2 2 1 − 2 1 1 2 2 2 2 1 1 = P J δ δ′ P) J 2 2 1 − 1 1 1 2 2 2 1 = F δ δ′ F 2 1 − 1 1 = 0

Hence q : V ker P 2. By Prop. A.4 in the Appendix, the fact that P 1 : (V , δ1) (V , δ 1) is a 2 → 2 1 1 → ′ ′1 quasi isomorphism implies that P n : (Sn(V ), δn) (Sn(V ), δ n) is a quasi isomorphism for all n n → ′ ′n ) ) n 1. A long exact sequence of cohomology shows that (ker P n, δn) is acyclic for all n 1, and ≥ ) ) n n ≥ in particular when n = 2. ) 2 2 2 Let (Map(V, ker P2 ), ∂) be the mapping complex as defined in Ex. 3.1. Since (ker P2 , δ2) is acyclic, Lemma 4.10 implies that the mapping complex is acyclic. It is easily verified that q2 i)s a 1- cocycle in this complex. Hence there exists a degree 0 map of graded vector spaces J 2 : V ker P 2 1 → 2 such that ∂J 2 = δ2J 2 J 2δ1 = J 2δ2 δ2J 1 = q 1 2 1 − 1 1 2 1 − 1 1 2

2 2 Thus (δJ)1 = (Jδ)1 and we have co)mpleted our base case. ) k k k k k Let)n > 2 and suppose that for all 1 < k < n that J1 is defined, (δJ)1 = (Jδ)1, and Pk J1 = 0.

We follow the same strategy as the base case. That is, we define a d)egree 1 map of graded vector n spaces and show that it is a 1-cocycle in (Map(V, ker Pn ), ∂). Let

n n 1 − q := J nδk δnJ k n k 1 − k 1 *k=2 *k=1 n ) n n We first show that qn lands in ker Pn . First observe that for k < n, Pn Jk = 0. Indeed by 31 equation (3) we have

P nJ n(x x ) = P n J n1 (x ) J nk (x ) n k 1 ∨ · · · ∨ k n 1 1 ∨ · · · ∨ 1 k 4 n1+ +n =n 5 ·*·· k = P n1 J n1 (x ) P nk J nk (x ) n1 1 1 ∨ · · · ∨ nk 1 k n1+ +n =n ·*·· k Since k < n, each summand will have a term with n 2 for some i = 1, , k. The inductive i ≥ · · · ni ni n n hypothesis implies that Pni J1 = 0, and hence Pn Jk = 0 for all k < n. Additionally, note that the n n n n fact that P is strict and commutes with the differentials implies that Pn δk = (P δ)k = (δ′P )k = n k δ′k Pk . Therefore, ) )

n n 1 − P nq = P nJ nδk P nδnJ k n 2 n k 1 − n k 1 *k=2 *k=1 n n 1 ) n n k − n k k = P J δ δ′ P J n k 1 − k k 1 *k=2 *k=1 n n n n 1 1 = P J δ δ′ P J n n 1 − n 1 1 n n n 1 = F δ δ′ F n 1 − n 1 = 0

Hence q : V ker P n. n → n n By Prop. A.9 in the Appendix, qn is a cocycle in the mapping complex. Thus, since (Map(V, ker Pn ), ∂) is acyclic, there exists a degree 0 map J n : V ker P n such that ∂J n = q . Therefore (δJ)n = 1 → n 1 n 1 n n n (Jδ)1 and Pn J1 = 0, and we have completed our inductive step. ) n Hence, if J is the unique lift of J1 n 1 to a morphism of graded algebras, then the above { } ≥ implies that J is a morphism of cdgas, and so we have P J = F with J a cofibration and P a weak equivalence.

Corollary 4.12. For every CE algebra (S(V ), δ) there exists a factorization of the fold map : (S(V ) ∇ ⊗ S(V ), δ ) (S(V ), δ) into a cofibration followed by a weak equivalence. Thus for every object in ⊗ → CEAlg there exists a cylinder object.

Remark 4.13. For the convenience of the reader we will explicitly describe the cylinder object 32 resulting from the application of Prop. 4.11 to the fold map : (S(V ) S(V ), δ ) (S(V ), δ). ∇ ⊗ ⊗ → Let Cyl(V ) := S(V ) S(I(V V )). That is, Cyl(V ) is the symmetric algebra on the graded ⊗ ⊕ vecto9r space 9

V 1 0 0 V 2 V 2 n = 1 V I(V V )n = / ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 0 10V n V n V n V n+1 V n+1 n 2 ⊕ ⊕ ⊕ ⊕ ≥ 0 We denote the given factorization by 20

(i ,i ) (S(V ) S(V ), δ ) 0 1 (Cyl(V ), δ) s (S(V ), δ) ⊗ ⊗ −→ −→ 9 3 The arity 1 differential on Cyl(V ) is given by

9 1 1 δ1(v, x, y, a, b) = (δ1(v), a, b, 0, 0)

3 The arity 1 structure maps of i0 and i1 are

(v, 0, 0, δ1(v), 0) v = 1 1 1 | | i01(v) = / 0 10(v, v, 0, δ1(v), 0) v 2 1 | | ≥ 0 2(v, 0, 0, 0, δ1(v)) v = 1 1 1 | | i11(v) = / 0 10(v, 0, v, 0, δ1(v)) v 2 1 | | ≥ 0 Finally s is the strict morphism defined20by the projection of the first coordinate.

Our main theorem (Thm. 4.18) in conjunction with Lemma 4.3 imply that an arbitrary morphism in

CEAlg can be factored into a cofibration followed by a weak equivalence.

4.2.2 Pushouts in CEAlg

We begin by recalling the construction of pushouts in cdga. Then we provide an alternative de- 1 scription of the pushout in Chp≥roj of a cofibration along an arbitrary morphism which we will lift to CEAlg. We then use a version of Vallette’s [17] "strictification" lemma, due to Rogers [14, Lemma

3.9], to prove that the pushout in CEAlg of an arbitrary (acyclic) cofibration along any morphism is an (acyclic) cofibration. 33

F G Let A A A be a diagram in cdga. The pushout of this diagram is ′ ←− −→ ′′ G A A′′

F ι′′ ι A ′ A A ′ ⊗A ′′ where A A is the quotient of A A by the ideal generated by F (a) 1 1 G(a) a A . ′ ⊗A ′′ ′ ⊗ ′′ { ⊗ − ⊗ | ∈ } The maps ι and ι are the canonical inclusions followed by the quotient map, i.e. ι (a ) = a 1 ′ ′′ ′ ′ ⊗ and ι (a ) = 1 a . ′′ ′′ ⊗ ′′

1 Pushout in Chp≥roj f g 1 Let (D , d ) (D, d) (D , d ) be a diagram in Ch≥ where f is a cofibration. Recall ′ ′ ←− −→ ′′ ′′ proj that D D := D D /I where I = (f(z), g(z)) D D z D and let π : (D ′ ⊕D ′′ ′ ⊕ ′′ { − ∈ ′ ⊕ ′′ | ∈ } ′ ⊕ D′′, d ) (D′ D D′′, d ) denote the projection of complexes. Since f is injective in all degrees ⊕ → ⊕ ⊕ n 2 there exists a partial section σ : D D of graded vector spaces such that ≥ ′ →

0 x = 1 fσ(x) = / | | 0 10x x 2 | | ≥ 0 We now define another positively graded vector2space D by

1 1 ) 1 (D′ D′′ )/I n = 1 Dn := / ⊕ 0 n n 10ker σ D′′ n 2 ) ⊕ ≥ 0 where I1 = (f(z), g(z)) z D1 . W20e also define a pair of degree 0 linear maps k : D D { − | ∈ } ′ ⊕ ′′ → D and l : D D D by → ′ ⊕D ′′ ) ) π1(x, y) (x, y) = 1 k(x, y) = / | | 0 10(x fσ(x), gσ(x) + y) (x, y) 2 − | | ≥ 0 20 34

τ(x, y) (x, y) = 1 l(x, y) = / | | 0 10(x, y) (x, y) 2 | | ≥ 1 0 1 1 1 where π is the restriction of the projectio20n above to D′ D′′ , and τ is a section of π in the ⊕ 1 sense that τ is a k-linear map such that π τ = idD D . We will then define a differential d on D ′⊕ ′′ as d := kd l. ⊕ ) ) d is in fact a differential. First observe since τ is a section of π, we there exists some z D1 ) ∈ such that l(x, y) = (x, y) + (f(z), g(z)) for all (x, y) D1 . In fact, for all (x, y) D D , ) − ∈ ∈ ′ ⊕ ′′ there exists a z D such that ∈ )

lk(x, y) = (x, y) + (f(z), g(z)) − where z is given explicitly as z = σ(x) for elements of degree n 2. Since f and g are cochain − ≥ maps it is easy to show that kd (f(z), g(z)) = 0 ⊕ − Therefore, on degree 1 elements

d2(x, y) = kd lkd l(x, y) ⊕ ⊕ ) = kd lkd (x, y) + (f(z), g(z)) ⊕ ⊕ − = kd lkd (&x, y) ' ⊕ ⊕

= kd lk(d′x, d′′y) ⊕

= kd (x, y) + (f(z′), g(z′)) ⊕ − & ' = kd (d′x, d′′y) ⊕ = 0 and on elements of degree n 2 a similar calculation shows that d2 = 0. Thus (D, d) is a cochain ≥ complex. ) ) )

We now show that k : (D′ D′′, d ) (D, d) and πl : (D, d) (D′ D D′′, d ) are cochain ⊕ ⊕ → → ⊕ ⊕ ) ) ) ) 35 maps. By (4.2.2) and (4.2.2) we have

dk(x, y) = kd lk(x, y) = kd (x, y) + (f(z), g(z)) = kd (x, y) ⊕ ⊕ − ⊕ & ' ) Hence k is a cochain map. Observe that (4.2.2) implies that πlk = π. Thus since π is a cochain map

πld = πlkd l = πd l = d πl, so πl is also a cochain map. ⊕ ⊕ ⊕ 1 )It is not hard to check that the following diagram in Chp≥roj commutes.

g (D, d) (D′′, d′′)

f kι′′ ι′′

kι′ (D′, d′) (D, d) (9) πl ) ) (D′ D D′′, d ) ⊕ ⊕

ι′

Thus it remains to show that πl is unique. Suppose that .: (D, d) (D′ D D′′, d ) is another → ⊕ ⊕ map that fits in this diagram. Note that in degree 1, D1 = (D D )1 and ′)⊕D) ′′ ) .kι′(x) = .(x, 0) = (x, 0)

.kι′′(y) = .(0, y) = (0, y)

Hence . is the identity in degree 1 and so is πl. In degrees n 2 we have ≥

.kι′(x) = .(x fσ(x), gσ(x)) = (x, 0) −

.kι′′(y) = .(0, y) = (0, y)

Thus since x ker σ we have .(x, y) = .(x fσ(x), gσ(x) + y) = (x, y) = πl(x, y). Therefore ∈ − πl is unique, and (9) is indeed a pushout. Moreover, since f is injective in degree n 2, kι is too. ≥ ′′

Pushouts of strict cofibrations in CEAlg 1 We lift the above construction of the pushout in Chp≥roj to CEAlg.

