# nLab geometric infinity-function theory

## Theorems

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Introduction

This entry is about the work by David Ben-Zvi, John Francis and David Nadler on application of a (∞,1)-categorical realization of geometric function theory to extended quantum field theory in the context in which Jacob Lurie proved the cobordism hypothesis.

So far this work is presented in the two articles

This entry is supposed to be the $n$Lab-working area for a ” Journal Club – Geometric $\infty$-function theory “. The corresponding discussion page at the $n$-Café is Journal Club – Geometric ∞-Function Theory

The idea is to

• jointly discuss at the $n$Café section-by-section these articles; to get an idea for what’s going on;

• add here, step by step, links to keywords appearing in these sections, and create the corresponding entries describing them.

• This is supposed to be a recursive and iterative process, which, if successful, will eventually create here a useful repository of entries that describe and explain the various aspects of the topic.

# Basic idea in three words

Geometric $\infty$-function theory is about the ∞-categorification of the following basic fact of matrix calculus: for $X$ and $Y$ finite sets, for $k$ a field and for $C\left(X\right)$ and $C\left(Y\right)$ the $k$-vector spaces of $k$-valued functions on $X$ and $Y$, respectively, we have natural isomorphisms

$C\left(X\right)\otimes C\left(Y\right)\simeq C\left(X×Y\right)\simeq \mathrm{LinMap}\left(C\left(X\right),C\left(Y\right)\right)\simeq C\left(X{\right)}^{*}\otimes C\left(Y\right)$C(X) \otimes C(Y) \simeq C(X \times Y) \simeq LinMap(C(X), C(Y)) \simeq C(X)^* \otimes C(Y)

of finite-dimensional vector spaces.

In geometric $\infty$-function theory one replaced the finite sets here with generalized spaces called perfect derived stacks, and $k$-valued functions by something like $k$-vector bundles on these.

That’s it, essentially. The point is that this simple statement then turns out to be a powerful organization and unification tool for lots of structures appearing in representation theory and functorial quantum field theory.

# Timeline

We will try to proceed as follows: we go through the sections of the two articles, step by step, possibly several steps for one section. Each week on Monday, one of us produces a “report” on the section he or she was assigned to read.

This “report” would try to give a rough idea of what is going on in a given section. A report may be anything from a heap of questions (likely) to a complete detailed rederivation of all the details (maybe not quite as likely, but let’s not exclude it!) The more questions, the more we all get involved, which is the whole point of doing this online.

I (Urs) am imagining that whatever the report is like, it consists of

• a bulleted list of whatever needs to be listed, with links to whatever deserves to be linked to, here in this $n$Lab-entry, in the following list of sections;

• a comment to the blog entry that maybe copies this entire content but at least alerts the blog readers about the new material now to be found here, accompanied by some comments as seems necessary.

The idea is that we have discussion on the blog but distill whatever we can into the $n$Lab here.

Here is the list of reports, as planned so far:

# Idea

The central structural theorem of TQFT, the cobordism theorem states that (∞,n)-categorical extended topological quantum field theories are entirely determined by their value in the point, which has to be an object in an (∞,n)-category with high dualizability properties.

Very generally, the point of geometric $\infty$-function theory is to construct and study concrete realizations of the FQFTs guaranteed to exist by this theorem.

The central tool, from which this entry draws its title, is an (∞,1)-categorical version of geometric function theory:

the rough idea is that the data the FQFT assigns to a manifold $\Sigma$ is a collection $Z\left(\Sigma \right)$ of $\infty$-functions – the physical fields – on $\Sigma$, or, more generally, sections of some $\infty$-bundle on $\Sigma$, and that the morphism $Z\left(\Sigma \right):Z\left({\Sigma }_{\mathrm{in}}\right)\to Z\left({\Sigma }_{\mathrm{out}}\right)$ assigned by the FQFT to a cobordism $\Sigma :{\Sigma }_{\mathrm{in}}\to {\Sigma }_{\mathrm{out}}$ is obtained from a pull-push-operation on the objects of $Z\left({\Sigma }_{\mathrm{in}}\right)$ through a span to obtain objects in $Z\left({\Sigma }_{\mathrm{out}}\right)$.

This picture arises naturally and is maybe best understood in terms of FQFTs that arise as σ-models, namely which are represented by a target space object $P$:

assume that $P$ is some kind of generalized space (which will usually mean: an (∞,1)-sheaf, see motivation for sheaves, cohomology and higher stacks for motivation of this point) and regard the manifolds $\Sigma$ as special cases of generalized manifolds.

Writing $\left[-,P\right]$ for the internal hom in the given context, every cobordism cospan is sent to a span

$\left[-,P\right]:\left(\begin{array}{ccc}& & \Sigma \\ & ↗& & ↖\\ {\Sigma }_{\mathrm{in}}& & & & {\Sigma }_{\mathrm{out}}\end{array}\right)↦\left(\begin{array}{ccc}& & \left[\Sigma ,P\right]\\ & {}^{{p}_{1}}↙& & {}^{{p}_{2}}↘\\ \left[{\Sigma }_{\mathrm{in}},P\right]& & & & \left[{\Sigma }_{\mathrm{out}},P\right]\end{array}\right)\phantom{\rule{thinmathspace}{0ex}}.$[-, P] : \left( \array{ && \Sigma \\ & \nearrow && \nwarrow \\ \Sigma_{in} &&&& \Sigma_{out} } \right) \mapsto \left( \array{ && [\Sigma, P] \\ & {}^{p_1}\swarrow && {}^{p_2}\searrow \\ [\Sigma_{in}, P] &&&& [\Sigma_{out}, P] } \right) \,.

If now $C\left(\left[{\Sigma }_{\mathrm{in}},P\right]\right)$ denotes some sensible collection of $\infty$-functions on the mapping space $\left[{\Sigma }_{\mathrm{in}},P\right]$, there will be an ∞-categorical pull-push operation

${\int }_{{p}_{2}}{p}_{1}^{*}\left(-\right):C\left(\left[{\Sigma }_{\mathrm{in}},P\right]\right)\to C\left(\left[{\Sigma }_{\mathrm{out}},P\right]\right)$\int_{p_2} p_1^*(-) : C([\Sigma_{in}, P]) \to C([\Sigma_{out}, P])

generalizing the analogous operation as described as groupoidification.

As described at geometric function theory, such pull-push operations can naturally be regarded as vast generalization of familiar matrix calculus, including in particular operations like Fourier–Mukai transformations.

## Summary

In IntTrans the basic machinery of these $\infty$-categorical pull-push operations is established.

In CharTheo the particular case of an FQFT is considered whose defining assigment to the point is the differential graded Hecke category

${H}_{G}=D\left(B\\left(G/B\right)\right)$H_G = D( B \backslash (G/B))

of $B$-equivariant D-modules on the flag variety $G/B$ of a complex reductive group $G$ with Borel subgroup $B$.

(… more to say here …)

# Keyword list

The following is supposed to going to be a list of linked keywords corresponding section-by-section to the Ben-Zvi/Francis/Nadler articles above.

## Integral Transforms

### 1. Introduction

I am going to use this space (the introduction) as my sandbox and a place to begin a conversation on things I want to talk about it. This way things do not muck up areas that Urs and others have made nice and they can be transported to the appropriate areas later.

• Drinfeld centers

Comment by David Ben-Zvi from n-category cafe

One quick comment: the Drinfeld center is not a “sub” in any reasonable sense (except in the loose sense that it’s a categorical limit). This is easy to see already for the category of representations of a finite group, where the Drinfeld center (or modules for the Drinfeld double) is the category of G-equivariant vector bundles on the group G (I think they also go by the name Yetter-Drinfeld modules?) Likewise the center of a ring in the derived world (ie Hochschild cohomology) is not really a subring - even for a commutative ring, its derived center doesn’t map in injectively (for a smooth commutative ring the Hochschild-Kostant-Rosenberg theorem tells us that Hochschild cohomology is the exterior algebra on derivations of the ring). I think we have to abandon the notions of sub and quotient in the homotopical world and stick to notions like (homotopy) limit and colimit.

I want to start replying to/understanding this comment. I have some reading to do first so I will just leave this query box as is for now.

### 2. Preliminaries

Recall that the goal of geometric $\infty$-function theory is

• to establish a good higher categorical version of linear algebra (“integral transform” = “higher matrix multiplication”!)

