K–Theory for Triangulated Categories 3\frac{1}{2}(A): A Detailed Proof of the Theorem of Homological Functors

Amnon Neeman (amnon.neeman@anu.edu.au)

The Australian National University

Abstract. Let A and B be abelian categories. Let H : A ⟶ B be a bounded δ– functor. We prove that H induces a natural map in higher K–theory. From a more precise analysis of the proof, we deduce that it is possible to define a K–theory of the bounded derived category of A, which contains Quillen’s K–theory of A as a retract.

AMS subject classification: 18F25, 18E30, 19D99

I would like to dedicate this article to Daniel Quillen and to Robert Thomason. This year we celebrate Quillen’s 60th birthday, and mourn the fifth aniversary of Thomason’s untimely death. Thomason helped me a tremendous amount, in the early days of this article. Quillen helped Thomason, when the étale descent theorem was still very new. Both Quillen and Thomason are outstanding mathematicians. They are also kind and decent human beings. Each went out of his way to help a young mathematician, whose work had been overlooked.

Table of Contents

0 Introduction 2 1 Simplicial preliminaries 14 2 The proof of Proposition 2.3 20

Step 2.1: 29 Step 2.2: 32

3 First reductions in the proof of Proposition 3.1 32

Step 3.1: 34 4 The homotopy fiber in Proposition 3.3 46

Step 4.1: 48 5 A reminder concerning homotopy cartesian squares 55 6 Outline of the proof of Proposition 4.2 57

Step 6.1: 57 Step 6.2: 64 Step 6.3: 76

\begin{matrix}c \\ ◯ \end{matrix} 2000 Kluwer Academic Publishers. Printed in the Netherlands.

0. Introduction

Let A be an abelian category. In [14], Quillen defines a K–theory spectrum associated to A, denoted K(A). This association is functorial. Given an exact functor A ⟶ B of abelian categories, there is an induced map K(A) ⟶ K(B). What we will prove here is the following generalisation.

Theorem. Let H : A ⟶ B be a bounded δ–functor. Perhaps we should remind the reader what this means.

1. For all i ∈ ℤ we have an additive functor H^{i} : A ⟶ B.

2. For all short exact sequences in A

0 ⟶ A′ ⟶ A ⟶ A′′ ⟶ 0

there is a long exact sequence in B

H^{i−1}(A′′) ⟶ H^{i}(A′) ⟶ H^{i}(A) ⟶ H^{i}(A′′) ⟶ H^{i+1}(A′)

which is natural in the short exact sequences.

3. For any object A ∈ A, all but finitely many of the H^{i}(A)’s vanish. This last hypothesis is the boundedness.

Then H induces a map in K–theory; there is a natural map K(H) : K(A) ⟶ K(B) induced by H, with the obvious restriction to K_{0} = π_{0}K(A).

The “naturality” of the map K(H) means explicitly that it is natural with respect to composing H with exact functors, on either side. It also means that it agrees with the usual map, when H is an exact functor. And that it agrees with the classically defined map of Waldhausen, when that map makes sense; see the example below.

Let us tell the reader the significance of the theorem. Perhaps this is best illustrated with a familiar example.

Suppose f : X ⟶ Y is a proper morphism of smooth, Noetherian schemes. Then the pushforward map Rf_{∗} is a δ–functor; it takes a coherent sheaf on X to a graded coherent sheaf on Y, and takes short exact sequences of coherent sheaves on X to long exact sequences on Y. From our Theorem, it follows that there is an induced pushforward map in K–theory.

From the definition of these maps [to be given later], it is easy to see that the map above agrees with the natural map defined by Waldhausen. Of course, to define the Waldhausen map

K(f) : K(X) ⟶ K(Y)

K–Theory for Triangulated Categories 3 \frac{1}{2} (A) 3

using Waldhausen’s techniques, one needs to construct an exact functor of Waldhausen categories. The significance of our result, when applied to the familiar example above, is that the exact functor of Waldhausen categories is for the birds. One does not need it to define the map in K–theory.

This theorem will be the major result of the article. It is possible to generalise this to exact categories. Suppose A and B are exact categories. A chain complex in B

⋯ ⟶ X_{i} ⟶ X_{i+1} ⟶ ⋯

is defined to be acyclic or exact if

1. every map

X_{i} ⟶ X_{i+1}

can be decomposed as

X_{i} ⟶ K_{i} ⟶ X_{i+1}

with X_{i} ⟶ K_{i} an admissible epi, and K_{i} ⟶ X_{i+1} an admissible mono.

2. The sequences K_{i} ⟶ X_{i+1} ⟶ K_{i+1} are admissible short exact sequences.

Then one can define bounded δ–functors A ⟶ B exactly as for abelian categories. The following theorem is true

Theorem. Any bounded δ–functor A ⟶ B, where A and B are exact categories, induces a map in K–theory.

The theorem is true, but will not be proved in the article. The emphasis of this article, true to its title, is on keeping the proof short and simple. When necessary, we are quite willing to sacrifice some generality, in the interest of transparency. What we do not sacrifice is careful detail. The proof here attempts to be complete and self–contained, to allow the reader to scrupulously check the results. For a proof of the better theorem, about exact as opposed to abelian categories, the reader is referred to [9], [10], [11], [12] and [13].

This is perhaps the place to mention, that a sketch of all the theorems and proofs of this article may be found in [9] and [10]. The intention of this article is to focus narrowly on some (but not all) of the results in [9] and [10], and offer detailed and very complete proofs. Whenever anyone found a point obscure, I chose to err on the side of including an explanation in the manuscript. The manuscript therefore was greatly influenced by many people, and I would like to give my thanks to all of them.

It is perhaps relevant to know that nothing remotely like this result was known before. Ever since Quillen’s localisation and additivity theorems in [14], one has known that there is a relation between K–theory and exact sequences. This relation was beautifully and extensively developed in Waldhausen’s work [19]. But none of these theorems come even close to constructing, out of any δ–functor, an induced map in K–theory. There are examples where this can be applied, to deduce the existence of very surprising maps in higher K–theory.

The claim in the title, that we will see a “detailed proof” of a narrow theorem, is an example of truth in advertising. But the reader may well wonder where triangulated categories come in. The answer is they do not. The theorems of this article are purely about abelian categories. Triangulated categories only provide a certain motivation. The motivation they provide will perhaps become clear in the remainder of the Introduction.

Up until now, the Introduction has assumed no prior knowledge. But since we will now talk about where triangulated categories come in, we will need to assume that the reader knows what they are. We will also assume more familiarity with the basics of K–theory and of simplicial sets.

Let us next give a simplicial version of the theorems above. From now on, we will work with spaces rather than spectra. We leave it to the reader to check that the natural infinite loop structure is preserved by our construction. The space K(A) is usually given simplicially; there are by now many homotopy equivalent simplicial descriptions of it. For the purpose of this article, it will be convenient to work with the following.

A simplicial model for a delooping of the space K(A) is given by the

realisation of a bisimplicial set we will denote A [figure] . An (r, s)–

simplex in A [figure] is a diagram

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}]

where all squares are bicartesian. For readers unfamiliar with this simplicial set, let us observe that in this diagram every object is a subquotient of X_{0s}. In fact, if r = s the terms X_{ij} with i + j = s, that is the diagonal terms of the square, form a sequence of subquotients of X_{0s}. All the other information in the diagram amounts to choices, unique up to canonical isomorphism, of objects in A isomorphic to the intermediate subobjects and quotients. Thus we easily identify a

map from the diagonal realisation of the simplicial set A [figure] to

Quillen’s Q–construction. It is well–known that this map is a homotopy equivalence; see for instance Proposition 1.4 on page 1178 of [6].

Let T be a triangulated category. It is possible to define an asso-

ciated K–theory space, which we will denote T [figure] . One of the

unsatisfactory facets of the theory, as it now stands, is that there are

a number of competing possible definitions for T [figure] , and the

theorems that have so far been proved are somewhat sensitive to the definition. To keep the presentation here as non–technical as possible,

let us mention only one of the possible candidates for T [figure] .

DEFINITION 0.1. Let T be a triangulated category. Then T [figure]

is a bisimplicial set, and an (r, s)–simplex in T [figure] is a commu-

tative diagram

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}]

of objects and morphisms in T , together with a coherent differential ∂ : X_{rs} ⟶ ΣX_{00}. A coherent differential ∂ : X_{rs} → ΣX_{00} is a map such that for any 0 ≤ i ≤ i′ ≤ r, 0 ≤ j ≤ j′ ≤ s,

X_{ij} \underrightarrow{( \begin{matrix}α \\ −β \end{matrix} )} X_{i′j} ⊕ X_{ij′} \underrightarrow{( γ δ )} X_{i′j′} ⟶^{ϕ} ΣX_{ij}

is a triangle, where ϕ : X_{i′j′} → ΣX_{ij} is the composite

X_{i′j′} → X_{rs} ⟶^{∂} ΣX_{00} → ΣX_{ij},

while α : X_{ij} ⟶ X_{i′j}, β : X_{ij} ⟶ X_{ij′}, γ : X_{i′j} ⟶ X_{i′j′} and δ : X_{ij′} ⟶ X_{i′j′} are the maps given by the diagram defining the simplex.

In this article, we will prove the following theorem:

Theorem I.4.8. If A is a small abelian category, then the map

A [figure]  i  \overline{}D^{b}(A) [figure]

gr84.tex; 19/06/2000; 14:06; p.6

is a split inclusion in homotopy. There is a map

D^{b}(A) [figure]  p  \overline{}A [figure]

which is left inverse to it. That is, p ∘ i is homotopic to the identity.

In fact, we can give the homotopy inverse to i fairly explicitly. It is given by the homology functor. Precisely, let Gr(A) be the category of graded objects in A. Thus, an object of Gr(A) is a functor from the discrete category ℤ to the category A; in other words, to give such an object is to give an object of A for every integer in ℤ, positive or negative.

The category Gr(A) contains a full subcategory Gr^{b}(A), of all bounded graded objects of A. An object of Gr(A) lies in Gr^{b}(A) if for all but a finite number of integers in ℤ, the corresponding object of A is zero.

Let X be an object of Gr(A). Then X is a sequence of objects of A, indexed by ℤ. Let n ∈ ℤ be an integer. Then H^{n}(X) will be the way we refer to the n^{th} part of X. Given a morphism f : X ⟶ Y in Gr(A), H^{n}(f) : H^{n}(X) ⟶ H^{n}(Y) will be the n^{th} piece of this map. With this notation, Gr^{b}(A) ⊂ Gr(A) can be denoted

Gr^{b}(A) ≡ {X ∈ Gr(A) | H^{n}(X) = 0 for all but finitely many n ∈ ℤ} .