Proposition 4.14. Let F : (S(V ), δ) (S(V ′), δ′) be a strict cofibration of CE algebras and → G: (S(V ), δ) (S(V ), δ ) be an arbitary morphism in CEAlg. Then the pushout of F along → ′′ ′′ 36

G exists in CEAlg.

F 1 G1 Proof. We begin by applying the above construction to the diagram (V , δ 1) 1 (V, δ1) 1 ′ ′1 ←− 1 −→ (V , δ 1). That is, we construct a partial section σ : V V of F 1 and define a graded vector space ′′ ′′1 ′ → 1 V as 1 1 1 (V ′ V ′′ )/I n = 1 ) V n = / ⊕ 0 n n 10ker σ V ′′ n 2 ) ⊕ ≥ where I1 = (F 1(z), G1(z)) z V01 . Observe that for degree reasons G(z) = G1(z) for all { 1 − 1 | ∈ 2 } 1 z V 1and since F is strict I1 = F (z) 1 1 G(z) z V 1 . We will now lift k and l ∈ { ⊗ − ⊗ | ∈ } n to algebra morphisms. We define K : S(V ′) S(V ′′) S(V ) to be the unique lift of K1 n 1 ⊗ → { } ≥ 1 where K1 = k and ) 0 (x, y) = 1 n | | K1 (x, y) = / 0 101 Gnσ(x) (x, y) 2 ⊗ 1 | | ≥ 0 1 We define L: S(V ) S(V ′) S(V ′′) a20s the strict morphism with L = l. → ⊗ 1 We now will )show that the following diagram commutes

G S(V ) S(V ′′)

ι F Kι′′ ′′

Kι′ S(V ′) S(V ) πL ) S(V ) S(V ) ′ ⊗S(V ) ′′

ι′

Since (9) commutes, it suffices to check the commutativity of the higher arity structure maps of the diagram above. Moreover, for degree reasons, it suffices to check the commutativity of the higher arity structure maps only on elements of degree n 2. Let z V with z 2. Then for n 2 we ≥ ∈ | | ≥ ≥ have

n n 1 1 n 1 n n n n (Kι′F ) (z) = K ι′ F (z) = 1 G (σF (z)) = 1 G (z) = K ι′′ G (z) 1 1 1 1 ⊗ 1 1 ⊗ 1 n n 1

n n n n However, since Kι′′ is strict, Kn ι′′nG1 (z) = (Kι′′G)1 (z). Hence the inner square commutes.

πLKι′′ = ι′′ because (9) commutes and Kι′′, πL, and ι′′ are all strict. Finally since Kι′(x) = 37

(x F σ(x)) 1 + 1 Gσ(x) for all x V with x 2, we conclude that πLKι = ι and that − ⊗ ⊗ ∈ ′ | | ≥ ′ ′ the diagram above commutes. 1 Similar to the construction in Ch≥ , we define the differential on S(V ) by δ := Kδ L where proj ⊗

δ is the differential on S(V ′) S(V ′′) induced by the tensor product. An analogous computation ⊗ ⊗ ) ) will show that δ δ = 0. ◦ Next we show that K and πL are cdga morphisms. Note that for (x, y) V V with degree ) ) ∈ ′ ⊕ ′′ n 2, we have ≥

δK(x, y) =Kδ LK(x, y) ⊗ ) =Kδ (x, y) F σ(x) 1 + 1 Gσ(x) ⊗ − ⊗ ⊗ , - =Kδ (x, y) + K F δσ(x) 1 + 1 Gδσ(x) ⊗ − ⊗ ⊗ , - =Kδ (x, y) ⊗

For (x, y) = 1 we have LK(x, y) = (x, y) + (F 1(z), G1(z)) for some z V 0. We leave it to | | 1 − 1 ∈ 1 1 the reader to show that Kδ (F1 (z), G1(z)) = 0 by using the following facts ⊗ −

2 1 2 1 2 2 2 2 2 1 2 2 δ′′ G (z) = G δ (z) + G δ (z) δ′′ G (z) = G δ (z) + G δ (z) 1 1 1 1 2 1 − 2 1 1 1 2 1 K2 F 2δ2(z) 1 = K2 1 G2δ2(z) 2 2 1 ⊗ 2 ⊗ 2 1 , - , - K2 F 1δ1(z) 1 = K2 1 G1δ1(z) + K2 1 G2δ1(z) = K2 1 G2δ1(z) 1 1 1 ⊗ 1 ⊗ 1 1 2 ⊗ 1 1 2 ⊗ 1 1 , - , - , - , - Hence for all (x, y) V 1 V 1, ∈ ′ ⊕ ′′

δK(x, y) = Kδ LK(x, y) ⊗ 1 1 ) =Kδ (x, y) + (F1 (z), G1(z)) ⊗ − , - =Kδ (x, y) ⊗

To see that πL is a cdga morphism observe that πLK = π. Thus πLδ = πLKδ L = πδ L = ⊗ ⊗ δ πL. Moreover πL is the unique lift of the unique linear map πl in (9) and is therefore unique. ⊗ )

Corollary 4.15. Let F : (S(V ), δ) (S(V ′), δ′) be a strict (acyclic) cofibration and G: (S(V ), δ) → → (S(V ′′), δ′′) be an arbitrary morphism in CEAlg. Then the map induced by the pushout of F along 38

G is an (acyclic) cofibration.

Proof. The functor of indecomposables, T ∗, sends the pushout square constructed in Proposition

1 1 1 4.14 to a pushout square in Chp≥roj. Since F1 is an (acyclic) cofibration in Chp≥roj and pushouts in 1 Chp≥roj preserve (acylic) cofibrations, the arity 1 structure map of the induced CE algebra morphism, 1 1 (Kι ) , is an (acyclic) cofibration in Ch≥ . Hence, Kι : (S(V ), δ ) (S(V ), δ) is an (acyclic) ′′ 1 proj ′′ ′′ ′′ → cofibration. ) )

Pushouts of arbitrary (acyclic) cofibrations in CEAlg We begin with the "strictification" lemma which we will use to reduce the pushout of an arbitrary

(acyclic) cofibration to the pushout of a strict (acyclic) cofibration.

Lemma 4.16. Let F : (S(V ), δ) (S(V ′), δ′) be an arbitrary cofibration in CEAlg. Then there → exists a CE algebra (S(V ), δˆ) and an isomorphism Φ: (S(V ), δ ) (S(V ), δˆ) such that ΦF is ′ ′ ′ → ′ 1 1 strict with (ΦF )1 = F1 .

Proof. By hypothesis F 1 is injective in all degrees n 2. Again, we choose a partial section of 1 ≥ graded vector spaces σ : V V such that ′ →

0 x = 1 1 | | σF1 (x) = / 0 10x x 2 | | ≥ 0 We define a degree 0 linear map Φ1 : V V 20to be the identity, so that Φ1F 1 = F 1. It remains 1 ′ → ′ 1 1 1 to construct the higher arity structure maps of Φ so that (ΦF )n = 0 for all n 2, and we do so by 1 ≥ induction on n.

Let n = 2. Define Φ2 := Φ2F 2σ 1 − 2 1

2 Now observe that since F1 is zero on degree 1 elements, we have

(ΦF )2 = Φ2F 2 + Φ2F 1 = Φ2F 2 Φ2F 2σF 1 = 0 1 2 1 1 1 2 1 − 2 1 1 39

k k Let n > 2, and suppose for all 1 < k < n that Φ1 is defined and (ΦF )1 = 0. Then define

n Φn := ΦnF nσ 1 − k 1 *k=2 n n Thus Φ1 is well defined and (ΦF )1 = 0. n Let Φ: S(V ′) S(V ′) be the unique lift of Φ1 n 1 to a morphism of commutative graded → { } ≥ 1 algebras. Since Φ1 is an isomorphism, we can use the construction in Prop. 4.8 to build an inverse 1 for Φ. We use this inverse to define the differential δˆ := Φδ′Φ− . This clearly defines a differen- tial and promotes Φ: (S(V ), δ ) (S(V ), δˆ) to an isomorphism of CE algebras. Moreover, by ′ ′ → ′ 1 1 construction ΦF is strict with (ΦF )1 = F1 .

Proposition 4.17. Let F G (S(V ′), δ′) (S(V ), δ) (S(V ′′), δ′′) ←− −→ be a diagram in CEAlg where F is an (acyclic) cofibration and G is arbitrary. Then,

1. the pushout of this diagram exists in CEAlg

2. the morphism induced by the pushout of F along G is also an (acyclic) cofibration.

Proof. Suppose F is an (acyclic) cofibration. Then the strictification lemma implies that there exists a differential δ on S(V ) and a CE algebra isomorphism Φ: (S(V ), δ ) (S(V ), δ) such that the ′ ′ ′ → ′ 1 1 composition o3f F with Φ is strict with (ΦF )1 = F1 . Hence ΦF is a strict (acycl3ic) cofibration. Prop. 4.14 implies that there exists of pushout square in CEAlg of the form

G (S(V ), δ) (S(V ′′), δ′′)

ΦF j′′

j′ (S(V ′), δ) (S(V ), δ)

3 ) ) Note that j′′ is an (acyclic) cofibration by Cor 4.17. We now show that the following diagram is a 40 pushout square which will complete our proof.