• such that interesting classes of σ-model extended TQFTs ${Z}_{P}$ are represented by generalized spaces $P$

• in that the “higher linear map” ${Z}_{P}\left(\Sigma \right)$ assigned by the QFT to a cobordism cospan $\begin{array}{ccc}& & \Sigma \\ & ↗& & ↖\\ {\Sigma }_{\mathrm{in}}& & & & {\Sigma }_{\mathrm{out}}\end{array}$ is given by the span $\begin{array}{ccc}& & \left[\Sigma ,P\right]\\ & ↙& & ↘\\ \left[{\Sigma }_{\mathrm{in}},P\right]& & & & \left[{\Sigma }_{\mathrm{out}},P\right]\end{array}$.

This clearly requires that

One point made by the Ben-Zvi/Francis/Nadler work is that a good working context of higher category theory in which a good notion of geometric $\infty$-function theory can be set up nicely is the context assembled and developed in the PhD thesis of Jacob Lurie, consisting of

Especially for the newcomer and non-expert it must be understood that the plethora of high-powered terminology appearing here is conceptually simplified and unified by their description in the higher categorical context – so you gain by trying to learn this stuff here first before going into the standard literature:

This is the reason why we bother with geometric $\infty$-function theory and devote a detailed discussion to it: geometric $\infty$-function theory carries the promise of getting close to the sought-for Lawvere-ification of quantum field theory – providing its natural language.

#### 2.1 $\infty$-categories

The ∞-categories that we are dealing with here are

There are several reasons for

in the present context, the main one being that they allow to make precise the ideas summarized in the

In principle one will want to eventually understand geometric function theory in the context of more general higher category theory, in particular for (∞,n)-categories, but a great deal is already gained by just (hah!) looking at (∞,1)-categories – not the least because (only) for them a working well-developed full theory exists at the moment, developed by Andre Joyal and further developed by Jacob Lurie:

This subsumes and unifies notably a wealth of more-or-less ad-hoc constructions that have been known for a bit longer. In particular the theory of model categories is realized as a way to, well, model $\left(\infty ,1\right)$-categories:

In particular every

is naturally a

##### 2.1.1 Enhancing triangulated categories

The aim of geometric $\infty$-function theory is to develop a good $\infty$-categorical generalization of the simple notion of

• sets of functions on spaces

to

• $\infty$-categories of $\infty$-functions” on ”$\left(\infty \right)$-spaces”

essentially following the general philosophy of space and quantity.

Whatever the answer is, the collection of such $\infty$-functions should be

• abelian

and

• monoidal

in a suitable sense.

From various examples it has become clear that the right vertical categorification of a function $X\to ℂ$ is a functor $X\to {\mathrm{Vect}}_{ℂ}$, which – if sufficiently well behaved – we may regard as a vector bundle on $X$.

Indeed, just as functions $\left[X,ℂ\right]$ form a vector space, functors $\left[X,{\mathrm{Vect}}_{ℂ}\right]$ naturally form a 2-vector space.

Such a 2-vector space is in particular an abelian category. This is one of the ways in which we expect $\infty$-functions to form an abelian collection.

Experience shows that the right (or at least a very good) $\infty$-vertical categorification of abelian category is a

It turns out that just as model categories and related homotopical categories are best thought of as, well, models for $\left(\infty ,1\right)$-categories, so various constructions in homological algebra – and in the end really all of modern homological algebra – is best thought of as models for stable (∞,1)-categories.

This concerns notably

and

which, in turn, are already enhanced triangulated categories: namely dg-enriched versions thereof.

At this point you are urged to really have a look at the entry on stable (∞,1)-category and marvel about the fact that

• the definition of stable $\left(\infty ,1\right)$-category is short, simple and transparent.

It’s the most obvious thing in the world. And yet, it turns out that the rather involved definitions of derived triangulated category follow from this simple definition when one decides to look at just the 1-categorical shadow given by the homotopy category of the stable (∞,1)-category.

This is a general pattern here:

• $\left(\infty ,1\right)$-categorical notions – and in particular quasi-categorical notions – are conceptually simple and lend themselves to the formulation and description of higher categorical situations;

• but for concrete constructions in terms of them there is a wealth of tools with different areas of applicability, many of which have been understood and developed as theories in their own right for a considerable time.

#### 2.2 monoidal $\infty$-categories

A central aspect of geometric $\infty$-function theory is that we regard the collection $C\left(X\right)$ of $\infty$-functions assigned to a generalized space $X$ (concretely modeled as $\mathrm{QC}\left(X\right)$, see section 3 below) not just as a higher vector space but naturally as a higher algebra. Indeed, the central two theorems/properties of geometric function theory concern the interplay between the geometry of intersections or pullbacks

of generalized spaces and the algebra of tensor products

$C\left({Y}_{1}\right){\otimes }_{C\left(X\right)}C\left({Y}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$C(Y_1) \otimes_{C(X)} C(Y_2) \,.

In order to formulate this, one needs a good general theory of higher algebra. Just as ordinary algebra takes place inside a monoidal category, higher algebra takes place in a monoidal (∞,1)-category:

an “$\infty$-monoid” or “$\infty$-algebra” (to distinguish from the traditional A∞ algebra, which is supposedly a special case) is an algebra/monoid object in a monoidal (∞,1)-category.

In order to characterize the $\left(\infty ,1\right)$-categories $C\left(X\right)$ as algebra objects in such a sense we profit from the ease with which quasi-categories naturally reflect on themselves and allow us with comparative ease to talk about the (∞,1)-category of (∞,1)-categories $\left(\infty ,1\right){\mathrm{Cat}}_{1}$.

Since $\left(\infty ,1\right){\mathrm{Cat}}_{1}$ itself is a symmetric monoidal (∞,1)-category, we will essentially identify the $\infty$-algebras $C\left(X\right)$ as algebra objects in $\left(\infty ,1\right){\mathrm{Cat}}_{1}$.

But there is actually a slight technical simplification: we don’t deal with all $\left(\infty ,1\right)$-categories, but just with presentable (∞,1)-categories. See there for the (long) list of nice properties and characterization of presentable $\left(\infty ,1\right)$-categories.

So

• presentable $\left(\infty ,1\right)$-categories form the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories $\mathrm{Pr}\left(\infty ,1\right){\mathrm{Cat}}_{1}^{L}$;

• the $\infty$-algebras of $\infty$-functions $C\left(X\right)$ are algebra objects in $\mathrm{Pr}\left(\infty ,1\right){\mathrm{Cat}}_{1}^{L}$.

Notice that this means in particular that the “additive” structure on $C\left(X\right)$ is taken to be nothing but the $\left(\infty ,1\right)$-categorical colimit operation inside $C\left(X\right)$: this is the operation with respect to which (∞,1)-functors in $\left(\infty ,1\right){\mathrm{Cat}}_{1}^{L}$ are linear, and with respect to which the tensor product is bilinear.

In terms of this higher algebra we will obtain the two central (defining) theorems of geometric $\infty$-function theory.

###### Theorem

Fundamental theorem of geometric $\infty$-function theory

For $X\to Y←X\prime$ morphisms of nice generalized spaces (perfect ∞-stacks) and for $\infty$-functions $C\left(-\right)=\mathrm{QC}\left(-\right)$ (given by the assignment of $\left(\infty ,1\right)$-categories of quasicoherent sheaves) we have

• $\infty$-matrices (integral transforms) are $\infty$-functions on fiber products in that the following equivalence holds: $C\left(X{×}_{Y}X\prime \right)\simeq C\left(X\right){\otimes }_{C\left(Y\right)}C\left(X\prime \right)$

• $\infty$-linear maps are given by $\infty$-matrices in that also the following equivalence holds: $C\left(X{×}_{Y}X\prime \right)\simeq {\mathrm{Fun}}_{C\left(Y\right)}\left(C\left(X\right),C\left(X\prime \right)\right)$;

The following discussion aims to describe the role played by ∞-stacks and their morphisms in the context of geometric $\infty$-funcion theory.

We place ourselves in the context of an (∞,1)-category $H$ of (possibly derived) $\infty$-stacks, i.e. of (∞,1)-sheaves.

Recall what this means:

The fundamental datum that we fix once and for all and which determines what $H$ is like is a choice of category $S$ equipped with the structure of a site. $H$ is going to be the $\left(\infty ,1\right)$-category of generalized spaces which may be probed by objects of $S$ in a way that is consistent with the way one glues objects in $S$ using its structure of a site. More generally, $S$ itself may be an $\left(\infty ,1\right)$-category.

