## October 21, 2008

### Codescent and the van Kampen Theorem

#### Posted by Urs Schreiber

It seems threre is a nice general picture which exhibits close relations between the following items

- fundamental $\infty$-groupoids

- co $\infty$-stacks

- codescent

- natural differential geometry

- the van Kampen Theorem.

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 $\infty$-stacks and $\infty$-co-stacks. For definiteness, here is the setup:

the setup

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

As a model for nice $\infty$-categories I choose strict $\infty$-categories internal to $Spaces$, denoted $\omega Categories(Spaces)$. (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 $\infty$-bundles

There is a dull but important functor $P_0(-) : Spaces \to \omega Categories(Spaces)$ 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

$C TrivBund(- ) := hom(P_0(-), C) : Spaces^{op} \to \omega Categories(Spaces) \,,$ where $hom$ 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 $hom(P_0(X), C)$ 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, $C = \mathbf{B}G$, 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 $\{0,1,2,..n\}$, 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 $P_\omega^\geq([n]) \subset P_\omega([n])$ 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 $[n] \leftarrow P_\omega^\geq([n]) \to P_\omega([n])$ 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 $P_\omega : \Delta \to \omega Categories \,,$ for each simplex a big fat $\omega$-category modelling the fundamental $\infty$-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 $A : Spaces^{op} \to \omega Categories(Spaces)$ 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 $\omega Categories(Spaces)$ with respect to the Crans-Gray tensor product $\otimes$) $Desc(Y, A) := \int_{n \in \Delta} hom( P_\omega([n]), A(Y^{[n+1]})) \,,$ 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 $A(X) \to Desc(Y, A)$ 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 $H(-, A) := colim_{Y \in Covers(X)} Desc(Y, A) \,.$ This assignment $H(-,A) : Spaces^{op} \to \omega Categories(Spaces)$ 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 $\Pi : Spaces \to \omega Categories(Spaces)$ 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 $C TrivBund_\Pi(- ) := hom(\Pi(-), C) : Spaces^{op} \to \omega Categories(Spaces) \,,$

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 $Ho(\omega Categories(Spaces))$, namely $\omega$-functors out of $\omega$-categories weakly equivalent to the spaces $X$ (the “hypercovers” if also fibrations).

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

I think that

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

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

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

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

(So my second question: is this right?)

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

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

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

- $Codesc(Y, P_2)$ 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

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

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 $Codesc(Y, P_0) \stackrel{\simeq}{\to} X$ from the Czech groupoid to the underlying space is an old hat;

- that we have a weak equivalence $Codesc(Y, P_1) \stackrel{\simeq}{\to} P_1(X)$ 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 $Codesc(Y, P_2) \stackrel{\simeq}{\to} P_2(X)$ 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 $\Pi_\omega : Spaces \to \omega Categories$ is an $\omega$-co-stack with the above definition? )

homotopy

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

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 $(U_1 \sqcup U_2) =: Y \to X$ 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) $\pi(U_1) \sqcup_{\pi_1(U_1 \cap U_2)} \pi(U_2) \stackrel{\simeq}{\to} \pi(X) \,.$

The left hand side here is the pushout $\array{ \pi_1(U_1 \cap U_2) &\to& \pi_1(U_2) \\ \downarrow && \downarrow \\ \pi_1(U_2) &\to& \pi(U_1) \sqcup_{\pi_1(U_1 \cap U_2)} \pi(U_2) } \,.$

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 $\array{ \Pi_1(U_1 \cap U_2) &\to& \Pi_1(U_2) \\ \downarrow && \downarrow \\ \Pi_1(U_2) &\to& \Pi(U_1) \sqcup_{\Pi_1(U_1 \cap U_2)} \pi(U_2) }$ 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.

Posted at October 21, 2008 6:36 PM UTC

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### Re: Codescent and the van Kampen Theorem

This is related:

Carlos Simpson: A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen (pdf ).

On p. 2 this mentions the fundamental (“Poincaré”) $n$-groupoid $\Pi_n(X)$ of a space $X$ and defines “nonabelian cohomology” on $X$ with values in $A$ as $hom(\Pi_(X),A)$ (what I called flat differential nonabelian cohomology above). More on that in section 10, p. 63

Theorem 9.1 on p. 59 is the van Kampen theorem generalized to this case, saying that for $Y \to X$ a cover $\Pi_n(X)$ is isomorphic to the strict pushout $\Pi_n(Y) \sqcup_{\Pi_n(Y^{[2]})} \Pi_n(Y)$.

