The n-Category Café

Some literature.

I am not sure I can give a complete historical account, but one of the early influential articles on generalized differential cohomology, whose first section provides a good introduction, is

D. S. Freed
Dirac charge quantization and generalized differential cohomology
(arXiv).

As far as I am aware, the most recent and sophisticated development of this idea is that described in section 4 of

M. J. Hopkins and I. M. Singer
Quadratic functions in geometry, topology and M-theory
(arXiv).

The title of that article points to what has been one of the main motivations for considering this theory, namely functions that pair two “$n$-things with connection” and serve as, in particular, symplectic forms on the space of all these, thus providing a way to perform geometric quantization of background $n$-fields.

For ordinary differential cohomology ($n$-gerbes with connection) this quantum theory of $n$-connections was developed in two articles by Freed, Moore and Segal, which I once tried to summarize here

Freed, Moore, Segal on p-Form Gauge Theory, I
Freed, Moore, Segal on p-Form Gauge Theory, II .

The basic idea.

Maybe the quickest and most elegant way to describe the idea is this:

For every generalized cohomology theory $\Gamma$, there is a natural homomorphism

$$\Gamma^\bullet(X) \to H^\bullet(X, \mathbb{R}) \otimes \Gamma^\bullet(pt)$$

which sends each generalized cohomology class of a space $X$ to a differential form representing it.

For ordinary cohomology this is just the ordinary image of integral cohomology classes in deRham cohomology.

The corresponding differential generalized cohomology now is the collection of pairs, consisting of a generalized cohomology class together with a specific representative differential form.

This can be expressed as saying that the differential cohomology theory $A^\bullet_\Gamma$ is the pullback of the diagram

$$\array{ && \Omega^\bullet_{closed}(X)\otimes \Gamma^\bullet(pt) \\ && \downarrow \\ \Gamma^\bullet(X) &\to& H^\bullet(X, \mathbb{R}) \otimes \Gamma^\bullet(pt) }$$

namely

$$\array{ A^\bullet_\Gamma(X) &\to& \Omega^\bullet_{closed}(X)\otimes \Gamma^\bullet(pt) \\ \downarrow && \downarrow \\ \Gamma^\bullet(X) &\to& H^\bullet(X, \mathbb{R}) \otimes \Gamma^\bullet(pt) } \,.$$

More precisely, this pullback really has to be read as a weak pullback (homotopy pullback).

That means that as we chase a pair

$$(\omega, F) \in \Gamma^\bullet(X) \times \Omega^\bullet_{closed}(X)\otimes \Gamma^\bullet(pt)$$

consisting of a cohomology class and a closed curvature form from the top left to the bottom right of the diagram, the result along the two different ways need not be equal

$$F = \omega_{\mathbb{R}}$$

but may be just cohomologous

$$F - \omega_{\mathbb{R}} = \partial h \,.$$

That $h$ is the connection. In the case that $\Gamma^\bullet(-) = H^\bullet(-,\mathbb{Z})$ is ordinary integral cohomology, that $h$ is the (Cheeger-Simons) differential character proper, with $\omega$ regarded as its image in cohomology, and $F$ its curvature.

By comparison with the familiar cases, where this is what they are:

for instance an ordinary line bundle with connection on $X$ with Chern class $\omega$ and curvature 2-form $F$ the parallel transport of a connection is an element $h \in C^1(X,\mathbb{R})$ and we have

$$\partial h = \omega_{\mathbb{R}} - F$$

(maybe up to a sign…) Similarly for higher abelian gerbes with connection aka higher Cheeger-Simons differential characters.

Relation to $L_\infty$-connections and $n$-Transport

As emphasized in section 4 of Quadratic functions, a class of a generalized cohomology theory here is best thought of in terms of a map

$$X \to S_\Gamma$$

from $X$ into the spectrum representing the cohomology theory.

Let’s think about this $n$-categorically, in order to make contact with our way of talking:

for our base space $X$, choose a good cover $Y \to X$ and the corresponding Čech groupoid $Y^\bullet$.

An $n$-bundle on $X$ is a morphism

$$g : Y^\bullet \to \mathbf{B} G \,,$$

where $G$ is some $n$-group.

For instance $n$th integral cohomology is obtained by setting

$$G = B^{n-1}U(1)$$

while K-theory is obtained by setting, essentially,

$$G = U(\infty) \simeq U_K$$

or similar.