It is actually not hard to say this in a precise and fairly explicit way.

First the precise way.

• the first datum is: $S$ a small (∞,1)-category;

• let then $\mathrm{PSh}\left(S\right):=\mathrm{Funct}\left({S}^{\mathrm{op}},\infty \text{-}\mathrm{Grpd}\right)$ be the $\left(\infty ,1\right)$-category of (∞,1)-functors from the opposite (∞,1)-category ${S}^{\mathrm{op}}$ to the $\left(\infty ,1\right)$-category of $\infty$-groupoids.

• the second datum is: $i:H↪\mathrm{PSh}\left(S\right)$ a full (∞,1)-subcategory? of $\mathrm{PSh}\left(S\right)$ such that the inclusion is a geometric morphism of
(∞,1)-topoi, i.e. such that $i$ has a left exact left adjoint $\left(-{\right)}^{*}:\mathrm{PSh}\left(S\right)\to H$.

The same in one sentence: $H$ is a sub-(∞,1)-topos of the (∞,1)-topos of (∞,1)-presheaves on $S$.

Or equivalently: $H$ is an (∞,1)-category of (∞,1)-sheaves on $S$.

Now the fairly explicit way:

There are various different but equivalent explicit realizations, or models, for (∞,1)-categories. Of these consider quasi-categories, which are
simplicial sets that are weak Kan complexes, and SSet-enriched categories. These are related by the homotopy coherent nerve functor:

$N:\mathrm{SSet}\text{-}\mathrm{Cat}\to \mathrm{SSet}$N : SSet\text{-}Cat \to SSet

$F:\mathrm{SSet}\to \mathrm{SSet}\text{-}\mathrm{Cat}$F : SSet \to SSet\text{-}Cat

that sends every simplicial set to the SSet-enriched category freely generated from it. One can think for $S$ a quasi-category of $F\left(S\right)$ as a semi-strictification of it, in which composition of morphisms along 0-cells is strictly associative.

It is convenient and usual to switch back and forth between these two models. Quasi-categories tend to give rise to conceptually more transparent definitions, while SSet-categories tend to be more convenient for many explicit computations.

To get back to our (∞,1)-category of (∞,1)-sheaves $H$, we describe it in terms of these models as follows.

• $S$ is some small simplicial set which we can assume to be a weak Kan complex;

• $\infty \text{-}\mathrm{Grpd}:=N\left(\mathrm{Kan}\right)$ is the simplicial set which is the image under the homotopy coherent nerve functor of the full SSet-subcategory of SSet on Kan complexes;

• $\mathrm{PSh}\left(S\right)=\mathrm{SSet}\left({S}^{\mathrm{op}},\infty \text{-}\mathrm{Grpd}\right)$ is just the simplicial set of simplicial maps between the simplicial sets ${S}^{\mathrm{op}}$ and $\infty \text{-}\mathrm{Grpd}\right)$.

For explicitly constructing $H$ as a full sub-$\left(\infty ,1\right)$-category of $\mathrm{PSh}\left(S\right)$ in practice one usually switches from the quasi-category picture to the SSet-enriched category picture. So assume $H$ explicitly to be an SSet-enriched category and $i:H\to F\left(\mathrm{PSh}\left(S\right)\right)$ the suitable inclusion SSet-functor.

Notice that its left adjoint $\left(-{\right)}^{*}:\mathrm{PSh}\left(S\right)\to N\left(H\right)$ is (∞,1)-sheafification: it sends every (∞,1)-presheaf to the corresponding (∞,1)-sheaf. In other words, this is $\infty$-stackification. Describing the full $\infty$-stackification of a given $\left(\infty ,1\right)$-presheaf explicitly is usually hard. Moreover, it is usually much more than one wants to actually know.

Therefore a common approach to constructing $H$ is as an SSet-enriched category which has precisely the same objects as $F\left(\mathrm{PSh}\left(S\right)\right)$ has, but where every object is $\left(\infty ,1\right)$-equivalent to its $\left(\infty ,1\right)$-sheafification. This is in particular what happens in the most developed explicit model for $H$, due to K. Brown, A. Joyal, J. Jardine and others, in which (for $S$ an ordinary 1-category) $H$ is constructed as the canonical SSet-enrichment of the model category structure on $\mathrm{Funct}\left({S}^{\mathrm{op}},\mathrm{SSet}\right)$.

For such a model of $H$, let $A$ be an object, i.e. an $\left(\infty ,1\right)$-presheaf which need not be an $\left(\infty ,1\right)$-sheaf/$\infty$-stack, and let $X$ be any other object, then the simplicial set

$H\left(X,A\right)$H(X,A)

(which, if it is not Kan already, we are to think of as representing the $\infty$-groupoid obtained from its Kan-fibration replacement)

is equivalent to $H\left(X,\overline{A}\right)$, where $\overline{A}$ is the $\infty$-stackification of $A$.

This is most familiar in detail, if maybe not conceptually, in the context of abelian sheaf cohomology: an (ordinary) sheaf $A$ with values in non-negatively graded chain complexes is, by the Dold-Kan correspondence, a special case of an $\left(\infty ,1\right)$-presheaf with values in $\infty \text{-}\mathrm{Grpd}$. Computing the abelian sheaf cohomology of $A$ on $X$ can be understood as being the computation of the $\infty$-stackification of $A$ evaluated on $X$.

Moreover, for a fixed $\left(\infty ,1\right)$-presheaf $A$, homming into it yields a functor

$H\left(-,A\right):{H}^{\mathrm{op}}\to \mathrm{SSet}$H(-, A) : H^{op} \to SSet

which, being $\mathrm{SSet}$-valued, we are entitled to call an $\left(\infty ,1\right)$-presheaf on $H$. In as far as $H$ is thought of as an $\left(\infty ,1\right)$-category of $\infty$-stacks, this is an $\left(\infty ,1\right)$-presheaf or $\infty$-prestack on $\infty$-stacks.

It is natural, suggestive and common to write $A\left(X\right):=H\left(X,A\right)$, following the guide of the Yoneda lemma, even if $X$ is far from being representable.

Let for instance ${\mathrm{Vect}}_{\infty }\in H$ be an $\left(\infty ,1\right)$-presheaf which assigns to each test domain $U\in S$ an $\infty$-groupoid ${\mathrm{Vect}}_{\infty }\left(U\right)$ of some kind of $\infty$-vector bundles on $U$. Then ${\mathrm{Vect}}_{\infty }$ is, regarded as a generalized space modeled on $X$, the classifying space of $\infty$-vector bundles, essentially by definition.

So for $X\in H$ any other $\infty$-stack/generalized space,

${\mathrm{Vect}}_{\infty }\left(X\right):=H\left(X,{\mathrm{Vect}}_{\infty }\right)$Vect_\infty(X) := H(X, Vect_\infty)

is the (simplicial set whose Kan-fibrant replacement is) the $\infty$-groupoid of (some kind of) $\infty$-vector bundles of on $X$.

If the $X$ here is an $\infty$-groupoid valued presheaf which really takes values in higher groupoids and not just in sets, then $X$ is actually to be regarded as a generalized $\infty$-groupoid itself, of course. One often thinks of such $X$ as orbifolds. If we write ${X}_{0}↪X$ for the presheaf of objects in $X$, then

• ${\mathrm{Vect}}_{\infty }\left({X}_{0}\right)$ is the $\infty$-groupoid of $\infty$-vector bundles on ${X}_{0}$

• ${\mathrm{Vect}}_{\infty }\left(X\right)$ is the $\infty$-groupoid of equivariant $\infty$-vector bundles on ${X}_{0}$,

where the equivariance in question is that encoded by the inclusion ${X}_{0}↪X$.

This is notably relevant for the fundamental ∞-groupoid:

assume that our site $S$ is monoidal and equipped with a cosimplicial object, i.e. a functor ${\Delta }_{S}:\Delta \to S$ from the simplex category, such that ${\Delta }_{S}\left[0\right]$ is a generator of $S$, then there is canonically the functor

$\Pi :S\to H$\Pi : S \to H

which sends each test object ${X}_{0}$ to the $\left(\infty ,1\right)$-presheaf that sends each domain $V$ to the $V$-family version of the singular simplicial complex of $U$:

$\Pi \left({X}_{0}\right):U↦S\left(V\otimes {\Delta }^{•},{X}_{0}\right)\phantom{\rule{thinmathspace}{0ex}}.$\Pi(X_0) : U \mapsto S(V \otimes \Delta^\bullet, X_0) \,.

