LECTURE 2: WALDHAUSEN's S-CONSTRUCTION 1. Introduction

Home , Spectrum (topology), Waldhausen category

LECTURE 2: WALDHAUSEN’S S-CONSTRUCTION

AMIT HOGADI

1. Introduction In the previous lecture, we saw Quillen’s Q-construction which associated a topological space to an exact category. In this lecture we will see Waldhausen’s construction which associates a spectrum to a Waldhausen category. All exact categories are Waldhausen categories, but there are other important examples (see2). Spectra are more easy to deal with than topological spaces. In the case, when a topological space is an inﬁnite loop space (e.g. Quillen’s K-theory space), one does not loose information while passing from topological spaces to spectra.

2. Waldhausen’s S construction 1. Definition (Waldhausen category). A Waldhausen category is a small category C with a distinguished zero object 0, equipped with two subcategories of morphisms co(C) (cofibrations, to be denoted by ) and ω(C) (weak equivalences) such that the following axioms are satisfied: (1) Isomorphisms are cofibrations as well as weak equivalences. (2) The map from 0 to any object is a cofibration. (3) Pushouts by cofibration exist and are cofibrations. (4) (Gluing) Given a commutative diagram

C / A / / B

∼ ∼ ∼ C0 / A0 / / B0 where vertical arrows are weak equivalences and the two right horizontal maps are cofibrations, the induced map on pushouts [ [ C B → C0 B0 A A0 is a weak equivalence. By a cofibration sequence in C we will mean a sequence A B C such that A B is a cofibration and C is the cokernel (which always exists since pushout by cofibrations exist). 1 2 AMIT HOGADI

2. Example. Every exact category is a Waldhausen category where cofibrations are inflations and weak equivalences are isomorphisms. However there are other important examples of Waldhausen’s category which are not exact categories. If A is an abelian category, then the category of bounded chain complexes in A where cofibrations are monomorphism of complexes and weak equivalences are isomorphisms is also a Waldhausen’s category. 3. Definition. A spectrum is a sequence of pointed simplicial sets (or topolog- 1 ical spaces) E = {En}n≥0 together with maps S ∧ En → En+1. Morphisms of spectra is defined in an obvious manner. Given a spectrum E, we define the i-th homotopy group of E to be lim π E −→ i+n n n where the transition maps are

1 ∧S 1 πi+n(En) −→ πi+n+1(S ∧ En) → φi+n+1(En+1). Note that spectra can have negative homotopy groups. Morphism of spectra in- duce map on all homotopy groups. A morphism of spectra is called a π∗ equivalence if the induced map on all homotopy groups is an isomorphisms. The set of π∗-equivalences can be inverted to get a category called the homotopy category of spectra. This is a triangulated category. 4. Definition (Waldhausen’s S-functor). Waldhausen defined a functor from the category of small Waldhausen categories to the category of simplicial Waldhausen categories as follows. Let C be a Waldhausen category with 0 as the distinguished object. We now define a simplicial category with cofibrations S•C as follows. For n ≥ 0, SnC is the category whose objects are sequences

0 = A0 A1 ··· An together with choices Ai/j of the cokernels of Ai Aj for every 0 ≤ i ≤ j ≤ j and where Ai/i is the chose distinguished object. Note that by universal property of cokernels, we have an ’upper-triangular’ commutative diagram with 0 on the diagonal:

0 A1 A2 ··· An

A1/1 A2/1 ··· An/1

. . 0 . .

An/n = 0 LECTURE 2: WALDHAUSEN’S S-CONSTRUCTION 3

Thus: S0C is the zero category. S1C is the category C we think of objects of C as coﬁbrations 0 A from 0. S2C is the category of extensions, i.e. coﬁbration sequences.

We want to give {SnC} the structure of a simplicial category. For a monotonic map f :[m] → [n], define a functor ∗ f : SnC → SmC by sending 0 = A0 A1 ··· An, +choices of Ai/j to 0 = Af(0)/f(0) ··· Af(m)/f(0) where if f(i) = f(j), we define the cofibration Af(i)/f(0) Af(j)/f(0) to be the identity, and if f(i) < f(j) we define it to be the composition of the given maps

Af(i)/f(0) A(f(i)+1)/f(0) ··· Af(j)/f(0).

For every 0 ≤ i ≤ j ≤ m we deﬁne the quotient Ai/j to be Af(i)/f(j). It is straight- forward to check that this gives S•C the structure of a simplicial category.

Each SnC is a Waldhausen category where a morphism A• → B• is a weak equivalence if each Ai → Bi is a weak equivalence. It is a coﬁbration if for every 1 ≤ i ≤ j ≤ k ≤ n, in the diagram

Aj/i / / Ak/i / Ak/j

Bj/i / / Bk/i / Bk/j A → B , A → B and A S B → B are cofibrations in C. j/i j/i k/j k/j k/i Aj/i j/i k/i 0 5. Definition. Define S C = C. Now for every r > 0, we can iterate the S• functor to get an r-simplicial category SrC[−, −, ..., −]. We let ωSrC denote the r-simplicial subcategory of weak equivalences. Define r K(C)r := |NωS C|. Note that the nerve of an r-simplicial category is a r + 1-simplicial set. By || we mean the topological realization of this r + 1-simplicial set. We will confuse this with the diagonal of the r + 1 simplicial set NωSrC. 1 We want to define a map from S ∧ K(C)n → K(C)n+1 which depends on the following observation.

6. Remark. Let X•,• be a bisimplicial set. Let d(X•,•) denote its diagonal. Let X•,1 denote the simplicial set, whose last coordinate is fixed to be 1. Then one has a canonical map ∆1 × X•,1 → d(X•,•) 4 AMIT HOGADI defined as follows. For m ≥ 0, an m simplex of ∆1 × X•,1 is by definition a pair f ([m] → [1], α) where α ∈ Xm,1. We send this to an m simplex of d(X•,•) via the map id×f ∗ Xm,1 −→ Xm,m.

Now, K(C)n is homeomorphic to the diagonal of the n + 1-simplicial set NωSnC[−, −, ..., −] where the last simplicial direction is given by the nerve N, and the first n n n given by S . By definition, the n-simplicial category S1(S C) is nothing but n+1 S C[−, −..., −, 1]. But since S1 applied to any category, gives back the category itself, we have an identification of n-simplicial categories SnC[−, ...−] = Sn+1C[−, −, ..., −, 1]. Applying the Nerve construction to the subcategory of equivalences we get an identification d(NωSnC)[m] = NωSn+1C[m, m, ..., m, 1, m]. This is then used to define a map as explained in Remark6

∆1 × K(C)n → K(C)n+1.

Since the spaces K(C)n+1 have a unique zero simplex (as S0(C) is the trivial category), it is easy to check that the above map factors through 1 S ∧ K(C)n → K(C)n+1 thus deﬁning a spectrum K(C), which will be called the K-theory spectrum of C.