Signature Redacted" Certified By

Day convolution and the Hodge filtration on THH by Saul Glasman V-- Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of z CD

Doctor of Philosophy in Mathematics QLL = <~ - at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2015 @Massachusetts Institute of Technology 2015. All rights reserved. Parts of this thesis were first published in Mathematical Research Letters in 2015, published by International Press.

A uthor ...... Department of Mathematics April 29, 2015 Signature redacted" Certified by.... e ...... I...... Clark Barwick Cecil and Ida B. Green Career Development Associate Professor Thesis Supervisor

Accepted by. Signature redacted...... William Minicozzi Chairma Depart ent Committee on Graduate Theses

Day convolution and the Hodge filtration on THH by Saul Glasman

Submitted to the Department of Mathematics on April 29, 2015, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics

Abstract

This thesis is divided into two chapters. In the first, given symmetric monoidal oc- categories C and D, subject to mild hypotheses on D, we define an oc-categorical analog of the Day convolution symmetric monoidal structure on the functor category Fun(C, D). In the second, we develop a Hodge filtration on the topological Hochschild homolgy spectrum of a commutative ring spectrum and describe its elementary properties.

Thesis Supervisor: Clark Barwick Title: Cecil and Ida B. Green Career Development Associate Professor

3 4 Acknowledgments

This thesis, as well as the author's mathematical career, owes an inestimably enor- mous debt to nearly five years spent in the company of Clark Barwick's deep insight and inexhaustibly patient guidance. Clark proposed both of the projects contained in this thesis and taught me everything I needed to see them to completion. I'd like to thank MIT for providing a hospitable working environment and a the MIT-Harvard topology bloc for being an absolute powerhouse. There's nowhere like Boston to be a topology student. I'd also like to thank Jacob Lurie for writing the Two Indispensibles, Higher Topos Theory and Higher Algebra, and thus promoting quasicategory theory to something that anyone sufficiently motivated could use to do real work.

5 6 Contents

1 Introduction 9

2 Day convolution for oo-categories 13

2.1 The Day convolution symmetric monoidal oo-category ...... 13 2.2 The symmetric monoidal Yoneda embedding ...... 24

3 A spectrum-level Hodge filtration on topological Hochschild homol-

ogy 31 3.1 Preliminaries on twisted arrow categories, ends and coends ...... 31 3.2 Tensor products of commutative ring spectra with spaces ...... 38 3.3 The Hodge filtration ...... 42

3.4 M ultiplicative structure ...... 47

3.5 The layers of the filtration and Adams operations ...... 51

A Commutative algebras and their modules 57

7 8 Chapter 1

Introduction

In this thesis, we seek to begin the development of a theory of functorial, spectrum- level Hodge filtrations on topological Hochschild homology (THH) of commutative ring spectra and its cousins, such as TR and TC.

What is a Hodge filtration? The original Hodge filtration is a filtration on the de

Rham complex of a commutative ring R

Q*(R) -( --. -+( R*(R)) " 4 - (Q*(R)) = R where the complex (Q*(R)) " is the "stupid truncation" of Q*(R), which is not a homotopy invariant of * (R) but depends on R itself: it has the same terms as Q* (R) in degrees at most n, and is zero in degrees greater than n. This filtration gives rise to a spectral sequence known as the Hodge to de Rham spectral sequence. In a celebrated sequence of papers beginning with [71, Deligne proved that for a C-algebra

R, the spectral sequence degenerates at E2 , giving rise to a mixed Hodge structure on the de Rham cohomology of R.

In the special case where R is regular, the Hochschild-Kostant-Rosenberg theorem

[121 gives an isomorphism of graded abelian groups

HInH,(R) Loda ~ * (R) where HH* is Hochschild homology. In [14], Loday interpreted the Hodge filtration as

9 the -y-filtration associated to a A-ring structure that exists on the Hochschild homology of an arbitrary commutative ring R, thus generalizing the Hodge filtration in an algebraic direction. Rationally, this filtration is canonically split, and it coincides with the decomposition by eigenspaces of the Adams operations that exist on any rational A-ring.

Later, Pirashvili [24], working rationally, used functor homology to give a general- ization of the rational Hodge decomposition to what he called "higher-order Hochschild homology". In the present work, we take Pirashvili's approach and run with it in a homotopy-theoretic direction, using a homotopy coend formula for topological

Hochschild homology and its "higher-order" variants to give spectrum-level filtrations with properties analogous to those of Loday and Pirashvili's original examples.

We work with quasicategories throughout, and we will liberally reference Lurie's blockbuster volumes [15] and [17]. The structure of the paper is as follows. Chapter

3 is devoted to the construction and properties of the Hodge filtration. In order to investigate the multiplicative properties of this filtration, however, we must construct an oc-categorical analog of the Day convolution. This goal, which we accomplish in

Section 2.1, is the subject of the more technical Chapter 2, and is of independent interest. We'll recall some classical facts about this construction.

Let (C, Oc) and (D, ®-) be two symmetric monoidal categories such that D admits all colimits. In [6], Day equips the functor category Fun(C, D) with a "convolution" symmetric monoidal structure: If F, G : C -+ D are functors, then their convolution product F ODay G is defined as the left Kan extension of 0D o (F x G) : C x C -> D along Oc : C x C - C. According to [6, Example 3.2.2], the commutative monoids for the convolution product are exactly the lax monoidal functors from C to D. We show in Proposition 2.1.10 that the analogous proposition holds for oc-categories - that is, E, algebras for the Day convolution product are lax symmetric monoidal functors, which is to say oc-operad maps in the sense of [17, Definition 2.1.2.7] - thus answering a question of Blumberg.

In Section 2.2, we construct the Yoneda embedding for a symmetric monoidal

10 category C as a symmetric monoidal functor into the presheaf category

Fun(CP, Top) equipped with the Day convolution product, and deduce that our construction agrees in this special case with the Day convolution constructed by Lurie in [17, Corollary 6.3.1.121.

The study of THH is far from the only application of this version of the Day convolution. An important special case is the tensor product of Mackey functors; see, for example, [23]. In a recent paper [31, Barwick develops a theory of higher- categorical Mackey functors in order to study equivariant K-theory. In forthcoming work of the author with Barwick, Dotto, Nardin and Shah [4], an equvariant theory of the Day convolution is developed, building on the material in this thesis.

The stage is now set for Chapter 3. In Section 3.1, we collect some useful facts about ends and coends in the oo-categorical context. In Section 3.2, we discuss tensor products of commutative ring spectra with spaces. In Section 3.3, we define our central object of study: for each simplicial set X, we define a Hodge-like filtration on the functor

CRing -+ CRing: R - R 0 X where CRing is the category of commutative ring spectra. Keeping X free, we explore the geometric structure of our filtration.

In Section 3.5, we specialize to X = S1 , so that

R OX - THH(R).

Here we identify the graded pieces of the filtration and show that they are eigenspectra for the Adams operations V, as they must be if we are to sensibly call our filtration a

Hodge filtration. In addition, we believe that our filtration coincides with the TAQ- filtration investigated by Minasian [221 and McCarthy-Minasian [20]. We will take

11 up this questions in a future paper. Finally, in Appendix A, we give an account of modules over commutative algebras in quasicategories that fills a minor gap in the literature.

In a forthcoming paper [10], we will describe how to lift the Hodge filtration on

THH to a filtration by cyclotomic spectra, and thus obtain a filtration on topological cyclic homology TC as well as its antecedents TR and TF. We suspect this filtration to be related to a filtration on TF, with deep connections to p-adic geometry, whose existence was conjectured by Scholze [26], and we plan to investigate this link.

We also hope to show that the cyclotomic trace from K-theory to TC makes the weight filtration on K-theory (various versions of which are explicated beautifully in [11]) compatible with the Hodge filtration on TC, ideally by showing that the trace is a map of spectral A-rings (whatever these are) and by framing the filtrations on each side as 7-filtrations. In between, we plan to give explicit computations of the filtrations we define in interesting cases.

12 Chapter 2

Day convolution for oc-categories

2.1 The Day convolution symmetric monoidal 00-

category

Throughout this thesis, F will denote the category of finite sets, or by abuse of notation, the nerve of that category. Likewise, F, will denote the category of finite pointed sets and pointed maps or its nerve. Let C -+ F, and D* -+ F, be symmetric monoidal oo-categories (see [17, Definition 2.0.0.71). To sidestep potential set-theoretic issues, we'll fix a strongly inaccessible uncountable cardinal A and assume that both C® and DO are A-small. If k : K -+ F is a map of simplicial sets, then we denote by C' the pullback

k KJ In particular, if f is any morphism in F, CO is the pullback

Cf>C 11 A- T*

13 and if S is an object of T, C' is the fiber of C over S.