The category Gr(A) admits an endomorphism Σ, called the suspension functor. It is given by the formula H^{n}(ΣX) = H^{n+1}(X). This functor takes the subcategory Gr^{b}(A) ⊂ Gr(A) to itself. The restriction of Σ : Gr(A) ⟶ Gr(A) to the subcategory Gr^{b}(A) ⊂ Gr(A) is a functor; we call it also the suspension functor, and denote it Σ : Gr^{b}(A) ⟶ Gr^{b}(A). The inverse of Σ, both on the category Gr(A) and on the subcategory Gr^{b}(A), will be denoted Σ^{−1}.

It is possible to define a K–theory for Gr^{b}(A). Precisely, we define

DEFINITION 0.2. The symbol Gr^{b}(A) [figure] stands for a bisim-

plicial set, and an (r, s)–simplex in Gr^{b}(A) [figure] is a commutative

diagram

gr84.tex; 19/06/2000; 14:06; p.7

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}]

of objects and morphisms in Gr^{b}(A), together with a coherent differential ∂ : X_{rs} ⟶ ΣX_{00}. We assume that for every integer q, and for any 0 ≤ i ≤ i′ ≤ r, 0 ≤ j ≤ j′ ≤ s,

H^{q}(X_{ij}) ↓

H^{q}(X_{i′j}) ⊕H^{q}(X_{ij′}) → H^{q}(X_{i′j′}) ⟶^{ϕ}  H^{q+1}(X_{ij})
↓
H^{q+1}(X_{i′j}) ⊕ H^{q+1}(X_{ij′})

is an exact sequence. As in Definition 0.1, ϕ : X_{i′j′} → ΣX_{ij} is assumed to be the composite

X_{i′j′} → X_{rs} ⟶^{∂} ΣX_{00} → ΣX_{ij}.

REMARK 0.3. It should be noted that the construction has some

symmetries. The “transpose” of an (r, s)–simplex in Gr^{b}(A) [figure]

is an (s, r)–simplex. That is, if we replace X_{ij} by X_{ji} we have a perfectly

well–defined simplex. Similarly, a simplex in Gr^{b}(A) [figure] can also

be viewed as a simplex in  Gr^{b}(A^{op}) [figure] , where A^{op} is the

opposed category of A; simply read the simplex starting with X_{rs}. We will call this symmetry the “duality”. Later on in the theory, we will construct simplicial sets which are not their own transposes and duals; but transposition and duality are nevertheless useful symmetries to keep in mind. Once we have shown that a simplicial map or homotopy is well–defined, so are its transpose and dual.

Of course, there is a natural map

D^{b}(A) [figure]  H  \overline{}Gr^{b}(A) [figure]

induced by the cohomology functor. The functor H sends X ∈ Ob {D^{b}(A)} to the graded object H(X), that is the graded object in Gr^{b}(A) which is H^{n}(X) in degree n. The reason H(X) is bounded is that X ∈ D^{b}(A), and hence only finitely many of the cohomology objects H^{n}(X) can be non–zero. We can form the composite

A [figure]  i \overline{}D^{b}(A) [figure]  H \overline{}Gr^{b}(A) [figure]

What we will really prove in this article is

Theorem I.4.8. If A is a small abelian category, then the composite

A [figure]  i \overline{}D^{b}(A) [figure]  H \overline{}Gr^{b}(A) [figure]

induces a homotopy equivalence.

It immediately follows that if we compose {H ∘ i}^{−1} ∘ H, we get a map p

D^{b}(A) [figure]  p = \{H ∘ i\}^{−1} ∘ H \overline{}A [figure]

which is homotopy left inverse to

A [figure]  i  \overline{}D^{b}(A) [figure]

gr84.tex; 19/06/2000; 14:06; p.9

Thus the version of Theorem I.4.8 given earlier in the Introduction follows immediately from the more recent version.

Note also that given a δ–functor A ⟶ B of abelian categories, there is an induced simplicial map

A [figure]  \overline{}Gr^{b}(B) [figure]

which, by Theorem I.4.8 loops to a map K(A) ⟶ K(B). From this we deduce the unlabeled Theorem, stated at the beginning of the article.

Now we will say something about the proof. We introduce full subcategories Gr_{[m,n]}(A) ⊂ Gr^{b}(A) with the definition

Gr_{[m,n]}(A) = {X ∈ Gr^{b}(A) | H^{q}(X) = 0 unless m ≤ q ≤ n} .

One defines the simplicial set Gr_{[m,n]}(A) [figure] to be the subset of

Gr^{b}(A) [figure] where all the objects are assumed to lie in the subcate-

gory Gr_{[m,n]}(A). In other words, an (r, s)–simplex in Gr_{[m,n]}(A) [figure]

is a commutative diagram

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}]

of objects and morphisms in Gr_{[m,n]}(A), together with a coherent differential ∂ : X_{rs} ⟶ ΣX_{00} satisfying the exactness hypothesis of Definition 0.2. Note that ΣX_{00} is not assumed to be in the category Gr_{[m,n]}(A). The category Gr_{[m,n]}(A) is of course not closed with respect to the suspension functor Σ. We assume only that X_{ij} is an object of Gr_{[m,n]}(A), and nothing about its suspensions.

In this entire article, the category A will be held fixed. Therefore, we will permit ourselves to forget it in the notation. The symbols

Gr[figure: [m,n]]  Gr_{[m,n]}(A) [figure]

will be considered synonyms. In particular, if m = n = 0, the simplicial

set Gr[figure: [0,0]] is almost the same as the simplicial set A [figure] . In

either case, a simplex is a diagram

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}]

of bicartesian squares in A, and the only difference between the sim-

plicial sets is that in A [figure] there is a restriction placed on the

morphisms; the vertical ones are assumed epi, the horizontal ones mono. This should also begin to explain the hieroglyphics we are adopting to denote the simplicial sets; the arrows in the superscript of the A inside the box denote the restriction placed on the horizontal and vertical morphisms.

Clearly, the simplicial set Gr^{b}(A) [figure] is an increasing union

of the subsets Gr[figure: [m,n]] . Theorem I.4.8 is therefore an immediate

consequence of two propositions:

Proposition 2.3. The natural inclusion

A [figure]  \overline{}[figure]  Gr[figure: [0,0]]

induces a homotopy equivalence. Proposition 3.1. For any integers m′ ≤ m ≤ n ≤ n′, the natural inclusion

Gr[figure: [m,n]]  \overline{}[figure]  Gr[figure: [m′,n′]]

induces a homotopy equivalence.

In proving Proposition 3.1, it clearly suffices to consider the case where either n = n′ or m = m′; the two cases being dual to each other, we assume n = n′. It also suffices to assume m = m′ + 1. Finally, by translating using the suspension functor, we may reduce ourselves to the case m′ = 0. The actual Proposition we will prove is

Proposition 3.1. For any integer n ≥ 1, the natural inclusion

Gr[figure: [1,n]]  \overline{}[figure]  Gr[figure: [0,n]]

induces a homotopy equivalence.

The proof of Proposition 2.3 turns out to be very easy; we will without difficulty prove a suitable generalisation in Section 2. The proof of Proposition 3.1 is trickier, and will occupy the remainder of this article, as well as its sequel.

In this article, we only give the skeleton of the proof of Proposition 3.1. That is, up to seven technical lemmas, we give a complete proof. The seven technical lemmas are proved in the sequel, K–theory for triangulated categories 3\frac{1}{2}(B). It seems to make the exposition much more transparent, when we split up the proof in this way. The point, which we try to highlight, is that the seven technical lemmas of K–theory for triangulated categories 3\frac{1}{2}(B) have nearly identical proofs.

It should be noted that much better results are true. They may be found in [9], [10], [11], [12] and [13]. However, I have it on good authority that these articles are nearly impenetrable. So far three referees have tried to read them. Thomason and Deligne found them confusing to say the least. Franke read the articles much later, after substantial revisions, and it was mostly thanks to his endorsement that they were published.

My first attempt to present the result dates back to September 1988. For the first two years, nobody believed the result, and no one was willing to waste his/her time to find the mistake. In the spring of 1990 Thomason became the referee. Unlike the earlier referees, Thomason was determined to find the mistake. He spent the next two years studying my work. Finally, in May 1992, we met in Paris and went carefully over the proof of one of the results in K–theory for triangulated categories III, which have since been published in [12] and [13]. After two grueling weeks, Thomason told me I could quote him as saying that he has checked one version of the result, and it is correct.

In late June 1992 I sent the articles to Deligne. He rejected them in November 1992, saying he had “… strong doubts about the correctness both of the proofs and of the statements… ”. In particular, he wrote that Theorem 4.8 on page 63 of K–theory for triangulated categories I (now published as [9] and [10]) was “at least confusing”, probably wrong.

In response, I wrote a direct, narrow article, proving Theorem 4.8 on page 63. I submitted the article to Deligne in May 1993. In December 1993 it was rejected. This time, Deligne did not referee it himself. He quoted the referee as saying “the result is certainly interesting, and if correct deserves publication”. Thus the narrow article also failed to be convincing.

In March 1994, Deligne was kind enough to spend three days listening to me present orally a proof of the same contentious Theorem 4.8 on page 63. This article is essentially a written account of that proof. The author is very grateful to Thomason, Deligne, Franke and several anonymous referees. It is admirable that they spent the time studying an argument that they were reluctant to believe.

1. Simplicial preliminaries

Our lemmas in this article all concern simplicial maps. The Lemmas break into two main types. Sometimes we show that a map is a homotopy equivalence, sometimes we show that it is a quasi–fibration. Since this is all–pervasive in the article, the following, introductory section is to remind the reader about some elementary simplicial facts.

Given two simplicial sets, a simplicial map between them is a map taking n–simplices in one to n–simplices in the other, and commuting with the face and degeneracy maps. Let S. and T. be simplicial sets. A homotopy from the realisation of S. to the realisation of T. is a map

|S.| × I ⟶ |T.|

where |S.| is the geometric realisation of S., |T.| is the geometric realisation of T., and I is the unit interval. To do this simplicially, we need to give a simplicial model for I. The model we give is the nerve of the category 1. It is the category

0 ⟶ 1.

The nerve of this category is the simplicial set whose n–simplices are functors

n ⟶ 1

where n is the category

0 ⟶ 1 ⟶ 2 ⟶ ⋯ ⟶ n.

To give a functor n ⟶ 1 is nothing more than to give an integer i with −1 ≤ i ≤ n, and send the integers {0, 1, ⋯, i} to 0, while the integers {i + 1, i + 2, ⋯, n} map to 1.

