The n-Category Café

What I’ll say

- is about a phenomenon I have been thinking about ever since the start of this blog, originally, when I ran into it, using unusual terminology #, later understanding its relation to terms other people invented #;

- builds on stuff I am currently thinking about with Hisham Sati, Zoran Škoda and Danny Stevenson (mentioned the day before #)

- and was triggered the way I present it now today from a discussion with Thomas Nikolaus, a young colleague here in Hamburg.

It’s about generalized cohomology and generalized homotopy in the sense of $\mathrm{\infty }$-stacks and $\mathrm{\infty }$-co-stacks. For definiteness, here is the setup:

the setup

Let $\mathrm{Spaces}$ be a category of suitable spaces that we want to consider. For the following this can be any topos with a faithful functor $\mathrm{Manifolds}\to \mathrm{Spaces}$. So let’s fix one once and for all.

As a model for nice $\mathrm{\infty }$-categories I choose strict $\mathrm{\infty }$-categories internal to $\mathrm{Spaces}$, denoted $\omega \mathrm{Categories}(\mathrm{Spaces})$. (This is much less drastic for applications than it may seem: by freeing the strictly associative and unital compositions we can regard all things simplicial as $\omega$-categories: the free $\omega$-categories on the corresponding simplicial spaces.)

trivial $\mathrm{\infty }$-bundles

There is a dull but important functor $$P_0(-):\mathrm{Spaces}\to \omega \mathrm{Categories}(\mathrm{Spaces})$$ which sends each space $X$ to the $\omega$-category $P_0(X)$ with $X$ as its space of objects and all morphisms identities.

Dull as it is, this gives, for all $\omega$-categories $C$, rise to the assignment

$$C\mathrm{TrivBund}(-):=\mathrm{hom}(P_0(-),C):{\mathrm{Spaces}}^{\mathrm{op}}\to \omega \mathrm{Categories}(\mathrm{Spaces}),$$ where $\mathrm{hom}$ is the internal hom in $\omega$-cateories: objects are $\omega$-functors, morphisms are lax natural transformations, etc. I’ll assume $C$ to be an $\omega$-groupoid (all compositions in $C$ have strict inverses) so that this becomes pseudo natural transformations and and I don’t have to worry about lax vs. oplax.

Indeed, for $X$ a space we can think of $\mathrm{hom}(P_0(X),C)$ as the $\omega$-category of trivial $C$-principal $\omega$-bundles on $X$: objects are the trivial $C$-principal $\omega$-bundles (if $C$ is not a groupoid but a group, $C=BG$, then there is just a single such, of course), morphisms are the homomorphisms between these, etc.

What are the non-trivial $C$ $\omega$-bundles, then? We glue them from trivial ones. As glue, we use fat points.

fat points as glue: unorientals

A “fat point” is an $\omega$-category which is a cofibrant replacement of the point. For every interger $n$, let $P_{\omega }([n])$ denote the $\omega$-category which is the universal resolution of the indicrete 1-groupoid on $[n]$: so $P_{\omega }([n])$ has as objects the integers $\{\mathrm{0,1,2},..n\}$, as morphisms finite sequences of jumps between these, as 2-morphisms finite sequences of jumps between such jumps, as 3-morphisms – and so on. The notation suggests: $P_{\omega }([n])$ is the fundamental $\omega$-groupoid of the discrete but contractible space with $(n+1)$-points.

I have come to think of $P_{\omega }([n])$ as the $n$th unoriental, since it is much like Street’s $n$th oriental – but without the very orientation that prevents orientals from being resolutions of the point. If there is need to disabuse me from that habit, hopefully some reader will do so.

There is hence a close relation between the $n$th unoriental and the $n$-simplex. There is the obvious sub-$\omega$-category $P_{\omega }^{\ge }([n])\subset P_{\omega }([n])$ whose 1-morphisms are not allowed to go from an integer to a smaller integer. This induces naturally an $\omega$-anafunctor from the poset category $[n]$ to the $n$-th unoriental $$[n]\to P_{\omega }([n])$$ given by the span $$[n]\leftarrow P_{\omega }^{\ge }([n])\to P_{\omega }([n])$$ with the left leg an acyclic fibration (with respect to the “folk” model structure). The upshot is:

unless I am mixed up, this induces naturally on the unorientals the structure of a cosimplicial $\omega$-category $$P_{\omega }:\Delta \to \omega \mathrm{Categories},$$ for each simplex a big fat $\omega$-category modelling the fundamental $\mathrm{\infty }$-groupoid of that simplex regarded as a contractible discrete space.