The inclusion of the space of objects in $\Pi \left({X}_{0}\right)$ is just ${X}_{0}↪\Pi \left({X}_{0}\right)$. From the above we now have

• $H\left({X}_{0},{\mathrm{Vect}}_{\infty }\right)=:{\mathrm{Vect}}_{\infty }\left({X}_{0}\right)$ is the $\infty$-groupoid of $\infty$-vector bundles on ${X}_{0}$

• $H\left(\Pi \left({X}_{0}\right),{\mathrm{Vect}}_{\infty }\right)$ =: ${\mathrm{Vect}}_{\infty }\left(\Pi \left(X\right)\right)$ is the $\infty$-groupoid of $\Pi$-equivariant $\infty$-vector bundles on ${X}_{0}$.

By unwrapping what ”$\Pi$-equivariance” means, one finds that this may equivalently be stated as

• $H\left(\Pi \left({X}_{0}\right),{\mathrm{Vect}}_{\infty }\right)$ =: ${\mathrm{Vect}}_{\infty }\left(\Pi \left(X\right)\right)$ is the $\infty$-groupoid of flat $\infty$-vector bundles with connection on ${X}_{0}$.

It should be noted here that we can combine different levels of equivariance:

we may left Kan extend $\Pi :S\to H$ along the inclusion $S↪H$ to a functor $\Pi :H\to H$, so that for $X$ itself n $\infty$-orbifold of sorts, $\Pi \left(X\right)$ is the fundamental $\infty$-groupoid of that. We still have a canonical inclusion $X↪\Pi \left(X\right)$. If moreover ${X}_{0}↪X$ is the inclusion from above, we have

• $H\left({X}_{0},{\mathrm{Vect}}_{\infty }\right)=:{\mathrm{Vect}}_{\infty }\left({X}_{0}\right)$ is the $\infty$-groupoid of $\infty$-vector bundles on ${X}_{0}$

• $H\left(X,{\mathrm{Vect}}_{\infty }\right)=:{\mathrm{Vect}}_{\infty }\left(X\right)$ is the $\infty$-groupoid of equivariant $\infty$-vector bundles on ${X}_{0}$;

• $H\left(\Pi \left(X\right),{\mathrm{Vect}}_{\infty }\right)$ =: ${\mathrm{Vect}}_{\infty }\left(\Pi \left(X\right)\right)$ is the $\infty$-groupoid of equivariant $\infty$-vector bundles on ${X}_{0}$ with flat connection.

Morphisms from the fundamental ∞-groupoid are also called local systems.

#### 2.4 derived loop spaces

Of particular interest in this study of geometric $\infty$-function theory is the behaviour of $\infty$-functions on loop spaces. The $\left(\infty ,1\right)$-category $C\left(\Lambda X\right)$ of $\infty$-functions on the free loop space $\Lambda X$ of a sufficiently nice generalized space (a perfect ∞-stack) $X$ turns out to be the ∞-trace? or ∞-center? of that of $X$

$C\left(\Lambda X\right)\simeq \mathrm{Tr}\left(C\left(C\right)\right)\simeq \mathrm{Tr}\left(CX\right)\phantom{\rule{thinmathspace}{0ex}}.$C(\Lambda X) \simeq Tr( C(C)) \simeq Tr (C X) \,.

which in turn are identified with the $\infty$-version of ∞-Hochschild homology?

$\mathrm{Tr}\left(C\left(X\right)\right):={\mathrm{HH}}_{*}\left(C\left(X\right)\right):=C\left(X\right){\otimes }_{C\left(X\right)\otimes C\left(X{\right)}^{\mathrm{op}}}C\left(X\right)$Tr(C(X)) := HH_*(C(X)) := C(X) \otimes_{C(X)\otimes C(X)^{op}} C(X)

and ∞-Hochschild cohomology?

$Z\left(C\left(X\right)\right):={\mathrm{HH}}^{*}\left(C\left(X\right)\right):={\mathrm{End}}_{C\left(X\right)\otimes C\left(X{\right)}^{\mathrm{op}}}\left(C\left(X\right)\right)$Z(C(X)) := HH^*(C(X)) := End_{C(X) \otimes C(X)^{op}}(C(X))

of $X$.

All these statements, powerful as they are, become trivialities, due to the naturality of the language that we are using, once we realize the following:

homotopy pullbacks and loop space objects

one of the central crucial facts of higher category theory is that

the fiber product of a point with itself is not the point, but the loop space object based at the point:

let $x:*\to X$ be a morphism from the terminal object $*$ to some object $X$ in some $\infty$-category, then the $\infty$-categorical pullback of $x$ along itself is the based loop space object ${\Omega }_{x}X$.

$\begin{array}{ccc}{\Omega }_{x}X& \to & *\\ ↓& ⇓& {↓}^{x}\\ *& \stackrel{x}{\to }& X\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \Omega_x X &\to& * \\ \downarrow &\Downarrow& \downarrow^x \\ * &\stackrel{x}{\to}& X } \,.

This should be intuitively quite clear: the $\infty$-pullback commutes only up to a 2-cell, a homotopy from the constant map on $x$ to the constant map on $x$. But such a homotopy is nothing but a loop in $X$, so ${\Omega }_{x}X$ is the object that remembers all possible ways to make this diagram commute up to a 2-cell.

Notice, by the way, that this phenomenon is the source of all “long exact sequences” that you’ll ever run into: if we are for instance in a stable (∞,1)-category, so that the terminal object $*$ is even a zero object, $*=0$, then a sequence

$A\to B\to C$A \to B \to C

is exact (a fibration sequence) if the first morphism is the kernel of the second, meaning that we have a pullback

$\begin{array}{ccc}A& \to & 0\\ ↓& ⇓& ↓\\ B& \to & C\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ A &\to& 0 \\ \downarrow &\Downarrow& \downarrow \\ B &\to& C } \,.

But then, since $\infty$-pullback squares compose just as ordinary pullback squares do, further computing the kernel of the kernel $A\to B$ does not produce 0, as it would in an ordinary abelian category, but produces loops in $C$

$\begin{array}{ccccc}\Omega C& \to & A& \to & 0\\ ↓& ⇓& ↓& ⇓& ↓\\ 0& \to & B& \to & C\end{array}$\array{ \Omega C &\to& A &\to& 0 \\ \downarrow &\Downarrow& \downarrow &\Downarrow& \downarrow \\ 0 &\to& B &\to& C }

since the total (outer) diagram is of the form

$\begin{array}{ccc}\Omega C& \to & 0\\ ↓& ⇓& ↓\\ 0& \to & C\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \Omega C &\to& 0 \\ \downarrow &\Downarrow& \downarrow \\ 0 &\to& C } \,.

So

• the $\infty$-kernel of an $\infty$-kernel is not 0, but is loops.

Continuing this way one obtains long exact sequences

$\begin{array}{ccc}\Omega B& \to & 0\\ {↓}_{\overline{\Omega }f}& & ↓\\ \Omega C& \to & A& \to & 0\\ ↓& ⇓& ↓& ⇓& ↓\\ 0& \to & B& \stackrel{f}{\to }& C\end{array}$\array{ \Omega B &\to& 0 \\ \downarrow_{\bar \Omega f} && \downarrow \\ \Omega C &\to& A &\to& 0 \\ \downarrow &\Downarrow& \downarrow &\Downarrow& \downarrow \\ 0 &\to& B &\stackrel{f}{\to}& C }

induced from the single map $f:B\to C$.

For the purposes of geometric $\infty$-function theory what is more relevant than this construction of based loop spaces as kernels is the construction of unbased loop space objects. By a similar reasoning, one finds that the free loop space $\Lambda X$ of a generalized space $X$ is the $\infty$-pullback of the morphism $X\stackrel{\mathrm{Id}×\mathrm{Id}}{\to }X×X$ along itself

$\begin{array}{ccc}\Lambda X& \to & X\\ ↓& & {↓}^{\mathrm{Id}×\mathrm{Id}}\\ X& \stackrel{\mathrm{Id}×\mathrm{Id}}{\to }& X×X\end{array}$\array{ \Lambda X &\to& X \\ \downarrow && \downarrow^{Id \times Id} \\ X &\stackrel{Id \times Id}{\to}& X \times X }

The intuitive reasoning is the same as before, only that now we don’t fix a single point. For more details and in particular for descriptions and examples of how to explicitly compute these loops space objects by homotopy limits see the examples listed at

Notice that when we speak of “homotopy” in the above, we mean categorical homotopies. These loop space constructions see only homotopies which actually exist as morphisms. If it is isn’t clear what is meant by that statement, see the discusssion at constant ∞-stack for the two different perspectives on a topological space, once as a categorically discrete but topologically non-discrete object, once as a topologically discrete but categorically non-discrete object:

In order for the above homotopy pullbacks to compute the intended loop spaces of topological spaces, these topological spaces need to be regarded as the topologically discrete(!) ∞-groupoids they correspond to, i.e. as constant ∞-stacks.

