c

2008 by Institut Mittag-Leffler. All rights reserved

## The localization sequence for the algebraic K -theory of topological K -theory

### by

Andrew J. Blumberg

Stanford University Stanford, CA, U.S.A.

Michael A. Mandell

Indiana University Bloomington, IN, U.S.A.

Introduction

The algebraic K-theory of ring spectra is intimately related to the geometry of high- dimensional manifolds. In a series of papers in the 1970’s and 1980’s, Waldhausen es- tablished the deep connection between theK-theory of the sphere spectrum and stable pseudo-isotopy theory. The sphere spectrum has an infinite filtration called the chro- matic filtration that forms a tower at each prime p. The layers in the chromatic tower capture periodic phenomena in stable homotopy theory, corresponding to the Morava K-theory “fields”. The chromatic viewpoint has organized the understanding of stable homotopy theory over the last thirty years.

Applying algebraicK-theory to the chromatic tower of the sphere spectrum leads to an analogous “chromatic tower” of algebraicK-theory spectra. Waldhausen conjectured that this tower converges in the homotopy inverse limit to the algebraicK-theory of the p-local sphere spectrum [10]. McClure and Staffeldt verified a closely related connective variant of this conjecture [6]. The bottom layer of the chromatic tower is K(Q), the algebraic K-theory of the rational numbers, which is intimately related to questions in arithmetic. Thus, theK-theory chromatic tower can be viewed as an interpolation from arithmetic to geometry.

In [2], Ausoni and Rognes describe an ambitious program for analyzing the layers in
thep-complete version of Waldhausen’sK-theory chromatic tower. The analysis is based
on descent conjectures coming from Rognes’ Galois theory ofS-algebras [8], which relate
the layers of the tower to the K-theory of the Morava E-theory ring spectra En and
to theK-theory of thep-completed Johnson–Wilson ring spectraE(n)^{∧}_{p}. The spectrum

The first author was supported in part by a NSF postdoctoral fellowship.

The second author was supported in part by NSF grant DMS-0504069.

E(n) is not connective, and no tools exist for computing theK-theory of non-connective
ring spectra. However,E(n) is formed from the connective spectrumBPhniby inverting
the elementvn in π_{∗}(BPhni), and Rognes conjectured the following localization cofiber
sequence.

Rognes conjecture. The transfer map K(BPhn−1i^{∧}_{p})!K(BPhni^{∧}_{p}) and the
canonical map K(BPhni^{∧}_{p})!K(E(n)^{∧}_{p})fit into a cofiber sequence in the stable category

K(BPhn−1i^{∧}_{p})−!K(BPhni^{∧}_{p})−!K(E(n)^{∧}_{p})−!ΣK(BPhn−1i^{∧}_{p}).

In the case n=0, the statement is an old theorem of Quillen [7], the localization
sequence K(Z/p)!K(Z^{∧}p)!K(Q^{∧}p). The conjecture implies a long exact sequence of
homotopy groups:

...−!K_{q}(BPhn−1i^{∧}_{p})−!K_{q}(BPhni^{∧}_{p})−!K_{q}(E(n)^{∧}_{p})−!...

...−!K_{0}(BPhn−1i^{∧}_{p})−!K_{0}(BPhni^{∧}_{p})−!K_{0}(E(n)^{∧}_{p})−!0.

This in principle allows the computation of the algebraicK-groups ofE(n)^{∧}_{p} from those
ofBPhni^{∧}_{p} andBPhn−1i^{∧}_{p}. Because theBPhniare connective spectra, theirK-theory
can be studied using TC, and the main purpose of [2] is to provide tools suitable for
computing TC(BPhni^{∧}_{p}), and in particular to obtain an explicit computation of the
V(1)-homotopy in the casen=1.

The case n=1 is the first non-classical case of the conjecture and is of particular interest. The spectraE(1) andBPh1iare often denoted byLand`and are closely related to topological K-theory. Here L=E(1) is the p-local Adams summand of (complex periodic) topological K-theory, KU, and `=BPh1i is the p-local Adams summand of connective topological K-theory, ku. The Rognes conjecture then admits a “global”

version in this case, namely that the transfer map K(Z)!K(ku) and canonical map K(ku)!K(KU) fit into a cofiber sequence in the stable category

K(Z)−!K(ku)−!K(KU)−!ΣK(Z).

This version of the conjecture appears in [3], where the relationship between the alge- braicK-theory of topologicalK-theory, elliptic cohomology, and the category of 2-vector bundles is discussed.

The purpose of this paper is to prove both the local and global versions of the Rognes conjecture in the casen=1. Specifically, we prove the following theorem.

Localization theorem. There are connecting maps, which together with the transfer maps and the canonical maps,make the sequences

K(Z^{∧}p)−!K(`^{∧}_{p})−!K(L^{∧}_{p})−!ΣK(Z^{∧}p),
K(Z(p))−!K(`)−!K(L)−!ΣK(Z(p)),
K(Z)−!K(ku)−!K(KU)−!ΣK(Z)
cofiber sequences in the stable category.

Hesselholt observed that the Ausoni–Rognes and Ausoni calculations of THH(`) and THH(ku) in [2] and [1] are explained by the existence of a THH version of this localization sequence along with a conjecture about the behavior of THH for “tamely ramified” extensions of ring spectra. A precise formulation requires a construction of THH for Waldhausen categories. We will explore this more fully in a forthcoming paper.

The localization theorem above is actually a consequence of a “d´evissage” theorem
for finitely generated finite stage Postnikov towers. For an S-algebra (A∞ ring spec-
trum) R, let PR denote the full subcategory of left R-modules that have only finitely
many non-zero homotopy groups, all of which are finitely generated overπ_{0}R. WhenR
is connective andπ_{0}Ris left Noetherian, this category has an associated WaldhausenK-
theory spectrum. Restricting to the subcategory ofS-algebras with morphisms the maps
R!R^{0} for whichπ_{0}R^{0} is finitely generated as a leftπ_{0}R-module, we regardK(P_{(−)}) as
a contravariant functorK^{0} to the stable category.

We use the notationK^{0}because of the close connection with Quillen’sK^{0}-theory, the
K-theory of the exact category of finitely generated left modules over a left Noetherian
ring. The analogousK-theory spectrum for chain complexes over the ring π0R is well
known to be equivalent toK^{0}(π_{0}R) [9, Theorem 1.11.7]. The following theorem is the
main result of this paper.