The codescent I was talking about would be supposed to be the weak pushout here, instead.

Posted by: Urs Schreiber on October 22, 2008 2:07 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I wrote

a bit of general nonsense manipulation shows that this $Codesc(Y,\Pi)$ is the coend $Codesc(Y, \Pi) = \int^{n \in \Delta} \Pi(Y^{[n+1]}) \otimes P_\omega([n])$

(so my second question: is this right?)

I realize that to get there I was using one assumptions which I thought is true, but I should better check.

Let “hom” be inner hom and take all $\omega$-categories to be $\omega$-groupoids.

I started with the definition $Desc(Y,hom(\Pi(-),C)) := \int_{[n] \in \Delta} hom(P_\omega([n]), hom(\Pi(Y^{[n+1]}),C))$ and then used the hom-$\otimes$–adjunction inside the internal hom (e.g. Kelly, (1.27)) to get $\cdots \simeq \int_{[n] \in \Delta} hom(P_\omega([n])\otimes \Pi(Y^{[n+1]}), C) \,.$ Then I assumed the contravariant internal hom behaves as the contravariant external one in that it takes colimits to limits to conclude that $\cdots \simeq hom( \int^{[n] \in \Delta} P_\omega([n])\otimes \Pi(Y^{[n+1]}), C) \,.$

As far as that is right (is it??, I need to go back and check some literature…) it must be natural in $C$ and hence by definition we’d have $Codesc(Y, \Pi) = \int^{[n] \in \Delta} P_\omega([n])\otimes \Pi(Y^{[n+1]}) \,.$

Posted by: Urs Schreiber on October 22, 2008 2:57 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I should add a remark on the relation between the orientals $O(\Delta^{[n]})$, those “unorientals” $P_\omega([n])$ and Ronnie Brown’s $n$-simplex crossed complex $\Pi \Delta^n \,,$ which is the fundamental $n$-groupoid on the standard $n$-simplex regarded as a filtered space (i.e. objects have to be the corners of $\Delta^n$, 1-morphisms the edges, 2-morphisms the faces, 3-morphisms the tetrahedra, and so on.)

I believe that

- where $O(\Delta^{[n]})$ is the free $\omega$-category on the $n$-simplex

- it should be true that $P_\omega([n])$ is the free weak $\omega$-groupoid on the $n$-simplex

- and that $\Pi \Delta^n$ is the free $\omega$-groupoid on the $n$-simplex.

(Here by “weak $\omega$-groupoid” I mean an $\omega$-category in which each cell is an $\omega$-equivalence, while by “$\omega$-groupoid” I mean an $\omega$-category in which each cell has a strict inverse.)

All of what I said above could be considered with $P_\omega([n])$ replaced by $\Pi \Delta^n$. That will in fact make many things easier.

For instance, using the explicit formulas that Ronnie Brown gives for the crossed complex corresponding to $\Pi \Delta^n$ it is easy to show that for $\mathbf{A} : Spaces^{op} \to \omega Groupoids$ a presheaf coming from a sheaf $[\mathbf{A}]$ of complexes of abelian groups (by Brown-Higgin’s equivalence of crossed complexes with $\omega$-groupoids) we have that $H(X,\mathbf{A}) := colim_Y Desc(Y,\mathbf{A}) = colim_Y \int_{[n] \in \Delta} hom(\Pi\Delta^{n}, \mathbf{A}(Y^{[n+1]}))$ is precisely Čech cohomology on $X$ with coefficients in the sheaf $[\mathbf{A}]$ of complexes of abelian groups $H(X,\mathbf{A})/_\sim = Č(X,[\mathbf{A}]) \,.$

If indeed $\Pi \Delta^n$ is the free $\omega$-groupoid on the $n$-th oriental (as I think it is), using an adunction $F : \omega Category \leftrightarrow \omega Groupoids : U$ then this would in turn show that Street’s original definition of descent in terms of orientals does reproduce Čech cohomology with coeffients in sheaves of complexes of groups, too. This is a statement which clearly was intended to be obtained from orientals, but which I haven’t seen stated, or even mentioned, anywhere (?)

Posted by: Urs Schreiber on October 24, 2008 1:03 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I have not yet understood all of what you are saying here, but it always seemed to me that there should be a simplish extension of the following well known argument to higher dimensions and the existence of the various forms of higher van Kampen theorem were evidence for Grothendieck’s dream’.