If we allow ourselves to confuse the category 1 with its nerve, that is the simplicial set just described, then a simplicial homotopy from S. to T. is just a simplicial map

S. × 1 ⟶ T.

In other words, to any n–simplex in the product S. × 1 we must assign an n–simplex in T.. An n–simplex in S. × 1 is a pair of an n–simplex in S. and an n–simplex in 1. As we have just learned, an n–simplex in 1 is nothing more than an integer i with −1 ≤ i ≤ n. Thus a simplicial homotopy assigns an n–simplex in T. to every pair consisting of an n–simplex in S. and an integer i.

Out of habit I prefer to think of simplicial homotopies slightly differently. To describe them, I prefer to give their images on slightly degenerate simplices. Let x ∈ S_{n} be an n–simplex. Then if i is an integer with 0 ≤ i ≤ n, let σ_{i} be the i^{th} degeneracy map. Then σ_{i}x is an n + 1 simplex, which is of course degenerate. The pair (σ_{i}x, i) is an n + 1–simplex in S. × 1, and I prefer to describe simplicial homotopies by giving their images on simplices such as these. Any simplex in S.×1 can be described as a face of one of my favorite simplices; giving the homotopy on those therefore clearly suffices. This way has the appeal that a simplicial homotopy sends an n–simplex x to n + 1 different (n+1)–simplices in T., namely one for every integer i with 0 ≤ i ≤ n. It is more natural to think of simplicial homotopies this way. A simplicial homotopy takes a simplex x to the image of x×I, and x×I subdivides into n + 1 simplices.

This article is a long sequence of lemmas. Each lemma proves one of two things. Either it proves that a map is a homotopy equivalence, or it proves that a map is a quasi–fibration. Almost in every case, we will study a map from an (m + n)–simplicial set to an n–simplicial set. What we mean by this is of course a map of (m + n)–simplicial sets, where on the target, m of the simplicial structures are degenerate.

The first observation is that the number of simplicial structures is really irrelevant. It is always possible to increase it by introducing some more degenerate structures, or to decrease it by diagonalising some of the structures. We henceforth stop explicitly counting the simplicial structures.

The prototype lemmas in this article are therefore:

Prototype Lemma 1 Let S.. be a bisimplicial set, T. a simplicial set. Let f : S.. ⟶ T. be a simplicial map. That is, it is a bisimplicial map, where T. is made into a bisimplicial set with one degenerate structure. The map f induces a homotopy equivalence.

Prototype Lemma 2 Let S.. be a bisimplicial set, T. a simplicial set. Let f : S.. ⟶ T. be a simplicial map. The map f induces a quasi–fibration.

The prototype proof in case (1) will be by Quillen’s Theorem A, whereas the prototype proof in case (2) will be by Quillen’s Theorem B. Let us briefly remind the reader.

Quillen’s Theorem A. Let S.. be a bisimplicial set, T. a simplicial set. Let f : S.. ⟶ T. be a simplicial map. Suppose that, after realising only the simplicial structure that is degenerate on T., the map f becomes a homotopy equivalence of simplicial spaces. Then the realisation of f is a homotopy equivalence. (For a proof, see Proposition A.1(ii) on page 308 of [15]).

Note that the realisation of T. with respect to the degenerate structure is discrete. To show that the realisation of f with respect to the structure degenerate on T. is a homotopy equivalence, one needs to show that the partial realisation of S.. has the homotopy type of a discrete simplicial space. More precisely, we need to show that the fibers over the points in T. are contractible. These fibers will henceforth be called the Segal fibers of the map f. Variant of Quillen’s Theorem B. Let S.. be a bisimplicial set, T. a simplicial set. Let f : S.. ⟶ T. be a simplicial map. Let us again realise only the structure which is degenerate in T.; then the realisation of S_{· n} is a union over all t_{n} ∈ T_{n} of the Segal fiber f^{−1}(t_{n}) of t_{n}. Suppose that for any face map ∂ : T_{n} ⟶ T_{n−1}, the induced map on Segal fibers

∂ : f^{−1}(t_{n}) ⟶ f^{−1}(∂t_{n})

is a homotopy equivalence. Then the realisation of f is a quasi–fibration. (This result follows easily from the unlabeled Lemma, end of Section 1 of [14], page 98 by the numbering at the bottom of the page. The page numbering of [14] is a little bizarre in that there are numbers both at the top and the bottom of each page, and the numbers do not agree. At the request of one of the numerous referees, I am including a proof, reducing the above to Quillen’s Theorem B. I am quite sure that this argument is not my own; I learned it somewhere. However, I cannot remember where, and could not easily find a reference). Proof of the Variant of Quillen’s Theorem B. First observe that, for any simplicial set X, its barycentric subdivision is the nerve of a category. Take the category where the objects are simplices, and a morphism x ⟶ y is a description of x as one of the faces of y. A simplex in the nerve of this category is a chain of simplices

x_{0} ⊂ x_{1} ⊂ ⋯ ⊂ x_{n}

and we think of it as the simplex built on the centers of these, in the barycentric subdivision.

Suppose now that we are given a simplicial map f : S.. ⟶ T., satisfying the hypothesis of the Variant of Quillen’s Theorem B. If we first separately barycentrically subdivide S.. in each simplicial direction, and then diagonally realise, we get the nerve of a category. We denote it S. Let the barycentric subdivision of T. be the nerve of a category T . There is a functor S ⟶ T , which induces the barycentric subdivision of the map f. We will denote this functor by

F : S ⟶ T .

We will show that the functor F satisfies the hypothesis of Quillen’s Theorem B. Now let us repeat what we have just said, with symbols.

Recall that the objects of the category Δ are

n = {0, 1, …, n},

and the morphisms are order–preserving maps. A bisimplicial set S_{..} is a functor

Δ^{op} × Δ^{op} ⟶ Sets.

In particular, for every pair of integers m, n ≥ 0, we have a set S_{mn}. We call the elements of S_{mn} the (m, n)–simplices of S_{..}.

An object in the category S is a simplex s ∈ S_{mn}. If s ∈ S_{mn} and s′ ∈ S_{m′n′} are two objects in S, then a morphism s′ ⟶ s is a pair of order–preserving inclusions

α : m′ ⟶ m, β : n′ ⟶ n

so that, under the map

α^{#} × β^{#} : S_{mn} ⟶ S_{m′n′}

s ∈ S_{mn} maps to s′ ∈ S_{m′n′}. The functor F : S ⟶ T sends the object s ∈ S_{mn} to the object f(s) ∈ T_{n}. It sends the morphism α×β above, to β. We wish to show that F satisfies the hypothesis of Quillen’s Theorem B.

For any object t ∈ T (that is, for a simplex t ∈ T.), we consider the comma category t\F. Its objects are pairs, consisting of an object s ∈ S, and a morphism t ⟶ F (s) in T . The barycentric subdivision of the Segal fiber is the subcategory of t\F, on which t = F (s), that is, on which the morphism t ⟶ F (s) is an equality. We denote the barycentric subdivision of the Segal fiber by

f^{−1}(t) ⊂ t\F.

We will show that the inclusion of the Segal fiber into t\F has a right adjoint. Hence the inclusion is a homotopy equivalence. We will see this as follows.

Let t ⟶ F (s) be an object in t\F. Then s is an (m, n)–simplex; that is, s ∈ S_{mn}. Also, F (s) is the n–simplex f(s) ∈ T_{n}. And the morphism t ⟶ F (s) corresponds to an inclusion

β : n′ ⟶ n

so that β^{#} : T_{n} ⟶ T_{n′} takes f(s) ∈ T_{n} to t ∈ T_{n′}. The object t ⟶ F (s) is contained in the (smaller) Segal fiber if β happens to be the identity. I assert that the inclusion of the Segal fiber has a right adjoint. Let me write down the counit of adjunction. The counit uniquely defines the adjoint; we leave it to the reader to check, that what we write down as counit does give an adjunction.

Let t ⟶ F (s) be an object in t\F. By the previous paragraph, s ∈ S_{mn}, t ∈ T_{n′}, and there exists an inclusion

β : n′ ⟶ n

with β^{#}[f(s)] = t. But then the pair

1 : m ⟶ m, β : n′ ⟶ n

gives a map

1^{#} × β^{#} : S_{mn} ⟶ S_{mn′}

and we will denote the image of s under 1^{#}×β^{#} by \overline{s}. From its definition above, we have a map in S

\overline{s} ⟶ s.

In T , we have a commutative diagram

\begin{matrix}t & = & F (\overline{s}) \\ || & & ↓ \\ t & ⟶ & F (s) \end{matrix}

This commutative diagram is a morphism in t\F, and is the counit of an adjunction. It follows that the inclusion of the Segal fiber f^{−1}(t) into t\F is a homotopy equivalence.

But now, for every morphism t′ ⟶ t in T, we have a commutative diagram

[figure: f^{−1}(t); t\F; f^{−1}(t′); t′\F]

The horizontal inclusions are homotopy equivalences by the above, while the map f^{−1}(t) ⟶ f^{−1}(t′) is a homotopy equivalence by the hypothesis of the Variant of Quillen’s Theorem B. Therefore the map

t\F ⟶ t′\F

is a homotopy equivalence. By Quillen’s Theorem B, F is a quasifibration. □

Both Quillen’s Theorem A and the Variant of Quillen’s Theorem B reduce statements about f to statements about the Segal fibers. We either have to show the Segal fibers contractible, or to show the face maps to be homotopy equivalences.

Let me say something concerning the way we will go about using Quillen’s Theorem B. In this article and its sequel, it is used exactly twice. Both times, the proof is structured as follows. First we find a fixed space X and a homotopy equivalence

X ⟶ f^{−1}(t_{n})

of X with the Segal fiber f^{−1}(t_{n}). This shows that the Segal fiber has a homotopy type independent of t_{n}. Then, to establish that the face maps are homotopy equivalences, we consider the composite

X ⟶ f^{−1}(t_{n}) ⟶^{∂} f^{−1}(∂t_{n}) ⟶ X.

This is a map from X to itself. We then prove it a homotopy equivalence.

In finding the space X and the homotopy equivalence X ⟶ f^{−1}(t_{n}), we generally begin with the identity on f^{−1}(t_{n}), and construct a sequence of homotopies connecting it to a map which factors as

f^{−1}(t_{n}) ⟶^{ϕ} X ⟶^{ψ} f^{−1}(t_{n}).

Thus ψ ∘ ϕ is homotopic to the identity. Then one proves directly that ϕ ∘ ψ is also homotopic to the identity.

