Fibration theorems for TQ-completion of structured ring spectra
aa r X i v : . [ m a t h . A T ] M a y FIBRATION THEOREMS FOR TQ -COMPLETION OFSTRUCTURED RING SPECTRA NIKOLAS SCHONSHECK
Abstract.
The aim of this short paper is to establish a spectral algebra analogof the Bousfield-Kan “fibration lemma” under appropriate conditions. Wework in the context of algebraic structures that can be described as algebrasover an operad O in symmetric spectra. Our main result is that completionwith respect to topological Quillen homology (or TQ -completion, for short)preserves homotopy fibration sequences provided that the base and total O -algebras are connected. Our argument essentially boils down to proving thatthe natural map from the homotopy fiber to its TQ -completion tower is apro- π ∗ isomorphism. More generally, we also show that similar results remaintrue if we replace “homotopy fibration sequence” with “homotopy pullbacksquare.” Introduction
This paper is written in the context of symmetric spectra [20, 24], and moregenerally, modules over a commutative ring spectrum R ; see [13] for another ap-proach to a well-behaved category of spectra. We consider any algebraic structurein the closed symmetric monoidal category ( Mod R , ∧ , R ) of R -modules that canbe described as algebras over a reduced operad O ; that is, O [0] = ∗ is the trivial R -module and hence O -algebras are non-unital (see, for instance, [8, 18]).Topological Quillen homology, or TQ -homology for short, is the precise O -algebraanalog of ordinary homology for spaces and is weakly equivalent to the stabilizationof O -algebras; see, for instance, [1, 2, 18, 21]. The TQ -completion of an O -algebra X ,denoted X ∧ TQ , is supposed to be the “part of the O -algebra that TQ -homology sees”([8, 17, 19]). Analogous to Bousfield-Kan’s Z -completion [5] of a space, X ∧ TQ is thehomotopy limit of the cosimplicial resolution built by iterating the unit map of themonad associated to the TQ -homology adjunction. We review these constructionsin Section 3, but to keep this paper appropriately concise, we freely use the notationin [8].Suppose we start with a fibration sequence F → E → B of O -algebras andconsider the associated commutative diagram of the form F ( ∗ ) (cid:15) (cid:15) / / E (cid:15) (cid:15) / / B (cid:15) (cid:15) F ∧ TQ / / E ∧ TQ / / B ∧ TQ (1)in Alg O . The aim of this short paper is to establish sufficient conditions on E → B such that the bottom row is also a fibration sequence. In other words, we areinterested in establishing a TQ -completion analog of the Bousfield-Kan “fibrationlemma” [5, II.2.2], under appropriate additional conditions on E → B . If we are in the special situation where E, B are TQ -complete (i.e., their coaugmentation mapsin (1) are weak equivalences), then this amounts to verifying that ( ∗ ) is a weakequivalence. The following theorem is our main result. Theorem 1.1 ( TQ -completion fibration theorem) . Let E → B be a fibration of O -algebras with fiber F . If E, B are -connected, then the TQ -completion map F ≃ F ∧ TQ is a weak equivalence; furthermore, the natural map from F to its TQ -completion tower is a pro- π ∗ isomorphism. This idea generalizes. Suppose we instead start with a fibration p : X → Y thatfits into a left-hand pullback square of the form A / / (cid:15) (cid:15) X p (cid:15) (cid:15) B / / Y A ∧ TQ / / (cid:15) (cid:15) X ∧ TQ (cid:15) (cid:15) B ∧ TQ / / Y ∧ TQ (2)in Alg O . We would like to establish sufficient conditions on the pullback data B → Y ← X such that the right-hand square of the indicated form is also a homotopypullback diagram. Similar to above, if B, X, Y are TQ -complete, then this amountsto verifying that the TQ -completion map A ≃ A ∧ TQ is a weak equivalence. Thefollowing theorem is a generalization of our main result. Theorem 1.2 ( TQ -completion homotopy pullback theorem) . Consider any pull-back square of the form A / / (cid:15) (cid:15) X p (cid:15) (cid:15) B / / Y (3) in Alg O , where p is a fibration. If B, X, Y are -connected, then the TQ -completionmap A ≃ A ∧ TQ is a weak equivalence; furthermore, the natural map from A to its TQ -completion tower is a pro- π ∗ isomorphism.Remark . It is probably worth pointing out that our strategy of attack workswith O -algebras replaced by pointed simplicial sets. In more detail: Consider anypullback diagram of the form (3) in pointed simplicial sets, where p is a fibration,and assume that A is connected. If B, X, Y are 1-connected, then the Bousfield-Kan Z -completion map A ≃ A ∧ Z is a weak equivalence; furthermore, the naturalmap from A to its Z -completion tower is a pro- π ∗ isomorphism. This provides anew proof of the result in Bousfield-Kan (see, for instance [5, III.5.3]) that such A are Z -complete.For technical reasons explained in Remark 3.5, we make the following assump-tion; for instance, it allows for an iterable point-set model of TQ -homology andhence an associated point-set model for the TQ -resolution of a cofibrant O -algebra.Note that to say an operad O is n -connected means that, for each r ≥
0, its con-stituent O [ r ] is n -connected. Assumption 1.4.