G (S(V ), δ) (S(V ′′), δ′′)

F j′′ (10)

j′Φ (S(V ′), δ′) (S(V ), δ)

Indeed, suppose that there exists (S(W ), δ ) CEAlg along)wi)th a pair of maps Γ : (S(V ), δ ) W ∈ ′ ′ ′ → (S(W ), δ ) and Γ : (S(V ), δ ) (S(W ), δ ) such that Γ F = Γ G. Then since Φ is invert- W ′′ ′′ ′′ → W ′ ′′ ible, there exists a unique morphism Γ: (S(V ), δ) (S(W ), δ ) such that TFDC → W G) ) ) (S(V ), δ) (S(V ′′), δ′′)

ΦF ι′′ Γ′′

ι′ (S(V ′), δ) (S(V ), δ) ! Γ ∃ 3 ) ) " (S(W ), δW )

1 Γ′Φ−

Thus Γ is the map that witnesses the universal property of (S(V ), δ) in diagram (10).

) ) )

Theorem 4.18. The category CEAlg admits the structure of a Brown category of cofibrant objects in which a morphism F : (S(V ), δ ) (S(W ), δ ) is : V → W

A weak equivalence iff F 1 : (V, δ 1) (W, δ 1) is a quasi isomorphism of cochain com- • 1 V 1 → W 1 plexes

A cofibration iff F 1 : (V, δ 1) (W, δ 1) is injective in all degrees greater than 1 • 1 V 1 → W 1

Proof. Because the weak equivalences and cofibrations are characterized by the by their image in 1 Chp≥roj via T ∗, axioms 1, 2, and 5 are satisfied. Prop. 4.17 satisfies axiom 3 and Cor. 4.12 satisfies axiom 4. 41

4.3 Homotopy category of Chevalley-Eilenberg algebras

In this section we show that the localization of a Brown category with respect to its weak equiva- lences, which we call the homotopy category, exists by explicitly constructing its sets of morphisms.

Since CEAlg is a category of cofibrant objects, we have provided the dual of the corresponding statements pertaining to categories of fibrant objects. However, the proofs of such statements will be omitted as they follow immediately from [5]. We then will describe certain sets morphisms in the homotopy category associated to CEAlg. Using this description we then will then show that the category of finite dimensional Lie algebras embeds faithfully into the homotopy category of CEAlg, and show that no such embedding exists into the homotopy category associated to the Bousfield

Guggenheim model structure on cdga.

Homotopy category of a Brown category

Definition 4.19. Let C be a Brown category. Then the homotopy category of C, written as Ho(C), is a category equipped with a functor γ : C Ho(C) such that for every functor F : C D that → → inverts weak equivalences, there exists a unique functor F : Ho(C) D such that F γ = F . → Within the abstract framework of Brown category there i)s a notion of homotopy tha)t is reminiscent of the usual notion of homotopy from topology.

Definition 4.20. Two morphisms f, g : X Y in a Brown category C are said to be homotopic iff → there exists a map H : Cyl(X) Y → for some cylinder object of X such that Hi0 = f and Hi1 = g. If f and g are homotopic we write f g. ≃ By combining cylinder objects as is done in [13] we can show that the relation defines an equiva- ≃ lence relation on homC(X, Y ). The following proposition describes some useful facts about homo- topy.

Proposition 4.21. Suppose that f g : X Y . Then ≃ →

1. for any morphism u: Y Z we have uf ug : X Z. → ≃ → 42

2. for any morphism u: W X and for any cylinder object Cyl(W ), there exists a an acyclic → cofibration k : Y Z such that kfu kgu: W Z via a homotopy H : Cyl(W ) Z. → ≃ → →

Proof. Follows from Proposition 1 in [5].

In order to get our hands on the homotopy category we first make an approximation. We define an equivalence relation on hom (X, Y ) by f g iff there exists a weak equivalence k : Y Z such C ∼ → that kf kg. Note that this relation identifies homotopic maps. By using the above proposition ≃ one can easily verify that this equivalence relation respects composition. Thus we define πC be the category with the same objects as C and whose morphisms are equivalence classes of morphisms in

C under the relation . We define the class of weak equivalences in πC to be the image of the the ∼ weak equivalences in C under the canonical functor C πC. → Proposition 4.22. The category πC admits a calculus of left fractions, in the sense of [7], with respect to the class of weak equivalences. That is,

1. Given any weak equivalence k : W X and any morphism f : W Y , there exists a weak → → equivalence k : Y Z and a morphism f : X Z such that the following diagram is ′ → ′ → homotopy commutative. f W Y

k k′

f ′ X Z

2. Given a weak equivalence k : W X and a pair of parallel morphisms f, g : X Y such → → that fk gk, there exists a weak equivalence k : Y Z such that k f k g. ≃ ′ → ′ ≃ ′

Proof. Follows from Proposition 2 in [5].

Fix an object Y πC. Consider the category whose objects are weak equivalences [k]: Y Y ∈ JY → ′ and whose morphisms are from [k′] to [k] are commutative diagrams in πC of the form

[f] Y ′ Y ′′ (11)

[k] [k′] Y 43

If X and Y are objects of C, let

[X, Y ] := lim homπC(X, Y ′) −→ where the directed limit is indexed by Y . ′ ∈ JY

Proposition 4.23. If C is a Brown category and X, Y C, then ∈

homHo(C)(X, Y ) ∼= [X, Y ]

Proof. Follows from Theorem 1 in [5].

Morphisms of Ho(CEAlg) into C(g) For the remainder of this thesis we will denote a CE algebra by S(V ) leaving the differential, multiplication and unit implicit. We will write C(g) for the CE algebra of a finite dimensional Lie algebra equipped with its usual differential (see Def. 3.23). Let πCEAlg be the quotient category of

CEAlg by the relation F G iff there exists a weak equivalence K such that KF KG. We write ∼ ≃ [F ] for the equivalence class represented by F in πCEAlg

In the following sequence of propositions we will obtain a tangible description of [S(V ), C(g)].

We begin with a few technical results.

Proposition 4.24. Let K : S(V ) S(W ) be an acyclic cofibration. Then there exists a morphism → Ψ: S(W ) S(V ) of CE algebras such that ΨK = id . → S(V )

Proof. Follows immediately from Prop. A.1 in the Appendix.

Corollary 4.25. Let C(g) be the Chevalley-Eilenberg algebra of a finite dimensional Lie algebra. Then for every weak equivalence K : C(g) S(W ) there exists morphism Ψ: S(W ) C(g) such → → that ΨK = idC(g).

Proof. Since C(g) is the symmetric algebra on a vector space concentrated in degree 1, all mor- phisms in CEAlg with domain C(g) are cofibrations. 44

Lemma 4.26. Let C(g) be the Chevalley-Eilenberg algebra of a finite dimensional Lie algebra. Suppose that the following diagram commutes in πCEAlg

[F ] S(W ) S(W ′)

[K] [K′] C(g) where K and K are weak equivalences. Then there exists morphisms Ψ: S(W ) C(g) and ′ → Ψ : S(W ) C(g) such that ΨK = id = Ψ K and [Ψ] = [Ψ F ] in πCEAlg. ′ ′ → C(g) ′ ′ ′

Proof. By Cor. 4.25 there exists left inverses Ψ: S(W ) C(g) and Ψ : S(W ) C(g) of K → ′ ′ → and K′ respectively. Since the diagram commutes in πCEAlg, there exists a weak equivalence G: S(W ) S(U) such that GK GF K. GK : C(g) S(U) is a weak equivalence, so ′ → ′ ≃ ′ → there exists a left inverse Γ: S(U) C(g). Thus, →

ΨK = id = ΓGK′ ΓGF K C(g) ≃

Hence, by part 2 of Prop. 4.22, there exists a weak equivalence G : C(g) S(U ) such that ′ → ′ G ΓGF G Ψ. Therefore [Ψ] = [ΓGF ] in πCEAlg. ′ ≃ ′ Now observe that ΓGK′ = idC(g) = Ψ′K′. Again by part 2 of Prop. 4.22, there exists a weak equivalence G : C(g) S(U ) such that G ΓG G Ψ . Then by Prop. 4.21 there exists a weak ′′ → ′′ ′′ ≃ ′′ ′ equivalence G : S(U ) S(U ) such that G G ΓGF G G Ψ F . Therefore since G G is ′′′ ′′ → ′′ ′′′ ′′ ≃ ′′′ ′′ ′ ′′′ ′′ a weak equivalence, [ΓGF ] = [Ψ′F ]. so [Ψ] = [Ψ′F ] in πCEAlg.

Proposition 4.27. Let C(g) be the Chevalley-Eilenberg algebra of a finite dimensional Lie algebra, and S(V ) be a CE algebra. Then,

[S(V ), C(g)] ∼= homπCEAlg(S(V ), C(g))

Proof. Let be the category whose objects are the weak equivalences [K]: C(g) S(W ) in JC(g) → πCEAlg and whose morphisms are defined as above in (11). For each [K] choose a weak ∈ JC(g) equivalence, K : C(g) S(V ) CEAlg, representing it, and for every weak equivalence K choose → ∈ 45 a section, i.e. a CE algebra morphism Ψ: S(W ) C(g) such that ΨK = id . Consider the → C(g) diagram hom (S(V ), ): Set. Then the collection πCEAlg − JC(g) →

Ψ : homπCEAlg(S(V ), S(W )) homπCEAlg(S(V ), C(g)) ∗ → JC(g) : ; defines a cocone with vertex homπCEAlg(S(V ), C(g)). Indeed the previous lemma implies that TFDC in πCEAlg [F ] S(W ) S(W ′)

[Ψ] [Ψ′] C(g) where F is a morphism from [K ]: C(g) S(W ) to [K]: C(g) S(W ) in . Since the ′ → → ′ JC(g) diagram lives in the category Set, its direct limit is given by the disjoint union

homπCEAlg(S(V ), S(W )) S(W ) 8∈JC(g) modulo some equivalence relation. Because all identity maps are weak equivalences, [id]: C(g) → C(g) , and hence hom (S(V ), C(g)) is a summand of colimit. Thus there exists ∈ JC(g) πCEAlg an inclusion hom (S(V ), C(g)) ↩ [S(V ), C(g)]. Using the fact that every section Ψ is a πCEAlg → morphism in from [id]: C(g) C(g) to [K]: C(g) S(W ), and that the identity is the only JC(g) → → possible section of id: C(g) C(g), one can show that the inclusion hom (S(V ), C(g)) ↩ → πCEAlg → [S(V ), C(g)] is the function witnessing the universal property of homπCEAlg(S(V ), C(g)).

