The agreement of Waldhausen and Segal’s approach

which means that φ(f) = f iber{K(A) → K^deg(S)} is the connected cover of the fiber of K(A)→K(B).

We may regard φ(f)as a simplicial space [q]7→φq(f) =f iber{K(A)→K(Sq)}. Then φ0(f) = 0 and πi(φ1(f)) = K_i+1^Stein(f). An analysis shows that d0 −d1+d2: π0(φ2(f)) → π0(φ1(f))is zero, whereasd0−d1+d2−d3:π0(φ3(f))→π0(φ2(f)) is surjective, so theE2

term of the spectral sequence associated to the simplicial space looks like 0 K₃^Stein(f)/? . . .

0 K₂^Stein(f)/? ? . . . 0 K₁^Stein(f) 0 ? . . .

This gives thatK₁^Stein(f)is correct, wheras K₂^Stein(f)surjects ontoπ2 of relative K-theory.

2 Agreement of the various K-theories.

This section aims at removing any uncertainty due to the many definitions of algebraic K-theory that we have used. In 2.1 we show that the approach of Waldhausen and Segal agree, at least for additive categories. In section 2.2 we show that Segal’s machine is an infinite delooping of the plus-construction, and show how this is related to group-completion. In 2.3 we give the definition of the algebraic K-theory space of an S-algebra. Forspherical group rings as in II.1.4.4.2, i.e., S-algebras of the form S[G] for G a simplicial group, we show that the algebraic K-theory space of S[G] is the same as Waldhausen’s algebraic K-theory of the classifying space BG. Lastly, we show that the definition of the algebraic K-theory of anS-algebra as defined in chapter II is the infinite delooping of the plus-construction.

2. AGREEMENT OF THE VARIOUS K-THEORIES. 119 simpy natural transformations of such diagrams. For instance, HC(1¯ +)is isomorphic to C, whereas HC(2¯ +)consists of pushout diagrams

0 −−−→ c{0,1}



y y c{0,2} −−−→ c{0,1,2}

.

We see that HC(k¯ +) is equivalent as a category to C^×k via the map sending a functor c∈ obHC(k¯ +) to c{0,1}, . . . , c{0,k}. However, C^×k is not necessarily functorial in k, making HC¯ the preferred model for the bar construction of C.

Also, this formulation of HC¯ is naturally isomorphic to the one we gave in II.3, the advantage is that it is easier to compare with Waldhausen’s construction.

Any functor from Γ^o is naturally a simplicial object by precomposing with the circle S¹: ∆^o → Γ^o (after all, the circle is a simplicial finite pointed set). We could of course precompose with any other simplicial finite pointed set, and part of the point about Γ-spaces was that if M was a functor from Γ^o to sets, then {m7→M(S^m)} is a spectrum.

2.1.2 The relative H-construction.¯

In order to compare Segal’s and Waldhausen’s contructions it will be convenient to have a concrete model for the homotopy fiber ofH¯ applied to an exact functorC → Dof categories with sum (or more generally, a monoidal functor of symmetric monoidal categories). To this end we define the simplicial Γ-category CC→D by the pullback

CC→D(X) −−−→ HD(P S¯ ¹∧X)

 y

 y HC(S¯ ¹∧X) −−−→ HD(S¯ ¹∧X)

.

Here P S¹ is the “path space” on S¹ as defined in appendix A.1.7: (P S¹)q = S_q+1¹ where the ith face map is thei+ 1st face map inS¹, and where the zeroth face map ofS¹ induces a weak equivalence P S¹ → S₀¹ = ∗. The point of this construction is lemma 2.1.5 which displays it as a relative version of the H-construction, much like the usual construction¯ involving the path space in topological spaces.

Usually categorical pullbacks are of little value, but in this case it turns out that it is equivalent to the fiber product.

Definition 2.1.3 Let C₁ ^{f1 //}C₀^{oo f2} C₂ be a diagram of categories. The fiber product Q(f1, f2) is the category whose objects are tuples (c1, c2, α) where ci ∈ obC_i for i = 1,2 and α is an isomorphism in C0 fromf1c1 tof2c2; and where a morphism from (c1, c2, α)to

(d1, d2, β) is a pair of morphisms gi: ci →di for i= 1,2 such that f1c1

−−−→α f2c2 f1g1



y ^f2g2

 y f1d1

−−−→β f2d2

commutes.

Fiber products (like homotopy pullbacks) are good because of their invariance: if you have a diagram

C₁ ^{f1 //}

≃

C₀

≃

C₂

f2

oo

≃

C^′₁ ^f

1′

//C^′₀ C^′₂^f

2′

oo

where the vertical maps are equivalences, you get an equivalence Q

(f1, f2) → Q

(f₁^′, f₂^′).

Note also the natural mapF: C₁×_C0 C₂ →Q

(f1, f2)sending (c1, c2) to(c1, c2,1f1c1).

This map is occasionally an equivalence, as is exemplified in the following lemma. If C is a category, thenIsoC is the class of isomorphisms, and iff is a morphism, thensf is its source and tf is its target.

Lemma 2.1.4 Let C1 f1

//C0 C2

f2

oo be a diagram of categories, and assume that the map of classes

IsoC1

g7→(f₁g,sg)

−−−−−−→ IsoC0×obC0 obC1

has a section (the pullback is taken along source and f1). Then the natural map F: C1×C0

C2 →Q

(f1, f2) is an equivalence.

Proof: Letσ: IsoC0×obC0obC1 →IsoC1 be a section, and define G: Q

(f1, f2)→ C1×C0C2

by G(c1, c2, α) = (tσ(α, c1), c2) and G(g1, g2) = (σ(d1, β)g1σ(c1, α)⁻¹, g2). Checking the diagrams proves thatF and G are inverses up to natural isomorphisms built out of σ.