Throughout this paper, O denotes a reduced operad in the closedsymmetric monoidal category ( Mod R , ∧ , R ) of R -modules (see, for instance, [20, 24,25] ). We assume that O , R are ( − -connected, and that O satisfies the following IBRATION THEOREMS FOR TQ -COMPLETION 3 cofibrancy condition: Consider the unit map I → O ; we assume that I [ r ] → O [ r ] isa flat stable cofibration ( [18, 7.7] ) between flat stable cofibrant objects in Mod R foreach r ≥ . This is exactly the cofibrancy condition appearing in [8, 2.1] . Unlessstated otherwise, we work in the positive flat stable model structure [18] on Alg O . Relationship to previous work.
Ching-Harper prove in [8] that all 0-connected O -algebras are TQ -complete. However, it was known that this class does not rep-resent all TQ -complete O -algebras; for instance, one can show that any O -algebrain the image of U (see Section 3) is TQ -complete, by an extra codegeneracy ar-gument. We conjecture that any O -algebra with a principally refined Postnikovtower is TQ -complete, which would mirror the analogous result for Z -completion ofspaces. This paper is a first step in that direction. Acknowledgments.
The author wishes to thank John E. Harper for his supportand advice, Yu Zhang for many helpful conversations, and Jake Blomquist, DuncanClark, and Sarah Klanderman for useful discussions. The author would also liketo thank an anonymous referee for helpful comments on improving the expositionand clarity of the paper. The author was supported in part by National ScienceFoundation grant DMS-1547357 and the Simons Foundation: Collaboration Grantsfor Mathematicians
Outline of the main argument
We will now outline the proof of Theorem 1.1. It suffices to consider the caseof a fibration E → B in Alg O between cofibrant objects (otherwise, cofibrantlyreplace). The first step is (i) to build the associated cosimplicial resolutions of E, B with respect to TQ -homology by iterating the TQ -Hurewicz map id → U Q (seeSection 3) and (ii) to construct the coaugmented cosimplicial diagram F → ˜ F thatis built by taking (functorial) homotopy fibers vertically, followed by objectwise(functorial) cofibrant replacements. In this way, we obtain a commutative diagramof the form F (cid:15) (cid:15) / / ˜ F (cid:15) (cid:15) / / / / ˜ F (cid:15) (cid:15) / / / / / / ˜ F · · · (cid:15) (cid:15) E (cid:15) (cid:15) / / ( U Q ) E (cid:15) (cid:15) / / / / ( U Q ) E (cid:15) (cid:15) / / / / / / ( U Q ) E · · · (cid:15) (cid:15) B / / ( U Q ) B / / / / ( U Q ) B / / / / / / ( U Q ) B · · · (4)in Alg O , where the vertical columns are homotopy fibration sequences. Replacingif needed, we may also assume that ˜ F is a Reedy fibrant cosimplicial O -algebra. Remark . For ease of notational purposes, we usually suppress the codegeneracymaps in ∆-shaped diagrams appearing throughout this paper.