The argument starts with the classical vKT, say given by a pushout of group(oid)s, but if one wants to use groups then categorification should be done. For simplicity assume given an open cover with two open sets, and two covering spaces over them that restrict to isomorphic ones on the intersection. (Of course, we must keep that isomorphism.) Take now the category of sets and realise the two covering spaces as G-sets’ for the two fundamental groups as the two Gs. This gives a lax square of categories and functors, with inside it a pushout square of categories. The gluing of the two covering spaces to make one over the whole space is the induced functor from $\pi_1$ of the whole space to Sets.

There are some hidden traps in that but the argument intuitively works. Now assume we work with crossed modules / cat-groups or whatever you favorite model for the 2-types is. There are crossed module versions of the VKT, and the argument goes through with a bit more checking or so it would seem. Now there are vKTs for n-types (Brown Loday for instance) and although that is not optimised for the situation, it should be possible to extend that argument with $n-1$-stacks or whatever giving $n-1$’ functors to the category of weak $n-1$ categories (or similar).

This however looks more like descent than codescent to me, so I feel a bit confused.

If I understand what you are trying to do, my thought experiment justification for believing in Grothendieck’s dream is very closely related to what you are doing (if it was souped up to allow for differential structures etc, and with the extra knowledge of stacks etc that is now available.)

Posted by: Tim Porter on October 24, 2008 5:46 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Tim Porter wrote:

This gives a lax square of categories and functors,

[…]

This however looks more like descent than codescent to me

Thanks a lot for your comment! I believe I see what you are saying, but maybe we should draw these diagrams that we are talking about, to be sure that we have the same thing in mind.

So, let me first state the 0-dimensional version, to set us up.

Given a 0-category valued presheaf $\mathbf{A} : Spaces^{op} \to 0Cat$ and given a cover $Y \to X$ of Spaces, which gives rise to the simplicial space $Y \times_X Y \stackrel{\to}{\to} Y$ the descent 0-category is the equalizer $\array{ Desc(Y,\mathbf{A}) &\to& \mathbf{A}(Y) \\ \downarrow && \downarrow \\ \mathbf{A}(Y) &\to& \mathbf{A}(Y^{[2]}) } \,,$ where I write $Y^{[2]} := Y \times_X Y$ as usual. So $Desc(Y,\mathbf{A})$ is the collection of all those elements of $\mathbf{A}(Y)$ which glue on $Y^{[2]}$ in that their two possible pullbacks to $\mathbf{A}(Y^{[2]})$ coincide.

And $\mathbf{A}$ is a 0-stack aka sheaf if its value is equivalent (i.e. isomorphic) to its descent 0-category $\mathbf{A}(X) \simeq Desc(Y,\mathbf{A}) \,.$

The theory of descent for presheaves with values in higher categories is all about weakening this equalizer. For instance in the next degree, with $\mathbf{A} : Spaces^{op} \to 1Cat$ a category valued presheaf, the descent 1-category is “something like” the weak equalizer of $\mathbf{A}(Y^{[3]} \stackrel{\stackrel{\to}{\to}}{\to} Y^{[2]} \stackrel{\to}{\to} Y) = \mathbf{A}(Y^{[3]}) \stackrel{\stackrel{\leftarrow}{\leftarrow}}{\leftarrow} \mathbf{A}(Y^{[2]}) \stackrel{\leftarrow}{\leftarrow} \mathbf{A}(Y) \,.$

The (or at least one) way to make this “something like” precise is to say this weak equalizer is the end $Desc(Y, \mathbf{A}) = \int_{[n] \in \Delta} \mathrm{hom}( \Pi(\Delta^n), \mathbf{A}(Y^{[n+1]}) ) \,.$ The presheaf $\mathbf{A}$ is a stack if it is equivalent to its descent 1-category for all $Y$ $\mathbf{A}(X) \simeq Desc(Y,\mathbf{A}) \,.$

You know this. I am just stating it again to make sure that we know what we each are talking about.

Now to codescent: codescent is for co-presheaves:

so let now $\mathbf{B} : Spaces \to 1Cat$ be a 1-category valued co-presheaf. Its descent 1-category is “something like” the weak coequalizer $\array{ \mathbf{B}(Y^{[2]}) &\to& \mathbf{B}(Y) \\ \downarrow &\Downarrow^{\simeq}& \downarrow \\ \mathbf{B}(Y) &\to& Codesc(Y,\mathbf{B}) } \,.$ I have drawn it in this square pushout-kind form (omitting here the condition for the 2-cell on $Y^{[3]}$) to make the relation to van Kampen manifest. The details for this weak coequalizer diagram are in the appendix of Konrad and mine Parallel transport and functors.

