Day convolution and the Hodge filtration on THH
by
Saul Glasman
V--
z
Doctor of Philosophy in Mathematics
<~
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"
Accepted by.
........
I....................
.
.
Clark Barwick
Cecil and Ida B. Green Career Development Associate Professor
Thesis Supervisor
Certified by.... e .........
Signature redacted....................
Chairma
=
QLL
at the
William Minicozzi
Depart ent Committee on Graduate Theses
CD
-
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
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 occategories 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 enormous 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
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
9
A spectrum-level Hodge filtration on topological Hochschild homol31
ogy
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
7
.
.
.
.
3.1
57
8
Chapter 1
Introduction
In this thesis, we seek to begin the development of a theory of functorial, spectrumlevel 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))
where the complex (Q*(R))
homotopy invariant of
=R
" is the "stupid truncation" of Q*(R), which is not a
*(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
HIn
H,(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 generalization 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
along Oc : C x C
-
D
->
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 highercategorical 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 -
R0 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 TAQfiltration 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 00category
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 sym-
metric 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
Suppose k : K
-+
over S.
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.
Fun(C, D)O) is given by a morphism
f
:S
-+
Similarly, an edge of
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">
I
Fun(C, D)0
I
I;
An
ki
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
ft
F03> Do
jS
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'
with "The condition of Proposition 4.3.1.10 is vacuously satisfied for
7',
to X"
since there
are no nonidentity edges with source {1} in A 1 ".
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
I
A n-1
>Funs
If f
(CO, DOj)
: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 composition 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
=
9
1T
by [15, Lemma 2.4.2.71, it suffices to prove that
T'
such that
p(T') = p(T),
0 2T' =
f and
f
and
Dor
locally cocartesian;
is locally cocartesian. Choose
all of the edges of r' are locally cocartesian.
Since they are both locally cocartesian and have the same source,
to OT', and so
9
T
is equivalent to
T'.
In particular,
0
ir
r is equivalent
0T
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
Ff co
FT -
to C*. Thus Ff and F. together give a map Al
a map
T :
D)*,
which extends to
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
&2T
Fun(C,
-
is locally cocartesian. Since
T
0OT
and
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
cocartesian, by [15, Proposition 4.3.2.81. But since
02 T
&iT
is locally
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*
-4
D
induced by F3. Clearly replacing (Fj) 1z with
(F )/z : (Co)/z
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
is cofinal.
-
(CO)/z
An object of (CI)/z not in the image of Iz is a morphism q : X -+ Z
over gf in C*. But if X
defined by X
-+
-+
Y is a cocartesian edge of C* over
f,
then the 2-simplex
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
El
the proof.
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
a morphism in F,. Since the pushforward associated to
f
f
: S
-
T is
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'
induced by pushforward along
-
Co
f.
Let (F,),cso be an S-tuple of functors from C to D, and let F4 denote the composite
Then F decomposes as a product of maps
Fft
S : Cof'(t)
1
19
Doe
For each t, let F, be a p-left Kan extension of Fet along C
Ff = 1
f(t)
C't.
f,t
Then
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
associated to the inert map (n)
onto the
-+
: A()
-+ X(1)
(1) that picks out
j
is equivalent to projection
jth factor.
f
We'll retain the notation of the previous proof, but now assume that
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
the projection Fun(C, D)s -+ Fun(C, D)T.
f
is indeed
El
Proposition 2.1.10. A commutative monoid in Fun(C, D)® is exactly a lax symmetric 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)
l
which proves the result.
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.
simplicial set over (,)O
We regard (A 1 )O x (F,)
as a marked
via the composite
AyX (_F,),
.Lxd
A.~~-
(T,), X
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 q 3'cmm-anodyne inclusion
{(2)}
>(')I
'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
kEK
-+
colim
2
(YZ)EC
colim
2
(Y,Z)EC
XCC/X
(F(Y) 0 4(k)(Z))
XCC/X
(F(Y) 0 colim 4(k)(Z)).
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:
B,
OB
-+
XB
OB
and form the pullback
X
via the source map. Then there is a functor
(-),
which maps the object (x,
f)
X
XB OB -+ X
to f~x and makes the diagram
X
X XB OB -.
B
commute.