We will now show that the equivalence relation agrees with a simpler relation on hom (S(V ), C(g)). ∼ CEAlg Recall the construction in Remark 4.13 of the cylinder object

(i ,i ) (S(V ) S(V ), δ ) 0 1 (Cyl(V ), δ) s (S(V ), δ) ⊗ ⊗ −→ −→ 9 3 For an arbitrary CE algebra S(V ) and a Chevalley-Eilenberg algebra of a Lie algebra C(g), define a relation on the set hom (S(V ), C(g)) by, F G iff there exists a morphism H : Cyl(V ) CEAlg ≃Cyl → C(g) such that Hi0 = F and Hi1 = G. # 9 46

Proposition 4.28. Let C(g) be the Chevalley-Eilenberg algebra of a finite dimensional Lie algebra, and S(V ) be a CE algebra. If F, G: S(V ) C(g) are CE algebra morphisms, then F G iff → ≃ F G. Hence is an equivalence relation. ≃Cyl ≃Cyl # # Proof. Clearly F G implies that F G. Now suppose that F G. Then by Prop. 4.21 there ≃Cyl ≃ ≃ exists a weak equivalence K : C(g) S(W ) such that KF KG via Cyl(V ). By Cor. 4.25 there # → ≃ exists a Ψ: S(W ) C(g) such that ΨK = id . Hence F = ΨT F ΨT G = G via Cyl(V ). → C(g) 9≃ Therefore F G. ≃Cyl 9 #

Proposition 4.29. Let C(g) be the Chevalley-Eilenberg algebra of a finite dimensional Lie algebra, and S(V ) be a CE algebra. Then

[S(V ), C(g)] = hom (S(V ), C(g))/ ∼ CEAlg ≃Cyl # Proof. In the previous proposition we have shown that and agree on hom (S(V ), C(g)). ≃Cyl ≃ CEAlg We will now show and also agree. Then, ≃ ∼ #

hom (S(V ), C(g)) = hom (S(V ), C(g))/ πCEAlg CEAlg ≃Cyl # and Prop. 4.27 will yield the desired result.

Let F, G: S(V ) C(g) be morphisms of CE algebras. Suppose that F G. Then since → ≃ id (g) is a weak equivalence, F G. On the other hand suppose that F G. Thus there exists a C ∼ ∼ weak equivalence K : C(g) S(W ) such that KF KG. By Cor. 4.25 there exists a morphism → ≃ Ψ such that ΨK = id . Hence F = ΨT F ΨT G = G. C(g) ≃

Embedding LieAlg in Ho(CEAlg) We will now use Prop. 4.29 to show that the Chevalley-Eilenberg functor faithfully embeds the category of finite dimensional Lie algebras into Ho(CEAlg).

Proposition 4.30. Let C(h) and C(g) be Chevalley-Eilenberg algebras of finite dimensional Lie algebras. If F, G: C(h) C(g) are CE algebra morphisms then F G iff F = G. → ≃Cyl # 47

Proof. Clearly if F = G then F G. Now suppose that F G. Then there exists a ≃Cyl ≃Cyl homotopy H : Cyl(h) C(g) such that, Hi = F and Hi = G. Observe that Cyl(h) is the → # 0 1 # 1 1 symmetric alge9bra on h∗ concentrated in degree 1, and that i01 = i11 (see Remark 4.913). Since i0 and i1 must be strict for degree reasons, i0 = i1. Therefore F = Hi0 = Hi1 = G.

Corollary 4.31. Let h and g be finite dimensional Lie algebras. Then

homLieAlg(g, h) ∼= [C(h), C(g)]

Proof. Recall Prop. 3.25 which states that C : LieAlg CEAlg is full and faithful. Prop. 4.29 and → the prior proposition imply that

homLieAlg(g, h) ∼= homCEAlg(C(h), C(g)) ∼= [C(h), C(g)]

LieAlg does not embed in Ho(cdga)

Let g be the two dimensional k-vector space spanned by x, y . Define a bracket on g by { }

[a x + a y, b x + b y] = (a b a b )x 1 2 1 2 1 2 − 2 1

Let φ: g k be the linear transformation such that x 0 and y 1. If we consider k as an → ,→ ,→ abelian Lie algebra, then φ is a Lie algebra homomorphism. Moreover, C(φ): C(k) C(g) is a → quasi isomorphism of cdgas. It is not however a weak equivalence. Thus, in the homotopy category associated to the BG model structure on cdga we have the following bijection

C(φ)∗ homHo(cdga)(C(g), C(g)) homHo(cdga)(C(k), C(g)) −→

Let ψ , ψ : g g be two Lie algebra morphisms defined by 1 2 →

ψ1(ax + by) = by

ψ2(ax + by) = bx + by 48

Note that C(φ)∗(C(ψ1)) = C(φ)∗(C(ψ2)). Therefore the functor,

C : LieAlg Ho(cdga) → is not faithful. 49

5 Representability Theorems

In this section we will prove results that are analogous to E. Brown’s representability theorem for the cohomology of CW complexes [4] and the classification of principal G bundles over a CW complex.

Specifically, we will show that the cohomology functor Hn( , M): LieAlg AbGrp, where M is − → a trivial g-module and n 1, is representable as a functor Ho(CEAlg) AbGrp, and we will show ≥ → that set of all principal cofiber sequences with a specific base and cofiber can be classified by a set of morphisms in Ho(CEAlg). First we outline several homotopical constructions for a pointed Brown category.

5.1 Suspensions, cogroups, and principal cofiber sequences in Brown categories

Motivated by the homotopy theory of pointed CW complexes, Brown defines the notion of a loop functor and fiber sequences for pointed categories of fibrant objects in [5]. We dualize these con- structions to a suspension functor and cofiber sequences for a pointed category of cofibrant objects.

Then following [1] we introduce the notion of a principal cofiber sequence.

Suspension in Brown Categories

Definition 5.1. Let C be a pointed Brown category with zero object . The suspension of an object ∗ X C is defined to be the pushout of ∈

(i ,i ) Cyl(X) 0 1 X X ←− −→ ∗ 8 where Cyl(X) is any cylinder object of X. We denote the suspension of X by ΣX.

As shown in [5], different cylinder objects yield weakly equivalent suspensions. Thus, the assign- ment X ΣX defines a functor Σ : Ho(C) Ho(C). There is an alternative method to obtain the ,→ → suspension of an object X as a sequence of pushouts utilizing the cone of X.

Definition 5.2. Let C be a pointed Brown category with zero object . The cone of an object X C ∗ ∈ is defined to be the pushout of i Cyl(X) 0 X ←− −→ ∗ where Cyl(X) is any cylinder object of X. We denote the cone of X by Cone(X). 50

Remark 5.3. Let q : Cyl(X) Cone(X) be the map induced by the pushout. Then the composi- → tion qi : X Cone(X) is a cofibration and the pushout of 1 →

qi Cone(X) 1 X ←− −→ ∗ is ΣX. Indeed, this fact is easily verified by using the construction in Lemma 4.3 to factor the trivial map X , and the pushout pasting lemma. → ∗

Cogroup structure on the suspension The pinch map ΣX ΣX ΣX, makes the suspension of a pointed CW complex into a cogroup → ∨ object in the homotopy category. Similarly, in any pointed Brown category C, ΣX is a cogroup object in Ho(C) for all X B. The proof of this fact as well as the construction of the comultiplication ∈ can be found in [5, Sec. 4, Thm. 3]. However, we provide the definition of a cogroup object, as well as a useful proposition pertaining to cogroup objects.

Definition 5.4. Let C be a category with finite coproducts and terminal object C. A cogroup ∗ ∈ object in C is an object X equipped with three morphisms ∆: X X X, called comultiplica- → tion, and .: X , called the counit, and j : X X called inversi7on such that the following → ∗ → diagrams commute

1. Associativity ∆ X X X

∆ 7id ∆

∆ id ! X X X X X ! 7 7 7 2. Left/Right Unit laws

( id id ( X X X X ∗ ! ! ∗

7 ∆7 7

X 51

3. Invertibility (j, id) (id, j) X X X X

∆7 id id X

Proposition 5.5. Let C be a category with finite coproducts and terminal object. If X C is a ∈ cogroup object then hom (X, Y ) is a group for all Y C. C ∈

Proof. Let ∆: X X X be the comultiplication on X. Then for f, g : X Y , denote the → → morphism induced by the7universal property by (f, g): X X Y . Then (f, g)∆: X Y . → → This construction gives us a multiplication on homC(X, Y7). We direct the reader to Arkowitz’s book [2, Sec. 2.2] for the details.

Cofiber Sequences Let f : X Y be a cofibration in a pointed Brown category C and let the following be a pushout → square. X ∗ f

Y Cf

In [5] is shown that there exists a coaction in Ho(C) of ΣX on Cf .