Next, I try to repeat in terms of this language what you said in your message, and you please tell me if I got you right.

So now we consider $\mathbf{B} := \Pi_1$, the fundamental groupoid presheaf $\Pi : Spaces \to 1Groupoids$ which sends each space to its fundamental 1-groupoid. We want to check if this is actually a costack.

From van Kampen we know that $\Pi_1(X)$ happens to be the strict coequalizer of the above diagram $\array{ \Pi_1(Y^{[2]}) &\to& \Pi_1(Y) \\ \downarrow && \downarrow \\ \Pi_1(Y) &\to& \Pi_1(X) } \,.$ In particular, by the universal property of $Codesc(Y,\Pi)$ this gives us a universal morphism $Codesc(Y,\Pi) \stackrel{\simeq}{\to} \Pi(X) \,,$ which (as in lemma 2.something in Konrad and my article) one checks is a weak equivalence of categories.

So $\Pi_1$ is a costack!

One point I wanted to make in the above entry is that if we pass in the vanKampen setup from strict to suitably weak coequalizers, then we see that vanKampen is all about codescent.

(And I observed that there is a general formula for this weak coequalizer $Codesc(Y,\mathbf{B})$, namely the coend $Codesc(Y,\mathbf{B}) = \int^{[n] \in \Delta} \Pi(\Delta^n) \otimes \mathbf{B}(Y^{[n+1]})$)

But please let me know what you think.

Now there are vKTs for $n$-types (Brown Loday for instance)

Ah, maybe I don’t know this reference. I’ll try to find it. But maybe you could post the link? Thanks!

Posted by: Urs Schreiber on October 25, 2008 4:48 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

For information, here are the references for the Brown Loday higher vam Kampen theorem (taken from Ronnie’s publication list on the web)

R.Brown, J.-L.Loday, Van Kampen theorems for diagrams of spaces’, Topology, 26, 311-335, 1987.

R.Brown, J.-L.Loday, Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces’, Proc. London Math. Soc. (3), 54, 176-192, 1987. pdf

Ronnie has a discussion of these and lots more on his webpages for instance at

http://www.bangor.ac.uk/~mas010/hdaweb2.html

Posted by: Tim Porter on October 26, 2008 9:05 AM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I see what you are saying. Yes, from that point of view vKT is about codescent, but it is also about descent I think, possibly since when looking at the suitably weak morphisms from a model of an n-type to n-Cat or whatever, the variation is contravariant’ in the space part. This is flipping codescent in the n-groupoids to descent in the stacky things.

Posted by: Tim Porter on October 26, 2008 9:14 AM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I see what you are saying. Yes, from that point of view vKT is about codescent, but it is also about descent I think, possibly since when looking at the suitably weak morphisms from a model of an n-type to n-Cat or whatever, the variation is contravariant’ in the space part. This is flipping codescent in the n-groupoids to descent in the stacky things.

Okay, good. So let me see. Is what you just said maybe related to that computation which I mentioned:

consider the $\omega$-category presheaf $\mathbf{A} := hom(\Pi(-),C)$

of trivial $C$-principal $\omega$-bundles with “$\Pi$-connection”, for $C$ any $\omega$-groupoid and $\Pi$ any $\omega$-category valued co-presheaf (for instance the fundamental $\omega$-groupoid copresheaf). Then the descent $\omega$-category for $\mathbf{A}$ is corepresented by the codescent $\omega$-category

$Desc(Y, hom(\Pi(-),C)) \simeq hom(Codesc(Y, \Pi), C) \,,$ where $hom$ is the inner hom in $\omega$-groupoids.

Posted by: Urs Schreiber on October 26, 2008 11:11 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

It certainly looks that way. If, as I seem to remember, you are defining Descent and Codescent objects using end / coend formula, this should be a simple bit of fun with the end calculus. (The devil, as always, is in the details!)

This would justify completely your view of VKT as being exactly this statement. The one slight concern that I have is that the various forms of vKT that Ronnie has worked on needed some side conditions on the situation. As yet I do not quite see where they come in in this interpretation. They probably are absorbed into conditions on the cover $Y\to X$. What is your feelin about this?

Posted by: Tim Porter on October 27, 2008 8:48 AM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

this should be a simple bit of fun with the end calculus. (The devil, as always, is in the details!)

Yes. It’s a simple computation, displayed here, if I assume that the contravariant internal $hom(-,C)$ takes colimits to limits, and hence coends to ends.