Now, with the understanding of loop space objects as homotopy pullbacks understood, the above statement about $\infty$-Hochschild (co)homology becomes essentially a triviality:

we have

$\begin{array}{rl}C\left(\Lambda X\right)\simeq C\left(X{×}_{X×X}X\right)& \mathrm{by}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{loop}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{space}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{as}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{homotopy}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{pullback}\\ \cdots \simeq C\left(X\right){×}_{C\left(X\right)×C\left(X\right)}C\left(X\right)& \mathrm{by}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{fundamental}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{theorem}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{geometric}\infty -\mathrm{function}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{theory}\\ \cdots \simeq {\mathrm{HH}}_{*}\left(C\left(X\right)\right):=:\mathrm{Tr}\left(C\left(X\right)\right)& \mathrm{by}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{definition}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{trace}/\mathrm{Hochschild}\mathrm{homology}\end{array}$\begin{aligned} C(\Lambda X) \simeq C(X \times_{X \times X} X) & by\;\; loop \;\;space\;\; as \;\;homotopy\;\; pullback \\ \cdots \simeq C(X) \times_{C(X)\times C(X)} C(X) & by \;\;fundamental \;\;theorem \;\;of\;\; geometric \infty-function\;\; theory \\ \cdots \simeq HH_*(C(X)) :=: Tr(C(X)) & by \;\;definition \;\;of \;\;trace/ Hochschild homology \end{aligned}

or equivalently

$\begin{array}{rl}C\left(\Lambda X\right)\simeq C\left(X{×}_{X×X}X\right)& \mathrm{by}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{loop}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{space}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{as}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{homotopy}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{pullback}\\ \cdots \simeq {\mathrm{Fun}}_{C\left(X\right)}\left(C\left(X\right),C\left(X\right)\right)& \mathrm{by}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{fundamental}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{theorem}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{geometric}\infty -\mathrm{function}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{theory}\\ \cdots \simeq {\mathrm{HH}}^{*}\left(C\left(X\right)\right):=:Z\left(C\left(X\right)\right)& \mathrm{by}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{definition}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{center}/\mathrm{Hochschild}\mathrm{cohomology}\end{array}$\begin{aligned} C(\Lambda X) \simeq C(X \times_{X \times X} X) & by \;\;loop\;\; space\;\; as\;\; homotopy\;\; pullback \\ \cdots \simeq Fun_{C(X)}(C(X), C(X)) & by \;\;fundamental \;\;theorem \;\;of \;\;geometric \infty-function \;\;theory \\ \cdots \simeq HH^*(C(X)) :=: Z(C(X)) & by \;\;definition \;\;of \;\;center/Hochschild cohomology \end{aligned}

#### 2.5 ${E}_{n}$-structures

While for an ordinary monoids there is just one notion of commutativity (either it is or it is not commutative), already a monoidal category distinguishes between being just braided monoidal or fully symmetric monoidal.

This pattern continues, as expressed by the periodic table of k-tuply monoidal categories.

A higher category may be a k-tuply monoidal n-category or more generally k-tuply monoidal (n,r)-category for different values of $k$. The lowest value of $k=1$ (since for $k=0$ there is no monoidal structure at all) corresponds to monoidal product which is $\infty$-associative, i.e. associative up to higher coherent homotopies, but need not have any degree of commutativity.

One says that an $n$-category is symmetric monoiodal if it is “as monoidal as possible”, i.e. $\infty$-tuply monoidal. In particular, in Noncommutative algebra and Commutative algebra Jacob Lurie describes

It turns out that the monoidal $\left(\infty ,1\right)$-categories that we are concerned with here in general have a tuplicity of monoidalness (heh) in between 1 and $\infty$:

For each $1\le n\le \infty$ let ${E}_{n}$ denote the little n-disk operad whose topological space of ${E}_{n}^{k}$ of $k$-ary operations is the space of embedding of $k$ $n$-dimensional disks (balls) in one $n$-dimensional disk without intersection, and whose composition operation is the obvious one obtained from gluing the big outer disks into given inner disks.

In John Francis’ PhD thesis (reference EnAction below ) the theory of (∞,1)-categories equipped with an action of the ${E}_{n}$-operad is established, so that

• $\left(\infty ,1\right)$-categories with an ${E}_{1}$-action are precisely monoidal (∞,1)-categories – 1-fold monoidal $\left(\infty ,1\right)$-categories;

• $\left(\infty ,1\right)$-categories with an ${E}_{\infty }$-action are precisely symmetric monoidal (∞,1)-categories$\infty$-tuply monoidal $\left(\infty ,1\right)$-categories;

• $\left(\infty ,1\right)$-categories with an ${E}_{n}$-action for $1 are the corresponding $n$-tuply monoidal $\left(\infty ,1\right)$-categories in between.

Remark The second statement is example 2.3.8 in EnAction. The first seems to be clear but is maybe not in the literature. Jacob Lurie is currently rewriting Higher Algebra such as to build in a discussion of ${E}_{n}$-operadic structures in the definition of $k$-tuply monoidal $\left(\infty ,1\right)$-categories.

Now, since geometric $\infty$-function theory is indeed geometric, we obtain a simple but powerful statement about the $k$-tupliness (heh) of the monoidal structure on our $\left(\infty ,1\right)$-category of $\infty$-functions $C\left(X\right)$ of a space $X$:

As described above, by the fundamental theorem of geometric $\infty$-function theorey , higher traces on $C\left(X\right)$ corresponds to forming higher loop spaces of $X$. More generally, the ${E}_{n}$-center ${Z}_{{E}_{n}}\left(C\left(X\right)\right)$ of $C\left(X\right)$ may be taken to be $C\left(\left[{S}^{n},X\right]\right)$, where $\left[{S}^{n},X\right]$ is my notation for the $n$-sphere space of the generalized space $X$. But there is,

• by construction, a natural action of ${E}_{n+1}$ on $\left[{S}^{n},X\right]$;

• accordingly, a natural action of ${E}_{n+1}$ on $C\left(\left[{S}^{n},X\right]\right)$;

• accordingly, due to the fundamental theorem a natural action of ${E}_{n+1}$ on ${Z}_{{E}_{n}}\left(C\left(X\right)\right)$.

Again, due to the good formalism, this statement becomes almost a tautology. Notice that this statement is otherwise known as the Kontsevich conjecture, which categorifies the Deligne conjecture.

and in particular David Ben-Zvi’s comments to that.

Okay, this entire section here needs more details and in particular more links to nLab entries with more details.

### 3. perfect stacks

In this section of the paper, the notion of a perfect ∞-stack is defined, which is the general context in which the ideas of geometric $\infty$-function theory apply thoroughly without modifications.

In general, for any derived stack $X$ one can define the $\left(\infty ,1\right)$-category $\mathrm{QC}\left(X\right)$ of ‘functions on $X$’. One writes $X$ as a colimit of affine derived schemes $X\simeq {\mathrm{colim}}_{U\in {\mathrm{Aff}}_{/X}}U$ and then one sets

(1)$\mathrm{QC}\left(X\right)={\mathrm{lim}}_{U\in {\mathrm{Aff}}_{/X}}\mathrm{QC}\left(U\right)$QC(X) = lim_{U \in Aff_{/X}} QC(U)

where $\mathrm{QC}\left(U\right)$ for an affine stack $U=\mathrm{Spec}A$ is simply defined as ${\mathrm{Mod}}_{A}$.

One incarnation of derived stack in differential geometry is as an $\omega$-groupoid internal to differential graded manifolds. Since differential graded manifolds are ‘affine’, in the sense that knowledge of global functions is sufficient to determine the manifold, the limit construction above is not necessary. Hence, for such a derived stack (an $\omega$ groupoid $X$ internal to differential graded manifolds) we can more simply set

(2)$\mathrm{QC}\left(X\right)=\mathrm{smooth}\omega -\mathrm{functors}\mathrm{from}X\mathrm{to}\mathrm{chain}\mathrm{complexes}$QC(X) = smooth \omega-functors from X to chain complexes

since the latter plays the role of a ‘derived $ℂ$-module’.

