CATEGORICAL AND SIMPLICIAL CONSTRUCTIONS

AND DUALIZATION IN ALGEBRAIC K-THEORY

Alexander Nenashev

Abstract. Given an exact category A, we introduce a bisimplicial set W A which is self-dual and contains both the G-construction of A and its dual. We prove that the embeddings of G.A and G^{op}.A into W A are homotopy equivalences. If A is an exact category with duality, we calculate the induced action of duality on K_{1}(A). We also provide a survey of the self-duality property of some known constructions.

Exact categories with duality are an environment for the theory of Witt groups of schemes. On the other hand, one can study their K-theory. In summer ’99 Marek Szyjewski pointed out to me that it might be useful for his work on Witt groups [Sz1-2] to calculate the induced action of duality on K_{1} by using my description of K_{1} in terms of double short exact sequences [Ne2-4]. Though the answer

[A ⇉_{f_{2}}^{f_{1}} B ⇉_{g_{2}}^{g_{1}} C]^{∗} = ±[C^{∗} ⇉_{g_{2}^{∗}}^{g_{1}^{∗}} B^{∗} ⇉_{f_{2}^{∗}}^{f_{1}^{∗}} A^{∗}]  (0.1)

was clear up to sign from the very beginning, I had to examine which categorical and simplicial constructions were suitable to prove this in an easy way. This leads to the question which of the known constructions that one uses to define algebraic K-theory are self-dual. The first section of my paper is a survey on this matter.

The G-construction of Gillet and Grayson [GG] which proved to be a powerful tool for various objectives is obviously not self-dual. In Section 2, I introduce a bisimplicial set W which is a hybrid of G and G^{op} (NB: but not their composite) and is self-dual, G and G^{op} being embedded into W. Generalizing some techniques of my paper [Ne1], I prove that both embeddings are homotopy equivalences and W yields therefore one more model for the K-theory space of an exact category. Using W, I give a formal proof of (0.1) in Section 3. A generalization of W plays a crucial role in my computation of the action of λ^{2} on K_{1} [Ne5].

Acknowledgements. I started this work at the University of Bielefeld where I stayed with a research grant of the Alexander von Humboldt Foundation. Another part of it was done during my stay at the National University of Singapore. I am grateful to Anthony Bak (Bielefeld) and Jon Berrick (Singapore) for their interest to my work and various support, and to both universities for hospitality. I am greatly indebted to Marek Szyjewski for stimulating correspondence, for reading a preliminary version of this paper and correcting misprints, and for his help in typesetting some diagrams as xy-matrices.

Typeset by AMS-TEX

1. Exact categories with duality and a review of constructions

1.1. Generalities about dualization. Let A be a category. A duality on A is a functor ∗ : A ⟶ A^{op}, A ↦ A^{∗}, equiped with an isomorphism of functors

ω : Id_{A} ⟶^{≅} ∗ ∘ ∗, A ⟶^{ω_{A}} A^{∗∗}, such that A^{∗} ⟶^{ω_{A^{∗}}} A^{∗∗∗} and A^{∗∗∗} ⟶^{(ω_{A})^{∗}} A^{∗} are

inverse isomorphisms for any A ∈ A.

If A is an exact category, we will denote by A^{op} the dual exact category, i.e., a short exact sequence A ↣ B ↠ C in A (a s.e.s. for short) will be regarded as a s.e.s. A ↞ B ↢ C in A^{op} and vise versa. If the duality finctor ∗ : A ⟶ A^{op} is exact, we say that A is an exact category with duality.

Let ExCats denote the category of (small) exact categories, C be a category, and T : ExCats ⟶ C be a functor. In our examples, C will be the category of (small) categories, (multi)simplicial sets, or spaces, and T will be traditionally called a construction. We say that T is self-dual if there is a functorial isomorphism τ_{A} : T (A^{op}) ⟶^{≅} T (A) satisfying τ_{A^{op}} = τ_{A}^{−1}. If T is self-dual and ∗ : A ⟶ A^{op}

is an exact duality, then the composite map T A ⟶^{T (∗)} T (A^{op}) ⟶^{τ_{A}} T A is said to be the induced action of duality on T A. By abuse of notation, we will denote this induced action by ∗ : T A ⟶^{≅} T A. This isomorphism is obviously functorial in A if we restrict T to the category of exact categories with duality.