Suppose k : K -+ F is an arbitrary map of simplicial sets. We define a simplicial set Fun(C, D)O by the following universal property: there is a bijection, natural in k,

Funy. (K, Fun(C, D)®) -+ Funy.(C2, Dk), where Fun.F(-, -) denotes the simplicial set of maps over .T,.

Observation 2.1.1. A vertex of Fun(C, D)O) is a finite set S together with a functor

CO - Do, which is to say a functor

Fs : Cs -+ Ds) where CS denotes the product of copies of C indexed by S. Similarly, an edge of

Fun(C, D)O) is given by a morphism f : S -+ T in F, together with a functor

Ff : C-+ D

over A'. A section of the structure morphism Fun(C, D)O -+ F, corresponds to a map over F, from C* to DO.

Suppose D has all colimits. We seek to prove that Fun(C, D)O -+ T, is a cocartesian fibration. Prerequisitely:

Lemma 2.1.2. Fun(C, D)O - F, is an inner fibration.

Proof. Suppose we have 0 < i < n and a diagram

An ko"> Fun(C, D)0 I I An ki I;

14 Giving a lift of this diagram is equivalent to lifting the diagram

I I

In the statement of [15, Proposition 3.3.1.3], we can replace "cartesian" with "cocartesian" just by taking opposites. We deduce that the cofibration C -+ C is a categorical equivalence and therefore a trivial cofibration in the Joyal model structure, permitting the lift.

In particular, Fun(C, D)O is a quasicategory. We claim that the natural functor

Fun(C, D)® -+ J* is a cocartesian fibration. For this, it suffices to shaw that the functor is a locally cocartesian fibration [15, Definition 2.4.2.6] and that locally cocartesian edges are closed under composition [15, Lemma 2.4.2.7]. First we'll characterize the locally cocartesian edges:

Lemma 2.1.3. Fun(C, D)® is a locally cocartesian fibration, and a morphism (f:

S -+ T, F : CO - Df) of Fun(C, D)® is locally cocartesian iff the diagram

CO F03> Do jS ft P Col exhibits Ff as a p-left Kan extension of F [15, Section 4.3.2], where F is the composite of Fs : CO -+ DO with the natural inclusion DO c- DO.

Proof. Before trying too hard to prove this, it seems prudent to verify the following:

Lemma 2.1.4. The relative left Kan extensions arising in the statement of Lemma

2.1.3 actually exist.

Proof. Applying [15, Lemma 4.3.2.131, we must show that for each object X E C', the functor

(Co)lx -+ Do

15 admits a p-colimit. The proof of this is almost identical to that of [15, Corollary

4.3.1.11]; we simply replace the sentence "Assumption (2) and Proposition 4.3.1.10 guarantee that 7' is also a p-colimit diagram when regarded as a map from K' to X" with "The condition of Proposition 4.3.1.10 is vacuously satisfied for 7', since there are no nonidentity edges with source {1} in A1 ". l

Let e be the map An -+ Al which maps 0 to 0 and all other vertices to 1. A map

A0 -+ Fun(C, D)O lifting E whose leftmost edge is F1 gives a diagram

aA n-1 >Funs (CO, DOj) I If f A n-1 :Fun(CO, D*)

where the bottom horizontal map is the constant map at FO. By [15, Lemma 4.3.2.12], this diagram admits a lift. Since lifting left horn inclusions that factor through E is

sufficient to show that an edge is locally cocartesian, < is locally cocartesian.

Thus we have shown that Fun(C, D)@ is a locally cocartesian fibration, and for

each morphism f of TT, we can choose a locally cocartesian edge s over f which

corresponds to a relative left Kan extension as in Lemma 2.1.3. Suppose s' is another

locally cocartesian edge over f with the same source as s. Then s and s' are equivalent

as edges of Fun(C, D)O, and so they must correspond to equivalent functors C' -+

D'. Since one of these is relatively left Kan extended from CO, so must the other

be. This proves the converse.

The next proposition contains most of the technical work of this section. It

amounts to showing that relative left Kan extensions are compatible with compo-

sition on the base.

Proposition 2.1.5. In fact, given the existence of colimits in D, Fun(C, D)O -+ T,

is a cocartesian fibration.

Proof. To prove this, it is convenient to first establish the following variant of [15, Lemma 2.4.2.7]:

16 Lemma 2.1.6. Suppose p : X -+ S is a locally cocartesian fibration of oc-categories, and let f be a locally cocartesian edge of X. The following conditions are equivalent:

1. f is cocartesian.

2. For each 2-simplex o- of S such that 02o- = p(f), there is a 2-simplex c-' of X

such that p(o') = a, 02 a' is equivalent to f and both 01a' and ioo-' are locally cocartesian.

Proof. The implication 1 -> 2 is obvious, so let's prove the converse. Since the property of being cocartesian is closed under equivalence, we may assume that 09-'

is equal to f. Let T be a 2-simplex of X with a2T = f and Dor locally cocartesian;

9 by [15, Lemma 2.4.2.71, it suffices to prove that 1 T is locally cocartesian. Choose

T' such that p(T') = p(T), 02T' = f and all of the edges of r' are locally cocartesian.

Since they are both locally cocartesian and have the same source, 0 rT is equivalent

to 9 OT', and so T is equivalent to T'. In particular, 0 ir is locally cocartesian. l

Let 1 : L -+ F, and k : K -+ F. be maps of simplicial sets. If we have a map

L -+ Fun(C, D)O of simplicial sets over T, and an inclusion L C K over F, we

denote by Fk the induced map C* -+ D', and write Fk for Fjk. When it comes to

objects and morphisms, we'll sustain the same abuses we've been using so far: if I

is the inclusion of an object S of T., for instance, we'll write Fs for the morphism

C' - D'. We will abusively employ p without further decoration to denote

PL : D* -+ L.

Now we can prove Proposition 2.1.5. Suppose f : S -+ T and g : T -+ U are com- posable morphisms in F,. We let a: A2 -+ T, be the map recording the composition of f with g. Thus, for example, a morphism Ff over f is locally cocartesian iff Ff is a p-left Kan extension of F4 to C'.

Let S E F, and suppose Fs : Co -+ Do is a functor. We let Ff be a p-left Kan

extension of Fs to C*, and we let F be a p-left Kan extension of

17 FT - Ff co

to C*. Thus Ff and F. together give a map Al - Fun(C, D)*, which extends to a map T : A -+ Fun(C, D)* over a classifying a map F, : CO -+ DO. By Lemma

2.1.6, it suffices to prove that every edge of T is locally cocartesian. Since 0OT and

&2T are locally cocartesian by construction, it suffices to prove that 91T is locally cocartesian.

Next, if F, is a p-left Kan extension of F0 to C*, then certainly &iT is locally

cocartesian, by [15, Proposition 4.3.2.81. But since 02 T is locally cocartesian, this is true if Fa is a p-left Kan extension of F7 to C@, by the same proposition. This is what we shall prove.

The value of FU on Z c CO is by construction the p-colimit of the map

(Fug)/z : (CO)/z -4 DO* induced by F3. Clearly replacing (Fj) 1z with

(F )/z : (Co)/z -4 D

does not change the value of the colimit. On the other hand, F, is a p-left Kan extension of F7 to CO iff FU(Z) is a p-colimit of

(F7)/z : (CO) /z - Do.

Thus, by [15, Proposition 4.3.1.71, it suffices to prove that the natural inclusion

iz : (CT)/Z - (CO)/z

is cofinal. An object of (CI)/z not in the image of Iz is a morphism q : X -+ Z

over gf in C*. But if X -+ Y is a cocartesian edge of C* over f, then the 2-simplex

defined by X -+ Y -+ Z is initial among morphisms from q to an object in the image

of Iz. By Joyal's version of Quillen's theorem A [15, Theorem 4.1.3.1], this completes

18 the proof. El

Now we recall that there is a canonical product decomposition C' r Cs, by which we mean the cartesian power indexed by the set S' of non-basepoint elements of S.

With this in hand, we can make our main definition.

Definition 2.1.7. The Day convolution symmetric monoidal oo-category Fun(C, D)® is the full subcategory of Fun(C, D)O whose objects are those corresponding to functors F : Co -+ Do which are in the essential image of the natural inclusion

ts : Fun(C, D)s -+ Fun(Cs, Ds) - Fun(C", DO).

The fiber of this category over S E F is evidently Fun(C, D)5 , so the symmetric monoidal oo-category stakes are looking favorable. We need to verify a couple of things.

Lemma 2.1.8. Fun(C, D)O -+ F is a cocartesian fibration.

