A spectrum-level Hodge filtration on topological Hochschild homology

Abstract

We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum R, and more generally the factorization homology $R \otimes X$ for any space X, echoing algebraic constructions of Loday and Pirashvili. We give a geometric description of this filtration, investigate its multiplicative properties, and show that it breaks ${\text {THH}}$ up into common eigenspectra of the Adams operations.

Appendix: Commutative algebras and their modules

The purpose of this appendix is to develop a combinatorial theory of modules over commutative algebras in $\infty$-categories in the vein of the theory of modules over associative algebras set out in [16, §4.2, 4.3]. In fact, we’ll describe a commutative analogue $\mathcal {CM}^\otimes$ of the operads $\mathcal {LM}^\otimes$ and $\mathcal {BM}^\otimes$ of [16, §4.2.1] and [16, §4.3.1]. This should ease the task of constructing and manipulating such modules, and is thus likely to be of more general utility.

In particular, we prove Theorem 3.3 from above, which is tautological for 1-categories but more subtle for $\infty$-categories. Here it is restated for convenience. Recall that for a finite pointed set S, $S^o$ denotes the set of non-basepoint elements of S.

Theorem 7.1

Let $\mathbf {C}^\otimes$ be a symmetric monoidal $\infty$-category. A datum comprising a commutative algebra E in $\mathbf {C}$ and a module M over it—that is, an object of $\mathbf {Mod}^{\mathcal {F}_*}(\mathbf {C})$, the underlying $\infty$-category of Lurie’s $\infty$-operad $\mathbf {Mod}^{\mathcal {F}_*}(\mathbf {C})^\otimes$ [16, Definition 3.3.3.8]—gives rise functorially to a functor

$$\begin{aligned} A_{E, M} : \mathcal {F}_* \rightarrow \mathbf {C} \end{aligned}$$

such that

$$\begin{aligned} A_{E, M}(S) \simeq E^{\otimes S^o} \otimes M. \end{aligned}$$

We’ll prove Theorem 3.3 by first describing the $\infty$-operad $\mathcal {CM}^\otimes$ that parametrizes this data and then giving a map from $\mathcal {F}_*$ to the symmetric monoidal envelope of $\mathcal {CM}^\otimes$. We’ll work extensively with the model category of $\infty$-preoperads [16, §2.1.4], which we’ll denote $\mathcal {PO}$.

Definition 7.2

We define the operad $\mathcal {CM}^\otimes$ as (the nerve of) the 1-category in which

• an object is a pair (S, U) consisting of an object S of $\mathcal {F}_*$ and a subset U of $S^o$;

• a morphism from (S, U) to (T, V) is a morphism $f: S \rightarrow T$ in $\mathcal {F}_*$ such that $f(U) \subseteq V_+$ and for each $v \in V$, the set $U \cap f^{-1}(v)$ has cardinality exactly 1. In other words, f restricts to an inert morphism $U_+ \rightarrow V_+$.

It’s easily checked that the functor $\mathcal {CM}^\otimes \rightarrow \mathcal {F}_*$ that maps (S, U) to S makes $\mathcal {CM}^\otimes$ into an $\infty$-operad. The following result isolates the hard work involved in proving Theorem 3.3:

Proposition 7.3

We give $(\mathcal {F}_*)_{\langle 1 \rangle /}$ the structure of an $\infty$-preoperad by letting the target map $t: (\mathcal {F}_*)_{\langle 1 \rangle /}\rightarrow \mathcal {F}_*$ create marked edges. Define a map of $\infty$-preoperads

$$\begin{aligned} \phi : (\mathcal {F}_*)_{\langle 1 \rangle /}\rightarrow \mathcal {CM}^\otimes \end{aligned}$$

by

$$\begin{aligned} \phi (j : \langle 1 \rangle \rightarrow S) = {\left\{ \begin{array}{ll} (S, \{j(1)\}) &{} \text {if }j(1) \in S^o \\ (S, \emptyset ) &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

so that $(\mathcal {F}_*)_{\langle 1 \rangle /}$ is embedded as the full subcategory of $\mathcal {CM}^\otimes$ spanned by those objects (S, U) for which the cardinality of U is at most 1. Then $\phi$ is a trivial cofibration in $\mathcal {PO}$.