NIKOLAS SCHONSHECK
Applying holim ∆ (see [8, Section 8]) to the maps of ∆-shaped diagrams in (4),where we regard the left-hand vertical column as maps of constant ∆-shaped dia-grams, gives a commutative diagram in Alg O of the form F / / (cid:15) (cid:15) E / / ≃ (cid:15) (cid:15) B ≃ (cid:15) (cid:15) holim ∆ ˜ F / / E ∧ TQ / / B ∧ TQ where each row is a homotopy fibration sequence. The indicated maps are weakequivalences by [8, 1.6] since E, B are assumed to be 0-connected. It follows thatthe left-hand map F → holim ∆ ˜ F is a weak equivalence as well.The next step is to get the TQ -completion of F into the picture; the basic ideais to prove that F → holim ∆ ˜ F is weakly equivalent to the natural coaugmentation F → F ∧ TQ . Our strategy of attack is to objectwise resolve, with respect to TQ -homology, the upper horizontal diagram F → ˜ F ...( U Q ) F ( / / ...( U Q ) ˜ F / / / / ...( U Q ) ˜ F / / / / / / ...( U Q ) ˜ F · · · ( U Q ) F O O O O O O ( / / ( U Q ) ˜ F O O O O O O / / / / ( U Q ) ˜ F O O O O O O / / / / / / ( U Q ) ˜ F · · · O O O O O O ( U Q ) F O O O O ( / / ( U Q ) ˜ F O O O O / / / / ( U Q ) ˜ F O O O O / / / / / / ( U Q ) ˜ F · · · O O O O F ( ∗ ) ′ O O ( / / ˜ F ∗∗ ) O O / / / / ˜ F ∗∗ ) O O / / / / / / ˜ F · · · ( ∗∗ ) O O (5)in (4), and show that the maps ( ∗∗ ) induce weak equivalences after applyingholim ∆ (Propositions 4.8 and 5.5). Once this has been accomplished, we obtain acommutative diagram of the formholim ∆ ( U Q ) • +1 F ≃ / / holim ∆ × ∆ ( U Q ) • +1 ˜ FF ( ∗ ) ′ O O ≃ / / holim ∆ ˜ F ≃ O O (6)and conclude that the natural coagumentation map F ≃ F ∧ TQ ≃ holim ∆ ( U Q ) • +1 F is a weak equivalence. This proves the first part of Theorem 1.1.The second part of Theorem 1.1 requires additional work. In order to preciselyformulate this stronger result, we introduce the following two definitions. Definition 2.2.
A map of towers of O -algebras { X s } s → { Y s } s is a pro- π ∗ iso-morphism if the induced map { π n X s } s → { π n Y s } s IBRATION THEOREMS FOR TQ -COMPLETION 5 of (abelian) groups towers is a pro-isomorphism for each n ∈ Z . (Throughout thispaper, we assume all homotopy groups are derived [23, 24].) Remark . Given a pro- π ∗ isomorphism as above, it follows from the associatedlim short exact sequence that the induced map holim s X s ≃ holim s Y s is a weakequivalence; see, for instance, [8, Section 8]. Definition 2.4.
Define
Tot as the right derived functor of Tot in (
Alg O ) ∆ equippedwith the Reedy model structure. In other words, given a cosimplicial O -algebra X , we define Tot ( X ) to be Tot( RX ) where RX is the (functorial) Reedy fibrantreplacement of X in Alg O ∆ .Stated precisely, the second part of Theorem 1.1 asserts that the map of towers { F } s ( ∗ ) ′ −→ (cid:8) Tot s ( U Q ) • +1 F (cid:9) s is a pro- π ∗ isomorphism. To show that this assertion is true, note that the proofsof Propositions 4.8 and 5.5 imply that the tower maps (cid:8) ( U Q ) k F (cid:9) s → n Tot s ( U Q ) k ˜ F o s and n ˜ F n o s → n Tot s ( U Q ) • +1 ˜ F n o s are actually pro- π ∗ isomorphisms for each k, n ≥
0. Now consider the commutativediagram of towers of the form (cid:8)
Tot s ( U Q ) • +1 (cid:9) s / / n Tot s Tot s ( U Q ) • +1 ˜ F o s { F } s ( ∗ ) ′ O O / / n Tot s ˜ F o s O O (7)It follows from the tower lemma below that the horizontal and right-hand verticalmaps are pro- π ∗ isomorphisms, and hence the map ( ∗ ) ′ is as well. Proposition 2.5 (Tower lemma for O -algebras) . Suppose we are given a map fromthe Reedy fibrant cosimplicial O -algebra X to a tower of Reedy fibrant cosimplicial O -algebras { Y s } s X / / / / (cid:15) (cid:15) X / / / / / / (cid:15) (cid:15) X · · · (cid:15) (cid:15) (cid:8) Y s (cid:9) s / / / / (cid:8) Y s (cid:9) s / / / / / / (cid:8) Y s (cid:9) s · · · If (cid:8) X k (cid:9) s → (cid:8) Y ks (cid:9) s induces a pro- π ∗ isomoprhism for each fixed k , then { Tot n X } s → { Tot n Y s } s induces a pro- π ∗ isomorhpism for each fixed n .Proof. In the context of spaces, this is proven in [12, 1.4] and the same argumentremains valid in our setting. (cid:3)
NIKOLAS SCHONSHECK Background on TQ -homology and TQ -completion The purpose of this section is to briefly recall the definition of TQ -homology andits associated completion construction. For a more thorough introduction, usefulreferences include [8], [17], and [18]. Definition 3.1.