The effect is that an application of Quillen’s Theorem B, in this article and its sequel, always starts with a string of homotopies. The reason I have organised the proof this way is to highlight the homotopies, and in particular to highlight the fact that there are so few.

The way the pair of articles has been organised is that first we give the skeleton of the argument, presenting all the statements in logical order, and applying Quillen’s Theorem A only to statements whose proof is trivial. We define a proof to be trivial if it requires only one simplicial homotopy.

The “difficult” statements are all given first without proof. The proofs are all presented together in the sequel, K–theory for triangulated categories 3\frac{1}{2}(B). The “difficult” statements include, in particular, both applications of Quillen’s Theorem B.

To make it easy for the reader to see when a statement is being left to be proved later, all such statements are labeled “Step …”. They appear in the table of contents, so the reader can easily check back for the statement of any “Step”, that is check exactly what it was that was being left to prove later. In this way, hopefully the reading of the article will be made a little easier. First the reader gets an overview of the proof, and then come the difficult details.

2. The proof of Proposition 2.3

With the notation as in the Introduction, we now want to talk about monos and epis in the category Gr_{[m,n]}(A).

DEFINITION 2.1. A morphism X ⟶ Y in Gr_{[m,n]}(A) is called mono, and denoted

[figure: X; Y]

if the map H^{m}(f) : H^{m}(X) ⟶ H^{m}(Y) is a monomorphism.

DEFINITION 2.2. A morphism X ⟶ Y in Gr_{[m,n]}(A) is called epi, and denoted

[figure: X; Y]

if the map H^{n}(f) : H^{n}(X) ⟶ H^{n}(Y) is an epimorphism.

We can now consider an impressive array of simplicial sets; they are the

subsets of Gr[figure: [m,n]] where either the vertical or the horizontal arrows

are restricted to be either mono or epi. Thus in Gr[figure: [m,n]] both the

vertical and the horizontal morphisms are assumed mono. On the other

hand, Gr[figure: [m,n]] places no restriction on the horizontal morphisms,

while the vertical ones are assumed epi. In this section we will prove:

PROPOSITION 2.3. The following two natural inclusions induce homotopy equivalences

Gr[figure: [m,n]]  α  \overline{}[figure]  Gr[figure: [m,n]]

Gr[figure: [m,n]]  β  \overline{}[figure]  Gr[figure: [m,n]]

REMARK 2.4. If we take m = n = 0, then the fact that β ∘ α is a homotopy equivalence is the weak version of Proposition 2.3 that we needed in the Introduction. The stronger version proved here is of more than just idle interest; we will use the fact that β is a homotopy equivalence, when m and n are not both zero.

Proof of Proposition 2.3. Let us prove in detail that the inclusion

Gr[figure: [m,n]]  β  \overline{}[figure]  Gr[figure: [m,n]]

induces a homotopy equivalence. Then we will discuss how to modify the proof to show that so does α.

We introduce the trisimplicial set, which we denote

Gr[figure: [m,n]]  Gr[figure: [m,n]]

By definition, an (r, s, t)-simplex consists of a commutative diagram in Gr_{[m,n]}(A)

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}]

together with a coherent differential ∂ : Y_{rt} → ΣX_{00}. The i^{th} face maps are deleting the i^{th} column in the X square, the i^{th} column in the Y square, or the i^{th} row. There are three different commuting simplicial structures, hence a trisimplicial set. The degeneracy maps are the corresponding insertions of identities.

The idea is that the symbol for the simplicial set should also be its

definition. The simplicial set whose symbol is Gr[figure: [m,n]] Gr[figure: [m,n]]

is made by concatenating the simplicial sets Gr[figure: [m,n]] and Gr[figure: [m,n]] .

Thus the X’s lie in Gr[figure: [m,n]] , while the Y ’s are in Gr[figure: [m,n]] . The

two horizontal arrows joining the two boxes in the symbol

Gr[figure: [m,n]] Gr[figure: [m,n]] indicate what restriction, if any, is placed

on the horizontal morphisms connecting the X’s and Y ’s.

There are two projections out of our trisimplicial set. The first “forgets” the Y ’s, the second “forgets” the X’s. We will denote these maps

Gr[figure: [m,n]]  Gr[figure: [m,n]] g_{1} [figure]  g_{2} [figure] Gr[figure: [m,n]]  [figure] [figure]  [figure] [figure] Gr[figure: [m,n]]

K–Theory for Triangulated Categories 3 \frac{1}{2} (A) 23

We will show that these maps are homotopy equivalences. This immediately establishes that there is some homotopy equivalence of Gr[figure: [m,n]]

with Gr[figure: [m,n]] , but it is not completely clear that it is induced by the inclusion. The standard way to remedy this is to consider the larger commutative diagram

Gr[figure: [m,n]] Gr[figure: [m,n]] f_{1} [figure]  f_{2} [figure]^{Gr[figure: [m,n]] [figure] [figure]}  ^{Gr[figure: [m,n]] [figure] [figure]} [figure]^{Gr[figure: [m,n]]  Gr[figure: [m,n]]}  β [figure] g_{1} [figure]  g_{2} [figure] Gr[figure: [m,n]]  [figure] [figure]  [figure] [figure] Gr[figure: [m,n]]

We want to prove that the map β above is a homotopy equivalence. What we will show is that each of the maps f_{1}, f_{2}, g_{1} and g_{2} is a homotopy equivalence. But then the commutativity forces β to also be.

2.4.1. Notation. I hope that the notation is self explanatory. For example, the symbol

Gr[figure: [m,n]] Gr[figure: [m,n]]

stands for the trisimplicial set which consists of diagrams

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}]

We recall that the symbols inside the squares indicate the restrictions on the objects and the morphisms within the square (the Gr_{[m,n]} says the objects are restricted to lie in that category, while the two superscript arrows denote the restrictions on the horizontal and vertical morphisms). The arrows connecting the squares denote the restrictions on the horizontal morphisms connecting X’s and Y ’s.

It is very well known how to show that a map of the type

Gr[figure: [m,n]]  Gr[figure: [m,n]]  g_{2} \overline{}[figure] [figure] Gr[figure: [m,n]]

is a homotopy equivalence. The criterion is Quillen’s Theorem A; it suffices to show that each fiber is contractible. But let us explain this carefully once. The term

[figure] [figure] Gr[figure: [m,n]]  Gr[figure: [m,n]]

is naturally a bisimplicial set. Out of perversity, we view it as a trisimplicial set with a degenerate simplicial structure, in a way which makes

Gr[figure: [m,n]]  Gr[figure: [m,n]]  \overline{}[figure] [figure] Gr[figure: [m,n]]

a map of trisimplicial sets. By Tornhave’s theorem, we can geometrically realize the map in any order we wish. We wish to realize first the simplicial structure which is degenerate on

[figure] [figure] Gr[figure: [m,n]]

i.e. where the face maps are deleting a column in the left-hand square. There is an induced map of bisimplicial spaces. By a theorem of Segal, (Proposition A.1(ii) on page 308 of [15]), it suffices to show that this map is a homotopy equivalence. But on the right we get a discrete space consisting of the squares

[figure: ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}]

and the fiber over this square is the geometric realization of a simplicial set, which we will denote

Gr[figure: [m,n]]  Y

This is our notation for the simplicial set of diagrams

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}]

where the array of Y ’s is held fixed. The solid box around the Y ’s is to remind us that everything inside it is fixed. The integers r and t are held fixed, the Y_{ij}’s are fixed, and so are the maps among them, including the differential Y_{rt} ⟶ ΣY_{00}. We will call this simplicial set the Segal fiber of the map g_{2}. If we now argue similarly for the other maps in our diagram, we deduce that to prove our theorem it suffices to establish the contractibility of four simplicial sets.

2.4.2. To prove that f_{1} is a homotopy equivalence, it suffices to establish the contractibility of its Segal fiber,

X Gr[figure: [m,n]] .

2.4.3. To prove that f_{2} is a homotopy equivalence, it suffices to establish the contractibility of its Segal fiber,

Gr[figure: [m,n]] Y  .

2.4.4. To prove that g_{1} is a homotopy equivalence, it suffices to establish the contractibility of its Segal fiber,

X Gr[figure: [m,n]] .

2.4.5. To prove that g_{2} is a homotopy equivalence, it suffices to establish the contractibility of its Segal fiber,

Gr[figure: [m,n]]  Y  .

The reader will undoubtedly notice that the contractibility statements in 2.4.2, 2.4.3 and 2.4.4 are trivial. Nevertheless, the author wishes to be permitted to include a proof of the contractibility in 2.4.4. This is done mostly to introduce some handy notation and illustrate it in a particularly simple case.

2.4.6. Proof of the contractibility of

X Gr[figure: [m,n]] .

We remind the reader that a simplex in

X  Gr[figure: [m,n]]

is a commutative diagram

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}]

where the X’s are held fixed. There is only one simplicial structure, obtained by varying the value of the integer t. We want to prove the contractibility of the simplicial set. It suffices to construct a homotopy of the identity map with a collapsing map. A simplicial homotopy takes an integer i with 0 ≤ i ≤ t and a t–simplex

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{r0}; X_{rs}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{r0}; Y_{rt}]

to a (t + 1)–simplex. The contracting homotopy we have in mind takes the above pair, that is the integer i and the t–simplex, to

[figure: ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{s0}; X_{ss}; ⋮; ⋮; ⋯; ⋯; X_{0s}; X_{0s}; X_{ss}; X_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{0i}; Y_{0t}; Y_{si}; Y_{st}; _{(i+1)−times}; _{(t−i+1)−terms}; ; ]

The way to think of this homotopy is that, as i increases, 0 ≤ i ≤ t, this homotopy gradually replaces the Y ’s by X’s, contracting the simplex. This is the “contraction to the initial object”, which is so useful in the classical version of K–theory. The column of X’s on the right hand side will be denoted X_{E}, where the E stands for East. This is the east face of the square of X’s. Our shorthand for the homotopy is

X  X_{E} \overline{}Gr[figure: [m,n]]

2.4.7. We will use the subscripts E, W, N and S to refer to the East, West, North and South faces of a (fixed) simplex. The arrow in the above square is to remind us that the homotopy is creeping in from the left, gradually replacing the Y ’s with the X’s.