One should think about the condition in lemma 2.1.4 as a categorical equivalent of the Kan-condition in simplicial sets. This being one of the very few places you can find an error (even tiny and in the end totally irrelevant) in Waldhausen’s papers, it is cherished by his fans since in [234] he seems to claim that the pullback is equivalent to the fiber products iff1 has a section (which is false). At this point there is even a small error in [93, page 257], where it seems that they claim that the relevant map in lemma 2.1.4 factors throughf1.

Now, IsoHD(P S¯ ¹∧X) → IsoHD(S¯ ¹∧X)×_obHD(S¯ ¹∧X) obHD(P S¯ ¹∧X) has a section given by pushouts in the relevant diagrams. Hence CC→D(X) is equivalent to the fiber product category, and as such is invariant under equivalences. Consequently the natural map

CC→D(k+)q −−−→ C^×qk×_D×qk D^×(q+1)k∼=C^×qk× D^×k

2. AGREEMENT OF THE VARIOUS K-THEORIES. 121 is an equivalence. If we consider categories with sum and weak equivalences, we get a structure of sum and weak equivalence onCC→D as well, with

wCC→D =wHC¯ (S¹∧X)×_wHD(S¯ ¹∧X)wHD(P S¯ ¹∧X).

Notice also that the construction is natural: if you have a commuting diagram C −−−→ D

 y

 y C^′ −−−→ D^′

you get an induced mapCC→D →CC^′→D^′ by using the universal properties of pullbacks, and the same properties ensure that the construction behaves nicely with respect to composi-tion. FurthermoreC∗→D(1+)is isomorphic toD, so we get a mapD ∼=C∗→D(1+)→CC→D, and if we have mapsC → D → E whose composite is trivial, we get a mapCC→D(1₊)→ E. Recall that, if C is a category with sum (i.e., with finite coproducts and with a “zero object” which is both final and initial), then an exact functor to another category with sums D is a functor C → D preserving coproducts and the zero objects.

Lemma 2.1.5 Let C → D be an exact functor of categories with sum and weak equiva-lences. Then there is a stable fiber sequence

wHC →¯ wHD →¯ wH(C¯ C→D(1+)) Proof: It is enough to show that

wHD(S¯ ¹)→wH(C¯ C→D(1+))(S¹)→wH( ¯¯ H(C)(S¹))(S¹) is a fiber sequence, and this follows since in each degree n

wHD(S¯ ¹)→wH(C¯ C→D(1₊)_n)(S¹)→wH( ¯¯ H(C)(S¹)_n)(S¹) is equivalent to the product fiber sequence

wHD(S¯ ¹)→wH(D × C¯ ^×n)(S¹)→wH(C¯ ^×n)(S¹) and all spaces involved are connected.

We have a canonical map

HC(S¯ ¹)→SC

which in dimension q is induced by sending the sum diagramC ∈obHC¯ (q+) to c∈obSqC with cij =C{0,i+1,i+2,...,j−1,j} and obvious maps.

Scholium 2.1.6 For those familiar with algebraic K-theory, the additivity theorem for Waldhausen’s S-construction says that iS(SkC) → iS(C^×k) is a weak equivalence. We have not used this so far, but in the special case of additive categories it is an immediate consequence of theorem 2.1.7 below. The additivity theorem for Segal’s model, saying that iH(S¯ kC)→iH(C¯ ^×k) is a weak equivalence, is at the core of the proof of the theorem, and constitutes the second half of the proof.

Theorem 2.1.7 Let C be an additive category. Then the map iHC(S¯ ¹)→iSC

is a weak equivalence.

Proof: Since both BiHC¯ and BiSCare connected, the vertical maps in BiHC(S¯ ¹) −−−→ BiSC

≃



y ^≃y

Ω BiH( ¯¯ HC(S¹))(S¹)

−−−→ Ω BH(iS¯ C)(S¹) are equivalences by A.5.1.2, and so it is enough to prove that

BiH( ¯¯ HC(S¹))→BiH(SC)¯

is an equivalence, which again follows if we can show that for every q BiH( ¯¯ HC(q₊))→BiH¯(S_qC)

is an equivalence.

Essentially this is the old triangular matrices vs. diagonal matrices question, and can presumably be proven directly by showing that iSqC → iC^×q induces an isomorphism in homology after inverting π0(iSqC)∼=π0(iC^×q).

Assume we have proven that the projection iH(S¯ kC)→ iH(C¯ ^×k) is an equivalence for k < q (this is trivial for k = 0 or k = 1). We must show that it is also an equivalence for k = q. Consider the inclusion by zero’th degeneracies C → SqC (sending c to 0 ֌ 0 ֌ . . .֌0֌c), and the last face mapSqC→Sq−1C. We want to show that we have a map of fiber sequences

iH(C)¯ −−−→ iH(S¯ qC) −−−→ iH(S¯ q−1C)

 y

 y iH(C)¯ −−−→ iH(C¯ ^×q) −−−→ iH(C¯ ^×q−1) We do have maps of fiber sequences

iH(C)¯ −−−→ iH(S¯ qC) −−−→ iH(C¯ C→SqC(1+))

y y

iH(C)¯ −−−→ iH(C¯ ^×q) −−−→ iH(C¯ C→C^×q(1+))

and the only trouble lies in identifying the base spaces of the fibrations. We have a commuting square

iH(C¯ C→SqC(1+)) −−−→ iH(S¯ q−1C)



y ^≃y iH(C¯ C→C^×q(1+)) −−−→^∼ iH(C¯ ^×q−1)

2. AGREEMENT OF THE VARIOUS K-THEORIES. 123

In document The local structure of algebraic K-theory (sider 118-123)