Given an operad O , its τ O is the operad defined by( τ O )[ r ] := ( O [ r ] , for r ≤ , ∗ , otherwiseThe canonical map of operads O → τ O induces the following change of operadsadjunction, with left adjoint on top. Alg O ¯ Q / / Alg τ O ∼ = Mod O [1]¯ U o o (8)Here, ¯ Q ( X ) := τ O ◦ O ( X ) and ¯ U is the forgetful functor. It is proven in [16]and [18] that this is, in fact, a Quillen adjunction. Definition 3.2.
Let X be an O -algebra. The TQ -homology of X is the O -algebra TQ ( X ) := R ¯ U (cid:0) L ¯ Q ( X ) (cid:1) , where L and R indicate the appropriate derived functors.We would then like to form a cosimplicial (or Godement) resolution of the form X / / ( TQ ) X / / / / ( TQ ) X / / / / / / ( TQ ) X · · · (9)Although TQ ( X ) ≃ ¯ U ¯ Q ( X ) if X is cofibrant, the forgetful functor ¯ U need notsend cofibrant objects in Alg τ O to cofibrant objects in Alg O . Consequently, thereis no guarantee that ( TQ ) n X ≃ ( ¯ U ¯ Q ) n X for n ≥
2. The canonical cosimplicialresolution associated to (8) is therefore unlikely to be of the form (9).Because of this difficulty, an additional maneuver is required to construct aniterable point-set model for TQ ( X ). We follow [18, 3.16] to produce a rigidifiedversion of (9). First, factor the operad map O → τ O as O → J → τ O a cofibration followed by a weak equivalence. This induces (Quillen) adjunctions Alg O Q / / Alg J / / U o o Alg τ O o o (10)where Q ( X ) := J ◦ O ( X ) and U is the forgetful functor. Remark . The adjunction on the right is, in fact, a Quillen equivalence (see[18, 7.21]) and we therefore think of
Alg J as a “fattened up” version of Alg τ O .Furthermore, it follows that TQ ( X ) ≃ U Q ( X ) if X is a cofibrant O -algebra.The advantage of (10) is that the forgetful functor U sends cofibrant objects in Alg J to cofibrant objects in Alg O (see [18, 5.49]). For a cofibrant O -algebra X , wetherefore have weak equivalences of the form ( TQ ) n X ≃ ( U Q ) n X for all n ≥ X / / ( U Q ) • +1 X : ( U Q ) X / / / / ( U Q ) X / / / / / / ( U Q ) X · · · (11)is of the desired form (9). IBRATION THEOREMS FOR TQ -COMPLETION 7 Definition 3.4.
Let X be an O -algebra. The TQ -completion of X is the O -algebra X ∧ TQ := holim ∆ ( U Q ) • +1 ( X c ), where X c denotes the functorial cofibrant replace-ment of X in Alg O . Remark . We are now in a better position to explain the reasons for Assumption1.4, which are as follows. The connectivity assumption on O and R guarantees theresults of [7] used below are applicable, while the cofibrancy condition on O ensures[18, 5.49] that the forgetful functor Alg J → Alg O preserves cofibrant objects.4. Analysis of the horizontal direction
The purpose of this section is to analyze the maps ( O -algebra F → ˜ F . This is the content of Proposition 4.3 and is ac-complished by analyzing the corresponding coface cubes of E → ( U Q ) • +1 E and B → ( U Q ) • +1 B . We next show, in Proposition 4.6, that objectwise application ofthe TQ -homology spectrum functor preserves this cartesian estimate. Proposition4.8 then follows inductively.Our analysis in this section will involve a number of concepts from cubical ho-motopy theory. We provide an overview of the relevant details in Section 7. Definition 4.1.