Bruce: I’m shooting in the dark here with this $\omega$-groupoid sentence above. Am I right? What does that boil down to concretely?

A derived stack $X$ is perfect when $\mathrm{QC}\left(X\right)$ is ‘finitely generated’ in an appropriate sense. One first has to decide what means by a ‘finite object’, and then one must decide what it means for $\mathrm{QC}\left(X\right)$ to be ‘generated’ by these objects. There are various routes one could take, but happily in the context of perfect stacks these all turn out to be equivalent. (In fact, it seems more or less true that perfect stacks are precisely those stacks where these various requirements coincide).

### 4. Tensor products and integral transforms

Classical algebra is all about constructions in the category Ab of abelian groups. A ring $R$ in the usual sense is a monoid object in Ab, i.e. an object $R\in \mathrm{Ab}$ together with multiplication and unit morphisms $m:R\otimes R\to R$ and $\eta :ℤ\to R$ so that we have commutativity of appropriate diagrams expressing associativity and unity. Likewise, a right module $M$ over $R$ in the usual sense is an object $M\in \mathrm{Ab}$ together with an action morphism $a:M\otimes R\to M$ such that appropriate diagrams commute, and similarly for left modules.

Brave new algebra is about constructions in the stable (∞,1)-category of spectra ${S}_{\infty }$, which, like Ab, is closed symmetric monoidal (under smash product of spectra) and (co) complete. This means that we can consider algebra objects $R$ and commutative algebra objects $A$ in ${S}_{\infty }$, as well as modules over them.

Even more generally, one can develop a fearless new algebra in which we consider some closed, symmetric monoidal and (co)complete $\infty$-category $C$ and algebras and commutative algebras in it (see higher algebra). For our purposes, we take $C$ to be ${\mathrm{Pr}}^{L}$, the $\infty$-category of presentable (∞,1)-categories with morphisms given by colimit preserving functors. (More on the closed monoidal structure below.)

The particular algebra objects in ${\mathrm{Pr}}^{L}$ of interest to BZFN are stable (∞,1)-categories $\mathrm{QC}\left(X\right)$ of quasi-coherent sheaves on a perfect derived stack $X$ (the homotopy category of $\mathrm{QC}\left(X\right)$ being the good old-fashioned derived category of quasi-coherent sheaves). Here $\mathrm{QC}\left(X\right)$ is symmetric monoidal under the usual tensor product of quasi-coherent sheaves. Given two perfect derived stacks ${X}_{1},{X}_{2}$, consider the diagram

$\begin{array}{ccc}& & {X}_{1}×{X}_{2}\\ & {}^{{p}_{1}}↙& & ↘{}^{{p}_{2}}\\ {X}_{1}& & & & {X}_{2}\end{array}$\array{ && X_1 \times X_2 \\ & {}^{p_1}\swarrow && \searrow{}^{p_2} \\ X_1 &&&& X_2 }

Given an object $𝒫\in \mathrm{QC}\left({X}_{1}×{X}_{2}\right)$, we can define a functor ${\Phi }_{𝒫}:\mathrm{QC}\left({X}_{1}\right)\to \mathrm{QC}\left({X}_{2}\right)$ by pulling-back along ${p}_{1}$, tensoring with $𝒫$, and then pushing-forward along ${p}_{2}$. Thus given an object $ℱ\in \mathrm{QC}\left({X}_{1}\right)$, we have ${\Phi }_{𝒫}\left(ℱ\right):={{p}_{2}}_{*}\left({{p}_{1}}^{*}ℱ\otimes 𝒫\right)$. We think of this an integral transform of the sheaf $ℱ$ with respect to the kernel $𝒫$.

In fact, because of the naturality of the above operations, this process gives a functor

$\Phi :\mathrm{QC}\left({X}_{1}×{X}_{2}\right)\to {\mathrm{Fun}}^{L}\left(\mathrm{QC}\left({X}_{1}\right),\mathrm{QC}\left({X}_{2}\right)\right),$\Phi: QC(X_1 \times X_2) \rightarrow Fun^{L}(QC(X_1), QC(X_2)),

where ${\mathrm{Fun}}^{L}\left(\mathrm{QC}\left({X}_{1}\right),\mathrm{QC}\left({X}_{2}\right)\right)$ is the internal Hom in ${\mathrm{Pr}}^{L}$ and consists of colimit preserving functors and their natural transformations.

The main result of section 4 of BZFN (which has been called above the fundamental theorem of geometric $\infty$-function theory)is that this functor and its cousins are equivalences. As a slogan:

integral transforms = colimit preserving functors

This was first proved in the context of differential graded categories by Toën, building on work of Bondal, Orlov, and others. Note that one can define a functor $\Phi$ in the same way at the level of triangulated categories, but it is known to be badly behaved, and in fact could not be well-behaved, since we do not know how to make the category of triangulated categories into a closed symmetric monoidal category.

#### 4.1 Tensor products of $\infty$-categories

In order to be more precise, we need to look at the closed, symmetric monoidal structure on ${\mathrm{Pr}}^{L}$, as developed by Jacob Lurie DAG II.4 and DAG III.6. The internal hom between two presentable (∞,1)-categories $C,D$ is ${\mathrm{Fun}}^{L}\left(C,D\right)$, which consists of colimit preserving (∞,1)-functors. To construct it, one considers $C$ and $D$ as quasi-categories or weak Kan complexes. Functors from $C$ to $D$ are just maps of simplicial sets, so the (∞,1)-category of (∞,1)-functors from $C$ to $D$ is just the simplicial set of maps (internal Hom in simplicial sets) $\mathrm{Fun}\left(C,D\right)$. This is indeed an $\infty$-category again, whenever $D$ is an $\infty$-category/weak Kan complex. Now inside of $\mathrm{Fun}\left(C,D\right)$, we take the $\infty$-subcategory spanned by $0$-simplices representing colimit preserving functors to get ${\mathrm{Fun}}^{L}\left(C,D\right)$.

Chris: (If there is some inaccuracy noted by anyone, feel free to comment. I might have forgotten some fibrant or cofibrant replacement somewhere.)

For the tensor product of presentable (∞,1)-categories we construct the $\infty$-category $C\otimes D$ which is ‘the universal recipient of a bilinear functor’ from $C×D$. Here, we think of coproducts in $C$ and $D$ as addition, so if a functor $C×D\to E$ preserves colimits in each variable, then in particular it preserves coproducts and so is ‘bilinear’. Such a bilinear functor will factor uniquely (in a homotopic sense) through a universal bilinear functor $C×D\to C\otimes D$, just like for bilinear maps and tensor products of abelian groups.

Now given the above closed symmetric monoidal structure on ${\mathrm{Pr}}^{L}$ and since ${\mathrm{Pr}}^{L}$ also has limits and colimits, we can have all kinds of fun. For instance, given an algebra object $R\in {\mathrm{Pr}}^{L}$, a right $R$-module $M\in {\mathrm{Pr}}^{L}$, and a left $R$-module $N\in {\mathrm{Pr}}^{L}$, we can form their relative tensor product over $R$, $M{\otimes }_{R}N\in {\mathrm{Pr}}^{L}$ as the coequalizer of the pair of functors

$M\otimes R\otimes N\underset{\alpha }{\overset{\beta }{⇉}}M\otimes N$M \otimes R \otimes N \underoverset{\alpha}{\beta}{\rightrightarrows} M \otimes N

where $\alpha$ is the action of $R$ on the right of $M$ and $\beta$ is the action of $R$ on the left of $N$. (This generalizes the relative tensor product of modules over a ring in the category of abelian groups, which is defined to have exactly this coequalizing property.)

In section of 4.1 BZFN, there are various results, which are consequences of the (∞,1)-categorical Barr–Beck theorem.

Chris: They seem interesting and useful, but I don’t seem to need them just at the moment, so I’ll come back to them some other time.