Proof. We need to show that the target of any cocartesian edge of Fun(C, D)* whose

source is in Fun(C, D)O is also in Fun(C, D)*. To start, suppose f : S - T is a morphism in F,. Since the pushforward associated to f is compatible with the product decompositions on C' and Co, we have a decomposition

Cf LtQC7 tCT

where C' -+ A' is a cocartesian fibration classified by the map C' - Co induced by pushforward along f. Let (F,),cso be an S-tuple of functors from C to D, and let F4 denote the com-

posite

Then F decomposes as a product of maps

FftS : Co-f'(t) 1 Doe

19 be a p-left Kan extension of Fet along C (t C't. Then For each t, let F, f ) f,t Ff = 1 F, is a left Kan extension of F4 along CO --+ Co, and FT is in the essential image of Fun(C, D)T.

Proposition 2.1.9. Fun(C, D)* -+ T, is a symmetric monoidal oc-category.

Proof. Recall that a morphism f : S -+ T in TT is called inert if it induces a bijection

f0 : f-'(TO) -+ To.

By definition, a cocartesian fibration X -+ -F, is a symmetric monoidal category if for each n E N, the pushforwards associated to the n inert morphisms (n) -+ (1) exhibit X(n) as the product of n copies of X(1). If, as in our case, one already has an identification of X(n) with Xd) up one's sleeve, then another way to express this condition is to say that the pushforward

ij : A() -+ X(1) associated to the inert map (n) -+ (1) that picks out j is equivalent to projection onto the jth factor. We'll retain the notation of the previous proof, but now assume that f is inert.

Then for each t E T",

Cf, C x A'.

and so F, = F x A'. We conclude that the pushforward associated to f is indeed the projection Fun(C, D)s -+ Fun(C, D)T. El

Proposition 2.1.10. A commutative monoid in Fun(C, D)® is exactly a lax sym-

metric monoidal functor from CO to D*.

Proof. Recall that by definition, a lax symmetric monoidal functor from C® to D* is

a morphism of oo-operads from C® to D*; that is, it is a morphism of categories over

T, that preserves cocartesian edges over inert morphisms. What we must prove is

20 that that if s : T, --+ Fun(C, D)O is a commutative monoid - that is, a section of the structure map which takes inert morphisms to cocartesian edges - the corresponding functor

CO x.,. F* ~ C* -+ D* preserves cocartesian edges over inert morphisms. So if f is an inert map in F*, we must show that a functor C' -+ DO which corresponds to a cocartesian edge of Fun(C, DO) preserves cocartesian edges over f. By using the product decomposition of CO for f inert, we reduce to the following:

let G : C -+ D be a functor, and let F0 be the composite

C - D - D x A.

Then we must prove a functor F : C x A' -+ D x A' is a left Kan extension of F

(relative to A') iff it preserves cocartesian edges. Since G x A' is a left Kan extension

of F0 , the former condition merely states that F is equivalent to G x A', and G x A' clearly preserves cocartesian edges. To prove the converse, we find it most convenient

to deploy some machinery. By the opposite of [15, Proposition 3.1.2.3] applied to

the (opposite) marked anodyne map A' -+ (A')O and the cofibration 0 -+ C", the

inclusion of marked simplicial sets

C, -* (C x A)

is opposite marked anodyne and therefore a cocartesian equivalence in sSet+1 . This

means that F0 extends homotopy uniquely to a map of marked simplicial sets

(C x Al) -+ (D x A)

which proves the result. l

We'll now record a couple of technical lemmas for future use.

Lemma 2.1.11. The Day convolution of two strict symmetric monoidal functors is

21 strict symmetric monoidal.

Proof. Let's use the notation of the proof of [17, Proposition 3.2.4.31. We'll work in the model category (Set+)/p_, in which the fibrant objects are exactly symmetric monoidal categories with their cocartesian edges marked.

Let t : (2) -+ (1) be the active map. We regard (A1 )O x (F,) as a marked simplicial set over (,)O via the composite

AyX (_F,), .Lxd (T,), X A.~~-

Unwinding the definitions, we find that giving a pair of strict symmetric monoidal functors C* -+ DO together with a strict symmetric monoidal structure on their Day convolution is the same as giving a functor

((A')' x (-F*)') x(T*)o (CO)I -+ (D*)' of marked simplicial sets over (FT)O. Thus it suffices to show that the inclusion

i : ({0} x (-F*)') x (T->o (C')'"+((A)Y x (T,*)) x (.*)o (C')' is a trivial cofibration. But recall from the proof of [17, Proposition 3.2.4.3] that there is a left Quillen bifunctor

v/ : (Set')/p_ x (Set+)/43_ -+ (Set+)/Tp which takes a pair (X, Y) to the product X x Y regarded as a marked simplicial set over F* by composition with A. Now the conclusion follows from the fact that i arises as the product, under v, of (CO) with the q3'cmm-anodyne inclusion

{(2)} >(')I

22 Lemma 2.1.12. Suppose DO has the property that the tensor product commutes with colimits in each variable separately. Then the same is true of Fun(C, D)O for any CO.

Proof. Suppose q : K -- Fun(C, D) is a diagram and F : C -+ D is a functor. F and q define a functor V, : K -+ Fun(C, D) 2 = Fun(C, D)'), and we aim to show that the natural morphism G : colim ([tO) -+ p,(colim V) K K is an equivalence, where p : (2) -+ (1) is the active morphism. Evaluating each side on the object X E C specializes G to

Gx : colim colim (F(Y) 0 4(k)(Z)) 2 kEK (YZ)EC XCC/X

-+ colim (F(Y) 0 colim 4(k)(Z)). (Y,Z)EC 2 XCC/X kEK

By the hypothesis on D, we can pull out the colimit over K on the right and conclude that Gx is an equivalence.

In [17, Corollary 6.3.1.12], Lurie describes a Day convolution symmetric monoidal

structure on the category of presheaves on a small symmetric monoidal category, which for consistency ought to agree with our construction in the relevant case.

Proposition 2.1.13. Let C be a symmetric monoidal category and let P(C)* denote

the construction of 117, 6.3.1.121. Endow Top with its product symmetric monoidal

structure. Then there is a model for the symmetric monoidal category (COP)* such

that there is an equivalence of symmetric monoidal categories

P(C) - Fun(C P, Top)*.

In order to prove this, one only needs to show that Fun(CP, Top)* satisfies the

two criteria of [17, Corollary 6.3.1.12]. Criterion (2) follows immediately from Lemma

2.1.12, and Criterion (1) will be the subject of the next section.

23 With only a little more work, we'll be able to prove an analog of Proposition 2.1.13 for presheaves of spectra. In [18, 2.31, Lurie discusses a presentable stable symmetric monoidal oo-category Rep(I)O together with a symmetric monoidal stable

Yoneda embedding I -* Rep(I)O which induces an equivalence of categories

Fun, (Rep(I) , MD) -4 Fun(I, MD), where MO is any stable presentable symmetric monoidal oo-category and Fun*" is the category of symmetric monoidal functors which preserve colimits in the fibers.

Proposition 2.1.14. The Day convolution category Fun(IOP, Sp)O is equivalent as

a symmetric monoidal oo-category to the category Rep(I)*.

2.2 The symmetric monoidal Yoneda embedding

A functor

C* -+ Fun(CP, Top)0 .

is the same thing as a functor

CO x.7 (C"")O - TOpX

satisfying certain conditions; this in turn is the same as a functor

CO xF, (C"P)O x F Fx -+ Top

satisfying additional conditions, where Fx is the category of [17, Notation 2.4.1.21

(see also [17, Construction 2.4.1.4]). Constructing this functor is going to require a

small dose of extra technology. In Proposition 2.2.1, we'll describe a construction, due to Denis Nardin, of a strongly functorial pushforward for cocartesian fibrations.

Proposition 2.2.1. Let p : X -* B be any cocartesian fibration of oo-categories.

24 Let OB be the arrow category Fun(A1 , B), equipped with its source and target maps

s, t: OB -+ B, and form the pullback

X XB OB via the source map. Then there is a functor

(-), X XB OB -+ X

which maps the object (x, f) to f~x and makes the diagram

X XB OB -. X

commute.

Proof. Let 0' be the full subcategory of Ox spanned by the cocartesian arrows and

let p : Oc -+ OB be the projection. Then the essential point is that

(s,p) :Ox -+ X XB OB

is a trivial Kan fibration. Indeed, this follows from arguments made in [15, 3.1.2J, which we recall here for completeness: the square

I xI

-An X XI OB

25 determines the same lifting problem as the square

[(A")l X (A')'] H X [(A)b X f{0}] > X

(An)l X (AIl> BO of marked simplicial sets. But the left vertical map is marked anodyne [15, 3.1.2.3], so a lift exists.