Let E n +1 be the coface ( n +1)-cube associated to the coaugmentedcosimplicial O -algebra E → ( U Q ) • +1 E and define B n +1 similarly. Let ˜ F n +1 be thecoface ( n + 1)-cube associated to the coaugmented cosimplicial O -algebra F → ˜ F .The following proposition gives the uniform cartesian estimates on E n +1 and B n +1 (by setting k = 0) that we will ultimately use to analyze ˜ F n +1 . It is proven in[3, 7.1]; a special case is dealt with also in [8]. The proposition is a spectral algebraanalogue of Dundas’s [10, 2.6] higher Hurewicz theorem. Proposition 4.2 (Higher TQ -Hurewicz theorem) . Let k ≥ and X be a W -cubein O -algebras that is objectwise cofibrant. If X is (id + 1)( k + 1) -cartesian, then sois X → U Q X . The uniform cartesian estimates given by Proposition 4.2 applied to E n +1 , B n +1 imply a (slightly weaker) uniform cartesian estimate on ˜ F n +1 . Proposition 4.3.
Let n ≥ − . The coface ( n + 1) -cube ˜ F n +1 associated to F → ˜ F is id -cartesian.Proof. It follows from 4.2 that both E n +1 and B n +1 are ( n + 2)-cartesian and so[7, 3.8] the cube E n +1 → B n +1 is ( n + 1)-cartesian. This means that the iteratedhomotopy fiber [8, 2.6] of E n +1 → B n +1 is n -connected. Since this is weaklyequivalent to the iterated homotopy fiber of ˜ F n +1 , we conclude that ˜ F n +1 is ( n + 1)-cartesian. Repeating this argument on all subcubes completes the proof. (cid:3) The following two short lemmas are used in the proof of Proposition 4.6, whichstates that levelwise application of the TQ -homology functor preserves this cartesianestimate on ˜ F n +1 . Lemma 4.4.
Let k ∈ Z and let Y be a W -cube in J -algebras. If Y is k -cartesian,then so is U Y . NIKOLAS SCHONSHECK
Proof.
This is because U is a right Quillen functor and preserves connectivity of allmaps, since this connectivity is calculated in the underlying category Mod R . (cid:3) Lemma 4.5.
Let k ≥ − and let X be an objectwise cofibrant W -cube in O -algebras.If X is k -cocartesian, then so is Q X .Proof. If | W | = 0 or 1, note that an O -algebra (resp. a map between O -algebras) is k -cartesian if and only if it is k -connected. The result now follows from [18, 1.9(b)]and the observation that if X is cofibrant, then TQ ( X ) ≃ U Q ( X ). To show, moregenerally, that Q X is k -cocartesian, let P W be the poset of subsets V $ W . Byassumption, hocolim P W X → X W is a k -connected map of cofibrant objects, sohocolim P W Q X ≃ Q hocolim P W X → Q X W is also k -connected, by the first partof the proof. (cid:3) Proposition 4.6.
Let X be a W -cube in O -algebras. If X is objectwise cofibrantand is id -cartesian, then so is U Q X .Remark . Before we give the proof of Proposition 4.6 in full generality, here isthe argument assuming that X is a 2-cube, i.e., that W = { , } . In this case, X isthe commutative diagram X ∅ / / (cid:15) (cid:15) X { } (cid:15) (cid:15) X { } / / X { , } in Alg O , where each object is ( − U Q X are appropriately connected follows as inthe proof of Lemma 4.5. Let us now show that U Q X is 2-cartesian. The dualBlakers-Massey theorem of Ching-Harper [7, 1.9] implies that X is k -cocartesian,where k = min { k + 1 , k + k + 2 } = min { , } = 3By Lemma 4.5, this means that Q X is also 3-cocartesian. It is now important toobserve that Q X is a diagram in the stable category Alg J , so the fact that it is3-cocartesian implies it is 2-cartesian; see [7, 3.10]. Hence, by Lemma 4.4, U Q X isalso 2-cartesian.To see that U Q X is objectwise cofibrant, recall that Q is a left Quillen functorand that [18, 5.49] the functor U preserves cofibrant objects. Proof of Proposition 4.6.