Remark. Note that in general ∗ ∘ ∗ does not equal Id_{T}. Suppose that we have a notion of homotopy for arrows in C and that for any two exact functors F, F′ : A ⇉ B and an isomorphism α : F ⟶^{≅} F′, the arrows T (F), T (F′) : T (A) ⇉ T (B) are homotopic, the homotopy being natural in α. We will then say that T is a Waldhausen type construction, since the S-construction satisfies this property [Wa, Lemma 1.4.1], which was essentially used in [Wa]. One checks by using ω : Id_{A} ⟶^{≅} ∗ ∘ ∗ that, for such a construction, ∗ ∘ ∗ : T A ⟶ T A is homotopic to Id_{T A}.

1.2. The Q-construction. Quillen’s Q-construction is a functor Q : ExCats ⟶ Cats. Recall that Obj(QA) = Obj(A) and there are two dual ways of presenting a morphism in QA. By definition [Qu], an arrow from A to B in QA is given by data either of the form A ↞ X ↣ B or of the form A ↣ X′ ↞ B in A up to some equivalence, and one proceeds from one type of presentation to the other by taking pull-backs and push-outs correspondingly. When replacing A by A^{op}, the two ways of presenting arrows are just interchanged, which yields an isomorphism of categories QA^{op} ≅ QA [Qu, (5), Sect.2] and implies that Q is self-dual. To be strict, we cannot write QA^{op} = QA as we distinguish between the ‘upper’ and ‘lower’ presentations of arrows in QA.

An exact duality ∗ : A ⟶ A^{op} induces therefore a functor ∗ : QA ⟶ QA^{op} ⟶^{∼} QA and a map on classifying spaces ∗ : |N.QA| ⟶ |N.QA|. This yields a dualization map on the K-groups, ∗ : K_{m}(A) ⟶ K_{m}(A), where by definition K_{m}(A) = π_{m+1}(|N.QA|).

To any X ↣^{α} Y and Y′ ↠^{β} X′ in A, Quillen assigns the arrows

α_{!} = (X ↞^{1} X ↣^{α} Y) = (X ↣^{α} Y ↞^{1} Y) ∈ Hom_{QA}(X, Y)
β^{!} = (X′ ↞^{β} Y′ ↣^{1} Y′) = (X′ ↣^{1} X′ ↞^{β} Y′) ∈ Hom_{QA}(X′, Y′).

For A ∈ A and a zero object 0, put 0^{A} : 0 ↣ A and 0_{A} : A ↠ 0, then

0_{!}^{A} = (0 ↞ 0 ↣ A) = (0 ↣ A ↞^{1} A)
0_{A}^{!} = (0 ↞ A ↣^{1} A) = (0 ↣ 0 ↞ A),

and we assign to A the two-edge loop κ(A) in N.QA given by

[figure: 0; A; 0; •; 0_{A}^{!}; •; •; 0_{!}^{A}]

This yields an isomorphism K_{0}(A) ≅ π_{1}(N.QA). Assume that 0^{∗} = 0, then (0_{!}^{A})^{∗} =

(0 ↣ 0 ↞ A^{∗}) = 0_{A^{∗}}^{!} and (0_{A}^{!})^{∗} = (0 ↣ A^{∗} ↞^{1} A^{∗}) = 0_{!}^{A^{∗}}. It follows that the image of κ(A) under the map ∗ : N.QA ⟶ N.QA is the loop κ(A^{∗})^{−1}, whence in K_{0}(A) we have

[A]^{∗} = −[A^{∗}].  (1.1)

To calculate the dualization on K_{1}(A), recall [Ne2] that every element of K_{1}(A) can be represented by data of the form

l = (A ⇉_{f_{2}}^{f_{1}} B ⇉_{g_{2}}^{g_{1}} C) = (s_{1}, s_{2}),  (1.2)

where s_{1} = (A ↣^{f_{1}} B ↠^{g_{1}} C) and s_{2} = (A ↣^{f_{2}} B ↠^{g_{2}} C) are short exact sequences in A on the same objects. We say thet l is a double short exact sequence (d.s.e.s. for short).