This ought to be true. I need to go back to Kelly’s bock to check. Unless some expert just tells me…

The one slight concern that I have is that the various forms of vKT that Ronnie has worked on needed some side conditions on the situation. As yet I do not quite see where they come in in this interpretation. They probably are absorbed into conditions on the cover $Y \to X$. What is your feelin about this?

I am not sure. One thing that comes to mind is that for the codescent story, the fact that $\Pi_1(X)$ happens to be a strict pushout of the relevant codescent diagram is not relevant. In that to establish the weak equivalence $Codesc(Y, \Pi_1) \stackrel{\simeq}{\to} \Pi_1(X)$ one does not need to use or even mention any pushout properties of $\Pi_1(X)$. All that enters is that $Codesc(Y, \Pi_1)$ is, by definition, the weak pushout. So this says that $\Pi_1(X)$ is necessarily weakly equivalent to a weak pushout, but does not assert anything about it being a strict pushout.

So it might be that what Ronnie Brown discusses is a statement which is strictly stronger than just saying “$\Pi_1$ is a costack”.

(There is certainly an assumption on $Y \to X$, namely (if we are working with manifolds) that it is a surjective submersion. But since every cover by open subsets is a surjective submersion, this is satisfied.)

Posted by: Urs Schreiber on October 27, 2008 4:01 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

In that calculation of descent and codscent is there a total object’ and simplicial replacement version as well. That might be useful for interpreting things at least in low dimensions. For instance it might enable 2-dimensional versions of the theory to be more readily compared with the Brown Higgins theory.

Posted by: Tim Porter on October 27, 2008 4:46 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

In that calculation of descent and codscent is there a ‘total object’ and simplicial replacement version as well.

Not sure exactly what you mean by ‘total object’ here. If you mean an explicit description of the codescent $n$-groupoid $Codesc(Y, \Pi_n)$ for $n=1,2$, that’s in my articles with Konrad.

Concerning the simplicial version: I have the suspicion (we talked about that once already #) that simplicially the codescent object is obtained by

a) start with the simplicial $n$-category $\cdots \Pi_n(Y^{[3]}) \stackrel{\stackrel{\to}{\to}}{\to} \Pi_n(Y^{[2]}) \stackrel{\to}{\to} \Pi_n(Y) \,,$ then

b) take nerves in eauch degree to get a bisimplicial set, and then

c) take the codiagonal of that.

Last time when I asked about the codiagonal you pointed me to a thesis # which was very helpful in that it gave the formula for the codiagonal explicitly.

If one looks at that formula, one sees that “morally” it does the same thing as the coend formula for the Codescent object which I gave, in that it produces a something whose $k$-cells come from $r$-cells of $\Pi_n(Y^{[s+1]})$ for $r+s = k$.

I am expecting there to be some abstract nonsense way to see this relation and make it precise. But I haven’t found it yet.

Posted by: Urs Schreiber on October 27, 2008 7:57 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

For total objects the basic source was originally Bousfield and Kan.

They looked at a cosimplicial simplicial set as a functor from Delta to S= simplicial sets, then looked at the simplicial set of natural transformations from the Yoneda embedding to that functor. This gave a way to build a good simplicial set from a cosimplicial simplicial set. The term has since spread to other contexts, from a cosimplicial category Bourn and Cordier give an indexed limit which is the total category and which is I think the descent category in Duskin’s sense. I tend to use the term to mean any good functor which goes from BiS or Cosimp(S) to S or similar, so the codiagonal and diagonal functors are in this sense total complex functors. (This stuff is in the next part of the menagerie notes which I hope to finish soon. They should sketch out the links in this area.) The links with homotopy limits and colimits were explored by Bourn and Cordier in the 1980s and for homotopy ends and coends by Cordier and myself in our Trans Amer Math Soc article.

The simplicial and cosimplicial replacement functors should also be useful. These are given by Bousfield and Kan and are used in most of the papers mentioned above.

Posted by: Tim Porter on October 27, 2008 8:57 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Ah, now I see what you meant by “total object”.

Hm, so is this now supposed to lead me to the conclusion that, simplicially, the descent object is simply $Desc(Y,\mathbf{A}) = codiag(\mathbf{A}(Y^\bullet))$ for $\mathbf{A}$ a simplicial presheaf and $Y^\bullet$ a simplicial space (some hypercover, say) and that the codescent object is similarly simplicially simply $Codesc(Y,\mathbf{B}) = codiag(\mathbf{B}(Y^\bullet))$ where now $B$ is a simplicial co-presheaf and in both cases “codiag” is, as you say

any good functor which goes from $BiS$ or $Cosimp(S)$ to $S$ or similar

?