Now the diagram Oc x

top X xB 0 B X t4I B commutes, so composing s with any section of (s, p) gives the desired map. El

We'll also need to import a result from a recent paper with Clark Barwick and Denis Nardin [51:

Theorem 2.2.2. Let p : X -> B be a cocartesian fibration of oo-categories classified

by a functor F : B -+ Cat,. Let op : Cat, -+ Cat, be the functor that takes a

category to its opposite. Then there is a cocartesian fibration p' : X' -+ B classified

by op o F together with a functor MapB(-, - : X' XB X -+ Top such that for each

b c B, the diagram MapB(-,-) X' XB {b} XB X > Top

t- ,- Map(-,-) X P X Xb

homotopy commutes, where the vertical equivalence comes from the given identi-

fication of X' with X"". Moreover, using these identifications, for each morphism

f b -+ b' E B and for each (y, x) E x Xb, sends (up to equivalence) the

cocartesian edge from (y, x) to (F(f)(y), F(f)(x)) to the natural map Map(y, x) -+ Map(F(f)(y), F(f)(x)) induced by F(f).

26 Proof. Using the notation of [51, take X' - (Xv)OP and let MapB(-, -) be a functor classified by the left fibration M: ((X/B) -+ X' XB X of [5, 51. El

Observe that Lurie's category f1 [17, Notation 2.4.1.2] is the full subcategory of

O., spanned by the inert morphisms. Thus by Theorem 2.2.1 applied to C* x., (COP)®, we get a functor

o' : C* x.Kj (C P)® Xj FX -+ C® Xj (C"P)*.

Composing this with Map,(-, -) : C® xy (COP) - Top gives a functor

C*®: Xy (COP)* X, FX -* Top

which adjoints over to a functor

q : C® xy. (COP)* - To [17, Construction 2.4.1.41

which factors through Topx, by the properties of (-), and Mapy.(-, ). Adjointing again gives a functor

'0: C" -÷ Fun(COP, Top)®

We must check that V, factors through Fun(COP, Top)*. Indeed, for each S E F, and

(X,),,s. E C', we have

0((X,).,Es.) ~(Map(-, X.)).s-s.

We denote the ensuing functor

Y' :C -+ Fun(C"P, Top)®

and the final order of duty is to prove that Y@ is symmetric monoidal. Let :

(Zs)seso -+ (Wt)tETO be a cocartesian morphism of C® lying over the morphism

27 f : S -+ T of F., so that Wi ~ Zi, jef- 1(i) and let

Y : (C P)' -+ Top' be the induced morphism. For each (X)tETo c (COP)', let

0 K = (C P)) X>(F)g {f} and let p: K' -+ (COP)' be the natural inclusion, taking the cone point to (Xt)teTo. Then the condition we must verify is that

Ya0 p: K' -4 TopX is a colimit diagram relative to A', which is to say that the natural map

colim f,((Map(P,Z-))Eso) -+ (Map(Xt,Wt))tETo ((PS)SEso-+(Xt)tETo)EK is an equivalence. To prove this, we may as well take T = (1) and f to be active. Then we're really asking for the natural map

n nl colim ]I Map(P, Zj) -- Map(X, ( Zi) X-+P1OP20 -OPn. %=I i= I to be an equivalence for all n > 0, X E COP and (Zi)<

Define a category Dn by the pullback square

Dn ) (C",)/(zi,-..,,) t~ 14nOs CX/ fI C.

Then the functor on CI x c Cx/ taking X -+ P1 .. -D P, to H Map(P, Zj) is left

Kan extended from the constant functor D -- Top with image *, so we must show

28 the ensuing map of spaces

N(Dn) -+ Map(X, Zi)

is a weak equivalence. But replacing (C)/(z,...,z.) with its final object id(z,,...,z.) doesn't alter the homotopy type of the pullback, and the resulting pullback is Map(X, 0&"= Zi) by definition. Unwinding the sequence of morphisms shows that this is the equivalence desired.

We now turn to the proof of Proposition 2.1.14, which is largely a corollary of the above discussion.

Lemma 2.2.3. Let CI, DO and EO be symmetric monoidal oc-categories, and let

F : DI -+ E* be a symmetric monoidal functor which preserves colimits in each fiber. Then postcomposition with F determines a symmetric monoidal functor

(oF) : Fun(C, D)® -+ Fun(C, E)*.

Proof. Here's what this boils down to. Suppose A, B and C are oc-categories equipped with cocartesian fibrations to A', and we're given functors p : A -+ B and v : B -+ C both compatible with the projections to A'. Suppose that v preserves cocartesian edges and preserves colimits in the fibers, and moreover that p is the relative left Kan extension of its restriction to the fiber AO over {0} c A', by which we mean that the diagram Ao P" B

A >A' exhibits [ as a p-left Kan extension of yo in the sense of [15, Definition 4.3.2.2]. Then we want to show that the composition v o f is also the relative left Kan extension of its restriction to Ao.

29 Let a be an object of A1, and denote

(Ao)/a := AO XA A/a.

Denote by ta the natural map from (Ao)/a to B, and let

1 p, : (Ao)/a X A -+ B be an extension of [tpa to a morphism of cocartesian fibrations over A1 . Taking fibers over {1} gives a functor t' : (Ao)/a -+ B 1 , and it follows from the proof of [15, Corollary 4.3.1.111 (see also Lemma 2.1.4) that the condition on 1t is equivalent to the condition that p(a) is a colimit of pl for each a c A 1 . But the condition on v guarantees that v/ o / satisfies these conditions if p does. l

Proof of Proposition 2.1.14. Both categories certainly have underlying category Fun(I*P, Sp). By [18, Remark 2.3.101, we need only verify that Fun(IP, Sp)O satisfies the axioms characterizing Rep(I)0 . The first follows from Lemma 2.1.12, and the second follows from Proposition 2.1.13 together with Lemma 2.2.3, since the suspension spectrum functor is symmetric monoidal and preserves colimits. l

30 Chapter 3

A spectrum-level Hodge filtration on topological Hochschild homology

3.1 Preliminaries on twisted arrow categories, ends and coends

We'll first give a brief introduction to the definitions and basic properties of ends and coends of quasicategories; for a more thorough and classical treatment in the context of 1-categories, see [19, Chapter IX]. Let C be a presentable symmetric monoidal oc-category with colimits compatible with the symmetric monoidal structure - in particular, C has an internal hom right adjoint to the monoidal structure - and let I be a small oc-category.

Definition 3.1.1. The twisted arrow category Cf9 is defined by

(Oi)n:::: =12n+1 with faces and degeneracies given by

six dn-idn+1+iX

iX= Sn-i Sn+l+iX-

31 We'll adopt the following unorthodox notation:

(51:= ( )Op.

For a full discussion of twisted arrow categories, one should consult [16, 4.2] or [3, 2], though bear in mind that these two sources differ by an op in their definitions.

There is an evident functor 7 1t : 09 -+ IOP x I that takes an n-simplex x of 05 to xj[o,n] x XI[n+1, 2 n+l]. By [16, Proposition 4.2.3] and [16, Proposition 4.2.5], it is a left fibration classified by the functor IP x I -+ Top that takes X x Y to Hom(X, Y). Let

T : (I*P x I)0P -~ IP x I be the natural equivalence, and let R, denote the composite

:0. X (I xI)P +IOP XI.

If T is any functor from IOP x I to C, we can obtain a functor C9 -+ C by

precomposing with R, or a functor 01 -+ C by precomposing with RI.

Definition 3.1.2. The coend of T is defined by

T:= colim V o T. 61

Dually, we define the end of T by

T:= lim R, o T.

In particular, if F : IOP -+ C and G : I -+ C are functors, we can define a functor

Fx G : IOP x I -+ C x C by taking products. The symmetric monoidal structure on

C induces a functor 0 : C x C -+ C, and by postcomposition with 0 we obtain a functor F 0 G : I P x I - C

(this is the objectwise tensor product of functors, not to be confused with the Day

32 convolution product). The central current of this paper involves coends of the form

FOG which we'll abusively denote F0 G, but we'll first discuss an end of interest.

Proposition 3.1.3. Let D be any o0-category and let F, G : I -+ D be functors, both covariant this time. Let HomD : D4P x D -+ Top be the hom-space functor

(one model for this is mentioned above). Defining

Hom(F(-), G(-)) = HomD o (FP x G) o 'I 1 -W Top, there is a canonical equivalence

G) /, Hom(F(-), G(-)) ~Nat(F, which is functorial in both F and G.

Proof. This statement, which is elementary in the 1-categorical setting, becomes slightly tricky to prove for oo-categories, and we don't know of a proof elsewhere in the literature. We thank Clark Barwick and Denis Nardin for a helpful conversa- tion regarding this proof. The reader actually interested in understanding this proof should first note that it really exists at the level of combinatorics, not homotopy the-

ory: everything is going to be strictly defined, and we'll be fuelled by isomorphisms

of simplicial sets rather than equivalences.