Given a s.e.s. s = (A ↣^{f} B ↠^{g} C), consider the following diagram in QA:

\xymatrix{ &  & 0 \ar[d]^{0_{C}^{!}} \ar[dl]_{0_{B}^{!}} \\  & B & C \ar[l]_{e_{C,B}} \\ 0 \ar[ur]^{0_{!}^{B}} \ar[r]_{0_{!}^{A}} & A \ar[u]^{e_{A,B}} & 0 \ar[u]_{0_{!}^{C}} \ar[ul]_{e_{s}} \ar[l]^{0_{A}^{!}}}  (1.3)

where e_{A,B} = (A ↞^{1} A ↣^{f} B), e_{C,B} = (C ↞^{g} B ↣^{1} B), and e_{s} = (0 ↞ A ↣^{f} B). The

following assertions can be easily checked.

a) All the four triangles commute in QA and yield therefore 2-simplices in its nerve. Let τ(s) denote this triangular configuration of simplices in N.QA.

b) The boundary of τ(s) does not depend on f and g.

c) Thus, given a d.s.e.s. l as in (1.2), one can glue τ(s_{1}) and τ(s_{2}) to get a 2-spheroid σ(l) in N.QA.

d) Choose an orientation in (1.3). Then the image of τ(s) under the duality

map ∗ : N.QA ⟶ N.QA equals τ(s^{∗}), where s^{∗} = (C^{∗} ↣^{g^{∗}} B^{∗} ↠^{f^{∗}} A^{∗}), but with the opposite orientation (this amounts to taking pull-backs as in the proof of (1.1)). The same is true for the spheroids σ(l), and we have in K_{1}(A)

[l]^{∗} = −[l^{∗}],  (1.4)

with l^{∗} = (s_{1}^{∗}, s_{2}^{∗}).

Remark. In [Ne2], we assigned a loop μ(l) in G.A to a d.s.e.s. l and proved that every element of K_{1}(A) ≅ π_{1}(G.A) can be obtained in this way. I believe (in), but have never checked that the class of the loop μ(l) corresponds to the class of the 2-spheroid σ(l) via the isomorphisms π_{1}(G.A) ≅ K_{1}(A) ≅ π_{2}(N.QA). Thus, to be honest, (1.4) does not mean that we can calculate ∗ on all elements of K_{1}(A). Instead, we calculate ∗ on K_{1} in Section 3 in terms of G.A.

1.3. The S-construction. We consider the S-construction of Waldhausen ([Wa], [GG]) as a functor S : ExCats ⟶ SSets, the latter notation refers to simplicial sets. It is not self-dual ‘on the simplicial level’ as there is no isomorphism of simplicial sets S.A^{op} ⟶^{∼} S.A functorial in A. For, an n-simplex x in S.A^{op} is a diagram in A of the form

[figure: A_{n/n−1}; ⋯; x =; A_{2/1}; ⋯; A_{n/1}; A_{1/0}; A_{2/0}; ⋯; A_{n/0}]

The only reasonable way to get an n-simplex in S.A out of x is to take the transposed diagram

[figure: A_{1/0}; ⋯; x^{t} =; A_{n−1/n−2}; ⋯; A_{n−1/0}; A_{n/n−1}; A_{n/n−2}; ⋯; A_{n/0}]

But this does not yield a simplicial map since

d_{i}(x^{t}) = (d_{n−i}x)^{t}, s_{i}(x^{t}) = (s_{n−i}x)^{t}, 0 ≤ i ≤ n.  (1.5)

For any simplicial set X, let X̃ be the simplicial set with X̃_{n} = X_{n} and d_{i}, s_{i} replaced by d_{n−i}, s_{n−i} respectively. Formally speaking, X̃ is the composite of X : Δ^{op} ⟶ Sets and the endofunctor of Δ which takes a finite non-empty totally ordered set to the same set with the inverse order. There is no natural simplicial isomorphism X̃ ⟶ X, but there is a homeomorphism of geometric realizations |X̃| ⟶ |X| functorial in X which is induced by the maps

Δ^{n} ⟶ Δ^{n}, (s_{0}, ..., s_{n}) ↦ (s_{n}, ..., s_{0}).