Equipping such the $n$-bundle represented by such a cocycle with a connection amounts to picking a parallel transport $n$-functor

$$\mathrm{tra} : \Pi_{n+1}(Y^\bullet) \to \mathbf{B} E G \,,$$

where $E G = INN(G)$ denotes the inner automorphism $n+1$ group and where $\Pi_{n+1}(Y^\bullet)$ denotes the fundamental $(n+1)$-groupoid of $Y$ merged with jumps in the fibers.

This has to make the square

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y) &\stackrel{tra}{\to}& \mathbf{B} E G }$$

commute. If the fibers are $(n+1)$-connected, we can reformulate this elegantly as

$$\array{ \Pi_{n+1}^{vert}(Y) &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y) &\stackrel{tra}{\to}& \mathbf{B} E G }$$

The differential version of that (compare the slides I provided here)

is a diagram

$$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(g) \\ \uparrow& & \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(g) }$$

as described at great length in $L_\infty$-connections and applications (pdf, blog, arXiv).

There these diagrams are completed to

$$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(g) \\ \uparrow&& \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(g) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{\{P_i\}}{\leftarrow}& inv(g) }$$

which precisely amounts to picking up the characteristic curvature classes $\{P_i\}$ of the chosen connection which give the “differential” realization of the cohomology class $g$.

Noticing that

$$inv(g) = H^\bullet(B G, \mathbb{R})$$ for $G$ an ordinary compact group, we see that, rationally, the sequence of forms on the universal $G$-bundle

$$\array{ CE(g) \\ \uparrow \\ W(g) \\ \uparrow \\ inv(g) }$$

integrates to

$$\array{ \mathbf{B} G \\ \uparrow \\ \mathbf{B} E G \\ \uparrow \\ \mathbf{B} B G } \,,$$

where the last item has to be read as the one-object $\infty$-groupoid obatined from replacing the space $B G$ (which in general fails to have a group structure) with its rational approximation

$$B G \simeq \prod_k K(\mathbb{Q}, f(k)) \,,$$ (as on the bottom of p. 4 of Freed, Hopkins, Teleman, Twisted equivariant K-theory with complex coefficients) which happens to be an (abelian) $\infty$-group.

We see this way how our $\infty$-transport realizes the notion of differential cohomology in that completing the $n$-bundle (cohomology class) $$\Pi_{n+1}^{vert}(Y) \stackrel{g}{\to} \mathbf{B} G$$

to an $n$-bundle with connection

$$\array{ \Pi_{n+1}^{vert}(Y) &\stackrel{g}{\to}& \mathbf{B} G \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y) &\stackrel{tra}{\to}& \mathbf{B} E G \\ \downarrow && \downarrow \\ \Pi_{\infty}(X) &\stackrel{\{P_i\}}{\to}& \mathbf{B} B G }$$

picks up the differential characteristic forms $\{P_i\}$ representing our cohomology class in deRham cohomology.

Ordinary differential cohomology

Here is a more detailed description of the relation between $n$-transport/$L_\infty$-connections and “differential ordinary cohomology”, aka Cheeger-Simons differential cohomology

(this being essentially a review of what is described starting on slide 616)

When we restrict to ordinary differential cohomology ($n$-gerbes with connection) one nice side-effect is that we can work entirely with strict $n$-groupoids. Which makes many things more tractable.

For each $n$, there is the strict $n$-group

$$B^{n-1} U(1)$$

which is trivial in each degree except of its topmost one, where it has $U(1)$-worth of $(n-1)$-morphisms.

I write

$$\mathbf{B} B^{n-1} U(1)$$

for the corresponding one-object $n$-groupoid which is trivial everywhere except that it has $U(1)$ worth of $n$-morphisms.

For $X$ any smooth space and $\pi : Y \to X$ a good cover (or good surjective submersion, more generally), we can think of the Čech groupoid

$$Y^\bullet = (Y \times_X Y \stackrel{s,t}{\to} Y)$$

as a strict $n$-groupoid by throwing in all the higher $k$-simplices (passing to nerves this means that we look at the simplicial space induced by $Y \to X$ and truncate it beyond level $n$).