Let’s see, we should be able to check the relation of this to the globular formulas we had before, using the fact that the $\omega$-nerve for $\omega$-groupoids is $N(C) : \Delta^{op} \stackrel{\Pi(\Delta^{(-)})}{\to} \omega Groupoids^{op} \stackrel{Hom(-,C)}{\to} Sets$ with left adjoint $F : SimpSet \to \omega Groupoids$ given by $F(S) = \int^{n \in \Delta} S^n \times \Pi(\Delta^n) \,.$

Now, what was again the end/coend formula for the codiagonal?

Posted by: Urs Schreiber on October 27, 2008 9:22 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Let me add something more into this. The lovely result of Cegarra and Remedios suggests that codiagonal and diagonal are equivalent up to weak equivalence (in the simplicial case).

The codiagonal $\nabla$ is the right adjoint of Illusie’s total decalage (which is composition with the ordinal sum functor $\oplus: \Delta^{op} \times \Delta^{op} \to \Delta^{op}$). This gives it as a right Kan extension and hence gives an end formula. (I can give it if it would help.) It also has a description due to Artin and Mazur as follows:

$\nabla(Y)_n = \{y = (y_0,\ldots,y_n) \mid y_i \in Y_{i,n-i}, d_0^v y_i = d^h_{i+1}y_{i+1} for 0\leq i \leq n-1\}.$

For the face and degeneracy maps $d_i({y})=(d_i^v y_0,d_{i-1}^v y_1,\ldots, d_1^v y_{i-1}, d_i^h y_{i+1},\ldots, d_i^h y_n)$ and $s_i({y}) = (s_i^v y_0,s^v_{i-1} y_1, \ldots, s_0^v y_i,s_i^h y_i,s_i^h y_{i+1},\ldots, s_i^h y_n).$

Posted by: Tim Porter on October 27, 2008 9:48 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

On that question, I suspect the answer is almost’. I deliberately did not define good’!!! and sometimes you need a fibrant or cofibrant thingy to get an invariant answer, so in principal YES your idea looks right.

Posted by: Tim Porter on October 27, 2008 10:05 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Now, what was again the end/coend formula for the codiagonal?

Ah, found it:

$(\nabla X)_n = \int^{[p],[q]} \Delta([p] + [q], [n]) \times X_{p,q} \,.$ Hm, that must be a typo.(??) Let me assume the right expression is $(\nabla X)_n = \int^{[p],[q]} \Delta([n], [p] + [q]) \times X_{p,q}$ so that in the integrand the dependence of $(p,q)$ is correctly contra- and co-variant and the total dependence on $[n]$ contravariant.

So then for $\mathbf{B} : Spaces \to \omega Categories$ I want to check if first setting $X_{p,q} = N_q(\mathbf{B}(Y^{[q+1]}))$ and then computing the codiaginal of that and sending it back to $\omega$-categories by means of the left adjoint $F$ of the $\omega$-nerve $F\left( \int^{[p],[q]} \Delta([-], [p] + [q] ) \times N_p(\mathbf{B}(Y^{[q+1]})) \right) = \int^{[n],[p],[q]} \Delta( [n], [p] + [q]) \times N_p(\mathbf{B}(Y^{[q+1]})) \times \Pi(\Delta^n)$ is equivalent to the codescent coend we have been talking about.

That looks like I should use Fubini and write $\cdots = \int^{[p], [q]} \left( \int^{[n]} \Delta([n], [p]+ [q]) \times \Pi(\Delta^n) \right) \times N_p(\mathbf{B}(Y^{[q+1]})) \,.$ The term in brackets should be the free $\omega$-groupoid on the $[p+q]$-simplex, so just $\Pi(\Delta^{p+q})$ again, I suppose.(?) So $\cdots \simeq \int^{[p], [q]} \Pi(\Delta^{p+q}) \times N_p(\mathbf{B}(Y^{[q+1]})) \,.$ This makes me want to proceed with writing out the nerve as $\cdots \simeq \int^{[p], [q]} \Pi(\Delta^{p+q}) \times Hom(\Pi(\Delta^p), \mathbf{B}(Y^{[q+1]})) \,.$ Hm, this is not looking too bad, maybe. I want to end up with $\int^{[q]} \Pi(\Delta^q) \otimes \mathbf{B}(Y^{[q+1]}) \,.$

Hm…

(careful with the above computation, I am just playing around here.)