D´evissage theorem. Let R be a connective S-algebra (A_{∞} ring spectrum) with
π_{0}R left Noetherian. Then there is a natural isomorphism in the stable category

K^{0}(π0R)−!K^{0}(R),

where K^{0}(π0R)is Quillen’s K-theory of the exact category of finitely generated left π0R-
modules, and K^{0}(R) is the Waldhausen K-theory of the category of finitely generated
finite stage Postnikov towers of left R-modules, PR.

A longstanding open problem first posed explicitly by Thomason and Trobaugh [9, §1.11.1] is to develop a general d´evissage theorem for Waldhausen categories that

specializes to Quillen’s d´evissage theorem when applied to the category of bounded chain complexes on an abelian category. We regard the theorem described in this paper as a step towards a solution to this problem.

Acknowledgement. The authors would like to thank the Institut Mittag-Leffler for hospitality while writing this paper.

1. The main argument

In this section, we outline the proof of the d´evissage theorem in terms of a number of easily stated results proved in later sections; we then deduce the localization theorem from the d´evissage theorem. Although we assume some familiarity with the basics of Waldhausen K-theory, we review the standard definitions and constructions of Waldhausen [11] as needed. We begin with some technical conventions and a precise description of the Waldhausen categories we use.

Throughout this paper, R denotes a connective S-algebra with π0R left Noether- ian. We work in the context of Elmendorf–Kriz–Mandell–May (EKMM) S-modules, S-algebras, and R-modules [4]. Since other contexts for the foundations of a modern category of spectra lead to equivalentK-theory spectra, presumably the arguments pre- sented here could be adjusted to these contexts, but the EKMM categories have certain technical advantages that we exploit (see for example the proof of Lemma 4.2) and which affect the precise form of the statements below.

The input for WaldhausenK-theory is a (small) category together with a subcat- egory of “weak equivalences” and a subcategory of “w-cofibrations” satisfying certain properties. LetPR denote the full subcategory of leftR-modules that have only finitely many non-zero homotopy groups, all of which are finitely generated overπ0R. Although PR is not a small category, we can still construct a K-theory spectrum from it that is “homotopically small”, and we can find a small category with equivalent K-theory by restricting the sets allowed in the underlying spaces of the underlying prespectra;

see Remark 1.7 below for details. In what follows, let P denote PR or, at the reader’s preference, the small categoryPk (forklarge) discussed in Remark 1.7.

We makeP a Waldhausen category by taking the weak equivalences to be the usual weak equivalences (the maps that induce isomorphisms on all homotopy groups) and the w-cofibrations to be the Hurewicz cofibrations (the maps satisfying the homotopy extension property in the category of leftR-modules). Specifically, a mapi:A!X is a Hurewicz cofibration if and only if the inclusion of the mapping cylinder

M i=X∪_{i}(A∧I+)

in the cylinderX∧I+has a retraction in the category of R-modules. Some easy conse- quences of this definition are:

• The initial map∗!X is a Hurewicz cofibration for anyR-moduleX;

• Hurewicz cofibrations are preserved by cobase change (pushout);

• for a map of S-algebras R!R^{0}, the forgetful functor from R^{0}-modules to R-
modules preserves Hurewicz cofibrations.

Use of this type of cofibration was a key tool in [4] for keeping homotopical con- trol; a key fact is that pushouts along Hurewicz cofibrations preserve weak equivalences [4, Corollary I.6.5], and in particular Waldhausen’s gluing lemma [11,§1.2] holds.

Since we are thinking ofP in terms of finite Postnikov stages, it makes more sense
philosophically to work with fibrations: P^{op} forms a Waldhausen category with weak
equivalences the maps opposite to the usual weak equivalences, and with w-cofibrations
the maps opposite to the Hurewicz fibrations (maps satisfying the covering homotopy
property). Although not strictly necessary for the d´evissage theorem, the following the-
orem proved in§2 straightens out this discrepancy.

Theorem 1.1. The spectra K(P) and K(P^{op})are weakly equivalent.

This result is essentially a consequence of the fact thatP is a stable category, and
in particular follows from the observation that homotopy cocartesian and homotopy
cartesian squares coincide in P. As part of the technical machinery employed in the
proof of the d´evissage theorem, we introduce a variant of Waldhausen’s S_{}construction,
using homotopy cocartesian squares where pushout squares along w-cofibrations are used
in theS_{}construction. We denote this construction byS_{}^{0}. Theorem 1.1 follows from the
comparison of theS_{}^{0} construction with the S_{}construction in §2.

Let P_{m}^{n} for m6n denote the full subcategory of P consisting of those R-modules
whose homotopy groupsπ_{q}are zero forq >norq <m. In this notation, we permitm=−∞

and/orn=∞, so P=P_{−∞}^{∞} . Define a w-cofibration in P_{m}^{n} to be a Hurewicz cofibration
whose cofiber is still inP_{m}^{n}, or, equivalently, a Hurewicz cofibration inducing an injection
on πn. This definition makes P_{m}^{n} into a Waldhausen category with the usual weak
equivalences; it is a “Waldhausen subcategory” ofP [11, §1.2]. We prove the following
theorem in§3 and§4.

Theorem1.2.The inclusionP_{0}^{0}!P induces a weak equivalence of K-theory spectra.

LetE denote the exact category of finitely generated leftπ_{0}R-modules (or a skele-
ton, to obtain a small category). This becomes a Waldhausen category with weak
equivalences the isomorphisms and w-cofibrations the injections. Waldhausen’s “S_{}=Q”

theorem [11,§1.9] identifies the Waldhausen K-theory K(E) as K^{0}(π_{0}R). The functor

π0:P_{0}^{0}!E is an “exact functor” of Waldhausen categories: it preserves weak equiva-
lences, w-cofibrations and pushouts along w-cofibrations. It follows that π0 induces a
map fromK(P_{0}^{0}) toK(E)'K^{0}(π0R). We prove the following theorem in§4.

Theorem 1.3. The functor π_{0} induces a weak equivalence K(P_{0}^{0})!K(E).

Theorems 1.2 and 1.3 together imply the d´evissage theorem. ForK-theoretic rea-
sons, we should regard K^{0}(π0R)!K(P) to be the natural direction of the composite
zigzag, as this is compatible with the forgetful functorPHπ0R!PR(induced by pullback
along the map R!Hπ_{0}R). Although it appears feasible to construct directly a map of
spectraK^{0}(π_{0}R)!K(P) using a version of the Eilenberg–MacLane bar construction, the
technical work required would be unrelated to the arguments in the rest of this paper,
and so we have not pursued it.