The homotopy displayed above connects the identity with a map we will denote

[figure: X; X_{E}]

and this last map is clearly a contraction. □

The proofs of the contractibility in 2.4.2 and 2.4.3 are closely parallel. One uses the contracting homotopies

X  X_{E} \overline{}Gr[figure: [m,n]]

and

Gr[figure: [m,n]]  Y_{W} \overline{}Y

respectively to provide the contractions. The heart of the proof is therefore to establish the contractibility statement in 2.4.5. This contractibility statement is slightly sutbtler than the ones we have seen so far. We leave it as a Step, to be verified at the end. As we promised, all Steps are highlighted for easy reference. What we are leaving to be proved at the end is that the map g_{2} below induces a homotopy equivalence

Gr[figure: [m,n]]  Gr[figure: [m,n]]  g_{2}  \overline{}[figure] [figure] Gr[figure: [m,n]]

Before I label this as a step, let me dualise the statement. What we will prove is

Step 2.1. :

The natural map

gr84.tex; 19/06/2000; 14:06; p.29

Gr[figure: [m,n]]^{}^{}^{}^{}Gr[figure: [m,n]]  \overline{}^{[figure] [figure] Gr[figure: [m,n]]}

induces a homotopy equivalence.

We are not yet done with the proof of Proposition 2.3. Modulo Step 2.1, we now know that the map

Gr[figure: [m,n]]  β  \overline{}[figure]  Gr[figure: [m,n]]

is a homotopy equivalence. It remains to show that the map

Gr[figure: [m,n]]  α  \overline{}[figure]  Gr[figure: [m,n]]

also induces a homotopy equivalence. Let us first dualise this. We can equivalently show that

Gr[figure: [m,n]]  α^{op}  \overline{}[figure]  Gr[figure: [m,n]]

induces a homotopy equivalence. To do this, study the commutative diagram

gr84.tex; 19/06/2000; 14:06; p.30

Gr[figure: [m,n]] Gr[figure: [m,n]] f_{1} [figure]  f_{2} [figure]^{Gr[figure: [m,n]] [figure] [figure]}  ^{[figure] [figure] Gr[figure: [m,n]]} [figure]^{Gr[figure: [m,n]]  Gr[figure: [m,n]]}  α^{op} [figure] g_{1} [figure]  g_{2} [figure] Gr[figure: [m,n]]  [figure] [figure]  [figure] [figure] Gr[figure: [m,n]]

We wish to show that α^{op} is a homotopy equivalence. We will show it by establishing that f_{1}, f_{2}, g_{1} and g_{2} are. The Segal fiber of f_{1} is contracted by the homotopy

X  X_{E} \overline{}Gr[figure: [m,n]]

the Segal fiber of f_{2} is contracted by

Gr[figure: [m,n]]  Y_{W} \overline{}Y

and the Segal fiber of g_{1} is contracted by the homotopy

X  X_{E} \overline{}Gr[figure: [m,n]]

The only subtlety comes from the Segal fiber of g_{2}. Once again, we first dualise, then leave it as a Step. It remains to show that the map g_{2} induces a homotopy equivalence. Equivalently, it suffices to show that its dual does. We are therefore reduced to:

Step 2.2. :

The natural map

Gr[figure: [m,n]]^{}^{}^{}^{}Gr[figure: [m,n]]  \overline{}^{[figure] [figure] Gr[figure: [m,n]]}

induces a homotopy equivalence.

3. First reductions in the proof of Proposition 3.1

As indicated in the introduction, it remains to prove:

PROPOSITION 3.1. The natural inclusion

Gr[figure: [1,n]]  i  \overline{}[figure]  Gr[figure: [0,n]]

gr84.tex; 19/06/2000; 14:06; p.32

induces a homotopy equivalence.

Of course, the map i factors as a composite of two inclusions

Gr[figure: [1,n]]  α \overline{}[figure]  Gr[figure: [0,n]]  β  \overline{}[figure]  Gr[figure: [0,n]]

Note that the map α is well defined. Any morphism in Gr_{[1,n]} becomes a mono in Gr_{[0,n]}. If f : X ⟶ Y is a morphism in Gr_{[1,n]}, then H^{0}(f) : H^{0}(X) ⟶ H^{0}(Y) is most certainly a monomorphism in A, since H^{0}(X) = H^{0}(Y) = 0. By Proposition 2.3, we already know that β induces a homotopy equivalence. It therefore suffices to show that α is a homotopy equivalence.

The idea of the proof will be to study the commutative square

[figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]]  γ_{1} \overline{}[figure]  ^{[figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]} γ_{2}  γ_{3} [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]  α \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] Gr[figure: [0,n]]

gr84.tex; 19/06/2000; 14:06; p.33

Of course, I owe the reader a definition of the simplicial sets in this square. But observe for now that the bottom row needs no new definition; it is precisely (the transpose of) the map

Gr[figure: [1,n]]  α  \overline{}[figure]  Gr[figure: [0,n]]

We will ultimately prove that α is a homotopy equivalence by showing that γ_{1}, γ_{2} and γ_{3} are. Of these, γ_{1} is easy and will be done in this Section. Proving that γ_{2} is a homotopy equivalence is straight–forward, but technical. Hence we leave it as a Step. Might as well do so right now:

Step 3.1. :

The natural map

[figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

induces a homotopy equivalence.

The last remaining fact, that is the proof that γ_{3} is a homotopy equivalence, will occupy the rest of the pair of articles. So we recall that the remainder of this Section is devoted to proving that the map γ_{1} induces a homotopy equivalence.

Before we go on I owe the reader a definition of the simplicial

sets [figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]] and [figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]] . As the

shorthand for the simplicial set is meant to indicate, each of these sets is put together from three smaller pieces; the pieces are two rectangles and a strange–looking triangle.

In each case, a simplex is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}]

Thus a triangle in the symbol for the simplicial set indicates a triangle of Z’s, as it sensibly should. The 0 in the center of the triangle means that the only restriction placed on the Z_{ij}’s is that Z_{ii} = 0, that is the diagonal terms must vanish.

Of course, we assume that the diagram is a commutative diagram in Gr(A), but it is not immediately clear what should be meant by the coherence of the differentials. The requirement we place is the following. Every rectangle, embedded in the large diagram defining the simplex, should come with a differential, giving a long exact sequence in A. The coherence condition we impose is that when two rectangles are such that one lies inside the other, the differential on the large rectangle should induce the differential on the small, as in Definition 0.2.

36 Amnon Neeman

The remaining restrictions on the X’s and Y ’s are as indicated by the symbols inside the box. In the simplicial set [figure] 0 Gr[figure: [1,n]] Gr[figure: [1,n]] , a simplex is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}]

where the X’s and Y ’s are constrained to lie in Gr_{[1,n]}, and the only restriction on the morphisms is that the vertical morphisms between Y ’s are assumed epi. In the simplicial set [figure] 0 Gr[figure: [0,n]] Gr[figure: [0,n]] , a

simplex is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}]

where the X’s and Y ’s are free to lie in the larger category Gr_{[0,n]}, but now all vertical morphisms among X’s and Y ’s are assumed mono. The vertical morphisms connecting Y ’s are also assumed epi; an arrow

[figure: Y_{1}; Y_{2}]

is simultaneously mono and epi; it will be referred to as a “monoepi”. Recalling that every morphism in Gr_{[1,n]} is mono when viewed as a

38 Amnon Neeman

map in Gr_{[0,n]}, we see that the simplicial set [figure] 0 Gr[figure: [0,n]] Gr[figure: [0,n]] contains [figure] 0 Gr[figure: [1,n]] Gr[figure: [1,n]] ; it is less of a restriction to demand that a simplex lie in [figure] 0 Gr[figure: [0,n]] Gr[figure: [0,n]] . The map γ_{1} is the natural inclusion. Letting γ_{2} and γ_{3} be the natural projections, we obtain the commutative diagram referred to already

[figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]]  γ_{1} \overline{}[figure]  ^{[figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]} γ_{2}  γ_{3} [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]  α \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] Gr[figure: [0,n]]

We have left as a Step the fact that γ_{2} induces a homotopy equivalence. It remains to show LEMMA 3.2. In the diagram above, γ_{1} is a homotopy equivalence. PROPOSITION 3.3. In the diagram above, γ_{3} is a homotopy equivalence. Proof of Lemma 3.2. We must show that the map

[figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]]  γ_{1} \overline{}[figure]  [figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]

induces a homotopy equivalence. Consider the diagram

[figure] 0  Gr[figure: [1,n]] Gr[figure: [1,n]]  γ_{1} \overline{}[figure]  ^{[figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]} δ  δ′ [figure] 0  Gr[figure: [1,n]] [figure] [figure]  g \overline{}[figure]  [figure] 0  Gr[figure: [0,n]] [figure] [figure]

We will show that δ, δ′ and g are homotopy equivalences. The Segal fiber of δ is the simplicial set

K–Theory for Triangulated Categories 3 \frac{1}{2} (A) 41

[figure] Z  Y Gr[figure: [1,n]]

and it is contracted by the homotopy

[figure] Z  Y Y_{S} Gr[figure: [1,n]]

The Segal fiber for δ′ is the simplicial set

[figure] Z  Y Gr[figure: [0,n]]

and it is contracted by the homotopy

                                      gr84.tex; 19/06/2000; 14:06; p.41

[figure] Z  Y Y_{S} Gr[figure: [0,n]]

So far, this has proved that δ and δ′ are homotopy equivalences. We are therefore reduced to proving that so is g. That is, we want to show that the natural inclusion

[figure] 0  Gr[figure: [1,n]]  g  \overline{}[figure]  [figure] 0  Gr[figure: [0,n]]

induces a homotopy equivalence. To do this, we need the lemma:

LEMMA 3.4. In both the simplicial set [figure] 0 Gr[figure: [1,n]] and

[figure] 0 Gr[figure: [0,n]] , the objects Z in a typical cell

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}]

K–Theory for Triangulated Categories 3 \frac{1}{2} (A) 43

lie in the subcategory Gr_{[1,n]}. Furthermore, the vertical maps Z_{ij} ⟶ Z_{i′j} are epi. Proof. Because the simplicial set [figure] 0 Gr[figure: [1,n]] is contained in [figure] 0 Gr[figure: [0,n]] , it is enough to show that the statement holds for the larger [figure] 0 Gr[figure: [0,n]] . In other words, it suffices to show that a simplex in the simplicial set

[figure] 0 Gr[figure: [0,n]] , which is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}]

satisfies Z_{ij} ∈ Gr_{[1,n]} and Z_{ij} ⟶ Z_{i′j} is epi. We have a square

[figure: Z_{jj}; Y_{j0}; Z_{ij}; Y_{i0}]

Now recall that Z_{jj} = 0. We deduce a long exact sequence

H^{q}(Y_{i0}) ⟶^{ϕ_{q}} H^{q}(Y_{j0}) ⟶ H^{q+1}(Z_{ij}) ⟶ H^{q+1}(Y_{i0}) ⟶^{ϕ_{q+1}} H^{q+1}(Y_{j0})

and we are given that H^{q}(Y_{i0}) = H^{q}(Y_{j0}) = 0 unless 0 ≤ q ≤ n, while ϕ_{0} is a monomorphism and ϕ_{n} an epimorphism. It immediately follows that H^{q}(Z_{ij}) = 0 unless 1 ≤ q ≤ n. Thus Z_{ij} is indeed an object of Gr_{[1,n]}.