Objectwise cofibrancy is proven in the same way as inRemark 4.7. To show that
U Q X is id-cartesian, we induct on n . The cases | W | =0 , , X is a W -cube with | W | = n ≥ k -cubes with k < n . This verifies that U Q X isid-cartesian on all strict subcubes, so we must only further show that U Q X is itself n -cartesian.As in Remark 4.7, we first establish a cocartesian estimate on X , but now usethe higher dual Blakers-Massey Theorem of Ching-Harper. Adopting the notationof [7, 1.11], observe that each k V (the cartesianness of a particular | V | -dimensional IBRATION THEOREMS FOR TQ -COMPLETION 9 subcube of X ) is equal to | V | by assumption that X is id-cartesian. It follows thatfor any partition λ of W , we have | W | + X V ∈ λ k V = n + X V ∈ λ | V | = n + n = 2 n On the other hand, k W + | W | − n + n − n − X is (2 n − Q X is also (2 n − Q X is in the stable category Alg J , the proof of [7, 3.10] impliesthat Q X is (2 n − − n + 1 = n -cartesian. Therefore, by Lemma 4.4, U Q X is also n -cartesian. (cid:3) We are now in a position to prove the main result of this section.
Proposition 4.8.
Let n ≥ − and k ≥ . The coface ( n + 1) -cube associated to ( U Q ) k F → ( U Q ) k ˜ F is id -cartesian. In particular, the natural map ( U Q ) k F → holim ∆ ( U Q ) k ˜ F is a weak equivalence.Proof. The first part follows inductively from Propositions 4.3 and 4.6. The secondpart follows by observing that this cartesian estimate implies that the natural map(
U Q ) k F → holim ∆ ≤ n ( U Q ) k ˜ F is ( n + 1) connected (see Proposition 7.6), then usingthe associated lim short exact sequence. (cid:3) Remark . The increasing connectivity proven in Proposition 4.8 implies that,for all k ≥
0, the map of towers (cid:8) ( U Q ) k F (cid:9) s → n Tot s ( U Q ) k ˜ F o s is a pro- π ∗ iso-morphism. Remark . If one relaxes the connectivity assumptions on
E, B , but can stillshow that ˜ F n +1 is id-cartesian, then Proposition 4.8 remains valid. In this case,since the connectivities of E, B do not play a role in the following section, theconclusion of Theorem 1.1 also remains valid. We thank the referee for pointingthis out. 5.
Analysis of the vertical direction
The purpose of this section is to analyze the maps ( ∗∗ ) and, in particular, to proveProposition 5.5. The basic idea is to first show that, up to homotopy, there is anextra codegeneracy in each coaugmented cosimplicial diagram ˜ F n → ( U Q ) • +1 ˜ F n .This is accomplished by showing that each ˜ F n is weakly equivalent to an O -algebraof the form U Y and observing that the diagram
U Y → ( U Q ) • +1 U Y has an extracodegeneracy on the nose. A short spectral sequence argument then completes theanalysis.
Lemma 5.1.
For each n ≥ , there is a fibrant and cofibrant J -algebra G n with anatural zigzag of weak equivalences U G n ≃ ˜ F n in Alg O .Proof. We will prove the n = 0 case. The proof is essentially the same for n ≥ U past a homotopy limit, we have a natural zigzagof weak equivalences˜ F ≃ hofib( U QE → U QB ) ≃ U hofib( QE → QB )and the lemma follows by letting G be the functorial cofibrant replacement ofhofib( QE → QB ) in Alg J . (cid:3) Lemma 5.2. If Y is in Alg J , the diagram U Y → ( U Q ) • +1 U Y has an extra code-generacy.Proof.
One obtains an extra codegeneracy by defining s n = U ( QU ) n +1 Y U ( QU ) n ǫ → U ( QU ) n Y for all n ≥
0, where ǫ is the counit associated to the ( Q, U ) adjunction. (cid:3)
Lemma 5.3.
If the coaugmented cosimplicial O -algebra X − → X has an extracodegeneracy and X − is fibrant, then the natural map (cid:8) X − (cid:9) s → { Tot s X } s is apro- π ∗ isomorphism.Remark . In the proof below, we use the spectral sequence associated to a towerof fibrations of O -algebras. For details of the construction, see [8, 8.31]. It isessentially the same as the homotopy spectral sequence [5, X.6] of Bousfield-Kan;see also [14, VIII.1]. Proof.
Fix n ∈ Z and consider the coaugmented cosimplicial abelian group π n X − → π n X . The assumed extra codegeneracy implies that for any s ≥
0, we have π s π n X − ∼ = → π s π n X . It follows that there is an induced isomorphism on E pagesof the homotopy spectral sequences associated to (cid:8) X − (cid:9) s and { Tot s X } s . (Here,we are using the fact that, since X − is fibrant, the constant cosimplicial diagramwith value X − is Reedy fibrant.) The result now follows from [8, 8.36]. (cid:3) Proposition 5.5.