Next we deduce the localization theorem from the d´evissage theorem. Let Rbe one
of ku, `, or `^{∧}_{p}, and let β denote the appropriate Bott element in π_{∗}R in degree 2 or
2p−2. Then R[β^{−1}] isKU,L, or L^{∧}_{p}, respectively. For convenience, letZ denoteπ0R;

so Z=Z, Z(p), or Z^{∧}p, in the respective cases. Then for A=HZ, R, or R[β^{−1}], let CA

be the category of finite cell A-modules, which we regard as a Waldhausen category
with w-cofibrations the maps that are isomorphic to inclusions of cell subcomplexes,
and with weak equivalences the usual weak equivalences, that is, the maps that induce
isomorphisms on homotopy groups. As explained in [4], the Waldhausen K-theory of
CA is the algebraic K-theory of A. When A=HZ, the K-theory spectrum K(CA) is
equivalent to Quillen’s K-theory of the ring Z [4, Theorem VI.4.3]. In the case of
A=HZ or A=R, the K-theory spectrum K(CA) is equivalent to two other reasonable
versions of theK-theory ofA: theK-theory spectrum defined via the plus construction
of “BGL(A)” [4, §VI.7] and theK-theory spectrum defined via the permutative category
of wedges of sphereA-modules [4,§VI.6]. However, when A=R[β^{−1}], the construction
K(CA) and its variants are essentially the only known ways to define K(A).

LetCR[β^{−1}] denote the Waldhausen category whose underlying category isCR and
whose w-cofibrations are the maps that are isomorphic to inclusions of cell subcomplexes,
but whose weak equivalences are the maps that induce isomorphisms on homotopy groups
after inverting β. The identity functor CR!CR[β^{−1}] is then an exact functor, because
the weak equivalences inCR are in particular weak equivalences inCR[β^{−1}]. Let C_{R}^{β} be
the full subcategory ofCR of objects that are weakly equivalent inCR[β^{−1}] to the trivial
object ∗, that is, the finite cell R-modules whose homotopy groups become zero after
invertingβ. In other words, we considerC_{R}with two subcategories of weak equivalences,
one of which is coarser than the other, and the category of acyclic objects for the coarse
weak equivalences. This is the situation in which Waldhausen’s “fibration theorem”

[11, Theorem 1.6.4] applies. The conclusion in this case, restated in terms of the stable category, is the following proposition.

Proposition 1.4. There is a connecting map (of spectra) K(CR[β^{−1}])!ΣK(C_{R}^{β}),
which together with the canonical maps makes the sequence

K(C_{R}^{β})−!K(R)−!K(CR[β^{−1}])−!ΣK(C_{R}^{β})
a cofiber sequence in the stable category.

For the proof of the localization theorem, we need to identify K(C_{R}^{β}) as K(π0R),
K(CR[β^{−1}]) asK(R[β^{−1}]), and the maps as the transfer map and the canonical map.

We begin with the comparison of K(CR[β^{−1}]) with K(R[β^{−1}]). The localization
functorR[β^{−1}]∧R(−) restricts to an exact functor of Waldhausen categories

CR[β^{−1}]−!C_{R[β}^{−1}_{]}.

Waldhausen developed a general tool for showing that an exact functor induces a weak equivalence ofK-theory spectra, called the “approximation theorem” [11, Theorem 1.6.7].

In the case of a telescopic localization functor like this one, the argument of [10] adapts to show that the hypotheses of Waldhausen’s approximation theorem are satisfied. This then proves the following proposition.

Proposition 1.5. The canonical map K(R)!K(R[β^{−1}]) factors as the map
K(R)−!K(CR[β^{−1}])

in Proposition 1.4and a weak equivalence

K(C_{R}[β^{−1}])−!K(R[β^{−1}]).

Next we move on to the comparison of K(C^{β}_{R}) and K(π0R). Since finite cell R-
modules that areβ-torsion have finitely generated homotopy groups concentrated in a
finite range,C_{R}^{β} is a subcategory ofPR. Cellular inclusions are Hurewicz cofibrations, so
the inclusionC_{R}^{β}!PR is an exact functor of Waldhausen categories. On the other hand,
since the cofiber ofβ is a model for the Eilenberg–MacLaneR-moduleHπ_{0}R, an induc-
tion over Postnikov sections implies that every object in PR is weakly equivalent to an
object inC_{R}^{β}. Standard arguments (e.g., [4, Theorem VI.2.5] and the Whitehead theorem
for cell R-modules) apply to verify that the inclusion C_{R}^{β}!P_{R} satisfies the hypotheses
of Waldhausen’s approximation theorem [11, Theorem 1.6.7], and thus induces a weak
equivalence ofK-theory spectra.

The cofiber sequence of Proposition 1.4 is therefore equivalent to one of the form (cf. [3, Proposition 6.8ff])

K(P)−!K(R)−!K(R[β^{−1}])−!ΣK(P).

Applying the d´evissage theorem, we get a weak equivalence
K(HZ)'K^{0}(Z)'K(P)'K(C_{R}^{β}).

To complete the proof of the localization theorem, we need to identify the composite map K(HZ)!K(R) with the transfer map K(HZ)!K(R) induced from the forgetful functor. We begin by reviewing this transfer map.

The transfer map arises becauseHZ is weakly equivalent to a finite cellR-module;

as a consequence all finite cellHZ-modules are weakly equivalent to finite cellR-modules.

To put this into the context of Waldhausen categories and exact functors, letM_{R}(resp.

MHZ) be the full subcategory ofR-modules (resp.HZ-modules) that are weakly equiv- alent to finite cell modules. We make MR and MHZ Waldhausen categories with w- cofibrations the Hurewicz cofibrations and weak equivalences the usual weak equivalences.

Then the forgetful functor fromHZ-modules toR-modules restricts to an exact functor MHZ!MR. The inclusions CR!MR and CHZ!MHZ are exact functors. Again, a standard argument [4, Proposition VI.3.5] with Waldhausen’s approximation theorem shows that these inclusions induce weak equivalences ofK-theory spectra. The transfer mapK(HZ)!K(R) is the induced map

K(HZ)'K(MHZ)−!K(MR)'K(R).

The Waldhausen categoryPR is a Waldhausen subcategory ofMR, and the exact
functor C_{R}^{β}!MR lands in PR as does the exact functor MHZ!MR. We therefore
obtain the following commutative diagram of exact functors:

CHZ

C_{R}^{β} //

CR

M_{HZ} //P_{R} //M_{R}.