Proof. Let 0'
let p : Oc
-+
be the full subcategory of Ox spanned by the cocartesian arrows and
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,
which we recall here for completeness: the square
xI
I
-An
X
25
XI
OB
3.1.2J,
determines the same lifting problem as the square
[(A")l
X(A')'] H
(An)l
[(A)b X f{0}]
X
X
>
X
BO
(AIl>
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
0
X xB
X
B
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,
be the functor that takes a
-+ Cat,
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
X' XB
{b}
MapB(-,-)
XB
t-
X
,-
> Top
Map(-,-)
X P X Xb
homotopy commutes, where the vertical equivalence comes from the given identification of X' with X"". Moreover, using these identifications, for each morphism
f
b
-+
x Xb,
b' E B and for each (y, x) E
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(-, -)
-
classified by the left fibration M: ((X/B)
Observe that Lurie's category
f1
-+
X'
be a functor
El
X of [5, 51.
XB
[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
Composing this with Map,(-, -) : C® xy (COP)
C*
®:Xy
(COP)*
X,
FX
(C"P)*.
-
Top gives a functor
-*
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.
(Zs)seso
-+
(Wt)tETO
Let
:
be a cocartesian morphism of C® lying over the morphism
27
f
: S -+ T of F., so that
Wi ~
jef- 1 (i)
Zi,
and let
Y : (C P)'
be the induced morphism. For each
-+
(X)tETo c (COP)',
K = (C 0 P))
and let p: K'
Top'
X>(F)g
let
{f}
-+ (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
colim
X-+P1OP20 -OPn.
nl
]I Map(P, Zj) -- Map(X, (
%=
I
i=
Zi)
I
to be an equivalence for all n > 0, X E COP and (Zi)<<n E Cn.
Define a category Dn by the pullback square
Dn
t~
CX/
)
(C",)/(zi,-..,,)
14nOs
fI
C.
Then the functor on CI x c Cx/ taking X -+ P1
Kan extended from the constant functor D
28
.. -D
P, to
H
Map(P, Zj) is left
-- Top with image *, so we must show
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"
>A'
A
exhibits
[
B
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 A 1 , and denote
(Ao)/a := AO
XA
A/a.
Denote by ta the natural map from (Ao)/a to B, and let
p, : (Ao)/a
X
A1 -+
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.
30
l
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
xj[o,n]
x XI[n+1, 2 n+l].
7
1t : 09 -+ IOP x I that takes an n-simplex x of 05 to
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
F0 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
/, Hom(F(-), G(-))
~Nat(F, G)
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 conversation 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 theory: 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
I P x I
F
fop
J xD
CD
x
OP 9
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
DIP x D.
-0-
(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(-))
{id}
>
Fun(OI, S)
Fun(Or, 01)
-
-
>
Fun(0j, OD)
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
Tx
Fun(IP x I, D"P x D)
Fun(0I,
-+
34
OD)
Fun(Oi, D*P x D),
so that T is the category of commutative diagrams that look like
>O
0,
I*P x I
DOP x D.
-
We claim that there's a pullback square
T
Nat(F, G)
jInun(ID>
> Fun(IOP x 1, DOP x D).
{(FOP,G)}
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
set (call it U) of ways of extending (FOP, G) to a functor from (5 to
(
9
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)
where q-
A2k+1
=
x(qk x y)
_* 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>
A2k+1
35
A' X I
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
El
and G.
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.
Proposition 3.1.5. For any functor F : I
II
10 F
-
L
C,
colim F
where 1 is the constant functor at the unit of C.
Proof. We have
J
F~ colim 1 F(X)
X-4Ye0
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 + 2simplices 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
a map of simplicial sets. Then the pullback functor
adjoint
j! form
j*
j
: S -+ S' is
: sSet/s' -+ sSet/s and its left
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 :
replacement of r :
Z
-+
X the fibrant
po q in sSet/x; in particular, we have a covariant weak equivalence
X.
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
Yx 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
s
is a finite set S together with an element
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,MWe 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
S 1 [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
J
(by Proposition 3.1.5)
S A X*AE
XiS AAE
(by Proposition 3.1.4)
where
XIS(S) ~
colim
[n]EAoPS-+X[n]
S
~ colim SVXin]S
[n]EAOP
X[n]S
S A colim
AOP
~S A XS
~
Xs
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)
X*,
J
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
43
C" A AE-
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
p
and
SY=
Cf+1 A AE
(E 0 X)
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 Loday'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
<n,
so we are reduced to showing that
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.
l
This implies that the diagram is strongly cocartesian.