Now we need to show that Z_{ij} ⟶ Z_{i′j} is epi, whenever 0 ≤ i ≤ i′ ≤ j. But there is a square

[figure: Z_{i′i′}; Z_{i′j}; Z_{ii′}; Z_{ij}]

and recalling once again that Z_{i′i′} = 0, we have an exact sequence

H^{n}(Z_{ij}) ⟶^{ψ_{n}} H^{n}(Z_{i′j}) ⟶ H^{n+1}(Z_{ii′}) = 0

which establishes that ψ_{n} is an epimorphism. □ If we wanted to, and now we do, we could denote the simplicial set

[figure] 0 Gr[figure: [0,n]] by the symbol

[figure] Gr[figure: [1,n]]  Gr[figure: [0,n]]

In other words, we could allow the symbol for the simplicial set to carry more explicitly the restrictions on the objects and morphisms. When it is advantageous to remind ourselves of these restrictions, we will use the more explicit symbol. End of the proof of Lemma 3.2. Now recall that we wish to prove that the map g in the commutative square below

[figure]^{Gr[figure: [1,n]]}  Gr[figure: [1,n]]  g  \overline{}[figure]  ^{[figure]^{Gr[figure: [1,n]]}  Gr[figure: [0,n]]} d  d′^{[figure]^{Gr[figure: [1,n]]}  [figure] [figure]}  ^{[figure]^{Gr[figure: [1,n]]}  [figure] [figure]}

is a homotopy equivalence. It clearly suffices to show that d and d′ are homotopy equivalences. The Segal fiber of d is the simplicial set

[figure] Z  Gr[figure: [1,n]]

which is contracted by the homotopy

[figure] Z  Z_{E} \overline{}Gr[figure: [1,n]]

The Segal fiber of d′ is

[figure] Z  Gr[figure: [0,n]]

which is contracted by the homotopy

[figure] Z  Z_{E} \overline{}Gr[figure: [0,n]]

and this completes the proof of Lemma 3.2. □

4. The homotopy fiber in Proposition 3.3

It remains now only to prove Proposition 3.3, which states that the map

[figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]  γ  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [0,n]]

induces a homotopy equivalence. The proof we will give is based on a study of the Segal fiber. It turns out to be true that the map γ is a quasi–fibration. The Segal fibers are all homotopy equivalent, and in fact the face maps induce homotopy equivalences. This is true, but is difficult to prove directly. We will need slightly less.

The Segal fiber of γ is the simplicial set [figure] 0  Gr[figure: [0,n]] X  .

A simplex is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}]

where the square of X’s is held fixed. One could, of course, consider the special case where the square of X’s is very small. Let us take the special case of the Segal fiber over the (0,0)–simplex

0

In that case, a simplex in the Segal fiber is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0s}; Y_{ss}; 0]

We denote the simplicial set of such diagrams [figure: 0; Gr; [0,n]].

Ultimately, we will know that the map γ is a quasi–fibration, which means that for every X the simplicial sets

[figure] 0  Gr[figure: [0,n]] X  and  [figure: 0; Gr; [0,n]]

are homotopy equivalent. In fact, the face maps all induce homotopy equivalences. But at this stage we need less. What we leave as a Step is:

Step 4.1. :

Let γ be the map

[figure] 0  Gr[figure: [0,n]] Gr[figure: [0,n]]  γ  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [0,n]]

Up to homotopy, the identity on the Segal fiber of γ, that is the identity on the simplicial set

[figure] 0 Gr[figure: [0,n]] X,

factors through the simplicial set [figure: 0; Gr; [0,n]].

If we grant Step 4.1, then to prove the contractibility of the simplicial

set [figure] 0  Gr[figure: [0,n]] X  it suffice to show [figure: 0; Gr; [0,n]]

contractible. Thus Proposition 3.3, and hence the entire proof, is now reduced to showing [figure: 0; Gr; [0,n]] contractible. To prove the contractibility, we will have a lemma and a proposition: LEMMA 4.1. The natural inclusion

[figure: 0; Gr; [0,n]; α; 0; Gr; [0,n+1]]

is null homotopic; it factors, up to homotopy, through a contractible space. PROPOSITION 4.2. The natural inclusion

[figure: 0; Gr; [0,n]; α; 0; Gr; [0,n+1]]

is a homotopy equivalence.

Proof of Lemma 4.1. Recall that any morphism in Gr_{[0,n]} is epi when viewed as a morphism in Gr_{[0,n+1]}. It therefore follows that the map α above factors as

[figure: 0; Gr; [0,n]; 0; Gr; [0,n]; 0; Gr; [0,n+1]]

But the simplicial set [figure: 0; Gr; [0,n]] is contracted by the “con-

traction to the initial object”. Recall that a homotopy takes an integer i with 0 ≤ i ≤ s and an s–simplex to an (s+1)–simplex. Our particular homotopy takes the integer i and the s–simplex

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; Y_{0}; Y_{s}]

to the (s + 1)–simplex

[figure: ⋮; ⋯; ·; ·; ·; Z_{ii}; Z_{is}; Z_{ss}; ⋮; Y_{i}; Y_{s}; ⋮; ⋯; ·; ·; ·; 0; 0; 0; ⋮; ⋮; ⋯; ⋯; Σ^{−1}Y_{i}; Σ^{−1}Y_{s}; Σ^{−1}Y_{i}; Σ^{−1}Y_{s}; ⋮; 0; 0]

We need to tell the reader what the maps in the cell are, and to prove that it is a cell of a homotopy.

The interesting maps of degree zero (that is, maps which are neither differentials nor in the original diagram) are the maps Σ^{−1}Y ⟶ Z and Σ^{−1}Y ⟶ Y′. The maps Σ^{−1}Y ⟶ Y′ are 0. The maps Σ^{−1}Y ⟶ Z are defined to be the desuspensions of the given differentials Y ⟶ ΣZ of the simplex we began with.

Next, the differentials into objects of the form Σ^{−1}Y in the diagram are of two sorts. There are the differentials Z ⟶ Σ [Σ^{−1}Y ], and the differentials Y ⟶ Σ [Σ^{−1}Y′]. In either case, we define them to be the maps Z ⟶ Y and Y ⟶ Y′ of the original simplex.

This has defined all the maps and differentials of the proposed simplex in the contracting homotopy. The reader now needs to check that if we forget the differentials (that is, consider only the maps of degree 0), the diagram commutes. It needs to be checked that the composite of a differential with a map that is not a differential, in either order, is a differential. It needs also be checked that a differential between two objects in the same row or column must vanish. All this is straight forward, and is left to the reader.

To show that the simplex is well defined, we need to also prove that every square gives a long exact sequence in the abelian category A. At first, this may seem formidable, since we have to check many possible squares. But there is a helpful little lemma which we will now state, and which will be relied on a lot.

LEMMA 4.3. Suppose we are given a commutative diagram in Gr^{b}(A)

[figure: B; B′; B′′; A; A′; A′′]

together with a candidate coherent differential ∂ : B′′ ⟶ ΣA. Suppose we know that the composites

B′ ⟶ B′′ ⟶^{∂} ΣA ⟶ ΣA′

B′′ ⟶^{∂} ΣA ⟶ ΣA′′

B ⟶ B′′ ⟶^{∂} ΣA

A′′ ⟶ B′′ ⟶^{∂} ΣA

B′′ ⟶^{∂} ΣA ⟶ ΣB

all vanish. Then if any two of the following sequences is exact, so is the third

\begin{aligned}H^{q−1}(B′) & ⟶^{ϕ_{1}} & H^{q}(A) & → & H^{q}(A′) ⊕ H^{q}(B) & → & H^{q}(B′) & ⟶^{ϕ_{1}} & H^{q+1}(A) \\ H^{q−1}(B′′) & ⟶^{ϕ_{2}} & H^{q}(A) & → & H^{q}(A′′) ⊕ H^{q}(B) & → & H^{q}(B′′) & ⟶^{ϕ_{2}} & H^{q+1}(A) \\ H^{q−1}(B′′) & ⟶^{ϕ_{3}} & H^{q}(A′) & → & H^{q}(A′′) ⊕ H^{q}(B′) & → & H^{q}(B′′) & ⟶^{ϕ_{3}} & H^{q+1}(A′) \end{aligned}

where ϕ_{2} = ∂, and ϕ_{1} and ϕ_{3} are the obvious composites involving ∂.

Proof. The hypotheses are clearly enough to assure us that each of the three sequences

\begin{aligned}H^{q−1}(B′) & ⟶^{ϕ_{1}} & H^{q}(A) & → & H^{q}(A′) ⊕ H^{q}(B) & → & H^{q}(B′) & ⟶^{ϕ_{1}} & H^{q+1}(A) \\ H^{q−1}(B′′) & ⟶^{ϕ_{2}} & H^{q}(A) & → & H^{q}(A′′) ⊕ H^{q}(B) & → & H^{q}(B′′) & ⟶^{ϕ_{2}} & H^{q+1}(A) \\ H^{q−1}(B′′) & ⟶^{ϕ_{3}} & H^{q}(A′) & → & H^{q}(A′′) ⊕ H^{q}(B′) & → & H^{q}(B′′) & ⟶^{ϕ_{3}} & H^{q+1}(A′) \end{aligned}

is a chain complex; any two maps compose to give 0. Thus each of the three sequences above defines an object in D^{b}(A), the derived category of A. It is easy to check that the natural map

H^{q−1}(B′) ⟶^{ϕ_{1}} H^{q}(A) → H^{q}(A′) ⊕ H^{q}(B) → H^{q}(B′) ⟶^{ϕ_{1}} H^{q+1}(A)
↓  ↓  ↓  ↓  ↓
H^{q−1}(B′′) ⟶^{ϕ_{2}} H^{q}(A) → H^{q}(A′′) ⊕ H^{q}(B) → H^{q}(B′′) ⟶^{ϕ_{2}} H^{q+1}(A)

viewed as a morphism between objects in D^{b}(A), has for its mapping cone the object

H^{q−1}(B′′) ⟶^{ϕ_{3}} H^{q}(A′) → H^{q}(A′′) ⊕ H^{q}(B′) → H^{q}(B′′) ⟶^{ϕ_{3}} H^{q+1}(A′)