This induces the following commutative diagram of K-theory spectra, with the arrows marked “'” weak equivalences (see also Remark 1.7):

K(HZ)

'

K(C^{β}_{R})

'

//K(R)

'

K(M_{HZ})

' //K(P_{R}) //K(M_{R}).

Here the lower left mapK(MHZ)!K(PR) is a weak equivalence by the d´evissage theo- rem above, sinceMHZ=PHZ. This proves the following proposition.

Proposition 1.6. The transfer map K(HZ)!K(R) factors in the stable category
as a weak equivalence K(HZ)!K(C_{R}^{β})and the map K(C_{R}^{β})!K(R)in Proposition 1.4.

Finally, the localization theorem of the introduction follows immediately from Propositions 1.4, 1.5 and 1.6. We close this section with a remark on smallness.

Remark 1.7. The Waldhausen categories P_{R} and M_{R} discussed above can be re-
placed by small Waldhausen categories with equivalentK-theory spectra. For any cardi-
nalk, letP_{k}andM_{k} be the full subcategories ofP_{R}andM_{R}(respectively) consisting of
those objectsM such that the underlying sets of the underlying spaces of the underlying
prespectrum ofM are subsets of the power-set ofk. ThenP_{k} is a small category. When
kis bigger than the continuum and the cardinality of the underlying sets ofR, thenPk

andMk contain a representative of every weak equivalence class inPR andMR, respec- tively. Furthermore,PRandMRare closed under pushouts along Hurewicz cofibrations and pullbacks along Hurewicz fibrations, and closed up to natural isomorphism under the functors (−)∧X andF(X,−) for finite cell complexesX.

LetCP_{k} be the Waldhausen category whose objects are the objectsM ofP_{k}together
with the structure of a cell complex onM, whose morphisms are the maps ofR-modules,
whose w-cofibrations are the maps that are isomorphic to inclusions of cell subcomplexes,
and whose weak equivalences are the usual weak equivalences. LetCMkbe the analogous
category for Mk. When k is bigger than the continuum and the cardinality of the
underlying sets ofR, then, for any cardinalλ>k, the functorsCPk!PλandCMk!Mλ

are exact and satisfy the hypotheses of Waldhausen’s approximation theorem. It follows that these functors induce equivalences of K-theory spectra. The reader unwilling to consider the K-theory of Waldhausen categories that are not small can therefore use K(Pk) in place of K(PR),K(Mk) in place of K(MR), etc.

2. The S_{}^{0} construction and the proof of Theorem 1.1

For the Waldhausen categoriesP_{m}^{n}, the condition of being a w-cofibration consists of both
a point-set requirement (HEP condition) and a homotopical requirement (injectivity
on π_{n}). It is convenient for the arguments in the next section to separate these two
requirements. We do that in this section by describing a variantS_{}^{0} of theS_{}construction
defined in terms of “homotopy cocartesian” squares instead of pushouts of w-cofibrations.

For a large class of Waldhausen categories includingP,P^{op}, and the categoriesP_{m}^{n}, the
S_{}^{0} construction is equivalent to theS_{}construction and can therefore be used in its place

to construct algebraicK-theory. In fact, since the notions of homotopy cocartesian and
homotopy cartesian agree in P, Theorem 1.1, which compares the algebraic K-theory
ofP (defined in terms of cofibrations) with that of P^{op} (defined in terms of fibrations),
then follows as an easy consequence.

The hypothesis on a Waldhausen category we use is a weak version of “functorial factorization”. The Waldhausen category C admits functorial factorization when any mapf:A!B inC factors as a w-cofibration followed by a weak equivalence

A // //

f

55

T f ^{'} //B,

functorially inf in the category ArCof arrows inC. In other words, given the mapφof arrows on the left (i.e., the commuting diagram),

A ^{f} //

a

φ

B

b

A // //

a

T f ^{'} //

T φ

B

b

A^{0}

f^{0}

//B^{0} A^{0} // //T f^{0} _{'} //B^{0},

we have a map T φ that makes the diagram on the right commute and that satisfies
the usual identity and composition relations, Tid_{f}=id_{T f} and T(φ^{0}φ)=T φ^{0}T φ. For
example, the Waldhausen categoriesP andP^{op}admit functorial factorizations using the
usual mapping cylinder and mapping path-space constructions

M f= (A∧I+)∪_{A}B forP, N f=B^{I}^{+}×_{B}A forP^{op}. (2.1)
Functorial factorization generalizes Waldhausen’s notion of “cylinder functor satisfy-
ing the cylinder axiom”, but is still not quite general enough to apply to the Waldhausen
categoriesP_{m}^{n} forn<∞, where a map is weakly equivalent to a w-cofibration only when
it is injective onπn. This leads to the following definition.

Definition 2.2. LetC be a Waldhausen category. Define a mapf:A!B in Cto be a weak w-cofibration if it is weakly equivalent in ArC (by a zigzag) to a w-cofibration.

We say thatCadmitsfunctorial factorization of weak w-cofibrations (FFWC) when weak w-cofibrations can be factored functorially (in ArC) as a w-cofibration followed by a weak equivalence.

Recall that a full subcategoryBof a Waldhausen categoryCis called aWaldhausen subcategorywhen it forms a Waldhausen category with weak equivalences the weak equiv- alences of C, and with w-cofibrations the w-cofibrationsA!B inC (between objectsA

andB of B) whose quotientB/A=B∪A∗ is inB. In particular, it is straightforward to
check that a full subcategory of a Waldhausen category that is closed under extensions is
a Waldhausen subcategory; examples include the subcategoriesP_{m}^{n} ofP. We say that the
Waldhausen subcategoryB isclosed if every object ofC weakly equivalent to an object
ofBis an object ofB. The advantage of the hypothesis of FFWC over the hypothesis of
functorial factorization is the following proposition.

Proposition 2.3. If B is a closed Waldhausen subcategory of a Waldhausen cat- egory C that admits FFWC, then B admits FFWC. Moreover, a weak w-cofibration f:A!B in C between objects in B is a weak w-cofibration in B if and only if the map A!T f in its factorization in C is a w-cofibration in B.

Proof. Letf:A!B be a weak w-cofibration inC between objects inB. Thenf is
weakly equivalent by a zigzag inBto a w-cofibrationf^{0}:A^{0}!B^{0} in C,

A^{0}

f^{0}

' //A_{1}

f1

' ...

oo A_{n}^{'}oo ^{'} //

fn

A

f

B^{0} _{'} //B1 ...

oo ' oo ' Bn

' //B.