Definition 5.6. Let f : X Y be a cofibration in a pointed Brown category C. Then, →

f X Y C −→ −→ f along with the coaction C ΣX C , is said to be a cofiber sequence with base X and cofiber f → f Cf . Two cofiber sequences with bas7e X and cofiber Cf are said to be equivalent iff their respective 52 coactions agree and there exists a commutative diagram in Ho(C)

X Y Cf

id id

X Y ′ Cf in which the morphism Y Y is an isomorphism in Ho(C). → ′

Remark 5.7. Let qi1 : Z Cone(Z) be the cofibration defined in Remark. 5.3. Then for an → arbitrary morphism g : Z X we obtain a cofiber sequence →

g ) X Cone(Z) X ΣZ −→ −→ 8Z where g : X Cone(Z) X is the morphism induced by the pushout of → Z 7 qi g Cone(Z) 1 Z X ←− −→ " By the pushout pasting lemma and Remark 5.3, the cofiber of g is ΣZ.

f Definition 5.8 (Def. 2.8, [1]). A cofiber sequence X Y C is called a principal cofiber −→ → f sequence iff there exists an object Z C, called a classifying space, a morphism g : Z X, called ∈ → a classifying map, and two commutative diagrams in Ho(C) of the form ) g X Cone(Z) Z X ΣZ

id 7 φ (12)

f X Y Cf

ΣZ ΣX ΣZ

φ 7id φ ! Cf ΣX Cf in which the top row of (12) is the cofiber sequence ob7tained from g and the vertical maps are g f isomorphisms in Ho(C). We denote a principal cofiber sequence by Z X Y C . ) −→ −→ → f " Two principal cofiber sequences with classifying space Z, base X and cofiber Cf are said to be 53 equivalent iff their respective coactions agree and there exists a commutative diagram in Ho(C) of the form g Z X Y Cf " id id id

g′ Z X Y ′ Cf " in which Y Y is an isomorphism in Ho(C). → ′

5.2 Cohomology

Notation 5.9. Let M be a k-vector space and let n 1. We define K(M, n) to be the symmetric ≥ algebra on the graded vector space consisting of M ∗ concentrated in degree n, equipped with the trivial differential. That is, K(M, n) = (S(M ∗[n]), 0).

We begin this section by proving that the functor Hn( , M): LieAlg Set, where M is a trivial − → g-module and n 1, is represented by the the functor [K(M, n), ]: Ho(CEAlg) Set. We will ≥ − → then show that the functor [K(M, n), ] factors through the category of abelian groups, and that the − n bijection [K(M, n), C(g)] ∼= H (g, M) preserves the group structure.

Proposition 5.10. Let g be a finite dimensional Lie algebra and M be a finite dimensional vector space considered as a trivial g-module. There exists an bijection of sets

n H (g, M) ∼= [K(M, n), C(g)] for all n 1. ≥

Proof. Using Prop. 4.29, we identify [K(M, n), C(g)] with hom (K(M, n), C(g))/ . Let CEAlg ≃Cyl n 1 and let (hom (Λ g, M), d) be the cochain complex whose cohomology is H (g, M) (see Def. ≥ k ∗ ∗ # 3.27). Suppose that f : Λng M is a cocycle. Since M is a trivial module, →

df(x x ) = ( 1)i+jf([x , x ], , xˆ , , xˆ , , x ) = 0 1 ∧ · · · ∧ n+1 − − i j · · · i · · · j · · · n+1 *i

ker d: hom (g, M) hom (Λ2g, M) = hom (K(M, n), C(g)) (13) k → k ∼ CEAlg , - Observe that since M is a trivial g-module,

H1(g, M) = ker d: hom (g, M) hom (Λ2g, M) k → k

Moreover, K(M, 1) can be seen as the Chevalley-Eilenberg algebra of M considered as an abelian

Lie algebra. Thus by (13) and Prop. 4.30,

1 H (g, M) ∼= homCEAlg(K(M, 1), C(g)) ∼= [K(M, 1), C(g)]

Now let n 2. Suppose that F, G: K(M, n) C(g) are homotopic via Cyl(M, n). Then the ≥ → homotopy H : Cyl(M, n) C(g) can be expressed as the following commutative diagram, → 9

n 9 H1 M M M Λng ∗ ⊕ ∗ ⊕ ∗ n (0, idM M ) δn 1 ∗⊕ ∗ − n 1 H1 − M M Λn 1g ∗ ⊕ ∗ −

n n n n n n 1 where H1 (x, x, 0) = F1 (x) and H1 (x, 0, x) = G1 (x) for all x M ∗. Thus, δn 1H1 − (x, x) = ∈ − − n n n 1 n 1 F (x) G (x). Define a n 1 cochain h: Λ g M by h = (H − j) where j : M 1 − 1 − − → 1 ∗ ∗ → n M ∗ M ∗ is the k-linear map given by j(x) = (x, x) for all x M ∗. Therefore, if f, g : Λ g M ⊕ − ∈ → are the cocycles corresponding to F and G respectively, then f g = dh. − On the other hand suppose that f, g : Λng M are two cocycles such that f g = dh for some → − h: Λn 1g M. As above we will define F, G: K(M, n) C(g) to be the CE algebra morphisms − → → corresponding to f and g respectively, and we will build a homotopy H : Cyl(M, n) C(g) from F → to G. For degree reasons H has only two nontrivial structure maps, Hn : M M M Λng , 1 9∗ ⊕ ∗ ⊕ ∗ → ∗ 55 which we define as

n H1 (x, 0, 0) = G(x)

Hn(0, x, 0) = F (x) G(x) 1 − n H1 (0, 0, x) = 0

n 1 n 1 and H − : M M Λ g , which we define as 1 ∗ ⊕ ∗ → − ∗

n 1 H1 − (x, 0) = h∗(x)

n 1 H1 − (0, x) = 0 for all x M . We leave it to the reader to verify that H commutes with the differentials on ∈ ∗ Cyl(M, n) and C(g), which will complete the proof.

9

n This proposition implies that H (g, M) ∼= [K(M, n), C(g)] as sets. In order to strengthen this result to an isomorphism of groups, we will show that K(M, n) a canonical cogroup structure.

Suspension in CEAlg

Notation 5.11. Observe that CEAlg is a pointed category with zero object given by S(0), which we will denote by . ∗

Proposition 5.12. Let M be a finite dimensional vector space and let n 1. Then ΣK(M, n+1) = ≥ ∼ K(M, n) in Ho(CEAlg).

Proof. Let ΣK(M, n + 1) be the pushout of

(i ,i ) Cyl(M, n + 1) 0 1 K(M, n + 1) K(M, n + 1) ←− ⊗ −→ ∗ 9 For degree reasons, (i0, i1) must be strict. Thus by Prop. 4.14, ΣK(M, n + 1) is the symmetric 56 algebra on the graded vector space M where

! M ∗ i = n + 1 / i 0 M = 0M ∗ M ∗ i = n 0 ⊕ 10 ! 0 otherwise 0 0 0 and the differential δ on ΣK(M, n + 1)2is strict where

) δ1(x, y) = x y 1 − − ) for all x, y M . We define a strict CE algebra morphism F : ΣK(M, n + 1) K(M, n) with ∈ ∗ → F 1(x, y) = y in degree n and 0 elsewhere. Thus, F 1 : (M, δ1) (M [n], 0) is a quasi isomorphism 1 1 1 → ∗ of cochain complexes. Therefore, ΣK(M, n + 1) ∼= K!(M), n) in Ho(CEAlg).

Proposition 5.13. Let n 1. The comultiplication on K(M, n) induced by the suspension is the ≥ strict CE algebra morphism ∆: K(M, n) K(M, n) K(M, n) given by → ⊗

∆1(x) = x 1 + 1 x 1 ⊗ ⊗

Proof. Dualizing the sequence of pullbacks which yield a multiplication on the loop space of an object in [5, Sec. 4, Thm. 3], we obtain the desired result by a long but straightforward calculation.

Specifically we proceed by obtaining a cylinder object by pushing out K(M, n+1) K(M, n+1) ⊗ → Cyl(M, n + 1) with itself. Then we invoke the universal property of the pushout to construct a morphism from this new cylinder into ΣK(M, n + 1) ΣK(M, n + 1). Then after identifying 9 ⊗ ΣK(M, n + 1) with K(M, n) we reach the desired result.

Theorem 5.14. Let g be a finite dimensional Lie algebra and let M be a trivial g-module. Then as groups

n H (g, M) ∼= [K(M, n), C(g)] for all n 1. ≥ 57

Proof. Let F, G: K(M, n) C(g). Then by Prop. 5.5 the sum of F and G is given by (F, G)∆: K(M, n) → → C(g), where (F, G): K(M, n) K(M, n) C(g) is the morphism induced by the universal prop- ⊗ → erty of the coproduct. For degree reasons, the structure maps of (F, G)∆ are zero except in arity n.

The arity n structure map (F, G)n∆1 : M Λng is given by 1 1 ∗ → ∗

n 1 n n (F, G)1 ∆1(x) = F1 (x) + G1 (x)

Moreover, the assignment in Prop. 5.10 [K(M, n), C(g)] Hn(g, M) is given by F (F n) , → ,→ 1 ∗ and therefore respects the group structure.

5.3 Classification of principal cofiber sequences

In this section we prove that the principal cofiber sequences in CEAlg with classifying space K(M, n+

1), base C(g) and cofiber K(M, n) are classified by [K(M, n + 1), C(g)] for all n 1. We then ≥ introduce central extensions of Lie algebras and describe their relationship to these principal cofiber sequences when n = 1.

Remark 5.15. Given a cofiber sequence with base C(g), the coaction of ΣC(g) on the cofiber is trivial. Indeed, by Prop. 4.14, ΣC(g) = . Thus it is not necessary to show that the coactions of two ∗ cofiber sequences with base C(g) agree in order to conclude that they are equivalent.

Proposition 5.16. Let g be a finite dimensional Lie algebra, and let M be a finite dimensional vector space. The set of equivalent principal cofiber sequences in CEAlg with classifying space K(M, 2), base C(g), and cofiber K(M, 1) is in bijective correspondence with [K(M, 2), C(g)].