So that’s the glue we need.

gluing (aka descent)

As Dominic Verity a while ago kindly confirmed (the guess) and explained (the full precise statement) # (only that back then we were talking about orientals where I now take the liberty of using instead unorientals), with the $\omega$-category valued presheaf $$A:{\mathrm{Spaces}}^{\mathrm{op}}\to \omega \mathrm{Categories}(\mathrm{Spaces})$$ being any assignment of structures (such as $C$-principal $\omega$-bundles) to spaces, the collection of structures on $X$ obtained by gluing $A$-structures on a cover $Y\to X$ is the end (in categories enriched over $\omega \mathrm{Categories}(\mathrm{Spaces})$ with respect to the Crans-Gray tensor product $\otimes$) $$\mathrm{Desc}(Y,A):={\int }_{n\in \Delta }\mathrm{hom}(P_{\omega }([n]),A(Y^{[n+1]})),$$ where $Y^{[k]}$ denotes the $k$-fold fiberwise product of $Y$ with itself (let’s assume this exists, i.e $Y\to X$ is regular).

$\omega$-stacks

It may happen that $A$-structures are already sufficiently non-trivial that gluing them this way does not matter much: call $A$ an $\omega$-stack if for all $Y\to X$ we have that the canonical $\omega$-functor $$A(X)\to \mathrm{Desc}(Y,A)$$ is a weak equivalence of $\omega$-categories between $A$-structures on $X$ and glued $A$-structures on $Y$

cohomology

If $A$ is not an $\omega$-stack itself, maybe the collection of all possible ways to glue $A$-objects is. All possible ways here means that we allow all possible covers, so set $$H(-,A):={\mathrm{colim}}_{Y\in \mathrm{Covers}(X)}\mathrm{Desc}(Y,A).$$ This assignment $$H(-,A):{\mathrm{Spaces}}^{\mathrm{op}}\to \omega \mathrm{Categories}(\mathrm{Spaces})$$ sending a space $X$ to the collection of glued $A$-structures on $X$ is the A-cohomology of $X$.

(So here is a first question: what can we say about how often we need to apply $H(-,-)$ to obtain $\omega$-stackification? For any $\omega$-category-valued presheaf $A$, $H(-,A)$ is closer to being an $\omega$-stack, and $H(-,H(-,A))$ is even closer. And so on. Is there any abstract nonsense to guarantee that this stabilizes after a (prefereably short) while?)

differential cohomology

As I understand it now, what I have been trying to think about all along (from my first $n$-Café entry on) is the following special case:

let $$\Pi :\mathrm{Spaces}\to \omega \mathrm{Categories}(\mathrm{Spaces})$$ be a copresheaf of $\omega$-categories. Something which assigns to each space objects that like to be pushed forward.

We have seen one example already, the dull choice $\Pi :=P_0(-)$. But as the notation is supposed to suggest, there are other choice such as $\Pi :=P_n(-)$ and in particular $\Pi :={\Pi }_{\omega }(-)$: the copresheaf which sends every space $X$ to its fundamental $\omega$-groupoid ${\Pi }_{\omega }(X)$ whose $k$-morphisms are suitable classes of suitable images of the standard $k$-disk in $X$.

By just replacing $P_0$ by an arbitrary such $\Pi$ we get for every $\omega$-groupoid $C$ the $\omega$-category valued presheaf of trivial $C$-principal bundles with $\Pi$-connection $$C{\mathrm{TrivBund}}_{\Pi }(-):=\mathrm{hom}(\Pi (-),C):{\mathrm{Spaces}}^{\mathrm{op}}\to \omega \mathrm{Categories}(\mathrm{Spaces}),$$

For instance if $\Pi ={\Pi }_{\omega }$ is the fundamental $\omega$-groupoid, then this is flat $\omega$-bundles with connection. (Non-flat connections arise instead as the obstruction to the extension of $C$ cohomology to flat differential $C$-cohomology through the canonical inclusion $P_0(X)\to {\Pi }_{\omega }(X)$.)

Accordingly, call $$H_{\Pi }(-,C):=H(\Pi (-),C)$$ the differential $C$-cohomology with respect to $\Pi$.

co-gluing (aka codescent)

There are two complementary (dual even, as now described) ways to look at cohomology: in terms of descent and in terms of $\omega$-anafunctors aka morphisms in the homotopy category $\mathrm{Ho}(\omega \mathrm{Categories}(\mathrm{Spaces}))$, namely $\omega$-functors out of $\omega$-categories weakly equivalent to the spaces $X$ (the “hypercovers” if also fibrations).