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

Lemma 7.4

Suppose $q : \mathbf {E} \rightarrow \mathbf {B}$ is an inner fibration of $\infty$-categories, K a simplicial set and $r : K^\lhd \rightarrow \mathbf {B}$ a map such that for each edge $e : k_1 \rightarrow k_2$ of K and for each $l \in \mathbf {E}$ with $q(l) = r(k_1)$, there is a cocartesian lift of e to $\mathbf {E}$ with source l. Denote the cone point of $K^\lhd$ by c and suppose $d \in \mathbf {E}$ is such that $q(d) = r(c)$. Then there is a map $r^{\prime } : K^\lhd \rightarrow \mathbf {E}$ lifting r and taking every edge of $K^\lhd$ to a cocartesian edge of $\mathbf {E}$.

Proof

Clearly we can lift in such a way that the image of every edge of $K^\lhd$ with source c is cocartesian; let $r^{\prime }$ be such a lift. We claim that $r^{\prime }$ already has the desired property. Indeed, $r^{\prime }$ can be viewed as a section of the cocartesian fibration $q_{K^\lhd } : \mathbf {E} \times _{\mathbf {B}} K^\lhd \rightarrow K^\lhd$, and then the result follows from [14, Proposition 2.4.2.7].

Lemma 7.5

Let $p: \mathcal {O}^\otimes \rightarrow \mathcal {F}_*$ be any $\infty$-operad. For any set T, let $\mathcal {P}_T$ be the poset of subsets of T ordered by reverse inclusion, and let $\mathcal {P}^{\prime }_T = \mathcal {P}_T{\setminus }\{T\}$, so that $\mathcal {P}_T \cong (\mathcal {P}^{\prime }_T)^\lhd$. For each $S \in \mathcal {F}_*$, let $r_S : \mathcal {P}_{S^o} \rightarrow \mathcal {F}_*$ denote the obvious diagram of inert morphisms. Let $X \in \mathcal {O}^\otimes _S$, let $\rho : \mathcal {P}_{S^o} \rightarrow \mathcal {O}^\otimes$ be the cocartesian lift of $r_S$ with $\rho (S^o) = X$ whose existence is guaranteed by Lemma 7.4, and suppose $|S^o| > 1$. Then $\rho$ is a limit diagram relative to p.

Proof

We work by induction on the size of $S^o$; the case $|S^o| = 2$ is an immediate consequence of the $\infty$-operad axioms. For each $k \in \mathbb {N}$, let $\mathcal {P}^{\le k}_T$ denote the poset of subsets of T of cardinality at most k. Then the restriction of $\rho$ to $(\mathcal {P}^{\le 1}_{S^o})^\lhd$ is a p-limit diagram by the $\infty$-operad axioms. We now argue by induction on k that for each k with $1 \le k \le |S^o|-1$, the restriction of $\rho$ to $(\mathcal {P}^{\le k}_{S^o})^\lhd$ is a limit diagram. Indeed, for each such $k > 1$, the $|S^o| = k$ edition of the lemma implies that $\rho |_{\mathcal {P}^{\le k}_{S^o}}$ is p-right Kan extended from $\rho |_{\mathcal {P}^{\le k - 1}_{S^o}}$, since for any $U \subseteq S$ with $|U^o| = k$, the inclusion

$$\begin{aligned} \mathcal {P}^{\prime }_{U^o} = \mathcal {P}_{U^o}^{\le k - 1} \hookrightarrow (\mathcal {P}_{S^o}^{\le k - 1})_{U/} \end{aligned}$$

is an isomorphism. Comparing the p-right Kan extensions along both paths across the commutative diagram

gives the induction step, and thence the result.

Corollary 7.6

Retaining the notation of Lemma 7.5: any nontrivial subcube of $\rho$ is a p-limit diagram. That is, if $U_1$ and $U_2$ are two subsets of $S^o$ with $U_1 \subseteq U_2$ and $|U_2{\setminus } U_1| > 1$, then the restriction of $\rho$ to the subposet $\mathcal {P}_{U_1, U_2}$ spanned by those subsets V of $S^o$ with $U_1 \subseteq V \subseteq U_2$ is a p-limit diagram.

Proof

By restricting to $\mathcal {P}_{U_2}$, we may assume $U_2 = S^o$. Let

$$\begin{aligned} \mathcal {P}^{\prime }_{U_1, U_2} = \mathcal {P}_{U_1, U_2}{\setminus }U_2 \end{aligned}$$

and let $\mathcal {Q}$ be the closure of $\mathcal {P}^{\prime }_{U_1, U_2}$ under downward inclusion. Then $\mathcal {Q}$ contains $\mathcal {P}_{S^o}^{\le 1}$, so by the above discussion, $\rho$ is p-right Kan extended from $\mathcal {Q}$. But $\mathcal {P}^{\prime }_{U_1, U_2}$ is coinitial in $\mathcal {Q} = \mathcal {Q}_{U_2/}$, so we’re good.