Proof. To obtain a cofiber sequence from a CE algebra morphism, we follow Remark. 5.7. Let

C"one(M, 2) be the pushout of

i Cyl(M, 2) 0 K(M, 2) ←− −→ ∗ 9 and let Q: Cyl(M, 2) C"one(M, 2) be the morphism induced by this pushout. Since i is strict, → 0 Prop. 4.14 implies that C"one(M, 2) is the symmetric algebra on M M in degrees 1 and 2, with the 9 ∗⊕ ∗ only nontrivial differential given by the identity in arity 1, and that the composition Qi : K(M, 2) 1 → 58

C"one(M, 2) is the strict map given by (Qi )1(x) = ( x, x) in degree 2. Given a morphism in 1 1 − CEAlg, F : K(M, 2) C(g), we obtain a principal cofiber sequence by a sequence of pushouts. → Since Qi1 is a cofibration, we take the pushout of Qi1 along F . Because cofibrations are preserved by pushouts, we take the pushout of the induced map out of C(g) along the unique map into the zero object. The following diagram illustrates this construction.

F K(M, 2) C(g) ∗

Qi1 J

P C"one(M, 2) S(M ′) S(M ′′)

By Prop. 4.14,

g∗ M ∗ M ∗ i = 1 / ⊕ ⊕ i 0 (M ′) = 0M ∗ i = 2 0 10 0 otherwise 0 0 0 and the structure maps of the different20ial δ′ on S(M ′) are nontrivial in arity 1 and 2. The arity 1 structure map on degree 1 elements

1 δ′ : g∗ M ∗ M ∗ M ∗ 1 ⊕ ⊕ → is given by δ 1(g, x, y) = x y, and the arity 2 structure map on degree 1 elements F′ 1 − −

2 j k δ′ : g∗ M ∗ M ∗ Λ g∗ Λ (M ∗ M ∗) 1 ⊕ ⊕ → ⊗ ⊕ j+(k=2 is given by δ 2(g, x, y) = δ 2(g) 1+F 2(y) 1, where δ is the differential on C(g). On degree 2 F′ 1 g1 ⊗ 1 ⊗ g elements of M , δ 1 = 0 = δ 2. Moreover, morphism J : C(g) S(M ) is the strict map induced ′ F′ 1 F′ 1 → ′ by including g∗ into M ′. Hence, by Prop. 4.14,

M ∗ M ∗ n = 1 / ⊕ n 0 (M ′′) = 0M ∗ n = 2 0 10 0 otherwise 0 0 0 and the only nontrivial structure map of the2differential δ on S(M ) is given by, δ 1(x, y) = x y ′′ ′′ ′′1 − − 59 for all (x, y) (M )1. Thus as a graded vector space M = g [1] M , and P : S(M ) S(M ) ∈ ′′ ′ ∗ ⊕ ′′ ′ → ′′ is the strict CEAlg morphism given by projection.

We will now construct a cofiber sequence with base C(g) and cofiber K(M, 1) of the form

C(g) S((g∗ M ∗)[1]), δ K(M, 1) −→ ⊕ F −→ & ' We define the arity 2 structure map, δ 2 : g M Λjg ΛkM , of the differential F 1 ∗ ⊕ ∗ → j+k=2 ∗ ⊗ ∗ δ on S((g M)[1]) by % F ∗ ⊕

δ 2(g, x) = δ 2(g) 1 + F 2(x) 1 F 1 g1 ⊗ 1 ⊗

The other structure maps of δF must be trivial for degree reasons. The fact that F is a CEAlg 3 2 morphism implies that δF 2δF 1 = 0. Hence δF is a differential. Thus there exists a commutative diagram of the form

J P C(g) (S(M ′), δ′) S(M ′′)

id R′ R′′

C(g) S((g M )[1], δ K(M, 1) ∗ ⊕ ∗ F & ' where R and R are the strict CEAlg morphisms given by the assignments (g, x, y) (g, y) and ′ ′′ ,→ (x, y) y, respectively. Since R and R are weak equivalences, ,→ ′ ′′

F K(M, 2) C(g) S((g∗ M ∗)[1]), δ K(M, 1) −→ −→ ⊕ F −→ & ' is a principal cofiber sequence.

Suppose that F, G: K(M, 2) C(g) are homotopic via H : Cyl(M, 2) C(g). As shown in → → 1 the proof of Prop. 5.10, H : M ∗ M ∗ g∗ is a k-linear map such that 1 ⊕ → 9

δ 2H1(x, x) = F 2(x) G2(x) g1 1 − 1 − 1

We define a strict algebra morphism Φ: S((g M )[1]), δ S((g M )[1]), δ by ∗ ⊕ ∗ F → ∗ ⊕ ∗ G & ' & ' 60

Φ1(g, x) = (g + H1(x, x), x). Then TFDC 1 1 −

Φ2 Λjg ΛkM 2 Λjg ΛkM j+k=2 ∗ ⊗ ∗ j+k=2 ∗ ⊗ ∗

% 2 % 2 δF 1 δG1 1 Φ1 g M g M ∗ ⊕ ∗ ∗ ⊕ ∗ Hence Φ is a CEAlg morphism. Moreover, the following diagram commutes

K(M, 2) C(g) S((g M )[1], δ K(M, 1) ∗ ⊕ ∗ F & ' id id Φ id

K(M, 2) C(g) S((g M )[1], δ K(M, 1) ∗ ⊕ ∗ G & ' 1 Since Φ1 is an isomorphism of cochain complexes, Φ is an isomorphism in CEAlg and thus a weak equivalence. Therefore the principal cofiber sequences corresponding to F and G are equivalent, and the assignment

F F : K(M, 2) C(g) K(M, 2) C(g) S((g∗ M ∗)[1]), δ K(M, 1) → ,−→ −→ −→ ⊕ F −→ , - , & ' - is well defined. F On the other hand, let K(M, 2) C(g) S(V ) K(M, 1) be a principal cofiber se- −→ → → quence with classifying space K(M, 2), base C(g) and cofiber K(M, 1). We obtain a morphism in

[K(M, 2), C(g)] from a principal cofiber sequence by the assignment

F K(M, 2) C(g) S(V ) K(M, 1) F : K(M, 2) C(g) −→ −→ −→ ,−→ → , - , - By the definition of equivalence between principal cofiber sequences, this assignment is well defined.

It is easily verified that this assignment is an inverse to the assignment above. This completes our proof.

This result easily generalizes to the following theorem.

Theorem 5.17. Let g be a finite dimensional Lie algebra, and let M be a finite dimensional vector space. The set of equivalent principal cofiber sequences in CEAlg with classifying space K(M, n + 61

1), base C(g), and cofiber K(M, n) is in bijective correspondence with [K(M, n + 1), C(g)] for all n 1. ≥

Proof. The proof when n = 1 is given above and the proof for n 2 follows similarly. Given a ≥ morphism F : K(M, n + 1) C(g) we obtain a principal cofiber sequence via a sequence of → pushouts. This principal cofiber sequence will be equivalent to

F K(M, n + 1) C(g) S(g∗[1] M ∗[n]), δ K(M, n) −→ −→ ⊕ F → & ' 2 2 where the structure maps of δF are trivial except in arity 2 and n + 1. In arity 2, δF 1 is δg1 on g∗[1] n+1 n+1 and zero on M ∗[n], and in arity n + 1, δF 1 is F1 on M ∗[n] and zero on g∗[1]. The assignment of a principal cofiber sequence to a morphism in [K(M, n + 1), C(g)] is identical to the n = 1 case above.

Central extensions of Lie algebras

Definition 5.18. Let g be a Lie algebra and let M be a vector space. A central extension of g by M is a short exact sequence of Lie algebras

0 M h g 0 → → → → where M is given the structure of an abelian Lie algebra, and [m, x] = 0 for all m M and x h. ∈ ∈

We say that two central extensions of g by M are equivalent iff there exists a Lie algebra isomorphism such that TFDC M h g

id id

M h′ g

Proposition 5.19. Let g be a Lie algebra and M be a vector space considered as a trivial g-module. Then there exists a bijection

H2(g, M) = central extensions of g by M / ∼ { } ∼ 62 where the relation is given by equivalence of central extensions. ∼

Proof. We will provide the assignments that induce this bijection and direct the reader to [19, Thm.

7.6.3] for the details. Let homk(Λ∗g, M), d the complex defined in Def. 3.27 whose cohomology is H∗(g, M). Suppose tha&t ' 0 M h g 0 → → → → is a central extension. Since h g is surjective, we can choose a k-linear section σ : g h. Then → → the alternating bilinear function f : g g M given by × →

f(g, g′) = [σ(g), σ(g′)] σ([g, g′]) − induces a 2-cocycle f : Λ2g M. On the other hand, if we are given a 2-cocycle, f : Λ2g M, → → we can define a bracket on g M by ⊕

[(g, m), (g′, m)] = ([g, g′], f(g, g′)) which yields a central extension

0 M g M g 0 → → ⊕ → →

Proposition 5.20. Let g be a finite dimensional Lie algebra and let M be a finite dimensional vector space. Then the set of equivalent central extensions of g by M is in bijective correspondence with the set of equivalent principal cofiber sequences in CEAlg with classifying space K(M, 2), base C(g) and cofiber K(M, 1).

Proof. Let 0 M h g 0 → → → → be a central extension. If we apply the Chevalley-Eilenberg functor to this extension we obtain a cofiber sequence C(g) C(h) K(M, 1). Since h = g M as a vector space we can obtain a → → ∼ ⊕ 63 degree a linear map by the composition

δ 2 h1 j k pr 2 M ∗ ↩ g∗ M ∗ Λ g∗ Λ M ∗ Λ g∗ → ⊕ −→ ⊗ −→ j+(k=2 which we denote as F . Since M is in the center of h, the arity 2 structure map of the differential δh is ) δ 2(g, x) = δ 2(g) 1 + F (x) 1 h1 g1 ⊗ ⊗ Thus, since δ 3δ 2 = 0, we can define a CE algebra mor)phism F : K(M, 2) C(g) to be the h2 h1 → unique lift of F n = 2 n F1 = / 0 100) n = 2 ∕ 0 By the proof of Prop. 5.16 we know that 20

F K(M, 2) C(g) C(h) K(M, 1) −→ −→ −→ is a principal cofiber sequence.