It follows from (1.5) that we have a simplicial map

S.A^{op} ⟶^{∼} S̃.A, x ↦ x^{t}.

Combining it with |S̃.A| ⟶ |S.A|, we get the dualization map

|S.A^{op}| ⟶ |S.A|,  (1.6)

which means that the S-construction is dual on the level of spaces (geometric realizations).

Recall that N.QA is a subdivision of S.A [Wa, Sect. 1.9]. A s.e.s. s = (A ↣^{f} B ↠^{g} C) in A yields a 2-simplex t(s) in S.A. I leave it to the reader to check that τ(s) of (1.3) is exactly this subdivision of t(s), and to reprove (1.1) and (1.4) via the map (1.6).

1.4. The K-construction. Giffen [Gi] introduced two categorical constructions, KA and kA, and proved that their classifying spaces are homotopy equivalent to the loop space of |N.QA|. The big K is self-dual, the small k is not, thus we have in fact three functors k, k^{op}, K : ExCats ⟶ Cats. It is not known yet whether a good description for K_{1} can be obtained in terms of any of these constructions. However, it might be interesting to use K or k for some problems of the Witt theory for exact categories with duality. This will be an objective of our future joint work with M. Szyjewski. 1.5. The G-construction. Unlike Q- and S-, the G-construction is not self-dual, even on the level of spaces. For an n-simplex in G.A^{op} is a pair of diagrams of the form

[figure: A_{n/n−1}; A_{n/n−1}; ⋯; ⋯; A_{1/0}; ⋯; A_{n/0}; A_{1/0}; ⋯; A_{n/0}; A_{0}; A_{1}; ⋯; A_{n}; A′_{0}; A′_{1}; ⋯; A′_{n}]

in A subject to certain conditions (cf. [GG], [Ne2-4]). These data by no means represent a simplex in G.A. In order to study the duality maps on K-groups in terms of G, we need a functorial homotopy equivalence between G.A and G.A^{op}. One of the options is to embed both constructions into an appropriate space and prove that the embeddings are homotopy equivalences. We do it by introducing a bisimplicial set W A = W..A such that n ↦ W_{n,0} and n ↦ W_{0,n} can be identified with G.A and G.A^{op} respectively. This W -construction is self-dual in a sense applicable to bisimplicial sets. As a result, we can say that G is self-dual in the homotopy category of spaces.

The formal definitions and proofs will be given in Section 2. Here we give an idea as to what kind of data the low-dimensional simplices in W A can be represented by.

(i) A (0, 0)-simplex in W A is a pair of objects (A, A′) ∈ A × A (the same as a vertex in G.A or G.A^{op}).

(ii) A (1, 0)-simplex in W A connecting (A_{0}, A′_{0}) to (A_{1}, A′_{1}) is a pair of s.e.s’s (A_{0} ↣ A_{1} ↠ A_{1/0}, A′_{0} ↣ A′_{1} ↠ A_{1/0}) with the same cokernel object (which is the same as a 1-simplex in G.A).

(iii) A (0, 1)-simplex in W A connecting (A_{0}, A′_{0}) to (A_{1}, A′_{1}) is a pair of s.e.s’s (A_{0} ↞ A_{1} ↢ A_{1/0}, A′_{0} ↞ A′_{1} ↢ A_{1/0}) with the same kernel object (which is the same as a 1-simplex in G.A^{op}).

(iv) A (1, 1)-simplex is a pair of commutative diagrams

[figure: A_{0,1/0}; A_{1,1/0}; A_{1/0,1/0}; A_{0,1/0}; A_{1,1/0}; A_{1/0,1/0}; A_{0,1}; A_{1,1}; A_{1/0,1}; A′_{0,1}; A′_{1,1}; A_{1/0,1}; A_{0,0}; A_{1,0}; A_{1/0,0}; A′_{0,0}; A′_{1,0}; A_{1/0,0}]  (1.7)

with the same upper horizontal and right vertical s.e.s’s. The vertices of this ‘square’ are the pairs (A_{i,j}, A′_{i,j}) with i, j ∈ {0, 1}. Its four edges are given by the corresponding pairs of s.e.s’s in these diagrams.