So, define a codescent $\omega$-category $\mathrm{Codesc}(Y,\Pi )$ to be an $\omega$-category which corepresents the descent $\omega$-category $\mathrm{Desc}(Y,C{\mathrm{TrivBund}}_{\Pi })$ in the sense that we have an equivalence $$\mathrm{hom}(\mathrm{Codesc}(Y,\Pi ),C)\simeq \mathrm{Desc}(Y,C{\mathrm{TrivBund}}_{\Pi })$$ natural in $C$.

I think that

- by using the $\mathrm{hom}$-adjunction in $\omega \mathrm{Categories}(\mathrm{Spaces})$ with the Crans-Gray tensor product $\otimes$, and

- using that the contravariant $\mathrm{Hom}(-,Q)$ sends colimits to limits and hence coends to ends

a bit of general nonsense manipulation shows that this $\mathrm{Codesc}(Y,\Pi )$ is the coend

$$\mathrm{Codesc}(Y,\Pi )={\int }^{n\in \Delta }\Pi (Y^{[n+1]})\otimes P_{\omega }([n]).$$

(So my second question: is this right?)

At least in low dimensional examples I think I can check this:

- $\mathrm{Codesc}(Y,P_0)$ computed in the conext of 1-categories is the ordinary Czech groupoid of $Y$;

- $\mathrm{Codesc}(Y,P_1)$ for 1-categories is the “universal path pushout” described in appendix A of my first article with Konrad Waldorf (blog, arXiv);

- $\mathrm{Codesc}(Y,P_2)$ for 2-categories is the “covering of the path 2-groupoid” described in section 2.1 of my third article with Konrad (blog, arXiv).

$\omega$-co-stacks

So we should say: the $\omega$-category valued co-presheaf

$$B:\mathrm{Spaces}\to \omega \mathrm{Categories}(\mathrm{Spaces})$$ is an $\omega$-co-stack if for all regular $Y\to X$ we have $$B(X)\simeq \mathrm{Codesc}(Y,B).$$

I expect that ${\Pi }_{\omega }$, in particular, should be an $\omega$-co-stack, but I have a proof for this at the moment only for truncations to low dimensions:

- that we have a weak equivalence $\mathrm{Codesc}(Y,P_0)\stackrel{\simeq }{\to }X$ from the Czech groupoid to the underlying space is an old hat;

- that we have a weak equivalence $\mathrm{Codesc}(Y,P_1)\stackrel{\simeq }{\to }{P}_1(X)$ from the “paths in the Czech groupoid” to the underlying path groupoid is lemma 2.15 in my first with Konrad.

- that we have a weak equivalence $\mathrm{Codesc}(Y,P_2)\stackrel{\simeq }{\to }{P}_2(X)$ from the “2-paths in the Czech 2-groupoid” to the underlying path 2-groupoid is the statement in appendix B of my third with Konrad.

(so my third question: does anyone see a general nonsense way to prove, if true, that the fundamental path $\omega$-groupoid ${\Pi }_{\omega }:\mathrm{Spaces}\to \omega \mathrm{Categories}$ is an $\omega$-co-stack with the above definition? )

homotopy

Given this, and by direct duality with the definition of nonabelian cohomolgy for presheaves $A.:{\mathrm{Spaces}}^{\mathrm{op}}\to \omega \mathrm{Categories}$ by $H(-,A):={\mathrm{colim}}_Y\mathrm{Desc}(Y,A)$, it is tempting to define nonabelian homotopy $$H(B,-):\mathrm{Spaces}\to \omega \mathrm{Categories}$$ for co-presheaves $B:\mathrm{Spaces}\to \omega \mathrm{Categories}(\mathrm{Spaces})$ by $$H(B,-):={\lim }_Y\mathrm{Codesc}(Y,B).$$

Indeed, as I mention in a moment, the van Kampen theorem and its higher versions seem to be just this idea applied to the archetypical case of fundamental $n$-groupoids, $B={\Pi }_n$.