Applying functorial factorization inC, we get weak equivalences
B^{0}/A^{0} oo ^{'} T f^{0}/A^{0} ^{'} //T f1/A1 oo ^{'} ... T fn/An

oo ' ^{'} //T f /A,

which imply that the mapA!T fis a w-cofibration inBif and only iff:A!B is a weak w-cofibration inB.

In general, the weak w-cofibrations do not necessarily form a subcategory ofC; the composition of two weak w-cofibrations might not be a weak w-cofibration. However, in the presence of FFWC the weak w-cofibrations are well-behaved.

Proposition2.4. Let C be a Waldhausen category that admits FFWC. If f:A!B and g:B!C are weak w-cofibrations in C,then gf:A!C is a weak w-cofibration in C.

Proof. Applying functorial factorization inC, we can factorfandgas the composites
A // //T f ^{'} //B and B // //T g ^{'} //C.

The composite maph:T f!B!T gis a weak w-cofibration, as it is weakly equivalent in ArCtog. We can therefore apply the factorization functor tohto obtain a w-cofibration T f!T h and a weak equivalenceT h!T g. The composite w-cofibrationA!T f!T h is then weakly equivalent togf.

A square diagram in a Waldhausen category (for example) is called homotopy co- cartesian if it is weakly equivalent (by a zigzag) to a pushout square where one of the parallel sets of arrows consists of w-cofibrations. The corresponding set of parallel arrows in the original square then consists of weak w-cofibrations. When a Waldhausen cate- gory admits FFWC, there is a good criterion in terms of the factorization functorT for detecting homotopy cocartesian squares. We state it for a Waldhausen category that is saturated, i.e., one whose weak equivalences satisfy the “two-out-of-three property” (see, for example, [11,§1.2]). The proof is similar to that of Proposition 2.3.

Proposition2.5. Let C be a saturated Waldhausen category admitting FFWC with functor T. A commutative diagram

A ^{f} //

B

C //D

with f a weak w-cofibration is homotopy cocartesian if and only if the map T f∪AC!D is a weak equivalence.

Likewise, for saturated Waldhausen categories admitting FFWC, homotopy cocarte- sian squares have many of the usual expected properties. We summarize the ones we need in the following proposition; its proof is a straightforward application of the previous proposition.

Proposition 2.6. Let C be a saturated Waldhausen category admitting FFWC.

(i) Given a commutative cube

A^{0} //

BBB B^{0}

!!CCC

A //

B

C^{0} //

B

BB D^{0}

!!C

CC C //D

with the faces ABCDand A^{0}B^{0}C^{0}D^{0} homotopy cocartesian,if the maps A^{0}!A,B^{0}!B,
and C^{0}!C are weak equivalences, then the mapD^{0}!D is a weak equivalence.

(ii) Given a commutative diagram A //

B

//X

C //D //Y

with the square ABCD homotopy cocartesian, if either A!C is a weak w-cofibration or both A!B and B!X are weak w-cofibrations,then the square AXCY is homotopy cocartesian if and only if the square BXDY is homotopy cocartesian.

We use the concept of weak w-cofibration and the previous propositions to build a
homotopical variant of theS_{}construction. First, we recall theS_{}construction in detail.

Waldhausen’s S_{} construction produces a simplicial Waldhausen category S_{}C from a
Waldhausen category C and is defined as follows. Let Ar[n] denote the category with
objects (i, j), for 06i6j6n, and a unique map (i, j)!(i^{0}, j^{0}), for i6i^{0} andj6j^{0}. Then
S_{n}C is defined to be the full subcategory of the category of functorsA: Ar[n]!C (where
we writeA_{i,j} forA(i, j)) such that the following conditions are satisfied:

• A_{i,i}=∗for alli;

• the mapA_{i,j}!A_{i,k} is a w-cofibration for alli6j6k;

• the diagram

A_{i,j} //

A_{i,k}

Aj,j //Aj,k

is a pushout square for alli6j6k.

The last two conditions can be simplified to the hypothesis that each map
A_{0,j}−!A_{0,j+1}

is a w-cofibration and that the induced maps A0,j/A0,i!Ai,j are isomorphisms. This
becomes a Waldhausen category by defining a map A!B to be a weak equivalence
when each A_{i,j}!B_{i,j} is a weak equivalence in C, and to be a w-cofibration when each
A_{i,j}!B_{i,j} and each induced mapA_{i,k}∪Ai,jB_{i,j}!B_{i,k} is a w-cofibration inC.

Definition 2.7. LetC be a saturated Waldhausen category that admits FFWC. De-
fineS_{n}^{0}C to be the full subcategory of functorsA: Ar[n]!C such that the following con-
ditions are satisfied:

• the initial map∗!Ai,i is a weak equivalence for alli;

• the mapAi,j!Ai,k is a weak w-cofibration for alli6j6k;

• the diagram

A_{i,j} //

A_{i,k}

Aj,j //Aj,k

is a homotopy cocartesian square for alli6j6k.

We define a map A!B to be a weak equivalence when eachAi,j!Bi,j is a weak
equivalence inC, and to be a w-cofibration when each Ai,j!Bi,j is a w-cofibration inC
and each induced mapAi,k∪A_{i,j}Bi,j!Bi,k is a weak w-cofibration inC.

The following theorem in particular allows us to iterate theS^{0}_{}construction.

Theorem2.8. Let C be a saturated Waldhausen category that admits FFWC. Then
S_{n}^{0}C is also a saturated Waldhausen category that admits FFWC.

Proof. The proof that S_{n}^{0}C is a saturated Waldhausen category follows the same
outline as the proof thatSnC is a saturated Waldhausen category, substituting Proposi-
tions 2.5 and 2.6 for the homotopy cocartesian squares in place of the usual arguments
for pushout squares. To show thatS_{n}^{0}Cadmits FFWC, we construct a factorization func-
torT as follows. Given a weak w-cofibrationf:A!B in S_{n}^{0}C, let (T f)i,j=T(fi,j). This
then factorsf as a mapA!T f followed by a weak equivalenceT f!B; we need to check
that the map A!T f is a w-cofibration. Since, by construction, the maps Ai,j!T fi,j

are w-cofibrations, we just need to check that the mapsA_{i,k}∪Ai,jT f_{i,j}!T f_{i,k} are weak
w-cofibrations. Since, by hypothesis, f is a weak w-cofibration in S_{n}^{0}C, it is weakly
equivalent (by a zigzag) to a w-cofibrationf^{0}:A^{0}!B^{0} inS^{0}_{n}C. It follows that the maps

Ai,k∪A_{i,j}T fi,j−!T fi,k and A^{0}_{i,k}∪_{A}^{0}

i,jT f_{i,j}^{0} −!T f_{i,k}^{0}

are weakly equivalent (by the corresponding zigzag). SinceA^{0}_{i,j}!B_{i,j}^{0} is a w-cofibration,
the canonical map

A^{0}_{i,k}∪A^{0}_{i,j}T f_{i,j}^{0}

' //A^{0}_{i,k}∪A^{0}_{i,j}B_{i,j}^{0}

T f_{i,k}^{0}

' //B_{i,k}^{0}
is a weak equivalence. The mapA^{0}_{i,k}∪_{A}^{0}

i,jB_{i,j}^{0} !B_{i,k}^{0} is a weak w-cofibration by hypothe-
sis, and so the mapsA^{0}_{i,k}∪_{A}^{0}

i,jT f_{i,j}^{0} !T f_{i,k}^{0} andAi,k∪A_{i,j}T fi,j!T fi,k are therefore weak
w-cofibrations.