2. Simplicial technique and a self-dual version of the G-construction

2.1. Categories of arrows. Suppose P ∈ Δ, i.e., P is a finite non-empty totally ordered set. Denote by ArP the category of arrows in P. We use the notation j/i for the arrow (i ≤ j) ∈ ArP. For i ≤ j ≤ k, the sequence j/i → k/i → k/j is said to be a (formal) short exact sequence in ArP. Let A be an exact category with a distinguished zero object 0. Then a functor F : ArP → A is said to be exact if it satisfies F (i/i) = 0 for any i ∈ P and takes formal s.e.s’s in ArP to s.e.s’s in A. For P = [n], such a functor can be displayed as a commutative diagram

[figure: F (n/n − 1); ⋯; F (2/1); ⋯; F (n/1); F (1/0); F (2/0); ⋯; F (n/0)]

in which all the sequences of the form F (j/i) ↣ F (k/i) ↠ F (k/j) are s.e.s’s in A.

Let Ar^{op}P be the opposite category. A (formal) s.e.s. in Ar^{op}P is a sequence of the form j/i ⟵ k/i ⟵ k/j for any i ≤ j ≤ k in P. An exact functor F : Ar^{op}[n] → A is basically a commutative diagram

[figure: F (n/n − 1); ⋯; F (2/1); ⋯; F (n/1); F (1/0); F (2/0); ⋯; F (n/0)]

in which all the sequences of the form F (j/i) ↞ F (k/i) ↢ F (k/j) are s.e.s’s in A.

Note that the constructions S.A and S^{op}.A = S.A^{op} can be defined by

(S.A)[P] = Exact(ArP, A), (S^{op}.A)[P] = Exact(Ar^{op}P, A), P ∈ Δ,

where Exact refers to the set of exact functors in the above sense (compare to the diagrams in Sect. 1.3).

2.2. Strictly dominant and codominant functors. Let A and B be exact categories and F : A → B an exact functor. Let imF denote the class of those B ∈ B with B ≅ F (A) for some A ∈ A. We call an admissible monomorphism B ↣ B′ (resp. epimorphism B ↞ B′) an F -mono (resp. F -epi) if its cokernel (kernel) belongs to imF.

Definition 2.1. We say that F is strictly dominant or (+1)-dominant if for any f : B ↣ B′ in B there exists an object B′′ in B and a filtration B ↣ B_{1} ↣ ⋯ ↣ B_{n} = B′ ⊕ B′′ in which the composite map equals ( \begin{matrix}f \\ 0 \end{matrix} ) and all the arrows are F -monos.

We say that F is strictly codominant or (-1)-dominant if for any f : B′ ↠ B in B there exists an object B′′ in B and a cofiltration B ↞ B_{1} ↞ ⋯ ↞ B_{n} = B′ ⊕ B′′ in which the composite map equals (f, 0) and all the arrows are F -epis.

Main Example. F is said to be cofinal if for any B ∈ B there exists B′ with B ⊕ B′ ∈ imF. Notice that the functor diag : A → A × ⋯ × A (any number of factors) is cofinal. Denote by E_{n}(A) the category of exact sequences of length n in A. Then the functor E_{n}(A) → A × ⋯ × A that forgets the arrows is cofinal.

Lemma 2.2. Any cofinal functor F is both strictly (+1)- and (-1)-dominant

Proof. Given f : B ↣ B′, take an object B̃ with cokerf ⊕ B̃ ∈ imF. The one-step

filtration B \underrightarrow{( \begin{matrix}f \\ 0 \end{matrix} )} B′ ⊕ B̃ shows that F is strictly dominant. Dualize to prove that F is strictly codominant.

2.3. Cubes of categories. We regard the set [1] = {0, 1} as a category. By an n-dimensional cube X = X_{(n)} of (exact) categories we mean a functor from ([1]^{n})^{op} to the category of (exact) categories. For ε ∈ [1]^{n} we denote by X (ε) the category standing at the vertex ε. By a map of n-dimensional cubes we mean a natural (exact) transformation of functors. Given two cubes X_{(m)} and Y_{(n)}, we define an (m + n)-dimensional cube X × Y by

(X × Y)(ε_{1}, …, ε_{m+n}) = X (ε_{1}, …, ε_{m}) × Y(ε_{m+1}, …, ε_{m+n}).