Hence my fourth question: there is plenty of literature on cohomology of the above kind, in the existing literature usually with simplicial sets where I have $\omega$-categories, i.e cohomology of simplicial presheaves. I don’t think I have come acrosse the definition just proposed, of the dual concept: homotopy with values in co-presheaves in higher structures. Is there any literature along these lines?

natural differential geometry

Before getting to the van Kampen theorem, one aside on a geometric interpretation of $\omega$-category valued copresheaves:

we have seen that $\omega$-bundles and their pullbacks give nice standard examples of co-presheaves with values in $\omega$-categories. Do we have a similar nice picture for co-presheaves?

By a lucky coincidence, currently Martin Markl is staying in Hamburg and reminds us of his work on “natural differential geometry”, which I reported about in Markl on natural differential operators almost two years ago, when he was visiting last time.

As recalled there, in that context people study “natural bundles” which are (covariant!) functors from manifolds to bundles. They then prove a cute theorem which says that the category of natural bundles is canonically equivalent to the category of jet bundles, bundles assciated to the $k$-jet generalization of frame bundles – $k$-frame bundles – for all integer $k$.

Given that a $k$-frame is really essentially nothing but the infinitesimal version of a $k$-dimensional path, this seems to nicely harmonice with the co-presheaves $\Pi$ of $\omega$-categories of $\omega$-paths discussed above.

finally: the van Kampen theorem

If I understand well, the life work of Ronnie Brown and the work of some of his collaborators on Nonabelian algebraic topology was crucially motivated and grew out of attempts to better understand and generalize to higher fundamental groups the van Kampen theorem.

This theorem provides a way for computing the fundamental group ${\pi }_1(X)$ of a space $X$ from the fundamental groups ${\pi }_1(U_1)$ and ${\pi }_1(U_2)$ of two open subsets of that space covering it $(U_1\bigsqcup U_2)=:Y\to X$ by stating that the obvious projection out of the sum of these two groups “over” the fundamental group ${\pi }_1(U_1\cap U_2)$ of the double intersection onto ${\pi }_1(X)$ is an isomorphism (I am suppressing all basepoint information notationally) $$\pi (U_1){\bigsqcup }_{{\pi }_1(U_1\cap U_2)}\pi (U_2)\stackrel{\simeq }{\to }\pi (X).$$

The left hand side here is the pushout $$\begin{array}{ccc}{\pi }_1(U_1\cap U_2)& \to & {\pi }_1(U_2)\\ \downarrow & & \downarrow \\ {\pi }_1(U_2)& \to & \pi (U_1){\bigsqcup }_{{\pi }_1(U_1\cap U_2)}\pi (U_2)\end{array}.$$

While I had come across the term “van Kampen theorem” when reading Ronnie Brown’s work, I unfortunately had managed to avoid ever looking into any detail into what it is about. Until today, when Thomas Nikolaus told me about his impression that this theorem should be about codescent in the sense of gluing conditions for co-presheaves.

After a bit of discussion, it occurred to me that not only should this be the case, but also that this notion of codescent is precisely the one that I have already been talking about, in articles with Konrad and, actually, in terms of ends and coends just yesterday in Göttingen, in terms of objects co-representing the descent $\omega$–category (following Street, of course).

Just consider the following: Ronnie Brown apparently early on pointed out that a better way to formulate the van Kampen theorem is to pass from fundamental groups to fundamental groupoids. So consider the pushout of fundamental 1-groupoids $$\begin{array}{ccc}{\Pi }_1(U_1\cap U_2)& \to & {\Pi }_1(U_2)\\ \downarrow & & \downarrow \\ {\Pi }_1(U_2)& \to & \Pi (U_1){\bigsqcup }_{{\Pi }_1(U_1\cap U_2)}\pi (U_2)\end{array}$$ and compare with the pushout diagram diagram A.5, p. 47 in Konrad and mine article, whose weak pushout (think: homotopy colimit!) I claimed above is the codescent 1-groupoid for 1-paths, to notice: it’s the same diagram (generalized from just two patches to arbitrary covers and arbitrary surjective submersions).

This seems to show indeed: the van Kampen theorem is about the codescent condition of the $\omega$-costack ${\Pi }_{\omega }$ – or at least ${\Pi }_1$.

Whereby I arrive at my last question: does this sound right to the experts out there? if so: is this “well known”? What would be good related literature to look at?

Update: Some discussion of this stuff here is taking place in another thread:

Here is a question by David Corfield concerning the notion of “co-gluing”.

Here my reply (co-gluing = codescent).

Here Tim Porter has an interesting comment on the relevance of cohomotopy.