On the other hand, if we are given a principal cofiber sequence with classifying space K(M, 2), base C(g), and cofiber K(M, 1)

F K(M, 2) C(g) S(V ) K(M, 1) −→ −→ −→ it can be easily verified that it is equivalent to

F K(M, 2) C(g) (S(g∗ M ∗)[1]), δ ) K(M, 1) −→ −→ ⊕ F −→ where δF is the differential defined in the proof of Prop. 5.16. Then applying Remark 3.26 to the cofiber sequence C(g) (S(g M )[1]), δ ) K(M, 1) we obtain a central extension −→ ∗ ⊕ ∗ F −→

0 M g M g 0 → → ⊕ → →

Where the bracket on g M is given by ⊕

2 [(g′, m′), (g′, m′)] = [g, g′], F1 ∗(g, g′) , - 64

Moreover it is obvious the the notion of equivalence is the same between central extension and principal cofiber sequences.

Thus the notion of a central extension is recovered by the notion of a principal cofiber sequence in

Ho(CEAlg), and the correspondence between central extensions and the second cohomology group is related to the correspondence of Prop. 5.16 in the following theorem.

Theorem 5.21. Let g be a finite dimensional Lie algebra and let M be a finite dimensional vector space considered as a trivial g-module. Then the following collection of bijections commutes

Prop. 5.19 Central extensions of g by M / H2(g, M) { } ∼ ∼=

Prop. 5.20 = ∼ ∼= Thm. 5.14

Principal cofiber sequences with classifying space Prop. 5.16 [K(M, 2), C(g)] K(M, 2), base C(g), and cofiber K(M, 1) ∼ = " #$ ∼

Where the relations are given by equivalence of central extensions and equivalence of principal ∼ cofiber sequences.

Proof. The proof of this statement follows immediately from the construction of the bijections be- tween these sets. 65

6 Future Work

The immediate next step will be to generalize Thm. 5.21 to cohomology with nontrivial coefficients and more general extensions of Lie algebras. After this connection is solidified, this homotopical framework can be utilized to study cohomology with coefficients in non-abelian Lie algebras. 66

A Appendix

The following results are well known to experts, but are difficult to find in literature at the appropriate level of generality needed in this thesis. The proofs given here are due to Rogers [15].

A.1 Acyclic cofibrations in CEAlg admit left inverses

Proposition A.1. Let F : (S(V ), δ) (S(V ′), δ′) be an acyclic cofibration in CEAlg. Then there → exists a cdga morphism G: (S(V ), δ ) (S(V ), δ) such that GF = id . ′ ′ → S(V )

Proof. By definition, the chain map F 1 : (V, δ1) (V , δ 1) is an acyclic cofibration of com- 1 1 → ′ 1′ plexes (i.e., a weak equivalence that is injective in all degrees). Therefore there exists a chain map

σ : (V , δ 1) (V, δ1) such that σF 1 = id , and a chain homotopy h: V V [1] such that ′ 1′ → 1 1 V ′ → ′ id F 1σ = δ 1h + hδ 1. Thus we have an isomorphism of complexes V ′ − 1 1′ 1′

1 = F j : V ker σ ∼ V ′, (14) 1 ⊕ ⊕ −→ where j : ker σ V is the inclusion. We will extend j to a cdga map → ′

J : S(ker σ), δ˜ S(V ′), δ′ → & ' & ' where δ˜ := δ 1 . Indeed, let J 1 := j and for k 2 define 1 1′ |ker σ 1 ≥

k k k k J : ker σ S (V ′), J := δ′ h j. 1 → 1 ◦ ◦

By using the fact that h is a chain homotopy, and that δ δ = 0, a direct calculation shows that ′ ◦ ′ δ′J = Jδ˜. Since the chain map (14) is an isomorphism, it follows from Prop. 4.8 that

F J : S(V ) S(ker σ) S(V ′) ⊗ ⊗ → is a isomorphism of cdgas. Note that (F J) ι = F , where ι : S(V ) S(V ) S(ker σ) ⊗ ◦ S(V ) S(V ) → ⊗ is the inclusion. 67

Next, observe that the projection Pr: S(V ) S(ker σ) S(V ) defined as ⊗ →

x, if y = 1k Pr(x y) := / ⊗ 0 100, otherwise 0 is a morphism of cdgas. Finally, let G: S(V ) 20S(V ) be the composition G := Pr (F J) 1. ′ → ◦ ⊗ − Then we have

1 GF = Pr (F J)− (F J) ι = Pr ι = id , ◦ ⊗ ⊗ ◦ S(V ) S(V ) S(V ) & ' and this completes the proof.

A.2 Every weak equivalence in CEAlg is a quasi-isomorphism of cdgas

We first prove two simple lemmas that will allow us to avoid using calculations involving spectral sequences.

Let (A, d) be a Z-graded cochain complex equipped with a decreasing filtration of subcomplexes

A A · · · ⊇ Fn ⊇ Fn+1 ⊇ · · · satisfying the following two conditions:

1. The decreasing filtration is locally bounded on the left i.e. for all p Z there exists an integer ∈ ℓp such that Ap = Ap. Fℓp

2. The filtration is locally bounded on the right i.e, for all p Z there exists an integer rp such ∈ that Ap = 0 Frp { }

Let Gr(A) denote the associated graded cochain complex, i.e. the direct sum of quotient complexes:

Gr(A) := A/ A. Fn Fn+1 n ( 68

Recall that if A and B are filtered cochain complexes, then a chain map f : A B preserves the → filtration iff for all n Z ∈ f A B. Fn ⊆ Fn & ' A filtration preserving chain map induces a chain map on the associated graded complexes

Gr(f): Gr(A) Gr(B). → in the obvious way.

Lemma A.2. Let (A, d) be a Z-graded cochain complex of R-modules equipped with a decreasing filtration that is locally bounded on the left and right. If Gr(A) is acyclic, then A is acyclic as well.

Proof. Let a Ap be a degree p cocycle in A. Then there exists an ℓ such that a A and hence a ∈ ∈ Fℓ is a p-cocycle in the complex A/ A as well. Since Gr(A) is acyclic, A/ A is acyclic, Fℓ Fℓ+1 Fℓ Fℓ+1 therefore there exists b Ap 1 such that 1 ∈ Fℓ −

a db Ap. − 1 ∈ Fℓ+1

By construction a db is a degree p cocycle in A and hence a cocycle in the quotient A/ A. − 1 Fℓ+1 Fℓ+1 Fℓ+2 As before, since Gr(C) is acyclic, there exists b Ap 1 such that 2 ∈ Fℓ+1 −

a db db Ap. − 1 − 2 ∈ Fℓ+2

Continuing this way, we obtain a sequence b , b , . . . of elements in A such that for any n 0 we 1 2 ≥ have a (db + db + + db ) Ap. Since the filtration on A is locally bounded on the − 1 2 · · · n ∈ Fℓ+n right, the sequence is finite i.e. there exists an r such that

a (db + db + + db ) Ap = 0 − 1 2 · · · r ∈ Fr { }

Hence, a = r db A, and so a is a coboundary. i=1 i ∈ + Lemma A.3. Let (A, dA) and (B, dB) be a Z-graded cochain complex of R-modules equipped with a decreasing filtration that is locally bounded on the left and right. Let f : A B be a filtration → 69 preserving chain map. If the induced chain map on the associated graded cochain complexes

Gr(f): Gr(A) Gr(B) → is a quasi-isomorphism, then f is a quasi-isomorphism as well.

Proof. Let (cone(f), ∂) denote the cochain complex corresponding to the mapping cone of f, i.e.,

cone(f)p := A[ 1]p Bp = Ap+1 Bp, ∂(a, b) := d a, d b f(a) . − ⊕ ⊕ − A B − & ' The locally bounded filtrations on A and B induce a decreasing filtration on cone(f) :

cone(f) := A[ 1] B. Fn Fn − ⊕ Fn

Note that each filtered piece cone(f) is a subcomplex since f preserves the filtrations. Moreover, Fn a simple check shows that the induced filtration on cone(f) is locally bounded on the left and right.

Recall that the a chain map between complexes of R-modules is a quasi-isomorphism if and only if its mapping cone is acyclic [19, Cor. 1.5.4]. Hence, by hypothesis the cochain complex cone Gr(f) is acyclic. On the other hand, since quotients of complexes commute with direct sums, we ha&ve a na'tural isomorphism

cone Gr(f) ∼= Gr cone(f) . & ' & ' Hence, the associated graded complex Gr cone(f) is acyclic, and so Lemma A.2 implies that cone(f) is acyclic. Therefore, f is a quasi-i&somorphi'sm.

Next, recall that there is a functor

S : Ch∗ cdga, (15) R → which sends a chain map f : (V, d) (V , d ) to a morphism of cdgas → ′ ′

F := S(f): S(V ), d S(V ′), d′ S → S & ' & ' 70 where

F (x x x ) := f(x ) f(x ) f(x ) 1 ∨ 2 ∨ · · · ∨ n 1 ∨ 2 ∨ · · · ∨ n d (x x x ) := dx x x x S 1 ∨ 2 ∨ · · · ∨ n ± i ∨ 1 ∨ · · · ∨ i ∨ · · · ∨ n (16) *i 3 d′ (x′ x′ x′ ) := d′x′ x′ x x′ . S 1 ∨ 2 ∨ · · · ∨ n ± i ∨ 1 ∨ · · · ∨ i′ ∨ · · · ∨ n *i 3 Proposition A.4. Let (V, d) be a non-negatively graded cochain complex of vector spaces over a

field k of characteristic zero. There exists a natural isomorphism of graded commutative algebras

H S(V ), dS ∼= S H(V, d) . & ' & ' Proof. Since we are working over a field, the Künneth theorem implies that H T (V ), dT ∼= T H(V, d) , where T (V ) is the tensor algebra generated by V . Since we are workin&g over a fi'eld of&char- ' acteristic zero, taking co-invariants commutes with taking cohomology. Hence, H S(V ), dS ∼= S H(V, d) . See the proof of Prop. 2.1 in [12, Appendix B] for the complete details&. ' & ' Corollary A.5. Let k be a field with char k = 0. The functor (15) S : Ch∗ cdga preserves k → quasi-isomorphisms.