Let B be a symbol. For P ∈ Δ we denote by BP the disjoint union {B} ∪ P with B declared to be less than any element of P. Given P_{1}, …, P_{n} ∈ Δ and θ = (θ_{1}, …, θ_{n}) with θ_{i} = ±1, we introduce an n-dimensional cube of categories

Γ_{θ}(P_{1}, …, P_{n}) = [Ar^{θ_{1}}P_{1} → Ar^{θ_{1}}(BP_{1})] × ⋯ × [Ar^{θ_{n}}P_{n} → Ar^{θ_{n}}(BP_{n})],

where Ar^{θ_{i}} refers to Ar or Ar^{op} according to whether θ_{i} = +1 or −1. For ε ∈ {0, 1}^{n}, let Γ_{θ}(P_{1}, …, P_{n}; ε) be the category at the ε-vertex of the cube Γ_{θ}(P_{1}, …, P_{n}), i.e., Γ_{θ}(P_{1}, …, P_{n}; ε) is the direct product of n categories with i-th factor equal to Ar^{θ_{i}}P_{i} or Ar^{θ_{i}}(BP_{i}) accordingly to ε = 1 or 0.

Suppose we are given an n-dimensional cube of exact categories X. We generalize the notation of [Gr2, Sect. 4] and define the mapping cone construction C^{θ}X to be an n-fold multisimplicial set given by

C^{θ}X [P_{1}, …, P_{n}] = Exact(Γ_{θ}(P_{1}, …, P_{n}), X),

where “Exact” stands for the set of functors whose components at all vertices are polyexact, i.e. exact in each variable.

Notice that

G.A = C^{(+1)}[A ⟶^{diag} A × A], G^{op}.A = G.A^{op} = C^{(−1)}[A ⟶^{diag} A × A].

Regarding C^{θ}X as a multisimplicial exact category, we can apply the S- or S^{op}- construction degreewise to obtain an (n + 1)-fold multisimplicial set S^{ν}C^{θ}X with ν = ±1 respectively.

In [Ne1] we studied homotopy properties of the C- and SC-constructions. The technique of [Ne1] generalizes directly to C^{θ}, the main result being the following (cf. [Ne1, Prop. 1.6])

Proposition 2.3. Let X be an n-dimensional cube of exact categories and let θ = (θ_{1}, …, θ_{n}) ∈ {−1, 1}^{n}. Let i ∈ {1, …, n} and suppose that all the edge-fuctors of X in the i-th direction are strictly θ_{i}-dominant. Put θ^{(i)} = (θ_{1}, …, θ̂_{i}, …, θ_{n}). Then we have a homotopy fibration sequence

C^{θ}X → S^{θ_{i}}C^{θ^{(i)}}X (ε_{i} = 1) → S^{θ_{i}}C^{θ^{(i)}}X (ε_{i} = 0),

where X (ε_{i} = 1) and X (ε_{i} = 0) are the hyperfaces of X orthogonal to the i-th direction.

In the next subsection we will use the following corollary of this proposition which can be proved by the same argument as Corollary 1.8 in [Ne1].

Proposition 2.4. Let X → Y be a functor of n-dimensional exact cubes and suppose that the edge-functors of X and Y in the i-th direction are strictly θ_{i}- dominant. Let ε_{1}, …, ε_{i−1}, ε_{i+1}, …, ε_{n} ∈ {0, 1} and put

ε_{(0)} = (ε_{1}, …, ε_{i−1}, 0, ε_{i+1}, …, ε_{n})
ε_{(1)} = (ε_{1}, …, ε_{i−1}, 1, ε_{i+1}, …, ε_{n}).

Suppose that the square

S^{θ_{i}}X (ε_{(1)}) ⟶ S^{θ_{i}}Y(ε_{(1)})
↓  ↓
S^{θ_{i}}X (ε_{(0)}) ⟶ S^{θ_{i}}Y(ε_{(0)})

is homotopy cartesian for any choice of ε_{1}, …, ε_{i−1}, ε_{i+1}, …, ε_{n}. Then the map C^{θ}X → C^{θ}Y is a homotopy equivalence.