BothS_{}CandS_{}^{0}Cbecome simplicial Waldhausen categories with face map∂_{i}deleting
theith row and column, and degeneracy maps_{i} repeating theith row and column. For
eachn, we denote the nerve of the category of weak equivalences inS_{n}C byw_{}S_{n}C. Asn
varies, w_{}S_{}C assembles into a bisimplicial set, and we denote the geometric realization
by|w_{}S_{}C|. By definition, the algebraicK-theory space ofC is Ω|w_{}S_{}C|. The algebraic
K-theory spectrum is obtained by iterating theS_{}construction: itsnth space is|w_{}S_{}^{(n)}C|.

The following theorem therefore implies that algebraicK-theory can be constructed from
theS_{}^{0} construction.

Theorem 2.9. Let C be a saturated Waldhausen category that admits FFWC. The
inclusion of S_{}Cin S_{}^{0}Cis an exact simplicial functor and the induced map of bisimplicial
sets w_{}S_{}C!w_{}S_{}^{0}C is a weak equivalence.

Proof. The fact thatS_{}C!S_{}^{0}C is an exact simplicial functor is clear from the def-
inition of the simplicial structure and the weak equivalences and w-cofibrations in each
category. To see that it induces a weak equivalence on nerves, it suffices to show that
for eachn, the mapw_{}S_{n}C!w_{}S_{n}^{0}C is a weak equivalence. We do this by constructing a
functor Φ:S_{n}^{0}C!S_{n}C and natural weak equivalences relating Φ and the identity in both
S_{n}^{0}C andS_{n}C. InS^{0}_{n}C, we construct a zigzag of natural weak equivalences of the form

Φoo _{'}^{q} Θ ^{ε}

' //Id, and inSnC, we construct a weak equivalence Φ!Id.

We define the functor Θ inductively as follows. Given an object A in S_{n}^{0}C, let
Θ0,0A=A0,0 and letε0,0: Θ0,0A!A0,0 be the identity map. Given Θ0,jAand

ε_{0,j}: Θ_{0,j}A−!A_{0,j},

define Θ0,j+1A to beT f andε0,j+1:T f!A0,j+1 to be the canonical weak equivalence, wheref: Θ0,jA!A0,j+1is the composite ofε0,j with the mapA0,j!A0,j+1in the struc- ture of the functor A. As a consequence of the construction, we have w-cofibrations Θ0,jA!Θ0,j+1A. Next we construct the diagonal objects Θi,iA; given Θj,jAand

ε_{j,j}: Θ_{j,j}A−!A_{j,j},

define Θj+1,j+1A to be T f and εj+1,j+1:T f!A0,j+1 to be the canonical weak equiva- lence, forf: Θj,jA!Aj,j!Aj+1,j+1. Finally, for 0<i<j, let

Θi,jA= Θ0,jA∪Θ_{0,i}AΘi,iA

and let ε_{i,j}: Θ_{i,j}A!A_{i,j} be the map induced by the universal property of the pushout.

By Proposition 2.6, the mapsε_{i,j} are weak equivalences. Thus, Θ defines a functor from
S_{n}^{0}C to itself, andεdefines a natural transformation from Θ to the identity.

We define Φ andq by setting Φ_{i,i}A=∗,

Φi,jA= Θi,jA/ΘAi,i,

and lettingq_{i,j}: Θ_{i,j}A!Φ_{i,j}A be the quotient map. Since the initial map∗!Θ_{i,i}Ais a
weak equivalence, so is the final map Θ_{i,i}A!∗, and thus eachq_{i,j} is a weak equivalence.

The construction of Θi,jA as a pushout for 0<i<j implies that the induced map from
the quotient Φ0,jA/Φ0,iAto Φi,jAis an isomorphism. It follows that Φ defines a functor
fromS^{0}_{n}Cto SnC and a natural weak equivalenceqinS^{0}_{n}Cfrom Θ to Φ.

Finally, ifAis an object ofSnC, thenAi,i=∗for alli, and the mapε: ΘA!Afactors through the quotient ΦA. This defines a natural weak equivalence from Φ to the identity in SnC.

Theorem 1.1 is an easy consequence of Theorem 2.9: because a square in P is
homotopy cocartesian if and only if it is homotopy cartesian (homotopy cocartesian
in P^{op}), the categoriesS_{n}^{0}P andS_{n}^{0}P^{op}are (contravariantly) isomorphic, for eachn, by
the isomorphism that renumbers the diagram by the involution (i, j)7!(n−j, n−i). The
bisimplicial setsw_{}S_{}^{0}P andw_{}S_{}^{0}P^{op}then only differ in the orientation of the simplices,
and the spaces Ω|w_{}S^{0}_{}P|and Ω|w_{}S_{}^{0}P^{op}|are homeomorphic. Comparing iterates of the
S_{}^{0} construction and the suspension maps, we obtain an equivalence ofK-theory spectra.

3. A reduction of Theorem 1.2

The purpose of this section is to reduce Theorem 1.2 to Theorem 3.10 below. That theorem is similar in spirit to Theorem 1.3, and both are proved with closely related arguments in the next section.

We begin by reducing to the subcategory of connective objects.

Lemma 3.1. The inclusion P_{0}^{∞}!P induces an equivalence of K-theory.

Proof. SinceP=colimmP_{−m}^{∞} andS_{}P=colimmS_{}P_{−m}^{∞} , it suffices to show that the
inclusions P_{−m}^{∞} !P_{−m−1}^{∞} induce equivalences of K-theory. Suspension gives an ex-
act functor P_{−m−1}^{∞} !P_{−m}^{∞} . The composite endomorphisms on P_{−m}^{∞} and P_{−m−1}^{∞} are
the suspension functors, which on the K-theory spectra induce multiplication by −1
[11, Proposition 1.6.2], and in particular are weak equivalences.