Proof. Follows from the fact that the isomorphism in Prop. A.4 is natural.

Theorem A.6. If F : S(V ), δ S(V ′), δ′ is a weak equivalence in the category CEAlg, then it → is also a quasi-isomor&phism of'cdga&s. That is,'F induces an isomorphism of graded vector spaces:

= H S(V ), δ ∼ H S(V ′), δ′ −→ & ' & ' Proof. For any CE algebra S(V ), δ , there is a canonical descending filtration by linear subspaces & ' S(V ) = S(V ) S(V ) S(V ) F0 ⊇ F1 ⊇ F2 ⊇ · · · defined by “word-length”, i.e., ∞ S(V ) := Sk(V ). (17) Fn n k (≤ 71

If y = x x S(V ), then by definition 1 ∨ · · · ∨ m ∈ Fn

δy = δ (x ) x x x . ± 1 i ∨ 1 ∨ · · · ∨ i ∨ · · · ∨ m *i By definition of a CE algebra, V is concentrated in degree3s 1. Hence, for any x V , δ (x) ≥ ∈ 1 ∈ k k k S(V ) is a finite sum δx = k 1 δ (x), given by linear maps δ1 : V S (V ). It then follows ≥ → that the word length of δy is+greater than or equal to the word length of y. Hence, S(V ) is a Fn sub-cochain complex of S(V ).

By construction, the word-length filtration (17) is clearly locally bounded on the left. Now sup- pose y S(V )p is a element of degree p > 0, homogeneous with respect to word length. Since all ∈ generators (i.e., elements of V ) have degree 1, it follows that the word-length of y is less than or ≥ equal to p. Hence, S(V )p = 0, Fp+1 and therefore the filtration is locally bounded on the right.

Next, we analyze the associate graded complex Gr S(V ) . Observe that for each n 0, we ≥ have an isomorphism of graded vector spaces & '

S(V )/ S(V ) = Sn(V ), Fn Fn+1 ∼ and therefore Gr S(V ) ∼= S(V ) as graded vector spaces. Now consider the induced differential δ on S(V )/& S'(V ). If y = x x S(V ), then we have Gr Fn Fn+1 1 ∨ · · · ∨ m ∈ Fn

δy = δk(x ) x x x . ± 1 i ∨ 1 ∨ · · · ∨ i ∨ · · · ∨ m i 1 k 1 *≥ *≥ 3 And since im δk Sk(V ), we deduce that 1 ⊆

δ y + S(V ) = δ1(x ) x x x + S(V ). Gr Fn+1 ± 1 i ∨ 1 ∨ · · · ∨ i ∨ · · · ∨ m Fn+1 i 1 & ' *≥ 3 1 Hence, the differential δGr only depends on the differential d := δ1 on the cochain complex (V, d). Therefore, we have an isomorphism of cochain complexes

Gr S(V ) , δGr ∼= S(V ), dS , , & ' - & ' 72

where dS is the differential (16). Finally, suppose F : S(V ), δ S(V ), δ is a weak equivalence. For any generator x V → ′ ′ ∈ 1 k we have F (x) = F1 (x)+& k 2 F'1 (x)&, where the'finite sum on the right hand side is given by linear ≥ maps F k : V Sk(V ). +It follows that F respects the word length filtrations on S(V ) and S(V ). 1 → ′ ′ Moreover, this observation also implies that the following diagram of cochain complexes commutes

Gr(F ) Gr S(V ) , δGr Gr S(V ′) , δG′ r , - , - & ' & ' (18) ∼= ∼= 1 S(F1 ) (S(V ), dS) (S(V ′), dS′ )

1 where S(F1 ) is the morphism obtained by applying the functor (15) to the chain map

1 F : (V, d) (V ′, d′). 1 →

1 1 Since F is a weak equivalence, F1 is a quasi-isomorphism, by definition. Hence, S(F1 ) is quasi- isomorphism by Cor. A.5, and hence Gr(F ) is as well, by the commutativity of diagram (18). There- fore, Lemma A.3 implies that F is a quasi-isomorphism, and this completes the proof.

A.3 Obstruction theory for semi-free cdga morphisms

Let S(V ), δ and S(V ), δ be CE algebras. For each k 1 let Hom (V, Sk(V )) denote the ′ ′ ≥ k ′ coch&ain comp'lex wh&ose under'lining graded vector space consists of linear maps f k : V Sk(V ) → ′ of arbitrary degree, with differential

k k k f k k 1 ∂(f ) := δ′ f ( 1) f δ . k ◦ − − | | ◦ 1

Let (A, ∂A) denote the direct product of these cochain complexes i.e.

∞ k k A := Homk(V, S (V ′)) = Homk(V, k 1S (V ′)) ∼ ⊕ ≥ k 1 <≤ i k Given a map f : V i 1S (V ′) in A, we denote by f the composition with the projection → ⊕ ≥

k i k f := πkf, πk : i 1 S (V ′) S (V ′), ⊕ ≥ → 73

k and when convenient we identify f A with its corresponding infinite sequence (f )k 1 ∈ ≥ We observe that A is equipped with a canonical decreasing filtration of sub cochain complexes

A = A A F1 ⊇ F2 ⊇ · · · where A := f A f k = 0 k n 1 . Fn { ∈ | ∀ ≤ − } A simple calculation shows that for each n 1 we have isomorphisms of complexes ≥ n 1 − i A/ A = Hom V, S (V ′) , Fn ∼ k i=1 ( & ' and that (A, ∂A) is complete with respect to this filtration i.e.,

(A, ∂A) = proj lim A/ nA ∼ n 0 F ≥ We now introduce an auxiliary collection of degree 1 symmetric multi-linear maps

m L : A⊗ A, m 1. (m) → ∀ ≥

For m = 1, for all f A and for each k 1 we define ∈ ≥

k k L (f) : V S (V ′) (1) → k k k i deg f k 1 L (f) := δ′ f ( 1) f δ . (1) i − − 1 *i=1 For m 2, for all f , f , . . . , f A and for each k 1 we define ≥ (1) (2) (m) ∈ ≥

k k L (f , f , . . . , f ) : V S (V ′) (m) (1) (2) (m) → L (f , f , . . . , f )k := f i1 f i2 f im δm (m) (1) (2) (m) − (1) ∨ (2) ∨ · · · ∨ (m) 1 i1+i2+ +im=k *··· & ' We note that the above maps are compatible with the filtration on A in the following sense: For all m 1, and i , i , . . . , i 1 we have an inclusion ≥ 1 2 m ≥

L(m) i1 A i1 A im A i1+i2+ +im A. (19) F ⊗ F ⊗ · · · ⊗ F ⊆ F ··· & ' The multi-linear maps L(m) allow us to define the following polynomial function on the degree 74 zero elements of A:

κ: A0 A1 → (20) κ(f) := L(m)(f, f, . . . , f). m 1 *≥ m The infinite series in the definition of κ is well defined=, sin>c?e: (1@9) implies that κ is compatible with the filtration, A = A, and since A is complete with respect to the filtration. F1

Lemma A.7. Let Z(κ) A0 denote the zero locus of the polynomial κ. There is a bijection of sets ⊆

homCEAlg S(V ), S(V ′) ∼= Z(κ) & ' Proof. Let f A0. A simple unraveling the definition (20) of κ shows that for each n 1, the ∈ ≥ degree 1 linear map κ(f)n : V Sn(V ) is → ′ n n n n k i1 i2 i k κ(f) = δ′ f f f f k δ . k − ∨ ∨ · · · ∨ 1 k=1 k=1 i1+i2+ +i =n * * *··· k & ' Hence, for every linear map f A0 satisfying κ(f) = 0 there is a unique algebra map F : S(V ) ∈ → S(V ) satisfying δ F = F δ with F := F = f. ′ ′ 1 |V

It follows from the compatibility (19) of the multi-linear maps L(m) with the filtration that, for each n 0, the function κ induces a well defined map to the quotients ≥

κ¯ : A0/ A0 A1/ A1 n Fn → Fn κ¯ f + A0 := κ(f) + A1 n Fn Fn & ' Lemma A.8. For all f A0 the following identity holds ∈

L κ(f) L f, f, . . . , f, κ(f) = 0. (1) − (m+1) m 2 *≥ & m ' Proof. One inductively “climbs up the tower” = A0>/? @ A0 A0/ A0 and verifies · · · → Fn+1 → Fn → · · · that for each n 0, the analogous equality involving κ¯ is satisfied for all elements of A0/ A0. ≥ n Fn The proof then follows since A0 is the projective limit of the quotients A0/ A0. Fn

k k n 1 Proposition A.9. Let n > 1 and suppose a collection of degree 0 linear maps F : V S (V ) − { 1 → }k=1 75 satisfy the equations m m m k m k δk′ F1 = Fk δ1 (21) *k=1 *k=1 for all 1 m n 1. Then the degree 1 linear map q : V Sn(V ) defined as ≤ ≤ − n → ′ n 1 n − n k n k q := δ′ F F δ n k 1 − k 1 *k=1 *k=2 is a 1-cocycle in the cochain complex (A, ∂A).

k k n 1 n 1 k 0 Proof. The linear maps F : V S (V ) − represent a class f¯ = − F + A in the { 1 → }k=1 k=1 1 Fn quotient + n 1 0 0 − k 0 A / A = Hom (V, S (V ′)) . Fn ∼ k (k=1 The equations (21) are satisfied if and only if

κ¯ (f¯) = 0 κ(f) A1. n ⇔ ∈ Fn

On the other hand,the identity Lemma A.8 implies that

(L κ(f))n L f, f, . . . , f, κ(f) n = 0 (1) − (m+1) m 2 *≥ & m ' But since κ(f) A1, the above equality reduces=to >? @ ∈ Fn

n (L(1)κ(f)) = 0.

A direct calculation then shows that

n κ(f) = qn, and it follows from the definition of L(1) that

n (L(1)κ(f)) = ∂Aqn.

Hence, ∂Aqn = 0. 76

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