#### 4.2 Sheaves on fiber products

In our present context, we consider a morphism of perfect derived stacks $q:X\to Y$. By pulling-back along $q$ and tensoring, we make $M=\mathrm{QC}\left(X\right)$ into a $R=\mathrm{QC}\left(Y\right)$-module. To see this, note that the functor $\left(?\right)\otimes {q}^{*}\left(?\right):\mathrm{QC}\left(X\right)×\mathrm{QC}\left(Y\right)\to \mathrm{QC}\left(X\right)$ is indeed bilinear since pullback ${q}^{*}$ is the left adjoint of pushforward ${q}_{*}$, so preserves colimits, as does $\otimes$, being the left adjoint of the internal Hom of sheaves (‘sheaf’ or ‘local’ Hom). Thus by the universal property of the tensor product of $\left(\infty ,1\right)$-categories, we do indeed get an action functor $\mathrm{QC}\left(X\right)\otimes \mathrm{QC}\left(Y\right)\to \mathrm{QC}\left(Y\right)$.

Now given a pair of perfect derived stacks ${X}_{1},{X}_{2}$ over $Y$, we get two $R=\mathrm{QC}\left(Y\right)$-modules $M=\mathrm{QC}\left({X}_{1}\right)$ and $N=\mathrm{QC}\left({X}_{2}\right)$ (left and right don’t matter here, since $R=\mathrm{QC}\left(Y\right)$ is symmetric monoidal) and we can form their relative tensor product

$\mathrm{QC}\left({X}_{1}\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left({X}_{2}\right).$QC(X_1) \otimes_{QC(Y)} QC(X_2).

Now we can define a functor

$⊠:={{p}_{1}}^{*}?\otimes {{p}_{2}}^{*}?:\mathrm{QC}\left({X}_{1}\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left({X}_{2}\right)\to \mathrm{QC}\left({X}_{1}{×}_{Y}{X}_{2}\right).$\boxtimes:={p_1}^{*}? \otimes {p_2}^{*}?: QC(X_1) \otimes_{QC(Y)} QC(X_2) \rightarrow QC(X_1 \times_Y X_2).

(We see that the above functor is bilinear (since pullbacks and tensor products preserve colimits) and coequalizes the action of $\mathrm{QC}\left(Y\right)$ on $\mathrm{QC}\left({X}_{1}\right)$ and $\mathrm{QC}\left({X}_{2}\right)$ (by commutativity of the fibre product diagram), so $⊠$ does indeed define a functor.)

The first step in proving that ‘integral transforms=colimit preserving functors’ is to show that $⊠$ is an equivalence. Then one has to show that $\mathrm{QC}\left({X}_{1}\right)$ is self-dual as a $\mathrm{QC}\left(Y\right)$-module, and so conclude that

$\mathrm{QC}\left({X}_{1}\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left({X}_{2}\right)\simeq \mathrm{QC}\left({X}_{1}{\right)}^{*}{\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left({X}_{2}\right)\simeq {\mathrm{Fun}}_{Y}^{L}\left(\mathrm{QC}\left({X}_{1}\right),\mathrm{QC}\left({X}_{2}\right)\right).$QC(X_1) \otimes_{QC(Y)} QC(X_2) \simeq QC(X_1)^{*}\otimes_{QC(Y)} QC(X_2)\simeq Fun_{Y}^{L}(QC(X_1),QC(X_2)).

As an intermediate step, BZFN first establish an equivalence $⊠:\mathrm{QC}\left({X}_{1}{\right)}^{c}\otimes \mathrm{QC}\left({X}_{2}{\right)}^{c}\simeq \mathrm{QC}\left({X}_{1}×{X}_{2}{\right)}^{c}$, where the superscript $c$ denotes the $\infty$-subcategories of compact objects. (Here, we use the tensor product of small, stable idempotent complete $\infty$-categories $\mathrm{QC}\left({X}_{1}{\right)}^{c}\otimes \mathrm{QC}\left({X}_{2}{\right)}^{c}$, which is just like that for presentable $\infty$-categories except that bilinear functors are only required to preserve finite colimits.) Proposition 3.22 established that the external product $\otimes$ takes compact objects to compact objects, so we do get a functor as above, and that the category $\mathrm{QC}\left({X}_{1}×{X}_{2}{\right)}^{c}$ is generated by external products. To prove that $⊠$ is an equivalence, it is therefore sufficient to prove that for ${M}_{i},{N}_{i}\in \mathrm{QC}\left({X}_{i}\right)$, we have a natural isomorphism

${\mathrm{Hom}}_{{X}_{1}×{X}_{2}}\left({M}_{1}⊠{M}_{2},{N}_{1}⊠{N}_{2}\right)\simeq {\mathrm{Hom}}_{{X}_{1}}\left({M}_{1},{N}_{1}\right)\otimes {\mathrm{Hom}}_{{X}_{2}}\left({M}_{2},{N}_{2}\right),$Hom_{X_1 \times X_2}(M_1 \boxtimes M_2, N_1 \boxtimes N_2) \simeq Hom_{X_1}(M_1, N_1) \otimes Hom_{X_2}(M_2, N_2),

which is a nice exercise using the dualizability of the ${M}_{i}$ and the projection formula.

Having established the equivalence $⊠:\mathrm{QC}\left({X}_{1}{\right)}^{c}\otimes \mathrm{QC}\left({X}_{2}{\right)}^{c}\simeq \mathrm{QC}\left({X}_{1}×{X}_{2}{\right)}^{c}$, we can now establish the equivalence without the superscript $c$. Since (by definition of a perfect stack) $\mathrm{Ind}\left(\mathrm{QC}\left({X}_{i}{\right)}^{c}\right)\simeq \mathrm{QC}\left({X}_{i}\right)$ and the fact (Proposition 4.4) that $\mathrm{Ind}:\mathrm{Idem}\to {\mathrm{Pr}}^{L}$ from small idempotent complete stable $\infty$-categories to ${\mathrm{Pr}}^{L}$ is symmetric monoidal, we get that

$\mathrm{QC}\left({X}_{1}\right)\otimes \mathrm{QC}\left({X}_{2}\right)\simeq \mathrm{Ind}\left(\mathrm{QC}\left({X}_{1}{\right)}^{c}\otimes \mathrm{QC}\left({X}_{2}{\right)}^{c}\right)\simeq \mathrm{Ind}\left(\mathrm{QC}\left({X}_{1}×{X}_{2}{\right)}^{c}\right)\simeq \mathrm{QC}\left({X}_{1}×{X}_{2}\right).$QC(X_1) \otimes QC(X_2) \simeq Ind(QC(X_1)^{c} \otimes QC(X_2)^{c}) \simeq Ind(QC(X_1 \times X_2)^{c}) \simeq QC(X_1 \times X_2).

This gives the absolute version of the equivalence we want. To make it relative over $Y$, one has to think about how to actually compute $\mathrm{QC}\left({X}_{1}\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left({X}_{2}\right)$ and ${X}_{1}{×}_{Y}{X}_{2}$ in concrete terms. For the former, you use Barr-Beck and realize the category as modules for the monad ${\pi }_{*}{\pi }^{*}$, where $\pi :{X}_{1}{×}_{Y}{X}_{2}\to {X}_{1}×{X}_{2}$. For the latter, you realize the fibre product as the limit of a cosimplicial diagram. Then some comparision takes place. It would be hard to give a nicer, more concise explanation than in BZFN, so for further details, take a look.

Now given the equivalence $⊠:\mathrm{QC}\left({X}_{1}\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left({X}_{2}\right)\to \mathrm{QC}\left({X}_{1}{×}_{Y}{X}_{2}\right)$, it remains to see that given a morphism $p:X\to Y$ of perfect stacks, $\mathrm{QC}\left(X\right)$ is self-dual as a $\mathrm{QC}\left(Y\right)$-module. For this, we use the already established equivalence $\mathrm{QC}\left(X\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left(X\right)\simeq \mathrm{QC}\left(X{×}_{Y}X\right)$. To establish self-duality, we need to define a trace $\tau :\mathrm{QC}\left(X\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left(X\right)\to \mathrm{QC}\left(Y\right)$ and unit $u:\mathrm{QC}\left(Y\right)\to \mathrm{QC}\left(X\right){\otimes }_{\mathrm{QC}\left(Y\right)}\mathrm{QC}\left(X\right)$ so that the composition $\mathrm{id}\otimes \tau \circ u\otimes \mathrm{id}:\mathrm{QC}\left(Y\right)\to \mathrm{QC}\left(Y\right)$ is the identity. To do this, consider the relative diagonal morphism $\Delta :X\to X{×}_{Y}X$ and define $u={\Delta }_{*}{p}^{*}$ and $\tau :{p}_{*}{\Delta }^{*}$. Then a diagram chase and the base-change formula show that $u$ and $\tau$ satisfy the necessary conditions.