We also have P_{0}^{∞}=colimnP_{0}^{n} and S_{}P_{0}^{∞}=colimnS_{}P_{0}^{n}, which give the following
proposition.

Proposition3.2. The spectrum KP_{0}^{∞}is equivalent to the telescope of the sequence
of maps KP_{0}^{0}!...!KP_{0}^{n}!KP_{0}^{n+1}!....

The proof of Theorem 1.2 will then be completed by showing that the maps
P_{0}^{n}−!P_{0}^{n+1}

induce weak equivalences of K-theory spectra, which we do by studying the cofiber.

According to Waldhausen [11, Corollary 1.5.6], for a Waldhausen category C and a

Waldhausen subcategoryB, the spectrum-level cofiber ofKB!KCis theK-theory spec-
trum of the simplicial Waldhausen categoryF_{}(C,B), defined as follows. The Waldhausen
categoryFq(C,B) has as objects the sequences ofqcomposable w-cofibrations in C

C0 // //... // //Ci // //... // //Cq

such that the quotients Cj/Ci are in B for all 06i<j6q, and has commutative dia-
grams as morphisms. A map C!D is a weak equivalence when the maps Ci!Di are
weak equivalences inC for all i, and is a w-cofibration when the maps Ci!Di are w-
cofibrations for all i and the maps Cj∪C_{i}Di!Dj are w-cofibrations for all i<j. The
face map∂i deletes the objectCiand uses the composite w-cofibrationC_{i−1}!Ci+1, and
the degeneracy map si repeats the object Ci, inserting the identity map. We use the
following variant of this construction.

Definition 3.3. For a saturated Waldhausen category C that admits FFWC and a
closed Waldhausen subcategoryB, letF_{q}^{0}(C,B) denote the category that has as objects
the sequences ofq composable weak w-cofibrations inC,

C_{0}−!...−!C_{i}−!...−!C_{q},

such that for all 06i<j6qthere exist an objectB_{i,j} inBand a mapC_{j}!B_{i,j} for which
the square

Ci //

Cj

∗ //Bi,j

commutes and is homotopy cocartesian. A morphismC!DinF_{q}^{0}(C,B) is a commutative
diagram

C_{0} //

... //C_{i}

//... //C_{q}

D0 //... //Di //... //Dq.

We makeF_{q}^{0}(C,B) into a Waldhausen category by declaring a mapC!D to be a weak
equivalence when the maps Ci!Di are weak equivalences in C for all i, and to be a
w-cofibration when the mapsCi!Di are w-cofibrations for alliand the maps

Cj∪C_{i}Di−!Dj

are weak w-cofibrations for alli<j.

Note that in the above definition, the choice of “Bi,j” inBis not part of the structure
of the object C. The hypothesis involving the existence of the objectsBi,j is a precise
way of making sense of the condition that the “homotopy cofiber” ofCi!Cj lies inBfor
an arbitrary Waldhausen categoryC. We assembleF_{}^{0}(C,B) into a simplicial Waldhausen
category just as F_{}(C,B) above: the face map ∂i deletes the object Ci and uses the
composite weak w-cofibration Ci−1!Ci+1; the degeneracy map si repeats the object
Ci, inserting the identity map. The inclusion F_{}(C,B)!F_{}^{0}(C,B) is an exact simplicial
functor.

We have stated the definitions ofF andF^{0}for generalCandB, rather than just for
P_{0}^{n+1} andP_{0}^{n}, in order to apply a standard trick of Waldhausen [11] of commuting these
constructions past another construction, in this case theS_{}^{0} construction. When C is a
saturated Waldhausen category that admits FFWC, the Waldhausen categoryS_{p}^{0}Fq(C,B)
is isomorphic toFq(S_{p}^{0}C, S_{p}^{0}B) and the Waldhausen categoryS^{0}_{p}F_{q}^{0}(C,B) is isomorphic to
F_{q}^{0}(S_{p}^{0}C, S_{p}^{0}B). The argument for Theorem 2.9 also shows that the inclusionw_{}Fq(C,B)!
w_{}F_{q}^{0}(C,B) is a homotopy equivalence of simplicial sets for all q. Since S_{p}^{0}C is also a
saturated Waldhausen category that admits FFWC and S_{p}^{0}B is a closed Waldhausen
subcategory of S_{p}^{0}C, it follows that the inclusion w_{}Fq(S_{p}^{0}C, S_{p}^{0}B)!w_{}F_{q}^{0}(S_{p}^{0}C, S_{p}^{0}B) is
a homotopy equivalence of simplicial sets for all p and q. This proves the following
proposition.

Proposition3.4. Let Cbe a saturated Waldhausen category that admits FFWC and
B be a closed Waldhausen subcategory. Then the inclusion F_{}(C,B)!F_{}^{0}(C,B)induces a
weak equivalence of K-theory spectra. Thus,we have a cofibration sequence

KB −!KC −!KF_{}^{0}(C,B)−!ΣKB
in the stable category.

The remainder of the proof of Theorem 1.2 is to show that w_{}S_{}^{0}F_{}^{0}(P_{0}^{n+1},P_{0}^{n}) is
weakly contractible for all n>0. The first step is to eliminate one of the simplicial
directions.

Definition 3.5. Let Un+1 be the subcategory of P_{0}^{n+1} consisting of all objects and
those maps that induce an isomorphism on πn+1 and an injection inπn. Let uS_{p}^{0}P_{0}^{n+1}
denote the subcategory ofS_{p}^{0}P_{0}^{n+1} consisting of all objects and those mapsf inS_{p}^{0}P_{0}^{n+1}
such thatfi,j is inUn+1 for all 06i<j6p.

To avoid a possible point of confusion, note thatU_{n+1} and uS^{0}_{p}P_{0}^{n+1} are not cate-
gories of weak equivalences in the sense of Waldhausen. Rather, the point of the category
U_{n+1}is that a mapf:A!BinP_{0}^{n+1}has homotopy cofiber inP_{0}^{n}if and only iffis a map

inUn+1. Writingu_{}S_{p}^{0}P_{0}^{n+1}for the nerve ofuS_{p}^{0}P_{0}^{n+1}, the set ofq-simplicesuqS_{p}^{0}P_{0}^{n+1}is
precisely the set of objects ofF_{q}^{0}(S_{p}^{0}P_{0}^{n+1}, S_{p}^{0}P_{0}^{n}). The following lemma is a special case
of Waldhausen’s “swallowing lemma” [11, Lemma 1.6.5].