2.4. The W -construction. Consider the square of exact categories

W = W(A) = [\begin{matrix}A & ⟵^{Id} & A \\ ↓^{diag} & ^{Id}↓ \\ A × A & ⟵^{diag} & A \end{matrix}]

and let W A = C^{(+1,−1)}W(A), i.e., W A is the mapping cone construction which is ‘covariant’ in the horizontal and ‘contravariant’ in the vertical direction, applied to the square W(A). We also put

V(A) = [\begin{matrix}0 & ⟵ & 0 \\ ↓ & & ↓ \\ A × A & ⟵^{diag} & A \end{matrix}]

and V A = C^{(+1,−1)}V(A). Then, by Proposition 2.4 (with i = 1), the natural map V(A) ↪ W(A) induces a homotopy equivalence V A → W A. Notice that G.A = V_{·,0}(A) ↪ V (A) and that V (A) = iG.A in the notation of [Wa, Sect.1.4]. A corollary to Lemma1.4.1 in [Wa] asserts that S.A ↪ iS.A is a homotopy equivalence. The same argument works for the G-construction, whence the composite map

G.A = V_{·,0}A ↪ V A → W A

is a homotopy equivalence. Replacing V(A) and V A by

V′(A) = [\begin{matrix}A & ⟵ & 0 \\ ^{diag}↓ & & ↓ \\ A × A & ⟵ & 0 \end{matrix}]

and V′A = C^{(+1,−1)}V′(A) we get that

G.A^{op} = G^{op}.A = V′_{0,·}A ↪ V′A → W A

is a homotopy equivalence. We have proved

Theorem 2.5. The embeddings

G.A = W_{·,0}A ↪ W A and G^{op}.A = W_{0,·}A ↪ W A

are homotopy equivalences.

Remark. In [Ne5] we use the C^{(+1,−1)}-functor applied to the square

[\begin{matrix}SES(A) & ⟵ & SES(A) \\ ↓ & & ↓ \\ A × A × A & ⟵ & SES(A) \end{matrix}] ,

where SES(A) denotes the category of s.e.s’s in A and SES(A) → A × A × A takes A ↣ B ↠ C to (A, B, C), to calculate λ^{2} on K_{1}(A). The same argument shows that the resulting bisimplicial set is homotopy equivalent to C[SES(A) → A × A × A], which is the ‘triple’ G-construction [Gr1] and represents therefore the K-theory of A homotopy type.

3. Dualization on K_{1} by means of W

For any bisimplicial set X, let X̄ be the bisimplicial set X̄_{m,n} = X_{n,m}. There is a canonical homeomorphism of their geometric realizations ϕ_{X} : |X̄| ⟶ |X| which takes x × Δ^{m} × Δ^{n} to x × Δ^{n} × Δ^{m} for any x ∈ X_{n,m}. The W -construction is self-dual in the sense that W A^{op} = \overline{W A}, which yields a homeomorphism |W A^{op}| = |\overline{W A}| ⟶ |W A|.

A duality functor ∗ : A ⟶ A^{op} induces a bisimplicial map W A ⟶ W (A^{op}) = \overline{W A} under which G.A = W_{·,0} maps to G^{op}.A = W_{0,·}. We have therefore a commutative diagram of spaces

\xymatrix{|W A| \ar[r]^{∗} & |\overline{W A}| \ar[r]^{ϕ_{W A}} & |W A| \\ |G.A| \ar[u] \ar[rr]^{∗} &  & |G^{op}.A| \ar[u]}

This enables us to define the induced maps ∗ : K_{m}A ⟶ K_{m}A by identifying K_{m}A ≅ π_{m}(G.A) ≅ π_{m}(W A), where the latter isomorphism is given by the homotopy equivalence G.A ↪ W A of Theorem 2.5.