The final result from this section, Corollary 4.12, is useful for the applications to topological field theory:

Given a finite simplicial set $\Sigma$ a perfect stack $X$, we may form the mapping stack ${X}^{\Sigma }$, which is again perfect. Then there is an equivalence $\mathrm{QC}\left({X}^{\Sigma }\right)\simeq \mathrm{QC}\left(X\right)\otimes \Sigma$.

Chris:Haven’t thought this through. Someone may comment, or I’ll come back to it later.

#### 4.3 Geometric base stacks

The ‘fundamental theorem’ described above can be extended to the case where ${X}_{1}\to Y$ is a perfect morphism of geometric stacks (${X}_{1}$ and $Y$ need not be absolutely perfect) and ${X}_{2}\to Y$ is an arbitrary morphism of stacks.

Bruce: What’s a geometric stack?

Chris: A stack is geometric if it is quasi-compact (any open cover has a finite sub-cover) and the diagonal morphism is representable and affine, though that probably doesn’t help much. I don’t know much about stacks yet, but maybe someone else can explain this. I think the point is that one needs some hypotheses to actually prove stuff for stacks.

### 5. Applications

In this section, BZFN study the categorical center $Z$ and trace $\mathrm{Tr}$ of presentable mononoidal $\infty$-categories, which will be defined as analogues of some classical algebraic notions.

Two geometric cases are of special interest.

When the monoidal category is $\mathrm{QC}\left(X\right)$, then there are equivalences

$Z\left(\mathrm{QC}\left(X\right)\right)\simeq \mathrm{QC}\left(\mathrm{LX}\right)\simeq \mathrm{Tr}\left(\mathrm{QC}\left(X\right)\right),$\mathit{Z}(QC(X))\simeq QC(LX) \simeq \mathit{Tr}(QC(X)),

where $\mathrm{LX}=X{×}_{X×X}X$, the derived loop space of $X$.

When the monoidal category is $\mathrm{QC}\left(X{×}_{Y}X\right)$ equipped with convolution, then there is an equivalence

$Z\left(\mathrm{QC}\left(X{×}_{Y}X\right)\right)\simeq \mathrm{QC}\left(\mathrm{LY}\right)$\mathit{Z}(QC(X \times_Y X))\simeq QC(LY)

and once Grothendieck duality has been worked out for derived stacks, then there will be another equivalence

$\mathrm{QC}\left(\mathrm{LY}\right)\simeq \mathrm{Tr}\left(\mathrm{QC}\left(X{×}_{Y}X\right)\right).$QC(LY) \simeq \mathit{Tr}(QC(X \times_Y X)).

The categorical center is a generalization of lots of things: the center of an algebra, the Hochschild cochain complex of an algebra, the Drinfeld double of a monoidal category… It turns out to be an ${E}_{2}$-category, the $\infty$-categorical version of a braided monoidal category. That’s probably explained in section 6.

#### Some background

Let’s start with a good old-fashioned algebra $A$ (an algebra object in $\mathrm{Ab}$ of $\mathrm{Vect}$). We can form the center $Z\left(A\right)$ of $A$, which again is a commutative algebra. One can also form the endomorphisms of $A$ as a bimodule over itself: ${\mathrm{End}}_{A\otimes {A}^{\mathrm{op}}}\left(A\right)$. The natural homomorphism

$Z\left(A\right)\to {\mathrm{End}}_{A\otimes {A}^{\mathrm{op}}}\left(A\right)$Z(A) \rightarrow End_{A \otimes A^{op}}(A)

that sends a central element $x$ to multiplication by $x$ is an isomorphism of algebras (multiplication by $x$ is indeed a bimodule endomorphism since $x$ is central and every bimodule endomorphism $\phi$ is of this form since it is determined by $\phi \left(1\right)=x$).

If we consider $A$ as an algebra object in the (dg enhanced) derived category of $\mathrm{Ab}$, then we can also consider the derived endomorphisms of $A$ as an $A$-bimodule, which is a dg algebra (defined up to quasi-isomorphism) that we may think of as the derived center of $A$:

${\mathrm{RHom}}_{A\otimes {A}^{\mathrm{op}}}\left(A,A\right).$RHom_{A \otimes A^{op}}(A,A).

The cohomology of this dg algebra is a graded algebra known as the Hochschild cohomology of $A$. Its zeroth graded piece is of course just the classical center $Z\left(A\right)={\mathrm{End}}_{A\otimes {A}^{\mathrm{op}}}\left(A\right)$.

To compute ${\mathrm{RHom}}_{A\otimes {A}^{\mathrm{op}}}\left(A,A\right)$ in practice, one has take a projective resolution of the first entry $A$ as an $A$-bimodule and then write down the complex of $\mathrm{Hom}$s from this resolution to $A$. The standard such resolution is the ‘bar resolution’ of $A$, which is built using the functor

$L=A\otimes ?:\mathrm{Mod}{A}^{\mathrm{op}}\to \mathrm{Mod}A\otimes {A}^{\mathrm{op}},$L=A \otimes ?: Mod A^{op} \rightarrow Mod A \otimes A^{op},

which is left adjoint to the forgetful functor $R:\mathrm{Mod}A\otimes {A}^{\mathrm{op}}\to \mathrm{Mod}{A}^{\mathrm{op}}$. Then $\mathrm{LR}$ is a comonad on $\mathrm{Mod}A\otimes {A}^{\mathrm{op}}$ and so applying it to $A$ gives (in the usual way) an augmented simplicial object whose $k$th term is ${A}^{\otimes k+1}$. In particular, the $-1$st term is just $A$. By taking alternating sums of the face maps, one gets a complex ${C}_{*}\left(A\right)$ of free $A$-bimodules that can be shown to be exact and so provides a resolution of $A$. Then the $\mathrm{Hom}$ complex

${\mathrm{Hom}}_{A\otimes {A}^{\mathrm{op}}}\left({C}_{*}\left(A\right),A\right)$Hom_{A \otimes A^{op}}(C_*(A),A)

is known as the Hochschild cochain complex of $A$ and provides the standard model for the derived center of $A$.

Now for the categorical trace. Given two bimodules $M,N$ we can form their relative tensor product $M{\otimes }_{A}\otimes {A}^{\mathrm{op}}N$, giving a new bimodule. We think of this operation as a ‘bilinear pairing’ on the category of bimodules (remember, bilinear=colimit preserving). We could also take tensor product in the derived category of bimodules, which in practice requires that we resolve one of $M$ or $N$ by flat (or a fortiori free) $A$-bimodules, and so get a derived pairing of bimodules. In particular, we could take the derived pairing of $A$ with itself as an $A$-bimodule to get the Hochschild chain complex

$A{\otimes }_{A\otimes {A}^{\mathrm{op}}}{C}_{*}\left(A\right),$A \otimes_{A \otimes A^{op}} C_*(A),

whose homology is the Hochschild homology of $A$.

#### Centers and traces

The above constructions can be carried over to algebra objects $A$ in a closed symmetric monoidal $\infty$-category $𝒞$. In particular, we consider $C={\mathrm{Pr}}^{L}$, the $\infty$-category of presentable $\infty$-categories.

We define the categorical center to be the endomorphism object

${\mathrm{End}}_{A\otimes {A}^{\mathrm{op}}}\left(A\right)\in {\mathrm{Pr}}^{L},$End_{A \otimes A^{op}}(A) \in Pr^{L},

where we consider $A$ as module-category over itself on the left and on the right and we compute the internal $\mathrm{Hom}$ in ${\mathrm{Pr}}^{L}$ of $A$ with itself as bimodule.

Also we define the categorical trace to the the pairing object

$A{\otimes }_{A\otimes {A}^{\mathrm{op}}}A$A \otimes_{A \otimes A^{op}} A

of $A$ with itself as a bimodule, using the tensor product in ${\mathrm{Pr}}^{L}$ and then coequalizing the left and right actions of $A$ on itself to form the relative tensor product.

# Background literature

For the general (∞,1)-categorical formalism

For the stable aspects

For the monoidal aspects

For the general TQFT background and in particular see

In particular see also the beginning of section 4.1 there for more on ${E}_{n}$-monoidal $\left(\infty ,1\right)$-categories.

For more details on loop space objects for derived stacks

John Francis’ work on actions of the little k-cubes operad on $\left(\infty ,1\right)$-categories is here

For more related material see Northwestern TFT Conference 2009.

Revised on July 7, 2013 22:57:51 by Urs Schreiber (89.204.153.62)