Posted by: Urs Schreiber on October 27, 2008 10:21 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I assume that the contravariant internal hom(-,C) takes colimits to limits, and hence coends to ends

I think there is no problem as you just need to follow the formula through using adjointness of tensor with hom/cotensor. The equaliser, coequaliser construction of ends and coends should then do the trick. (I have been looking at this a bit recently with homotopy limits and colimits, which are very closely related to your constructions, and withing the simplicialy enriched settings I know there is nothing to worry about. In your slightly more structured case I am not 100% certain but nearly!)

Posted by: Tim Porter on October 27, 2008 4:53 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

if I assume that the contravariant internal $hom(-,C)$ takes colimits to limits, and hence coends to ends

Assuming you are working in a symmetric (or at least braided) monoidal closed category, this is true. We have

(1)$\mathbb{C}(A, hom(B, X))\;\cong\;\mathbb{C}(B, hom(A, X)),$

and so $hom(-,X)$ is self-adjoint on the right.

Posted by: Robin Houston on October 27, 2008 5:01 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Assuming you are working in a symmetric (or at least braided) monoidal closed category,

Yes, everything in this discussion is in the context of $\omega$-groupoids, equivalently: crossed complexes, which is symmetric monoidal closed.

Eventually I will want to see how far the constructions go beyond that to more general $\omega$-categories (for which the tensor is no longer symmetric), but let’s leave that for later…

We have

(1)$\mathbb{C}(A, hom(B, X))\;\cong\;\mathbb{C}(B, hom(A, X)),$

and so $hom(-,X)$ is self-adjoint on the right.

Hm, sorry for the dumb question, but what is $\mathbb{C}$ here? And how does this imply the desired statement?

Posted by: Urs Schreiber on October 27, 2008 7:40 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Sorry, Urs. I wrote that comment rather quickly from work, and I should have been clearer. Your talk of $\omega$-groupoids makes me a little nervous that what I’m saying isn’t applicable to your situation – I confess I haven’t followed this thread very closely – but I’ll try to spell it out clearly and we’ll see.

Suppose we have a symmetric monoidal closed category $\mathbb{C}$. Then there is a natural isomorphism

(1)$\mathbb{C}(A\otimes B,C)\cong\mathbb{C}(A, hom(B,C)).$

So the symmetry of tensor gives a natural isomorphism

(2)$\mathbb{C}(A,hom(B,C))\cong(\mathbb{C}(A\otimes B,C)\cong\mathbb{C}(B\otimes A, C)\cong\mathbb{C}(B, hom(A,C)).$

Put slightly differently, that’s a natural isomorphism

(3)$\mathbb{C}^{op}(hom(B,C),A)\cong\mathbb{C}(B,hom(A,C)),$

or in other words an adjunction between the functor

(4)$hom(-,C): \mathbb{C}\to\mathbb{C}^{op}$

and the functor

(5)$hom(-,C): \mathbb{C}^{op}\to\mathbb{C}.$

The upshot is that $hom(-,C): \mathbb{C}\to\mathbb{C}^{op}$ is a left adjoint, which therefore preserves coends. That means it will take a coend in $\mathbb{C}$ to a coend in $\mathbb{C}^{op}$, otherwise known as an end in $\mathbb{C}$.

That’s more or less what you wanted, I hope?

Posted by: Robin Houston on October 27, 2008 10:26 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Robin,

thanks a lot, that’s great.

Put slightly differently, that’s a natural isomorphism

(1)$\mathbb{C}^{op}(hom(B,C),A)\cong\mathbb{C}(B,hom(A,C)),$

It’s simple tricks like that which they forgot to teach me in school, so I need to learn them here in the café.

That’s more or less what you wanted, I hope?

Yes, exactly. That means I can check the second question in my above entry with YES. :-)

Your talk of $\omega$-groupoids makes me a little nervous that what I’m saying isn’t applicable to your situation

There is no need to worry here, i think. $\omega Groupoids$ is well known to be symmetric monoidal closed by a list of results by Brown and Higgins.

(And I think the reason is clear: $\omega Categories$ is biclosed, with the left hom being given by lax transformations and the right hom by op-lax transformations. If we are however in $\omega$-groupoids (i.e. all cells have strict inverses) then in both cases these are pseudo-transformations and we can interchange them.)

Posted by: Urs Schreiber on October 27, 2008 10:40 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

Oh, good!