For each n ≥ , the TQ -completion map ˜ F n ≃ ( ˜ F n ) ∧ TQ is a weakequivalence.Proof. First, note that both ˜ F n and U G n (as constructed in Lemma 5.1) are cofi-brant. By taking further functorial replacements, it follows from Lemma 5.1 thatthere is a natural zigzag of weak equivalences U G n ≃ ( U G n ) c ≃ ( ˜ F n ) c ≃ ˜ F n inwhich each object is cofibrant. This induces a zigzag of towers { U G n } s (cid:15) (cid:15) ≃ n ˜ F n o s ( ∗∗ ) (cid:15) (cid:15) { Tot s ( U Q ) • +1 U G n } s ≃ { Tot s ( U Q ) • +1 ˜ F n } s and, by Lemmas 5.2 and 5.3, the left-hand vertical map is a pro- π ∗ isomorphism.It follows that ( ∗∗ ) is a pro- π ∗ isomorphism as well. Hence, ˜ F n ≃ ( ˜ F n ) ∧ TQ is a weakequivalence (see Remark 2.3). (cid:3) TQ -completion of homotopy pullback squares In this section, we prove Theorem 1.2. The strategy of proof is essentially thesame as that used in the proof of Theorem 1.1. The new arguments given in thissection are needed to obtain an analogue of Proposition 4.3; this is the content ofProposition 6.1 below.As in the proof of Theorem 1.1, we may assume that
B, X, Y are cofibrant, andwe then build the associated cosimplicial resolutions of
B, X, Y with respect to TQ -homology; then take levelwise homotopy pullbacks to obtain a coagumented IBRATION THEOREMS FOR TQ -COMPLETION 11 cosimplicial diagram A → ˜ A . In other words, we obtain maps of coaugmentedcosimplicial O -algebras of the form (cid:16) A → ˜ A (cid:17) / / (cid:15) (cid:15) (cid:16) X → ( U Q ) • +1 X (cid:17) (cid:15) (cid:15) (cid:16) B → ( U Q ) • +1 B (cid:17) / / (cid:16) Y → ( U Q ) • +1 Y (cid:17) (12)such that on each fixed cosimplicial degree, one has a homotopy pullback diagram.For instance, in cosimplicial degrees 0, 1 we have homotopy pullback diagrams ofthe form ˜ A / / (cid:15) (cid:15) ( U Q ) X (cid:15) (cid:15) ( U Q ) B / / ( U Q ) Y ˜ A / / (cid:15) (cid:15) ( U Q ) X (cid:15) (cid:15) ( U Q ) B / / ( U Q ) Y (13)in Alg O , and these are coaugmented by the diagram in Theorem 1.2. For the samereasons as in the proof of Theorem 1.1, we may assume A → ˜ A is objectwisecofibrant, and that ˜ A is a Reedy fibrant cosimplicial O -algebra. Proof of Theorem 1.2.
Construct a diagram identical to (5). For the same reasonsas in Theorem 1.1, the maps ( ∗∗ ) induces pro- π ∗ isomorphisms after applying Tot s .The result now follows from Proposition 6.1 below and arguing as in the proof ofTheorem 1.1. (cid:3) Proposition 6.1 (cf. Proposition 4.3) . Let n ≥ − . The coface ( n + 1) -cubeassociated to A → ˜ A is id -cartesian.Remark . As in the case of Proposition 4.3, it may be helpful to first understanda low-dimensional example of Proposition 6.1 before attacking the proof in fullgenerality. Suppose we wish to show the 1-cube A → ˜ A is id-cartesian. Considerthe corresponding 1-cubes of A, X, Y to obtain a commutative diagram of the form A (cid:15) (cid:15) (cid:31) (cid:31) ❄❄❄ / / X (cid:15) (cid:15) (cid:31) (cid:31) ❄❄❄❄ ˜ A (cid:15) (cid:15) / / U QX (cid:15) (cid:15) B / / (cid:31) (cid:31) ❄❄❄❄ Y (cid:31) (cid:31) ❄❄❄❄ U QB / / U QY (14)in
Alg O . We will make frequent use of [7, 3.8] in the following analysis.Since the back and front faces of (14) are both homotopy pullback diagrams(i.e., infinitely caretesian), the entire 3-cube is also infinitely cartesian. The 1-cubes X → U QX and Y → U QY are both 2-cartesian (i.e., the maps are 2-connected)by Proposition 4.2 and so the right-hand face of (14) is 1-cartesian. Therefore,the left-hand face is 1-cartesian as well. Since B → U QB is 2-cartesian (also byProposition 4.2), we conclude that A → ˜ A is 1-cartesian. One then repeats thisargument on all subcubes of A → ˜ A , i.e., on the objects A and ˜ A . Proof of Proposition 6.1.