Proof of Proposition 7.3

Now let $\mathcal {O}^\otimes$ be any $\infty$-operad, and let $F : (\mathcal {F}_*)_{\langle 1 \rangle /}\rightarrow \mathcal {O}^\otimes$ be a morphism of $\infty$-preoperads. Consider the diagram

We claim that the $\mathcal {F}_*$-relative right Kan extension $\phi _* F$ along the dotted line exists [14, §4.3.2]. Indeed, let (S, U) be an object of $\mathcal {CM}^\otimes$, and let $\mathcal {Q}_{(S, U)}$ be the subposet of $\mathcal {P}_{S^o}$ spanned by subsets T such that

• $S^o{\setminus }U \subseteq T$, and

• $|T \cap U| \le 1$.

Then the natural map

$$\begin{aligned} j_{(S, U)} : \mathcal {Q}_{(S, U)}^\lhd \rightarrow \mathcal {CM}^\otimes \end{aligned}$$

which takes all edges of $\mathcal {Q}_{(S, U)}^\lhd$ to inert edges of $\mathcal {CM}^\otimes$ gives rise to a map

$$\begin{aligned} k_{(S, U)} : \mathcal {Q}_{(S, U)} \rightarrow (\mathcal {F}_*)_{\langle 1 \rangle /}\times _{\mathcal {CM}^\otimes } (\mathcal {CM}^\otimes )_{(S, U)/} \end{aligned}$$

and $k_{(S, U)}$ is easily observed to be coinitial. Thus it suffices to show that $F \circ k_{(S, U)}$ admits a p-limit. But $F \circ k_{(S, U)}$ can be embedded, up to equivalence, into the cube of inert edges

$$\begin{aligned} \rho : \mathcal {P}_{S^o} \rightarrow \mathcal {O}^\otimes \end{aligned}$$

such that

$$\begin{aligned} \rho _{S^o} = {^p}{\prod _{U} F(\langle 1 \rangle , \{1\})} \times {^p}{\prod _{S^o{\setminus }U} F(\langle 1 \rangle , \emptyset )} \end{aligned}$$

where ${^p}{\prod }$ denotes a product relative to p. By Corollary 7.6 together with the induction argument used in the proof of Lemma 7.5, we see that $\rho _{S^o}$ is a p-limit of $F \circ k_{(S, U)}$. So $\phi _* F$ exists [14, Lemma 4.3.2.13], and it is clear that $\phi _* F$ is a morphism of $\infty$-operads. Since any morphism of preoperads from $(\mathcal {F}_*)_{\langle 1 \rangle /}$ to an $\infty$-operad extends over $\mathcal {CM}^\otimes$, $\phi$ must be a trivial cofibration. $\square$

Now we’ll relate our construction to Lurie’s category of modules. Let $\mathbf {C}^\otimes$ be an $\infty$-operad. Employing the notation of [16, §3.3.3], we define

$$\begin{aligned} \mathbf {Mod}(\mathbf {C}) := \mathbf {Mod}^{\mathcal {F}_*}(\mathbf {C})^\otimes _{\langle 1 \rangle } \end{aligned}$$

with analogous definitions of $\widetilde{\mathbf {Mod}}(\mathbf {C})$ and $\overline{\mathbf {Mod}}(\mathbf {C})$.

Proposition 7.7

There is an equivalence of $\infty$-categories

$$\begin{aligned} \mathbf {Mod}(\mathbf {C}) \simeq {\text {Fun}}^\otimes (\mathcal {CM}^\otimes , \mathbf {C}^\otimes ). \end{aligned}$$