Hence if any two of the sequences is exact, so is the third. □

Next we wish to apply Lemma 4.3, to prove that all the squares in the diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{ii}; Z_{is}; Z_{ss}; ⋮; Y_{i}; Y_{s}; ⋮; ⋯; ·; ·; ·; 0; 0; 0; ⋮; ⋮; ⋯; ⋯; Σ^{−1}Y_{i}; Σ^{−1}Y_{s}; Σ^{−1}Y_{i}; Σ^{−1}Y_{s}; ⋮; 0; 0]

give rise to long exact sequences. To begin with, this is trivial unless we are considering a square where one row lies below i and one above i; that is, interesting squares have the form

Z ⟶ Z′
↑  ↑
Σ^{−1}Y ⟶ Σ^{−1}Y′

This square is confusing, so consider the larger diagram

Z ⟶ Z′ ⟶ Y′′
↑ ↑ ↑
Σ^{−1}Y ⟶ Σ^{−1}Y′ ⟶ 0

By Lemma 4.3, it clearly suffices to consider the squares

Z ⟶ Y′′  Z′ ⟶ Y′′
↑ ↑ and ↑ ↑
Σ^{−1}Y ⟶ 0  Σ^{−1}Y′ ⟶ 0

hence it suffices to consider diagrams of the form

Z ⟶ Y′
↑  ↑
Σ^{−1}Y ⟶ 0

If we find this to still be a little confusing, we can consider the larger diagram

Z_{ii} ⟶ Y
↑  ↑
Z ⟶ Y′
↑  ↑
Σ^{−1}Y ⟶ 0

and the fact that the square

Z ⟶ Y′
↑  ↑
Σ^{−1}Y ⟶ 0

gives an exact sequence in A will follow if the two squares

Z_{ii} ⟶ Y  Z_{ii} ⟶ Y
↑ ↑ and ↑ ↑
Z ⟶ Y′  Σ^{−1}Y ⟶ 0

do. The square

\begin{matrix}Z_{ii} & ⟶ & Y \\ ↑ & & ↑ \\ Z & ⟶ & Y′ \end{matrix}

is a square in our old simplex, and hence must give a long exact sequence in A. Recalling that Z_{ii} = 0, the other square is just

\begin{matrix}0 & ⟶ & Y \\ ↑ & & ↑ \\ Σ^{−1}Y & ⟶ & 0 \end{matrix}

and yields the exact sequence

H^{q}(Y) ⟶ H^{q}(Y) ⟶ 0 ⟶ H^{q+1}(Y) ⟶ H^{q+1}(Y).
□

It now only remains to prove Proposition 4.2. Before we go on, we would like to remind the reader briefly about homotopy cartesian squares.

5. A reminder concerning homotopy cartesian squares

Suppose we are given a commutative square of maps of topological spaces

\begin{matrix}X & ⟶ & Y \\ ↓ & & ↓ \\ X′ & ⟶ & Y′ \end{matrix}

and suppose for simplicity that Y′ is path connected. The commutative square is called homotopy cartesian if the natural map from X to the homotopy pullback is a weak homotopy equivalence. The most obvious example is if both vertical maps are homotopy equivalences.

Suppose we are given a commutative diagram

\begin{matrix}X & ⟶ & Y \\ ↓ & & ↓ \\ X′ & ⟶ & Y′ \\ ↓ & & ↓ \\ X′′ & ⟶ & Y′′ \end{matrix}

where for simplicity we assume Y′ and Y′′ path connected. If any two of the the three squares

X ⟶ Y  X ⟶ Y  X′ ⟶ Y′
↓  ↓  ↓  ↓  ↓  ↓
X′ ⟶ Y′  X′′ ⟶ Y′′  X′′ ⟶ Y′′

is homotopy cartesian, it is well known that so is the third. This is a fact which we will use repeatedly, so the author felt he should remind the reader.

A special case of interest is when Y is contractible; to say that the two squares

[figure: X; Y; X; Y]

[figure]
[figure: X′; f; Y′; X′′; f′; Y′′]

are homotopy cartesian is to say that the natural map from the homotopy fiber of f to the homotopy fiber of f′ is a weak homotopy equivalence. Thus if the homotopy fibers are the same via the natural map, it follows that the square

\begin{matrix}X′ & ⟶ & Y′ \\ ↓ & & ↓ \\ X′′ & ⟶ & Y′′ \end{matrix}

is homotopy cartesian.

NOTATION 5.1. Since we will be using this a lot, it seems right to be able to refer to the squares in the diagram

\begin{matrix}X & ⟶ & Y \\ ↓ & & ↓ \\ X′ & ⟶ & Y′ \\ ↓ & & ↓ \\ X′′ & ⟶ & Y′′ \end{matrix}

The three squares

X ⟶ Y  X ⟶ Y  X′ ⟶ Y′
↓  ↓  ↓  ↓  ↓  ↓
X′ ⟶ Y′  X′′ ⟶ Y′′  X′′ ⟶ Y′′

will be called, respectively, the top square, large square and bottom square.

6. Outline of the proof of Proposition 4.2

It remains to prove Proposition 4.2. For the reader’s convenience, let us repeat it.

Proposition 4.2 The natural inclusion

[figure: 0; Gr; [0,n]; α; 0; Gr; [0,n+1]]

is a homotopy equivalence.

We want to prove that the map α is a homotopy equivalence. First, we give a definition

DEFINITION 6.1. A map denoted by the arrow [figure] is a map in Gr_{[0,m]}, restricted to be in isomorphism on H^{0}.

The key to our proof will be the Step

Step 6.1. :

The natural projection

[figure] 0  Gr[figure: [0,n]] Gr[figure: [1,n]]  π  \overline{}[figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

is a quasi–fibration.

The homotopy fiber of the map π is therefore homotopy equivalent to any Segal fiber; in particular to the Segal fiber above the trivial

(0, 0)–cell 0. This fiber is just [figure: 0; Gr; [0,n]]. We therefore get

a commutative diagram whose columns are homotopy fibrations

[figure: 0; Gr; [0,n]]  α  \overline{}[figure]  [figure: 0; Gr; [0,n+1]]^{[figure] 0  Gr[figure: [0,n]] Gr[figure: [1,n]]}  \overline{}[figure]  ^{[figure] 0  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]} [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]  \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n+1]]

We wish to show that α is a homotopy equivalence. Equivalently, we can prove that the bottom square in the diagram is homotopy cartesian. That is exactly what we will do.

LEMMA 6.2. The projection

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure] 0  [figure] [figure] Gr[figure: [0,n]] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

induces a homotopy equivalence.

Proof. We factor the map as

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  f_{1}  \overline{}^{[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]} f_{2} [figure] [figure] 0  [figure] [figure] Gr[figure: [0,n]] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

gr84.tex; 19/06/2000; 14:06; p.59

and it suffices to prove that both f_{1} and f_{2} are homotopy equivalences.

The Segal fiber of f_{1} is the simplicial set [figure] Z  Y  Y′ Gr[figure: [1,n]] Gr[figure: [0,n]] X ,

and it is contracted by the homotopy

[figure] Z  Y  Y′ Y_{S}  Y′_{S} Gr[figure: [1,n]]  Gr[figure: [0,n]] X

K–Theory for Triangulated Categories 3 \frac{1}{2} (A) 61

For the map f_{2}, observe that its Segal fiber is the simplicial set

[figure] Z  Gr[figure: [1,n]]  Y X  ,

and the homotopy

[figure] Z  Z_{E} \overline{}Gr[figure: [1,n]]  Y X

contracts it away. Some explanation is in order here. We need to note that it really is true that the fiber over any [figure: Z; Y; X] is contractible. We must convince ourselves that the homotopy above is well defined. The point is that by Lemma 3.4, the Z’s are restricted. The fixed data is a diagram

[figure: ⋮; ⋯; ·; ·; ·; Z_{00}; Z_{0s}; Z_{ss}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0t}; Y_{s0}; Y_{st}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0t}; X_{r0}; X_{rt}]

and in Lemma 3.4 we learned that the Z’s are constrained to lie in Gr_{[1,n]} and the vertical maps among them are constrained to be epi. This proves that the homotopy sweeping Z_{E} to the right is indeed a contraction of every Segal fiber. □

CONCLUSION 6.3. Consider now the commutative diagram

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure]  ^{[figure] 0  Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]}^{[figure] 0  [figure] [figure] Gr[figure: [0,n]] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]}  \overline{}[figure]  ^{[figure] 0  [figure] [figure] Gr[figure: [0,n+1]] [figure] [figure]  [figure] [figure] Gr[figure: [1,n+1]]} [figure] [figure] [figure]  [figure] [figure]  [figure] [figure] [figure] [figure]  [figure] [figure] Gr[figure: [1,n]]  \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n+1]]

gr84.tex; 19/06/2000; 14:06; p.63

It is trivial to see that all these simplicial sets are connected. We already reduced ourselves to proving that the bottom square is homotopy cartesian; now Lemma 6.2 guarantees that in the top square the two vertical morphisms are homotopy equivalences. Therefore the top square is homotopy cartesian. To show that the bottom square is homotopy cartesian, we will prove that the large square is.

The next argument is technical, so we leave it as a Step

Step 6.2. :

The projection

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]

induces a homotopy equivalence.

CONCLUSION 6.4. Consider now the commutative diagram of connected spaces

[figure] 0  Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure]  ^{[figure] 0  Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]}^{[figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]}  \overline{}[figure]  ^{[figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]} [figure] [figure] [figure]  [figure] [figure]  [figure] [figure] [figure] [figure]  [figure] [figure] Gr[figure: [1,n]]  \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n+1]]

gr84.tex; 19/06/2000; 14:06; p.65

In Conclusion 6.3, we already reduced ourselves to proving that the large square is homotopy cartesian; now Step 6.2 guarantees that in the top square the two vertical morphisms are homotopy equivalences. Therefore the top square is homotopy cartesian. To show that the large square is homotopy cartesian, we will prove that the bottom square is.

LEMMA 6.5. The projection

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]

induces a homotopy equivalence.

Proof. The Segal fiber is contracted by the homotopy

Gr[figure: [1,n]] Z_{N} Z  Y X  □

CONCLUSION 6.6. Consider the commutative diagram of connected spaces

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]}^{[figure] [figure] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]}  \overline{}[figure]  ^{[figure] [figure] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]} [figure] [figure] [figure] [figure]  [figure] [figure] Gr[figure: [1,n]]  \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n+1]]

gr84.tex; 19/06/2000; 14:06; p.67

We need to know that the bottom square is homotopy cartesian. We know that the top square is, by Lemma 6.5. We will prove what we need by showing that the large square is homotopy cartesian.