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,<n, and so
the entire cube is right Kan extended from a,.I..
l
U
Corollary 3.3.8. C"(U) is a retract of C(U).
This gives a map q : a,(U) -+ XU. On the other hand, the projection , : Cx(U)
-
Proof. Clearly aIpu is strongly cartesian, and so Xu is the limit of a,0 p f..
. =aI,
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.
C%"(U) is (n
Then the projection 0 : Cx(U) -+
+ 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
c Z. Then the projection
an R-module which is k-connective for some k
(E
X; M) -+ (E ® X; M)<"
+ k)-connected, and so we have convergence:
is (n + 1
(E & X; M) ~lim (E 0 X; M)<n.
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(
+ k)-connected. This connectivity is preserved by taking the colimit.
is (n + 1
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 multi-
plicative (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) -+ Algy 0 ,(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)
49
->
CAlg(Sp).
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
oso
F :
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
50
->
Cx,
where * denotes Day convolution. But by definition, CIfl+" is final among objects of
Fun( 7 P, Sp) mapping to Cx whose restriction to
(JTP)<n
is zero. But C "* C*'"
also restricts to zero on this subcategory, so the diagram can be completed in an
l
essentially unique way.
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
HZ,C;"(U)
{EZ
0
IUI =
n
IUI < n
with E., acting by sign on E"Z.
CI"T([n]), the suspension spectrum of an
Proof. We need more notation. Let To
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
-
H,(T1
)
zi : Hi(To)
is an isomorphism. Observe that both groups are free of rank
map g :
[i]
(4). For every injective
--+ [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
V
T
E(sl)U
UC[n],IUI=i
homotopy commutes by functoriality and on taking the ith homology we get the
commutative diagram
Hi (TO)
V
Z
Hi (T1)
Z.
UC[n],IUI=i
The vertical and diagonal maps are isomorphisms, so that zi is an isomorphism. This
completes the proof.
El
C-' to F"P,
.
Continuing the notation of Proposition 3.5.1, we'll write L for the restriction of
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(-))
HZ A Hom(L'(-), L(-))
~o,
NTp,<n
Hom(HZ A L'(-), HZ A L(-))
~Nat (HZ A L',7 HZ A L).
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.
On to Adams operations. The formula
THH(R) = R 9 S1
53
El
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,
[r] : Csi
f
arises from the natural transformation
-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
0~n1
05
-4n
>
C,r]:
C;-
>
-C÷
C<i-1
" >[rn1
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 associative 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 parametrizes 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
each v
f
: S -+ T in F such that for
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®
by
O(j: (1) -+ S)
=
S
1
(S,0)
fJ()Es
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
set and r : K'
->
B is an inner fibration of oo-categories, K a simplicial
-+ 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'
map r' : K'
by c and suppose d E E is such that q(d) = r(c). Then there is a
-+ E lifting r and taking every edge of K
58
to a cocartesian edge of E.
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<,
l
and then the result follows from [15, Proposition 2.4.2.71.
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 -+
inert morphisms. Let X E 00, let p : Ps -
TT
denote the obvious diagram of
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
consequence of the oo-operad axioms. For each k E N, let
p
-
2 is an immediate
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
so
is p-right
Kan extended from pjpk-1. Comparing the p-right Kan extensions along both paths
across the commutative diagram
<
<k-1i
l
gives the induction step, and thence the result.
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
and U 2 \ UIl > 1, then the restriction of p to the subposet PU1,J
subsets V of S 0 with U1 g V C U2 is a p-limit diagram.
59
2
g
U2
spanned by those
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 -+
60
C®
such that
PS
=
]1 F((1), {1}) x
U
where
"H
171 F((1), 0)
So\U
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®,
#
l
must be a trivial cofibration.
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 1 x 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)
morphism
f
: S -+ T of finite sets inducing a bijection
flu
: U
-+
(T, V) is a
' 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.
definition, the symmetric monoidal envelope of Env(CM®) has objects
(S, U, f : So -+ T)
62
By
where T e F. Our isomorphism will map this object to the object
(T, (f~-1(t), f -1 (t) n U)teT)
El
of (T+) L
Corollary A.1M. A CM®-algebraparametrizing a commutative monoid E and a
module M in C* gives rise to a functor
AE,M(S)
Proof. We construct
AE,M
AE,M
hT* -+ C such that
~ Eos" 0 M.
by embedding T, as the full subcategory of F+ consisting
of objects (S, U) for which IUI = 1.
El
63
64
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