Proof

Let X be a simplicial set equipped with the constant map $1_X : X \rightarrow \mathcal {F}_*$ with image $\langle 1 \rangle$. One then has set bijections

$$\begin{aligned} {\text {Hom}}(X, \widetilde{\mathbf {Mod}}(\mathbf {C})) \cong {\text {Hom}}_{\mathcal {F}_*}(X \times (\mathcal {F}_*)_{\langle 1 \rangle /}, \mathbf {C}^\otimes ) \end{aligned}$$

and

$$\begin{aligned} {\text {Hom}}(X, \overline{\mathbf {Mod}}(\mathbf {C})) \cong {\text {Hom}}^\otimes (X \times (\mathcal {F}_*)_{\langle 1 \rangle /}, \mathbf {C}^\otimes ) \end{aligned}$$

where ${\text {Hom}}^\otimes$ denotes the set of $\infty$-preoperad maps; this is to say that we have an isomorphism of categories between $\overline{\mathbf {Mod}}(\mathbf {C})$ and the category ${\text {Fun}}^\otimes ((\mathcal {F}_*)_{\langle 1 \rangle /}, \mathbf {C}^\otimes )$ of $\infty$-preoperad maps from $(\mathcal {F}_*)_{\langle 1 \rangle /}$ to $\mathbf {C}^\otimes$. Moreover, when $\mathcal {O}^\otimes = \mathcal {F}_*$, the trivial Kan fibration $\theta$ of [16, Lemma 3.3.3.3] is an isomorphism, so there is no difference between $\overline{\mathbf {Mod}}(\mathbf {C})$ and $\mathbf {Mod}(\mathbf {C})$.

Finally, we claim that the restriction map

$$\begin{aligned} \phi ^* : {\text {Fun}}^\otimes (\mathcal {CM}^\otimes , \mathbf {C}^\otimes ) \rightarrow {\text {Fun}}^\otimes ((\mathcal {F}_*)_{\langle 1 \rangle /}, \mathbf {C}^\otimes ) \end{aligned}$$

is a trivial Kan fibration. Noting that the categorical pattern $\mathfrak {P}_0$ of [16, Lemma B.1.13] serves as a unit for the product of categorical patterns, we deduce from [16, Remark B.2.5] that the product map

$$\begin{aligned} \mathcal {PO} \times {\text {Set}}_\Delta ^+ \rightarrow \mathcal {PO} \end{aligned}$$

is a left Quillen bifunctor. This means, in particular, that for each n, the morphism of $\infty$-preoperads

$$\begin{aligned} ((\mathcal {F}_*)_{\langle 1 \rangle /}\times (\Delta ^n)^\flat ) \cup _{(\mathcal {F}_*)_{\langle 1 \rangle /}\times (\partial \Delta ^n)^\flat } (\mathcal {CM}^\otimes \times (\partial \Delta ^n)^\flat ) \rightarrow \mathcal {CM}^\otimes \times (\Delta ^n)^\flat \end{aligned}$$

is a trivial cofibration in $\mathcal {PO}$, which gives the result.

We now wish to characterize the symmetric monoidal envelope of $\mathcal {CM}^\otimes$.

Definition 7.8

Let $\mathcal {F}^+$ be the category whose objects are pairs (S, U), with S a finite set and $U \subseteq S$ a subset, and in which a morphism from $(S, U) \rightarrow (T, V)$ is a morphism $f : S \rightarrow T$ of finite sets inducing a bijection $f|_U : U \cong V$. The disjoint union (which, mind you, is definitely not a coproduct) makes $\mathcal {F}^+$ into a symmetric monoidal category $(\mathcal {F}^+)^\amalg$.

Proposition 7.9

The symmetric monoidal envelope $\text {Env}(\mathcal {CM}^\otimes )$ is isomorphic to $(\mathcal {F}^+)^\amalg$.

Proof

We briefly sketch the proof, which is a routine 1-categorical exercise. By definition, the symmetric monoidal envelope of $\text {Env}(\mathcal {CM}^\otimes )$ has objects

$$\begin{aligned} (S, U, f: S^o \rightarrow T) \end{aligned}$$

where $T \in \mathcal {F}$. Our isomorphism will map this object to the object

$$\begin{aligned} (T, (f^{-1}(t), f^{-1}(t) \cap U)_{t \in T}) \end{aligned}$$

of $(\mathcal {F}^+)^\amalg$.

Corollary 7.10

A $\mathcal {CM}^\otimes$-algebra parametrizing a commutative monoid E and a module M in $\mathbf {C}^\otimes$ gives rise to a functor $A_{E, M} : \mathcal {F}_* \rightarrow \mathbf {C}$ such that

$$\begin{aligned} A_{E, M}(S) \simeq E^{\otimes S^o} \otimes M. \end{aligned}$$

Proof

We construct $A_{E, M}$ by embedding $\mathcal {F}_*$ as the full subcategory of $\mathcal {F}^+$ consisting of objects (S, U) for which $|U| = 1$.