Recall [Ne2-4] that given a d.s.e.s. l = (A ⇉_{f_{2}}^{f_{1}} B ⇉_{g_{2}}^{g_{1}} C), we can associate to it a

3-edge loop

μ(l) = •^{(0,0)} ⟶^{e(A)} •^{(A,A)} ⟶^{e(l)} •^{(B,B)} ⟵^{e(B)} •^{(0,0)}

in G.A = W_{·,0}A, where e(A) = (0 ⟶ A ⟶^{1} A, 0 ⟶ A ⟶^{1} A) is the canonical edge

from the base point to (A, A) (the same for B) and e(l) = (A ↣^{f_{1}} B ↠^{g_{1}} C, A ↣^{f_{2}} B ↠^{g_{2}} C) is the edge from (A, A) to (B, B) given by l. Denote m(l) the class of μ(l) in π_{1}(G.A) ≅ K_{1}(A). According to [Ne2], every element of K_{1}(A) can be represented as m(l) for some l. The image of μ(l) under the dualization is the loop

μ(l)^{∗} = •^{(0,0)} ⟶^{e(A)^{∗}} •^{(A^{∗},A^{∗})} ⟶^{e(l)^{∗}} •^{(B^{∗},B^{∗})} ⟵^{e(B)^{∗}} •^{(0,0)}

in G^{op}.A = W_{0,·}A, where e(A)^{∗} = (0 ⟵ A^{∗} ⟵^{1} A^{∗}, 0 ⟵ A^{∗} ⟵^{1} A^{∗}) (the same for

B) and e(l)^{∗} = (A^{∗} ↞^{f_{1}^{∗}} B^{∗} ↢^{g_{1}^{∗}} C^{∗}, A^{∗} ↞^{f_{2}^{∗}} B^{∗} ↢^{g_{2}^{∗}} C^{∗}). Thus the calculation of m(l)^{∗}

amounts basically to constructing a homotopy in W A which transforms μ(l)^{∗} to a

loop in G.A = W_{·,0}A. Let l^{∗} = (C^{∗} ⇉_{g_{2}^{∗}}^{g_{1}^{∗}} B^{∗} ⇉_{f_{2}^{∗}}^{f_{1}^{∗}} A^{∗}) be the dual d.s.e.s.

Proposition 3.1. μ(l)^{∗} is homotopic to μ(l^{∗}) in W A.

Corollary 3.2. m(l)^{∗} = m(l^{∗}) for any d.s.e.s. l.

Lemma 3.3. For any X ∈ A, put e^{op}(X) = (0 ⟵ X ⟵^{1} X, 0 ⟵ X ⟵^{1} X) ∈ W_{0,1}A. Then the loop

•^{(0,0)} ⟶^{e(X)} •^{(X,X)} ⟵^{e^{op}(X)} •^{(0,0)}

in W A is contractible.

Proof of the lemma. A contracting 2-cell is given by the (1,1)-simplex

[figure: 0; X; X; 0; X; X; 0; X; X; 0; X; X; 0; 0; 0; 0; 0; 0]

Proof of the proposition. By the lemma, we can replace the edges e(A)^{∗} = e^{op}(A^{∗}) and e(B)^{∗} = e^{op}(B^{∗}) in μ(l)^{∗} by e(A^{∗}) and e(B^{∗}) respectively, and the edge e(C^{∗}) in μ(l^{∗}) by e^{op}(C^{∗}). It then follows that μ(l)^{∗}μ(l^{∗})^{−1} is homotopic to the contour of the (1,1)-simplex (cf. (1.7))

[figure: C^{∗}; 1; 1; C^{∗}; g_{1}^{∗}; 0; C^{∗}; 1; 1; C^{∗}; g_{2}^{∗}; 0; C^{∗}; g_{1}^{∗}; B^{∗}; f_{1}^{∗}; f_{1}^{∗}; A^{∗}; 1; C^{∗}; g_{2}^{∗}; B^{∗}; f_{2}^{∗}; f_{2}^{∗}; A^{∗}; 1; 0; A^{∗}; 1; A^{∗}; 0; A^{∗}; 1; A^{∗}]

Remark. It comes as no surprise that the formula of Corollary 3.2 differs by sign from the formula (1.4) obtained by means of the Q-construction. Note that the induced action of ∗ : |W A| → |W A| on K_{0}(A) ≅ π_{0}(W A) is given by [A]^{∗} = [A^{∗}], which also differs by sign from (1.1). The reason is that we have a certain choice in the definition of duality maps caused by the choice of a homotopy equivalence W A ≅ W A^{op} for instance. We could get another map by composing with an involution of W A which provides −1 on the homotopy groups, and get another sign in the formulas as a result. However I think that the definitions in the paper are natural and should be used as they are.

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