Then the descent data for a $B^{n-1} U(1)$-$n$-bundle (= abelian $(n-1)$-gerbe) is a strict $n$-functor

$$g : Y^\bullet \to \mathbf{B} B^{n-1} U(1)$$

and indeed equivalence classes of such $n$-functors realize the $n$-th integral cohomology of $X$:

$$EquivClasses(Hom_{n-Grpd}(Y^\bullet,\mathbf{B} B^{n-1}U(1))) = H^{n+1}(X,\mathbb{Z}) \,.$$

So far this is nothing but an $n$-functorial restatement of the standard fact about Čech cohomology.

Now we turn this into “differential” cohomology by allowing the functor to also act on $n$-dimensional volumes in $X$.

I write:

$$\Pi_{n+1}(X)$$

for the strict fundamental $(n+1)$-groupoid of $X$: $(k \leq n)$-morphisms are thin-homotopy classes of globular smooth $k$-volumes in $X$, $(n+1)$-morphisms are full homotopy classes of globular $(n+1)$-volumes.

Then it’s a theorem (the proof of which has appeared in the literature so far only for $n=1$ and $n=2$, but it’s clear how this continues) that smooth strict $(n+1)$-functors from this path groupoid to $\mathbf{B} B^n U(1)$ are the same as closed $(n+1)$-forms on $X$:

$$n\mathrm{Funct}^\infty(\Pi_{n+1}(X), \mathbf{B}B^n U(1)) = \Omega^{n+1}_{closed}(X) \,.$$

So the question of differential cohomology is how to relate a cocycle $n$-functor

$$g : Y^\bullet \to \mathbf{B} B^{n-1}U(1)$$

with a curvature $(n+1)$-functor

$$F : \Pi_{n+1}(X) \to \mathbf{B} B^n U(1) \,.$$

To relate these, we need a connection on an $n$-bundle whose integral class is given by $g$ and whose curvature is $F$. This works as follows:

There is a strict $(n+1)$-groupoid

$$\Pi_{n+1}(Y^\bullet)$$

whose $k$-morphisms are generated from

- globular $k$-paths in $Y$

- together with “jumps” between $k$-paths in the fiber of $Y$

modulo the obvious relations which say that it does not matter whether I first move smoothly in $Y$ and then jump in the fiber, or the other way round.

More technically, this is the weak pushout

$$\array{ \Pi_{n+1}(Y \times_X Y) &\stackrel{\pi_1}{\to}& \Pi_{n+1}(Y) \\ \downarrow^{\pi_2} && \downarrow \\ \Pi_{n+1}(Y) &\to& \Pi_{n+1}(Y^\bullet) }$$

(called $C(Y)$ in 0705.0452).

Then: $B^{n-1}U(1)$-$n$-bundles with connection are given by smooth $(n+1)$-functors

$$\mathrm{tra} : \Pi_{n+1}(Y^\bullet) \to \mathbf{B} E B^{n-1} U(1)$$

such that pulled back to the jumps in fibers they reproduce a $B^{n-1} U(1)$-cocycle, meaning that they can be completed to a square

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} B^{n-1}U(1) \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y^\bullet) &\stackrel{\mathrm{tra}}{\to}& \mathbf{B} E B^{n-1}U(1) \,. }$$

Here $E B^{n-1} U(1) := INN(B^{n-1} U(1))$ is the inner automorphism $(n+1)$-group of $B^{n-1} U(1)$ and the vertical arrows are the canonical inclusions.

Given that square, it so happens that the shifted part of $\mathrm{tra}$ descends down to $X$ in that we can further complete to a double square

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} B^{n-1}U(1) && (integral class) \\ \downarrow && \downarrow \\ \Pi_{n+1}(Y^\bullet) &\stackrel{\mathrm{tra}}{\to}& \mathbf{B} E B^{n-1}U(1) && (connection) \\ \downarrow && \downarrow \\ \Pi_{n+1}(X) &\stackrel{F}{\to}& \mathbf{B} B^n U(1) && (curvature) \,. }$$

This way the $n$-connection

$$[\mathrm{tra}] \in \bar H^{n+1}(X)$$

refines the integral class

$$[g] \in H^{n+1}(X,\mathbb{Z})$$

to a differential class with $(n+1)$-form curvature

$$F \in \Omega^{n+1}_{closed}(X) \,.$$

Differential K-theory

After the more detailed discussion above about the appearance of ordinary differential cohomology from the point of view of $\infty$-transport/$L_\infty$-connections, I’ll now say something about differential K-theory from that point of view.