I’ve just realised that essentially the same argument works for the (not necessarily symmetric) biclosed case too. If we have natural isomorphisms

(1)$\mathbb{C}(A\otimes B, C)\cong\mathbb{C}(A, hom(B,C))$

and

(2)$\mathbb{C}(A\otimes B,C)\cong\mathbb{C}(B,hom'(A,C))$

then the functor

(3)$hom(-,C): \mathbb{C}\to\mathbb{C}^{op}$

(4)$hom'(-,C): \mathbb{C}^{op}\to\mathbb{C},$

which means that both $hom(-,C)$ and $hom'(-,C)$ take coends in $\mathbb{C}$ to ends in $\mathbb{C}$.

Posted by: Robin Houston on October 27, 2008 11:12 PM | Permalink | Reply to this

### Re: Codescent and the van Kampen Theorem

I’ve just realised that essentially the same argument works for the (not necessarily symmetric) biclosed case too.

Ah, right. Very nice. Thanks, that was very helpful. Now in hindsight it looks so trivial!

All right, so this means that we have the equivalence $Desc(Y, hom(\Pi(-),C)) \simeq hom(Codesc(Y,\Pi),C)$ even for $C \in \omega Categories$.

I keep thinking that I should be looking for an analogous but dual formula, something like $Codesc(Y,hom(C^\infty(-),\Pi)) \simeq hom( ... )$ but I don’t really know yet how to fill in the blanks. Nor if this is really on the right track…

Posted by: Urs Schreiber on October 27, 2008 11:34 PM | Permalink | Reply to this
Read the post Local Nets and Co-Sheaves
Weblog: The n-Category Café
Excerpt: Co-sheaf condition (codescent) for Haag-Kastler nets of local quantum observables?
Tracked: November 14, 2008 3:12 PM

### Re: Codescent and the van Kampen Theorem

Let me come back to this relation between the higher van Kampen theorem and the fact that $\Pi_\omega$ should be an $\omega$-costack.

So, take $X$ to be some (filtered) space and $Y^\bullet \to X$ a (hyper)cover. Then we can form the codescent $\omega$-groupoid

$Codesc(Y^\bullet, \Pi_\omega) := \int^{[n] \in \Delta} \Pi_\omega(\Delta^n) \otimes_{CransGray} \Pi_\omega(Y^n) \,.$

Here $\Pi_\omega(\Delta^n)$ is the fundamental $\omega$-groupoid of the standard $n$-simplex regarded as a filtered space with the standard filtration, so this is just, I suppose, the free $\omega$-groupoid (strict everything) on the simplicial set underlying the standard $n$-simplex.

Now, unless I am (still) mixed up, all these $\Pi_\omega(\Delta^n)$ are weakly equivalent to the point $\Pi_\omega(\Delta^n) \stackrel{\simeq}{\to} \mathrm{pt} \,.$

Okay, now my question:

it would be tempting to reason that if for two functors $T , T' : \Delta^{op} \times \Delta \to \omega Groupoids$ such that there is a transformation $T \Rightarrow T'$ which is a weak equivalence of $\omega$-groupoids in each component, that then this translates to coends:

$\int^{[n] \in \Delta} T(n,n) \stackrel{\simeq}{\to} \int^{[n] \in \Delta} T'(n,n) \,.$

But IF we have somehting like that, then we would get

\begin{aligned} Codesc(Y^\bullet, \Pi_\omega) &:= \int^{[n] \in \Delta} \Pi_\omega(\Delta^n) \otimes \Pi_\omega(Y^\bullet) \\ &\stackrel{\simeq}{\to} \int^{[n] \in \Delta} pt \otimes \Pi_\omega(Y^n) \\ &= \int^{[n] \in \Delta} \Pi_\omega(Y^n) \\ &= colim_{n} \Pi_\omega(Y^n) \end{aligned}

On the last line this is just the strict colimit. So this would relate the “homotopy colimit” given by codescent to the strict colimit. The higher van Kampen theorem would say that the last line is indeed equivalent to $\Pi_\omega(X)$ $\cdots \simeq \Pi_\omega(X) \,.$

In total we would have obtained a weak equivalence $Codesc(Y^\bullet, \Pi_\omega) \stackrel{\simeq}{\to} \Pi_\omega(X)$ for each (hyper)cover $Y^\bullet \to X$. Which would mean that $\Pi_\omega$ is an $\omega$-costack.

Can some argument like this be made to work?

Posted by: Urs Schreiber on November 25, 2008 2:04 PM | Permalink | Reply to this

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