Lemma3.6. The inclusion

u_{}S_{p}^{0}P_{0}^{n+1}=w0F_{}^{0}(S_{p}^{0}P_{0}^{n+1}, S_{p}^{0}P_{0}^{n})−!w_{}F_{}^{0}(S_{p}^{0}P_{0}^{n+1}, S^{0}_{p}P_{0}^{n})
is a weak equivalence for all p.

Proof. LetuwrS_{p}^{0} denote the category whose objects are the sequences ofrcompos-
able weak equivalences between objects inS_{p}^{0}P_{0}^{n+1},

A_{0} ^{'} //... ^{'} //A_{i} ^{'} //... ^{'} //Ar,

and whose mapsA!B are the commutative diagrams A0

' //

u

... ^{'} //Ai
' //

u

... ^{'} //Ar

u B0 ' //...

' //Bi ' //...

' //Br

with each mapAi!Bi inuS_{p}^{0}P_{0}^{n+1}. Then, for eachr,wrF_{}^{0}(S_{p}^{0}P_{0}^{n+1}, S^{0}_{p}P_{0}^{n}) is the nerve
u_{}wrS_{p}^{0} of the categoryu_{}wrS_{p}^{0}, and the map in question is the inclusion of u_{}w0S_{p}^{0} in
u_{}w_{}S_{p}^{0}. Thus, to show that it is a weak equivalence, it suffices to show that each iterated
degeneracy u_{}w0S^{0}_{p}!u_{}wrS_{p}^{0} is a weak equivalence. The map u_{}w0S^{0}_{p}!u_{}wrS_{p}^{0} is the
nerve of the functor that sendsAto the sequence of identity mapsA=...=A. This is left
adjoint to the functoruw_{r}S_{p}^{0}!uw_{0}S_{p}^{0} that sends a sequenceA_{0}!...!A_{r} to its zeroth
objectA_{0}. It follows thatu_{}w_{0}S_{p}^{0}!u_{}w_{r}S_{p}^{0} is a homotopy equivalence.

At this point we could state Theorem 3.10, although it would appear somewhat
cryptic. To explain the form it takes, consider the further simplification of replacing
the upper triangular diagrams ofS_{p}^{0} with a sequence of composable maps (at the cost of
losing the simplicial structure inp). For a Waldhausen categoryC, letF_{p−1}C=F_{p−1}(C,C)
be the Waldhausen category whose objects are the sequences of p−1 composable w-
cofibrations inC. Then the functorwS_{p}C!wF_{p−1}C that sends (A_{i,j}) toA_{0,1}!...!A_{0,p}
is an equivalence of categories and therefore induces a weak equivalence of nerves. When
C is a saturated Waldhausen category that admits FFWC, we have the Waldhausen
category F_{p−1}^{0} C=F_{p−1}^{0} (C,C) and the analogous functorwS_{p}^{0}C!wF_{p−1}^{0} C induces a weak
equivalence of nerves. This argument does not apply to the map

u_{}S^{0}_{p}P_{0}^{n+1}−!u_{}F_{p−1}^{0} P_{0}^{n+1},

however. The reason is that whenA1!A2andB1!B2are weak w-cofibrations inP_{0}^{n+1}
(maps that are injective onπn+1), and

A1 //

u

A2 u

B1 //B2

is a commutative diagram with the vertical maps in Un+1, then the induced map on homotopy pushouts (with notation as in (2.1))

M(A_{1}!A_{2})−!M(B_{1}!B_{2})

is generally not a map in U_{n+1}. The key is to look instead at the “weak fibration”

analogue: if we assume instead thatA_{1}!A_{2}andB_{1}!B_{2}are maps inP_{0}^{n+1} that induce
epimorphisms on π0, then for a commutative diagram as above, the induced map of
homotopy pullbacks

N(A1!A2)−!N(B1!B2)

is a map inUn+1 by the five lemma. This suggests the following definition.

Definition 3.7. Let F_{q}^{f}P_{0}^{n+1} be the category where an object is a sequence of q
composable mapsA0!...!AqinP_{0}^{n+1}that induce epimorphisms onπ0, and a morphism
is a commutative diagram. Let uF_{q}^{f}P_{0}^{n+1} denote the subcategory containing all objects
ofF_{q}^{f}P_{0}^{n+1}, but consisting of only those morphismsA!B such thatAi!Bi is inUn+1

for alli.

For any objectA=(Ai,j) inS_{p}^{0}P_{0}^{n+1}, the mapsAi,p!Aj,p induce epimorphisms on
π0for all 06i<j6p−1, and so we get a functorφ:S_{p}^{0}P_{0}^{n+1}!F_{p−1}^{f} P_{0}^{n+1} that sendsAto
A0,p!A1,p!...!Ap−1,p. Moreover, since the squares inAare homotopy (co)cartesian,
we can recover A_{i,j} from φA, up to weak equivalence, as the homotopy fiber of the
map A_{i,p}!A_{j,p} for all 06i<j6p−1. Likewise, for any object (Ai) in F_{p−1}^{f} P_{0}^{n+1}, the
homotopy pullback of each map A_{i}!A_{j} is an object ofP_{0}^{n+1}, and the mapping path-
space construction analogous to the construction in the proof of Theorem 2.9 con-
structs a functor Φ:F_{p−1}^{f} P_{0}^{n+1}!S_{p}^{0}P_{0}^{n+1} and natural weak equivalences relating φΦ
and Φφ to the identity functors in F_{p−1}^{f} P_{0}^{n+1} and S_{p}^{0}P_{0}^{n+1}. As mentioned above, the
five lemma implies that a morphism A!B in S_{p}^{0}P_{0}^{n+1} is in uS_{p}^{0}P_{0}^{n+1} if and only if
φA!φB is inuF_{p−1}^{f} P_{0}^{n+1}; thus,φand Φ restrict to functorsφ:uS_{p}^{0}P_{0}^{n+1}!uF_{p−1}^{f} P_{0}^{n+1}
and Φ:uF_{p−1}^{f} P_{0}^{n+1}!uS_{p}^{0}P_{0}^{n+1}. As a consequence, we obtain the following proposition.

Proposition 3.8. The functor φ:uS_{p}^{0}P_{0}^{n+1}!uF_{p−1}^{f} P_{0}^{n+1} induces a weak equiva-
lence of nerves for each p.