Fix a base space $X$ and a surjective submersion $\pi : Y \to X$ as before, and write, also as before, $Y^\bullet$ for the corresponding Čech groupoid.

For $K^0$-classes we can get away with thinking of that as just an ordinary 1-groupoid, since $K^0$ is just about ordinary (1-)bundles.

A $K^0$-class of $X$ is represented by a functor

$$g : Y^{\bullet} \to (\mathbf{B} U) \times \mathbb{Z}$$

where $U = U(\infty)$.

To get started, let’s look at ordinary $U(n)$ first and start with just the cocycle for a rank $n$ vector bundle

$$g : Y^{\bullet} \to \mathbf{B} U(n) \,,$$

where, as before, $\mathbf{B} U(n)$ denotes the one-object groupoid with $U(n)$ worth of morphisms

Again, equipping that with a connection amounts to extending to a diagram

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} U(n) \\ \downarrow && \downarrow \\ \Pi_2(Y) &\stackrel{tra}{\to}& \mathbf{B} E U(n) }$$

of smooth 2-functors, where, also as before, $E U(n) := INN(U(n)) = (U(n) \stackrel{Id}{\to} U(n))$ is the inner automorphism 2-group of $U(n)$.

By the theorem in Smooth functors vs. differential forms, this diagram represents precisely a $U(n)$-bundle with connection.

Recall from the Lie picture that we want to complete further to

$$\array{ CE(u(n)) \\ \uparrow \\ W(u(n)) \\ \uparrow \\ inv(u(n)) }$$

with

$$inv(u(n)) = H^\bullet( B U(n),\mathbb{R}) = \wedge^\bullet( u_2, u_4, u_6, \cdots ) \,.$$

Hence $inv(g)$ can be regarded as the Chevalley-Eilenberg algebra of the Lie $\infty$-algebra

$$b^1 u(1) \oplus b^3 u(1) \oplus b^5 u(1) \oplus \cdots$$

The $\infty$-group integrating that is

$$\Pi_\infty(X_{inv(g)}) = B U(1) \times B^{3} U(1) \times B^5 U(1) \times \cdots$$

and hence we complete to

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} U(n) \\ \downarrow && \downarrow \\ \Pi_2(Y) &\stackrel{tra_\nabla}{\to}& \mathbf{B} E U(n) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{ch(\nabla)}{\to}& \mathbf{B} \prod_i B^{2i+1} U(1) } \,.$$

The bottom morphism picks up the characteristic classes, as always, which here is the Chern character.

While the top square lives in the world of strict smooth 2-categories and strict smooth 2-functors between them, the lower square needs to be read in weak $\infty$ something. I won’t attempt to discuss that in more detail and just assume we trust that this makes sense and exists as the integration of our corresponding Lie $\infty$ diagram

$$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A_{vert}}{\leftarrow}& CE(u(n)) \\ \uparrow && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& W(u(n)) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\stackrel{ch(F_A)}{\leftarrow}& inv(u(n)) } \,.$$

Then as we pass $U(n) \to U(\infty) = U$ we should get the “$n$-transport incarnation” of differential $K^0$-theory in that

$$\array{ Y^\bullet &\stackrel{g}{\to}& \mathbf{B} U \times \{k \in \mathbb{Z}\} && (K-class) \\ \downarrow && \downarrow \\ \Pi_2(Y) &\stackrel{tra_\nabla}{\to}& \mathbf{B} E U && (connection) \\ \downarrow && \downarrow \\ \Pi_\infty(X) &\stackrel{ch(\nabla)}{\to}& \mathbf{B} \prod_i B^{2i+1} U(1) && (Chern character) } \,.$$

And this implies that the connection form on $Y$ itself is a sum of a bunch of (higher) Chern-Simons forms.

Sorry, that’s maybe not too shocking a statement in a way, but I thought it deserves to be said.

Another interesting thing to think about is whether things would prettify here if we’d modeled differential K-classes more explicitly in terms of $\mathbb{Z}_2$-graded vector bundles with super-connections on them.

I once chatted about how there is a nice functorial (parallel transport-like) way to think of the required superconnections here in the entry

Quillen’s superconnections – Functorially .

Plugging the observations made there into the formalism discussed here might lead to pleasing results…