LEMMA 6.7. The projections

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]]  \overline{}[figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n]]

and

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]]  \overline{}Gr[figure: [1,n]] [figure] [figure] [figure] [figure] [figure] [figure]

both induce homotopy equivalences.

Proof. Trivial. The reader is only reminded to read carefully all the restrictions in the symbol of the simplicial set. Here all the objects lie in Gr_{[1,n]}, and all the morphisms are unrestricted. This distinguishes it from earlier simplicial sets, where some of the objects were permitted to be in Gr_{[0,n]}, but some restrictions were placed on the morphisms.□

CONCLUSION 6.8. Consider the following two commutative diagrams of connected spaces

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]]  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]} [figure]  [figure]^{Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]}  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]} [figure] [figure] [figure] [figure]  [figure] [figure] Gr[figure: [1,n]]  \overline{}[figure]  [figure] [figure] [figure] [figure] [figure] [figure] Gr[figure: [1,n+1]]

gr84.tex; 19/06/2000; 14:06; p.69

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [1,n]]  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]} [figure]  [figure]^{Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]}  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]} Gr[figure: [1,n]] [figure] [figure] [figure] [figure] [figure] [figure]  \overline{}[figure]  Gr[figure: [1,n+1]] [figure] [figure] [figure] [figure] [figure] [figure]

gr84.tex; 19/06/2000; 14:06; p.70

We know by Lemma 6.7 that in both diagrams the large squares are homotopy cartesian. We need to know that in the first diagram, the bottom square is homotopy cartesian. It suffices to show that the top square in the first diagram is. But the top square is the same in both diagrams; hence it suffices to show that the bottom square in the second diagram is homotopy cartesian. This is what we will do.

LEMMA 6.9. The projection

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] [figure] [figure]

induces a homotopy equivalence.

Proof. The Segal fiber is contracted by the homotopy

Z Y  X X_{S}^{>0} Gr[figure: [1,n]]

We should explain the homotopy. For and object X of Gr(A), the object X^{>0} is defined by the formula

H^{q}(X^{>0}) = \{ \begin{matrix}H^{q}(X) & if q > 0 \\ 0 & if q ≤ 0 \end{matrix}

The homotopy has for its cells diagrams

[figure: ⋮; ⋮; ⋯; ⋯; Z_{00}; Z_{0r}; Z_{q0}; Z_{qr}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0r}; Y_{p0}; Y_{pr}; ⋮; ⋮; ⋯; ⋯; X_{00}; X_{0s}; X_{p0}; X_{ps}; ⋮; ⋮; ⋯; ⋯; X_{00}^{>0}; X_{0s}^{>0}; X_{00}^{>0}; X_{0s}^{>0}; ⋮; ⋮; ⋯; ⋯; R_{00}; R_{0s}; R_{i0}; R_{is}; \} ( \begin{matrix}t−i+1 \\ times \end{matrix} ); \} ( \begin{matrix}i+1 \\ terms \end{matrix} ); \{; \{; □]

K–Theory for Triangulated Categories 3 \frac{1}{2} (A) 73

CONCLUSION 6.10. Consider the commutative diagram of connected spaces

Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] Gr[figure: [1,n]]  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] Gr[figure: [1,n+1]]}^{Gr[figure: [1,n]] Gr[figure: [1,n]] Gr[figure: [0,n]] [figure] [figure]}  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]] [figure] [figure]} Gr[figure: [1,n]] [figure] [figure] [figure] [figure] [figure] [figure]  \overline{}[figure]  Gr[figure: [1,n+1]] [figure] [figure]  [figure] [figure] [figure] [figure]

gr84.tex; 19/06/2000; 14:06; p.74

We need to show the large square homotopy cartesian. By Lemma 6.9, we know that the top square is. We will prove that the bottom square is homotopy cartesian.

Now, consider the commutative diagram

Gr[figure: [1,n]]  Gr[figure: [0,n]]  α  \overline{}[figure]  ^{^{Gr[figure: [1,n+1]]}  Gr[figure: [0,n+1]]} β  β′^{Gr[figure: [1,n]] Gr[figure: [1,n]]  Gr[figure: [0,n]]}  \overline{}[figure]  ^{Gr[figure: [1,n+1]] Gr[figure: [1,n+1]]  Gr[figure: [0,n+1]]} γ  γ′ Gr[figure: [1,n]] [figure] [figure] [figure] [figure]  \overline{}[figure]  Gr[figure: [1,n+1]] [figure] [figure] [figure] [figure]

where β is the inclusion of a Segal fiber of γ, more specifically the Segal fiber over the (0, 0)–simplex 0. Similarly for β′ and γ′. We need to show that the bottom square is homotopy cartesian. We will prove this by means of a Step and a lemma:

Step 6.3. :

The natural projection

Gr[figure: [1,n]]^{}^{}^{}^{}Gr[figure: [1,n]]  ^{}^{}^{}^{}Gr[figure: [0,n]]  \overline{}Gr[figure: [1,n]]^{}^{}^{}^{}[figure] [figure] ^{}^{}^{}^{}[figure] [figure]

induces a quasi–fibration.

This means, of course, that the homotopy fiber can be identified with the Segal fiber, and in particular with the Segal fiber over the special

(0, 0)–cell 0. This Segal fiber is the simplicial set ^{Gr[figure: [1,n]]} Gr[figure: [0,n]]

in the diagram above. Thus, it follows from Step 6.3 that the columns of the diagram are fibrations. To complete the proof that the bottom square is homotopy cartesian, it therefore suffices to establish the lemma:

LEMMA 6.11. The natural inclusion

Gr[figure: [1,n]]  Gr[figure: [0,n]]  α  \overline{}[figure]  ^{^{Gr[figure: [1,n+1]]}  Gr[figure: [0,n+1]]}

induces a homotopy equivalence.

Proof. Consider the composite

Gr[figure: [1,n]]  Gr[figure: [0,n]]  i  \overline{}[figure]  ^{^{Gr[figure: [1,n]]}  Gr[figure: [0,n]]} α [figure]^{^{Gr[figure: [1,n+1]]}  Gr[figure: [0,n+1]]}

We will find homotopy inverses for i and α ∘ i. Since then i and α ∘ i are homotopy equivalences, so is α.

In fact, we will do better. We will find a single homotopy which

will do the trick. It is a homotopy on ^{Gr[figure: [1,n+1]]} Gr[figure: [0,n+1]] which

deformation retracts it to ^{Gr[figure: [1,n]]} Gr[figure: [0,n]] , and which takes

Gr[figure: [1,n]] Gr[figure: [0,n]] to itself. Thus our one homotopy, together with

its restriction to ^{Gr[figure: [1,n]]} Gr[figure: [0,n]] , will do the trick.

Before we write down the homotopy, it might be helpful to explain what it does on the level of (1, 0)–simplices. Take a (1, 0)–simplex in

Gr[figure: [1,n+1]] Gr[figure: [0,n+1]] , that is a diagram

\begin{matrix}X′ & ⟶ & Y′ \\ ↑ & & ↑ \\ X & ⟶ & Y \end{matrix}

There corresponds to it a long exact sequence, which ends

⟶ H^{n}(X) ⟶ H^{n}(X′ ⊕ Y) ⟶ H^{n}(Y′) ⟶ H^{n+1}(X) ⟶

Define an object which we will call {^{t}Y }^{<n+1} by the formula

H^{q} (\{^{t}Y \}^{<n+1}) = \{ \begin{matrix}0 & if q > n \\ Im \{H^{n}(X) ⟶ H^{n}(Y)\} & if q = n \\ H^{q}(Y) & if q < n \end{matrix}

then there is clearly an exact sequence

⟶ H^{n}(X) ⟶ H^{n} (X′ ⊕ {^{t}Y }^{<n+1}) ⟶ H^{n} ({^{t}Y′}^{<n+1}) ⟶ 0

and a square

{X′}^{<n+1} ⟶ {^{t}Y′}^{<n+1}
↑  ↑
X^{<n+1} ⟶ {^{t}Y }^{<n+1}

In other words, if we replace H^{n}(Y) by the image of H^{n}(X) and truncate both X and Y below dimension n + 1, then we still end up with a perfectly well–defined (1, 0)–simplex. The idea is now to extend this construction to a simplicial homotopy.

Our one homotopy will be given as the composite of two simplicial

homotopies. Recall that a simplex in ^{Gr[figure: [1,n+1]]} Gr[figure: [0,n+1]] is a

diagram

[figure: ⋮; X_{0}; X_{r}; ⋮; ⋮; ⋯; ⋯; Y_{00}; Y_{0s}; Y_{r0}; Y_{rs}]

Let ^{t}Y_{ij} be defined as follows

H^{q}(^{t}Y_{ij}) = \{ \begin{matrix}H^{q}(X_{i}) & if q ≥ n + 1 \\ Im \{H^{q}(X_{i}) ⟶ H^{q}(Y_{ij})\} & if q = n \\ H^{q}(Y_{ij}) & if q ≤ n − 1 \end{matrix}

Then the first homotopy has for its cells diagrams

[figure: ⋮; X_{0}; X_{r}; ⋮; ⋮; ⋯; ⋯; ^{t}Y_{00}; ^{t}Y_{0i}; ^{t}Y_{r0}; ^{t}Y_{ri}; ⋮; ⋮; ⋯; ⋯; Y_{0i}; Y_{0s}; Y_{ri}; Y_{rs}; ; ; _{(i+1)−terms}; _{(s−i+1)−terms}]

Note in particular that H^{n+1}(^{t}Y_{jk}) = H^{n+1}(X_{j}) is independent of k, that is it is constant in rows of the simplex. The second homotopy has cells

[figure: ⋮; X_{i}; X_{r}; ⋮; ⋮; ⋯; ⋯; ^{t}Y_{i0}; ^{t}Y_{is}; ^{t}Y_{r0}; ^{t}Y_{rs}; ⋮; X_{0}^{<n+1}; X_{i}^{<n+1}; ⋮; ⋮; ⋯; ⋯; \{^{t}Y_{00}\}^{<n+1}; \{^{t}Y_{0s}\}^{<n+1}; \{^{t}Y_{i0}\}^{<n+1}; \{^{t}Y_{is}\}^{<n+1}; \} ( \begin{matrix}r−i+1 \\ terms \end{matrix} ); \} ( \begin{matrix}i+1 \\ terms \end{matrix} ); \{; \{; □]

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Address for Offprints: Center for Mathematics and its Applications School of Mathematical Sciences John Dedman Building The Australian National University Canberra, ACT 0200 AUSTRALIA