Denote by ˜ A n +1 , B n +1 , X n +1 , Y n +1 the coface ( n +1) cubesassociated to the coaugmented cosimplicial diagrams in (12). Let C be any subcubeof ˜ A n +1 , say of dimension k . Let C B , C X , C Y denote the corresponding subcubesand consider the commutative diagram of subcubes C / / (cid:15) (cid:15) C X (cid:15) (cid:15) C B / / C Y (15)As in Remark 6.2, the first step is to establish that the cube (15) is infinitelycartesian. This is accomplished by Lemma 6.3 below. Next, Proposition 4.2 impliesthat C X and C Y are both ( k + 1)-caratesian, so the cube C X → C Y is k -cartesian.Therefore, the cube C → C B is k -cartesian. Since C B is ( k + 1)-cartesian (also byProposition 4.2), we conclude that C is k -cartesian. (cid:3) Lemma 6.3.
For any subcube C of ˜ A n +1 , the cube constructed in (15) is infinitelycartesian.Proof. The proof is by induction on k . If k = 0, then C is a single object and thelemma follows by construction. If k ≥
1, we may write C as a map of ( k − D → E of ˜ A n +1 . Consider the commutative diagram ofsubcubes D (cid:15) (cid:15) (cid:31) (cid:31) ❄❄❄❄ / / D X (cid:15) (cid:15) (cid:31) (cid:31) ❄❄❄ E (cid:15) (cid:15) / / E X (cid:15) (cid:15) D B / / (cid:31) (cid:31) ❄❄❄ D Y (cid:31) (cid:31) ❄❄❄ E B / / E Y and note that this diagram is precisely (15). By induction, the back and front faces(which are themselves both ( k +1)-cubes) are both infinitely cartesian, so the wholecube is as well. (cid:3) Appendix: cubical diagrams
The purpose of this appendix is to briefly summarize the tools of cubical homo-topy theory used in this paper, particularly in Section 4. While these notions canbe defined in other settings, we have phrased them in the context of O -algebrasto keep this section appropriately focused. For the more interested reader, usefulreferences for cubical diagrams of spaces include [11, A.8], [15], and [22]. In thecontext of O -algebras, see [7] and [8]. Definition 7.1.
Let X be a W -cube of O -algebras indexed on the set W = [ n ],where [ n ] := { , , . . . , n } . Let P ([ n ]) be the poset of nonempty subsets of [ n ]. Wesay that X is k -cartesian if the natural map X ∅ → holim P ([ n ]) X is k -connected.The connectivity of X ∅ → holim P ([ n ]) X gives information about the cube X asa whole. One might also be interested in subcubes (see [4, 3.6] or [11, A.8.0.1]) of X . This motivates the following definition, which appears in [10] and [11]. IBRATION THEOREMS FOR TQ -COMPLETION 13 Definition 7.2.
Given a function f : N → Z , we say that a cube X of O -algebrasis f -cartesian if each d -dimensional subcube of X is f ( d )-cartesian; here, N denotesthe non-negative integers. Remark . For instance, to say that a cube X is id-cartesian means that each d -dimensional subcube of X is d -cartesian. Definition 7.4.
Let Z − d → Z be a coaugmented cosimplicial O -algebra. The coface ( n + 1) -cube X n +1 associated to Z − → Z is the canonical ( n + 1)-cubeconstructed using the cosimplicial relations d j d i = d i d j − for i < j . Remark . For instance, X has the form on the left, and X the form on theright. Z − d / / d (cid:15) (cid:15) Z d (cid:15) (cid:15) Z d / / Z Z − d (cid:15) (cid:15) d ! ! d / / Z d (cid:15) (cid:15) d ❇❇❇❇❇❇❇❇ Z d (cid:15) (cid:15) d / / Z d (cid:15) (cid:15) Z d / / d ! ! ❉❉❉❉❉❉❉❉ Z d ❇❇❇❇❇❇❇❇ Z d / / Z One of the reasons cubical diagrams are useful is that (as described above) theynaturally arise from cosimplicial diagrams, combined with the following fact, whichis proven in [6, Section 6], [9], and [26, 6.7].
Proposition 7.6.
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