The first idea is that f, Hom(F(-), G(-)) should be the category of ways of

completing the three-quarters-of-a-commutative square

33 01 OD Ci CD

I P x I F OP 9 fop x J x D to a commutative square; that is, it should be identified with the pullback

Fun(OI, CD) X Fn(6j,D-PxD) {(F" x G) o W7t}.

Indeed, let S be the pullback

S >OD

-0- DIP x D. (F-PxG)oH 1

Then S -+ (51 is a left fibration classifying the composite Hom(F(-), G(-)), and the space of sections of this fibration is a model for fL Hom(F(-), G(-)). This means that in the diagram

f, Hom(F(-), G(-)) > Fun(OI, S) - > Fun(0j, OD)

{id} Fun(Or, 01) - Fun(0j, DIP x D), the left-hand square is a pullback, and the right-hand square is certainly also a pullback. The composite pullback square gives the desired description of f, Hom(F(-), G(-)).

Let T denote the pullback

T x Fun(0I, OD)

Fun(IP x I, D"P x D) -+ Fun(Oi, D*P x D),

34 so that T is the category of commutative diagrams that look like

0, >O

I*P x I - DOP x D.

We claim that there's a pullback square

Nat(F, G) T

jInun(ID> {(FOP,G)} > Fun(IOP x 1, DOP x D).

Indeed, one can give an explicit isomorphism between the fiber of T over (FOP, G) and Nat(F, G). We'll give the bijection between the 0-simplices; the higher simplices follow identically but with more cumbersome notation. The aim here is to give a bijection between the set Nat(F, G)o of natural transformations from F to G and the

9 set (call it U) of ways of extending (FOP, G) to a functor from (5 to ( D.

First we'll define a : Nat(F, G) 0 -+ U. An element x E Nat(F, G) 0 is in particular a map x: A' x I -+ D. If y is a k-simplex of 6, - thus a 2k + 1-simplex of I - we'll define

a(x)(y) = x(qk x y) where q- A2k+1 _* A 1 is the map with

0 i< k qk(i) -- 1i > k.

The inverse 3 of a is given as follows. Suppose Q : 01 -+ 6D is a functor lifting

FOP x G, and let y = (yAl, yi) be a k-simplex of A1 x I. There's a unique factorization

Ak q> A' X I

A2k+1

35 where y. is formed by applying some degeneracies to yI, and the only purpose of the above diagram is to give a compact description of exactly which degeneracies these are. Then we'll define 3(Q)(y) to be the k-simplex of D resulting from removing the same degeneracies from the (2k +1 )-simplex Q(y').

Once the definitions are fully unwrapped, it's immediate that a and 3 land where they're supposed to, and it's easy to check that # o a and a o # are the identities. Composing the pullback square defining T with the one we've just derived, we get a pullback square

Nat(F, G) - Fun(0 1, D)

I I {(F, G)} > Fun(6j, DP x D).

But we already have a name for the fiber in this square: it's

/, Hom(F(-), G(-)).

This gives us our isomorphism

Nat (F, G) ~Hom (F (-), G(-)

Moreover, this entire argument is contravariantly functorial in I. In particular, we could replace I with I x A' for some n, which makes our equivalence functorial in F and G. El

Now let i : I -+ J be a functor of small oo-categories, and assume again that C is presentable. If F : J --+ C is a functor, then we write i*F for F o i. Our next goal is to describe an "adjunction" formula involving i* and the left Kan extension

I Fun(IP, C) - Fun(JP, C).

Proposition 3.1.4. Let G : II -+ C be a functor, and let i!G be a left Kan extension of G along i*; explicitly, on objects,

i!(G)(j) = colim G(z). zEI xjj

36 Then there is an equivalence

i*F G JFoiG which is functorial in F and G.

Proof. Let T be a test object of C. Then

Hom F 0 i!G, T) ~ Hom(ZIG 0 F, T)

~ Hom(iG, Hom(F(-), T))

~Nat(i'!G, Hom(F(-), T)) ~ Nat(G, i*Hom(F(-), T))

~Nat (G, Hom ("* F(-), T))

~ Hom(G, Hom(i*F(-), T))

~ Hom(G 0 i*F, T)

~H om (Ji*F 0 G, T),

and each equivalence is functorial in F and G, which gives the result. L

Proposition 3.1.5. For any functor F : I - C,

1 0 F colim F II

where 1 is the constant functor at the unit of C.

Proof. We have F~ colim F(X) J X-4Ye0 1

so it's enough to prove that the source map s : (5 -+ 1 is cofinal. By Joyal's version of Quillen's Theorem A [15, Theorem 4.1.3.11, this is the same as proving that for

each X c I, the category X1. = (9 x, -x/ is weakly contractible as a space.

37 We may take as a model for X1, the simplicial set whose n-simplices are 2n + 2- simplices of I with leftmost vertex X. Let X' be the subcategory whose n-simplices are those 2n +2 simplices a of I for which oj [0,n+1] is totally degenerate at X. X',, has the totally degenerate 2-simplex at X as a final object, so it suffices to show that the inclusion X' --+ X, is an equivalence. But the functor that sends a 2n + 2 simplex - of I to sn+ 1d'j+1 o is right adjoint to this inclusion.

3.2 Tensor products of commutative ring spectra with

spaces

Let E be a commutative ring spectrum. Given a simplicial set X, the tensor product or factorization homology EOX is by definition the colimit of the constant X-diagram in CALg valued at E; see [9] and [21, 1] for explorations of these ideas. We aim to give a topological version of the simplicial formula for E 0 X in [9, 3.11.

We'll need to make use of a technical base-change lemma due to Barwick. For S

a simplicial set, we denote by sSet/s the category of simplicial sets over S endowed with the covariant model structure [15, Proposition 2.1.4.7]. Suppose j : S -+ S' is a map of simplicial sets. Then the pullback functor j* : sSet/s' -+ sSet/s and its left adjoint j! form a Quillen pair [15, Proposition 2.1.4.10].

Lemma 3.2.1. Suppose Y' 9 > Y

is a strict pullback square of simplicial sets in which p, and therefore p', is smooth in the sense of [15, Definition 4.1.2.9]. Then the natural transformation

Lp' o Rg* 3Rf* o Lp

is an isomorphism of functors ho sSet/y - ho sSet/x,.

The proof is a near-verbatim reproduction of one communicated by Barwick:

38 Proof. Suppose q : Z -+ Y is a left fibration, and denote by T : Z -+ X the fibrant replacement of r : po q in sSet/x; in particular, we have a covariant weak equivalence

By [15, Proposition 4.1.2.151, both r and T are smooth. Hence by [15, Proposition

4.1.2.18], for any vertex x E X, the induced map ?I : Zx -+ Z s a weak equivalence of simplicial sets. Now since p' is a smooth map as well, it follows that the natural map

Z XX X' z xx X'

X' is a covariant weak equivalence if and only if, for any point E X', the induced map vi : (Z xx X')C -+ (Z xx X')C on the fiber over x is a weak equivalence of simplicial sets. But this is true, since we can identify (Z xx X')C with Zf(c) and (Z XX X') with Zf(C).

Corollary 3.2.2. Retaining the notation of Lemma 3.2.1, let C be a presentable oo-category and let k : Y -+ C be a functor. Then we have an equivalence of functors

X' -+ C

pig*k ~+ f*ptk where now (-)* is restriction and (-)! is left Kan extension.

Proof. By straightening, the case C = Top is equivalent to the statement of Lemma

3.2.1. We can immediately extend this to presheaf categories C = Fun(D, Top), since the square Y'x D -9 Y x D

X'x D - X x D also satisfies the hypotheses of Lemma 3.2.1 and since colimits in Fun(D, Top) are

39 computed objectwise. Finally, composing k with a localization functor doesn't change anything, and any presentable oo-category is a localization of a presheaf category. E

Now we return to the problem of describing EOX for a commutative ring spectrum

E and a simplicial set X, which we'll take to be finite. The functor X : A"P -+ 7

classifies a left fibration k -+ A0 P with finite set fibers, and X is weakly equivalent to X. It fits into a pullback square

J 9 > E.F

where EF -+ F is the universal left fibration with finite set fibers, classified by the

identity functor on T. An object of E is a finite set S together with an element

s c S, and a morphism from (S, s) to (T, t) is a set map f : S -+ T with f(s) = t.

Let k denote the constant functor EF -+ CALg valued at E. Since g*k is also a constant functor, we have equivalences

E0X EOX

~ colim p'g*k

~ colim X*pik. AOP

Let U : CAIg -+ Sp be the forgetful functor, and write AE := Uop!k. Since simplicial

realizations of commutative ring spectra are computed on underlying spectra [17, Corollary 3.2.3.21, we have

E 0 X ~ colim X*AE -*)

as spectra. This is our topological version of [9, Definition 21.

Note that ({1}, 1) is an initial object of EF, so that k is left Kan extended from

({1}, 1), and therefore p!k is left Kan extended from {1} E F. It has the property

40 that AE(S) = EAS. Note further that AE extends to a symmetric monoidal functor

AO: .F.7 -+ SPA which is in turn the same data as E itself, since a commutative ring spectrum is by definition a morphism of oo-operads T. -÷ Sp^ [17, Definition 2.1.3.1] and F' is the symmetric monoidal envelope of F, [17, Construction 2.2.4.11.

What can we do if there's a module in the mix? We'll use the following result, proved in Appendix A:

Theorem 3.2.3. Let C* be a symmetric monoidal oo-category. We'll denote the category of finite sets by F and the category of finite pointed sets by F,. A datum comprising a commutative algebra E in C and a module M over it - that is, an

object of Mod*(C), the underlying oo-category of Lurie's oc-operad Mod*(C)* [17, Definition 3.3.3.81 - gives rise functorially to a functor

AEM F* - C

such that

AE,M(S) ~ ES" 0 M.

Let X be a pointed finite simplicial set, thought of as a functor A0 P -+ F. By

analogy with (*), we'll define the tensor product of E with X with coefficients in M as

(E 0 X; M) := colim X*AE,M-

We expect this construction to agree, under appropriate circumstances, with the

factorization homology of the pointed space X regarded as a stratified space [2] with

coefficients in a factorization algebra constructed from E and M.

We note also that a cocommutative coalgebra spectrum P defines a functor Cp:

FW -+ Sp, again mapping a finite set S to the S-indexed smash power of P, but we

won't go into the coherency details of this because all such functors arising in the present work can be defined easily at the point-set level.

41 Example 3.2.4. When X = Si is the usual simplicial model for the circle, with

S1[n] {0, 1, --- , n}

E 0 S' returns the usual simplicial expression for THH(E). If M is an E-module, then (E 0 Sl; M) returns the usual simplicial expression for THH(E; M), where we point S' by the lone 0-simplex [211.

3.3 The Hodge filtration

We'll now exploit this lemmatic mass to derive a filtration of the topological Hochschild homology spectrum of a commutative ring spectrum E. More generally, this filtration exists on E 0 X for an arbitrary simplicial set X. In this section, we'll define this filtration, present a geometric model for it, and give a convergence result.

A key step is the following jugglement of coends:

Proposition 3.3.1.

E 0 X = colim X*AE AOP

J S A X*AE (by Proposition 3.1.5)

J XiS AAE (by Proposition 3.1.4) where

XIS(S) ~ colim S [n]EAoPS-+X[n]

~ colim SVXin]S

[n]EAOP X[n]S S A colimAOP

~S A XS

~ X s

42 and the functoriality in S is given by diagonal inclusions and projections; in other words, X!S is equivalent to the functor

CE.x : F"" -+ Sp

coming from the coalgebra structure on EOX.

Remark 3.3.2. This is closely analogous to Pirashvili and Richter's description of

Hochschild homology as a coend over the category of noncommutative finite sets in [25].

To lessen wrist fatigue, we'll usually wite Cx for Cgx.

Similarly, if X, is a pointed simplicial set, then

(E 0 X,; M) J X*, A AE,M

where

X,,iS(S) -_ E,+)Xf".

This is just [13, Proposition 4.8] in a slightly different language.

Now our filtration of E 0 X will come from a filtration of the functor Cx.

Definition 3.3.3. Let .T" denote the full subcategory of F spanned by those sets T for which

ITI < n.

Denote by C71 the restricted-and-extended functor

RanT!' ,oCx r< ,op.

The nth Hodge-filtered quotient of E & X is

f" (E O X) J C" A AE-

43 In the same breath, we may as well define

S' +'(E O X) := Fib [E OX -+ 5>E"(E & X)] and the graded subquotient

5> "(E 0 X) := Fib [S'>"(E aX) -+ j>5,-(E 0 X)].

Of course, if we define

q71+1 := Fib [Cx -Ci"] and

C-n := Fib [Cx" - C, -l1] then we have the equivalences

I Cf+1 A AE and SY= (E 0 X) p CA AE-

The story for the tensor product with coefficients is practically identical:

Definition 3.3.4. Let F;" denote the full subcategory of F, spanned by those objects T for which 1T'l < n, and define

Ce f := Ran 0. Cx. X; Pigiv

Then the Hodge filtered quotient 7t5"(E & X; M) is given by

J T*Ck2 A AE,M.

Remark 3.3.5. Everything here goes through identically if we work in the category of algebras over a fixed commutative ring spectrum K, and we get a Hodge filtration

44 of E OK X by K-modules.

When X = S', we consider this the appropriate spectrum-level analogue of Lo- day's Hodge filtration (see [27, 4.5.151.) We'll present evidence for this shortly, but first we'll showcase some of the geometry of this rather abstract-looking filtration. We'll abusively conflate X with its geometric realization.

Proposition 3.3.6. Let F be the category of finite sets and injections and define a functor of 1-categories

an:_F4O -+ Top by

an(U) {(x,) E XU I at most n coordinates of (x.) are not at the basepoint} with the obvious projections as functorialities. (It's logical to set the convention aco(U) = XU.) Then the restriction of CI' to .T'P is equivalent to E' o a.

Proof. First note that for each finite set U, the functor

admits a left adjoint, and is thus homotopy cofinal. Now let P5 be the poset of subsets of U of cardinality at most n, Pu the poset of all subsets of U, and P& the poset of proper subsets of U. The inclusion

is an equivalence of categories, and E+ oan and C7" clearly have equivalent restriction to

E' a,(U) -+lim Ej an(V) VE(pg")op is an equivalence for every U. We work by induction on the cardinality of U; for

45 UI < n there is nothing to prove.

Lemma 3.3.7. an Lpu is a strongly cocartesian U-cube.

Proof. Replace a,Jpu with the diagram of cofibrations that takes V C U to the subspace (V) = {(xu) I xu E X for u E V} of

{(XU) E (CX)U I at most n coordinates of (xu) are not at the basepoint} where the basepoint on CX is the basepoint on X, not the cone point. This is clearly homotopy equivalent to the original diagram. Moreover, if x E 3(V), then the subcube of PNTc spanned by those spaces containing x is a face of P ' containing V.

This implies that the diagram is strongly cocartesian. l

So V -+ E o a (V) is a cocartesian, and therefore cartesian, cube of spectra.

But by the induction hypothesis, anp is right Kan extended from aI,

Corollary 3.3.8. C"(U) is a retract of C(U).

Proof. Clearly aIpu is strongly cartesian, and so Xu is the limit of a,0 p f... =aI,

This gives a map q : a,(U) -+ XU. On the other hand, the projection , : Cx(U) -

C<"(U) comes from regarding Ela,(U) as the limit of (E'an)Ip a. Tracing through the universal properties shows that

V) 0 (Eg') : C5"(U) -+ C "(U) is homotopic to the identity. El

Corollary 3.3.9. Suppose X is connected. Then the projection 0 : Cx(U) -+

C%"(U) is (n + 1)-connected (by which we mean that the homotopy fiber of 4' is n-connected).

46 Proof. By Corollary 3.3.8, it suffices to show that # is n-connected. But if we model X by a CW-complex with exactly one 0-cell, which we take to be the basepoint, then

# is homotopic to the inclusion of the n-skeleton of the natural CW structure on XU. E

Corollary 3.3.10. Suppose R is a connective commutative ring spectrum and M is an R-module which is k-connective for some k c Z. Then the projection

(E X; M) -+ (E ® X; M)<" is (n + 1 + k)-connected, and so we have convergence:

(E & X; M) ~lim (E 0 X; M)

Proof. For each object f : S -4 T of O(9 *, the projection

Cx(S) A AE,M(T) -+ C n(S) A AE,M(

is (n + 1 + k)-connected. This connectivity is preserved by taking the colimit. 0

Remark 3.3.11. We'd have to be born yesterday to expect good convergence behavior from our filtration in nonconnective situations.

3.4 Multiplicative structure

The goal of this technical section is to pour some symmetric monoidality into our

theory of coends; in particular, we'll see that taking a coend against a symmetric

monoidal functor maps the Day convolution to the tensor product, thus evincing

intriguingly Fourier transform-like behavior. This will allow us to deduce that the

Hodge filtration on the tensor product of a space with a commutative ring is multiplicative (Corollary 3.4.2).

Let I be a small symmetric monoidal oo-category. Suppose we have a symmetric

monoidal enrichment (IP)* of I'P and a symmetric monoidal enrichment (01)0 of

47 0, such that 7(1 extends to a symmetric monoidal functor

(NY) - (&)o - (1"P)o x.F. Io; such enrichments are constructed in [51, but alternatively, if I comes from is a symmetric monoidal 1-category, these objects are easily constructed as 1-categories, and all of the examples in the present work will be of this form.

Now let F : I -+ Sp be a functor. Of course, F factors essentially uniquely as

I " Fun(IoP, Sp) + Sp where F preserves colimits. We observe that F is homotopic to the composite

o- : Fun(I"P, Sp) +Miun(IO, Sp) x Fun(I, Sp)

-+ Fun(IP x I, Sp x Sp)

-+ Fun((9, Sp) Colim * Sp. since a preserves colimits, and by the behavior of coends of representable functors, restricts to F on I.

Suppose that F extends to a symmetric monoidal functor FO : I* -+ Sp^. Then F

extends essentially uniquely to a symmetric monoidal functor F® : Fun(IOP, Sp)* -+

SpA compatible with FO. Thus coends against a symmetric monoidal functor F

behave like Fourier transforms: they interchange the Day convolution and the tensor

product.

Now we'll specialize to the case where I is the category F of finite sets. We'd like

to give a symmetric monoidal enrichment of Proposition 3.3.1, which we'll build by

composing several functors.

48 First, by [17, Corollary 2.4.3.10], we have an equivalence

a : Fun*((T"P), Top') -+ Top.

Let Alg_,0 ,(Top) be the category of lax symmetric monoidal functors from (TFP)u to TopX. We have an obvious full and faithful inclusion

I : Fun ((*P)l lTopx) -+ Algy0,(Top).

We claim that I is colimit-preserving; indeed, it suffices to show that I preserves sifted colimits and coproducts. Since sifted colimits of both symmetric monoidal and lax monoidal functors are computed in the target category, we know that 1 preserves sifted colimits. Now by

Proposition 2.1.10, we can identify Alg_ 0,(Sp) with the category

CAlg(Fun(Y*P, Sp))

of commutative algebras in the Day convolution symmetric monoidal category Fun(Y0 P, Sp)®.

Thus the coproduct in AlgOp (Sp) is given by Day convolution, and the result follows

from Lemma 2.1.11.

Next, composition with the symmetric monoidal functor (E')* induces a functor

s : Algyp (Top) -+ Algop (Sp)

We claim that s is colimit-preserving; once again, it suffices to show that s preserves

sifted colimits and coproducts. But s preserves sifted colimits because these are

again computed objectwise in the target, and it preserves coproducts by Lemma 2.2.3

because these are given by Day convolution.

Finally, the functor F0 induces a functor on commutative algebras, which we

abusively denote

F* : AlgrOp(Sp) -> CAlg(Sp).

49 FO preserves colimits: it preserves sifted colimits, once again, because these are evaluated in Sp, and it preserves coproducts because it is symmetric monoidal.

Consider the composite

F : o s o o Top -+ CAlg(Sp).

Fc is given by the formula Fo(X) f x A F

and it extends uniquely to a symmetric monoidal functor from Top to CAlg(Sp)^, since the symmetric monoidal structure is given by the coproduct on each side.

Observe that any F is equivalent as a symmetric monoidal functor to AR for some commutative ring spectrum R. By the characterization of colimit-preserving functors

from Top into a presentable oo-category, we have proven the following:

Theorem 3.4.1. There is an equivalence of symmetric monoidal functors Top -+ CAlg(Sp) SC(_) A AR ~ ft® (-).

Corollary 3.4.2. The Hodge filtration on ROX is compatible with the multiplication

in the sense that, in the diagram

(R &X)!n A (R 0X) -:m.....>(R&X I I (R®X)A(R®X) > RDX,

the dotted arrow exists functorially in R.

Proof. By Theorem 3.4.1, the diagram of solid arrows arises as the coend of AR

against the diagram

C"* Ci'Ci~mn

cx * Cx ~-> Cxux -> Cx,

50 where * denotes Day convolution. But by definition, CIfl+" is final among objects of

Fun( 7 P, Sp) mapping to Cx whose restriction to (JTP)

3.5 The layers of the filtration and Adams operations

For the remainder of this paper, we'll stick with X = S'. In this case we'll be able to give a description of the layers of our filtration, and as a side effect elucidate its compatibility with Adams operations.

Since Cj" and (a fortiori) C"1 are right Kan extended from T"'"P, so is the

fiber Cgin. Let us determine the homology, after restriction to " of C;,,:

Proposition 3.5.1. We have

= n HZ,C;"(U) {EZ IUI 0 IUI < n

with E., acting by sign on E"Z.

Proof. We need more notation. Let To CI"T([n]), the suspension spectrum of an

n-torus, and let T := ClsI"([n]), which is the suspension spectrum of a certain

(n - 1)-skeleton of an n-torus, as described in Proposition 3.3.6. Clearly the projection

z SC1 -+ C I"-1

is an equivalence when evaluated on sets of cardinality less than n, so it suffices to

determine the fiber of z on .F('),oP. Denote this fiber L; certainly H,(L) is Z with the

sign action of En, since this is Hri(TO). We aim to show that Hi(L) = 0 for i < n.

51 For each i < n, we have to show that the map

zi : Hi(To) - H,(T1 ) is an isomorphism. Observe that both groups are free of rank (4). For every injective map g : [i] --+ [n], we get projections

T0 , T - (S) which on Hi, by Proposition 3.3.6, induce the projection onto the Z-factor corresponding to g. The diagram

TO T

V E(sl)U UC[n],IUI=i homotopy commutes by functoriality and on taking the ith homology we get the commutative diagram Hi (TO) Z Hi (T1)

V Z. UC[n],IUI=i

The vertical and diagonal maps are isomorphisms, so that zi is an isomorphism. This completes the proof. El

Continuing the notation of Proposition 3.5.1, we'll write L for the restriction of

C-' to F"P, .

Lemma 3.5.2. The homology of L determines it up to equivalence.

Proof. Let

L' : Fah n r e w an+Sp be another functor together with an isomorphism between HZ A L' and HZ A L. We

52 claim that there is an equivalence

HZ A Nat(L', L) -+ Nat(HZ A L', HZ A L).

Indeed, by Proposition 3.1.3, we have maps

HZ A Nat(L', L) ~ HZ A Hom(L'(-), L(-))

~o, HZ A Hom(L'(-), L(-))

- Hom(HZ A L'(-), HZ A L(-)) NTp,

Since L' and L take values only in shifted spheres and contractible spectra, the map

HZ A Hom(L'(S), L(T)) -+ Hom(HZ A L'(S), HZ A L(T)) is an equvalence for all S and T, so all maps in the chain are in fact equivalences.

That means we have a class in HZoNat(L', L) corresponding to the given isomorphism between HZ A L' and HZ A L. Any natural transformation in the corresponding connected component induces an isomorphism on integral homology, and so is an equivalence.

This makes it easy to write down a point-set model for L, if one is so inclined.

We also get the following result, which will be useful in just a minute:

Corollary 3.5.3. Any natural endomorphism of L which acts by multiplication by an integer s on homology is equivalent to the multiplication-by-s endomorphism on L.

Proof. The proof is the same as that of Lemma 3.5.2. El

On to Adams operations. The formula

THH(R) = R 9 S1

53 immediately suggests a family of potentially interesting operations

{, r E Z}

given by r = id 0 c-r: R 0 S' -+ R 0 S'

where c. is a degree r endomorphism of S'. These are called the Adams operations,

and one can refer to [1, 11] for a detailed account of their properties and history.

From our coend formula point of view, f arises from the natural transformation

[r] : Csi -4 Csi

acting by multiplication by r on each torus. Note that on the top homology of an

n-torus, r acts by multiplication by rn. By naturality of Kan extensions, [r] descends

to a natural transformation

[5n : Cji"5 n i

and thus a natural operation

(/f )" :THHE" -+ THH:?"

Moreover, by naturality of the formation of homotopy fibers, we have a homotopy

commutative diagram

-4n 05 0~n1

C,r ]: >

C;- > -C÷ C[r n1 The induced natural transformation [r]L : L -+ L acts by multiplication by rr on

homology, and by Corollary 3.5.3, [r]L is multiplication by r". Since Cfi" is right Kan extended from L, we have [r] = r" : C-1" -+ C-*"

54 and so:

Proposition 3.5.4. Defining

(,Or)(n) : THH " a THH=" by naturality of homotopy fibers, we have

(0r)(n) = n.

Loday's Hodge filtration [14] is defined as the -y-filtration associated to a A-ring structure, and this kind of compatibility with the Adams operations is a prominent

characteristic of any -y-filtration [8, Proposition 3.1]. It should thus be a prerequisite

for any putative Hodge filtration on THH.

55 56 Appendix A

Commutative algebras and their modules

The purpose of this appendix is to develop a combinatorial theory of modules over commutative algebras in oo-categories in the vein of the theory of modules over asso- ciative algebras set out in [17, 4.2, 4.3]. In fact, we'll describe a commutative analog CM® of the operads LM' and BM* of [17, 4.2.1] and [17, 4.3.11. This should ease the task of constructing and manipulating such modules. In particular, we prove

Theorem 3.2.3 from above, which is tautological for 1-categories but more subtle for oo-categories. Here it is restated for convenience:

Theorem A.1. Let C* be a symmetric monoidal oo-category. A datum comprising a commutative algebra E in C and a module M over it - that is, an object of Mod* (C), the underlying oo-category of Lurie's oo-operad Mod-* (C)® [17, Definition 3.3.3.81

- gives rise functorially to a functor

AE,M I* - C such that

AE,M(S) ~ EOss 0 M.

We'll prove Theorem 3.2.3 by first describing the oo-operad CM® that para- metrizes this data and then giving a map from F, to the symmetric monoidal envelope

57 of CM®. We'll work extensively with the model category of oo-preoperads 117, 2.1.41, which we'll denote PO.

Definition A.2. We define the operad CM® as (the nerve of) the 1-category in which

" an object is a pair (S, U) consisting of an object S of F. and a subset U of S';

" a morphism from (S, U) to (T, V) is a morphism f : S -+ T in F such that for each v c V, the set U n f- 1 (v) has cardinality exactly 1.

It's easily checked that the functor CM® -+ F, that maps (S, U) to S makes CM*

into an oo-operad. The following result isolates the hard work involved in proving Theorem 3.2.3:

Proposition A.3. We give (F)(i)/ the structure of an oo-preoperad by letting the

target map t: (T*)(i)/ - F,, create marked edges. Define a map of oo-preoperads

# : (F,)(1)/ -+ CM®

1 fJ()Es O(j: (1) -+ S) = S (S,0) otherwise.

Thus we embed (F*)(I)/ as the full subcategory of CM® spanned by those objects

(S, U) for which the cardinality of U is at most 1. Then # is an trivial cofibration in PO.

The proof will take the form of a series of lemmas.

Lemma A.4. Suppose q : E -> B is an inner fibration of oo-categories, K a simplicial

set and r : K' -+ B a map such that for each edge e : k, -+ k 2 of K and for each

I E E with q(1) = r(ki), there is a cocartesian lift of e to E with source 1. Denote the cone point of K' by c and suppose d E E is such that q(d) = r(c). Then there is a

map r' : K' -+ E lifting r and taking every edge of K to a cocartesian edge of E.

58 Proof. Clearly we can lift in such a way that the image of every edge of K with source c is cocartesian; let r' be such a lift. We claim that r' already has the desired property.

Indeed, r' can be viewed as a section of the cocartesian fibration qK< : E x BK -+ K<, and then the result follows from [15, Proposition 2.4.2.71. l

Lemma A.5. Let p : 0* -+ F, be any oo-operad. For any set T, let PT be the poset of subsets of T ordered by reverse inclusion, and let PT = PT \ {T}, so that

PT 2 (PT)<. For each S E TF, let rs : Pso -+ TT denote the obvious diagram of inert morphisms. Let X E 00, let p : Ps - 00 be the cocartesian lift of rs with p(S) = X whose existence is guaranteed by Lemma A.4, and suppose 1S0 1 > 1. Then p is a limit diagram relative to p.

Proof. We work by induction on the size of SO; the case IS01 - 2 is an immediate consequence of the oo-operad axioms. For each k E N, let p k denote the poset of subsets of T of cardinality at most k. Then the restriction of p to (PS: )' is a p-limit diagram by the oc-operad axioms. We now argue by induction on k that for each

k with 1 < k < IS'l - 1, the restriction of p to (PsDok)' is a limit diagram. Indeed, for each such k > 1, the 1S01 = k edition of the lemma implies that pji, k is p-right so Kan extended from pjpk-1. Comparing the p-right Kan extensions along both paths across the commutative diagram

gives the induction step, and thence the result. l

Corollary A.6. Retaining the notation of Lemma A.5: any nontrivial subcube of

p is a p-limit diagram. That is, if U1 and U2 are two subsets of S' with U1 g U2

and U2 \ UIl > 1, then the restriction of p to the subposet PU1,J 2 spanned by those 0 subsets V of S with U1 g V C U2 is a p-limit diagram.

59 Proof. By restricting to Pu9 , we may assume U2 = So. Let

pf,2 =:U1 pu2 \ U2 and let Q be the closure of PG,, under downward inclusion. Then Q contains Ps, so by the above discussion, p is p-right Kan extended from Q. But PU1 p 2 is coinitial in Q = Qu2 , so we're good. El

Proof of Proposition A.3. Now let 0* be any oo-operad, and let F : (T,)(l)/ -+ 0® be a morphism of oc-preoperads. Consider the diagram

CM" F

We claim that the 7k-relative right Kan extension O*F along the dotted line exists [15, 4.3.2]. Indeed, let (S, U) be an object of CM®, and let Q(su) be the subposet of Pso spanned by subsets T such that

9 SO \ U C T, and

* IT n UI ; 1.

Then the natural map j(su) : Qsu) CM® which takes all edges of Q"sp) to inert edges of CM® gives rise to a map

k(su) : Q(SU) - (F*)(1)/ XCMo (CM*)(su)/ and k(su) is easily observed to be coinitial. Thus it suffices to show that F o k(su) admits a p-limit. But F o k(su) can be embedded, up to equivalence, into the cube of inert edges

p : PS -+ C®

60 such that

PS = ]1 F((1), {1}) x 171 F((1), 0) U So\U where "H denotes a product relative to p. By Corollary A.6 together with the induction argument used in the proof of Lemma A.5, we see that Ps is a p-limit of Fok(su). So #,F exists [15, Lemma 4.3.2.13], and it is clear that #,F is a morphism of oo-operads. Since any morphism of preoperads from (T.)(l)/ to an oo-operad extends over CM®, # must be a trivial cofibration. l

Now we'll relate our construction to Lurie's category of modules. Let CO be an oo-operad. Employing the notation of [17, 3.3.3], we define

Mod(C) := ModF*(C)') with analogous definitions of Mod(C) and Mod(C).

Proposition A.7. There is an equivalence of oo-categories

Mod(C) ~ Fun*(CM®,C®).

Proof. Let X be a simplicial set equipped with the constant map 1x X -+ F with image (1). One then has set bijections

Hom(X, Mod(C)) ~ Homj,(X x (-4) , C*)

and

Hom(X, Mod(C)) ~ Hom*(X x (F*)(1)/, C*)

where Hom® denotes the set of oc-preoperad maps; this is to say that we have an

isomorphism of categories between Mod(C) and the category Fun®((F*)(l)/, C®) of oo-preoperad maps from (F*)(i)/ to C®. Moreover, when 0 = F*, the trivial Kan fibration 0 of [17, Lemma 3.3.3.31 is an isomorphism, so there is no difference between Mod(C) and Mod(C).

61 Finally, we claim that the restriction map

< * : Fun (CM®,C") -+ Fun*((F*)(,)/, CO) is a trivial Kan fibration. Noting that the categorical pattern 1o of [17, Lemma B.1.131 serves as a unit for the product of categorical patterns, we deduce from [17, Remark

B.2.5] that the product map

PO x Set+ -PO is a left Quillen bifunctor. This means, in particular, that for each n, the morphism of oo-preoperads

((-F*)1)x (A")') U)((A") (CM® x (OA")1) -+ CM® x (A")l

is a trivial cofibration in PO, which gives the result.

We now wish to characterize the symmetric monoidal envelope of CM'.

Definition A.8. Let T+ be the category whose objects are pairs (S, U), with S a

finite set and U C S a subset, and in which a morphism from (S, U) -+ (T, V) is a morphism f : S -+ T of finite sets inducing a bijection flu : U ' V. The disjoint union (which, mind you, is definitely not a coproduct) makes F+ into a symmetric

monoidal category (.F+)".

Proposition A.9. The symmetric monoidal envelope Env(CM*) is isomorphic to

(-F+)"-

Proof. We briefly sketch the proof, which is a routine 1-categorical exercise. By

definition, the symmetric monoidal envelope of Env(CM®) has objects

(S, U, f : So -+ T)

62 where T e F. Our isomorphism will map this object to the object

(T, (f~-1(t), f -1 (t) n U)teT)

of (T+) L El

Corollary A.1M. A CM®-algebraparametrizing a commutative monoid E and a module M in C* gives rise to a functor AE,M hT* -+ C such that

AE,M(S) ~ Eos" 0 M.

Proof. We construct AE,M by embedding T, as the full subcategory of F+ consisting of objects (S, U) for which IUI = 1. El

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