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Stream: theory: category theory

Topic: simplicial profunctors

Tim Hosgood (Apr 15 2020 at 18:33):

Something I've come across a few times now is what I've been calling "simplicial profunctors", for lack of a better name. I was hoping that somebody could tell me a bit more about what exactly these things are! Rather than try to describe them in full generality, I'll just explain the main example that I've been working with.

Let $X$ be a topological space, and $X_\bullet$ its Čech nerve (with respect to some cover). Write $\Delta^p$ to mean the (topological) $p$-simplex. Then I'm interested in functors $F\colon X_\bullet\times\Delta^\bullet\to\mathsf{Set}$. This looks a bit like a profunctor to me (somebody who knows not much about profunctors), because the nerve is simplicial, and the simplex is cosimplicial, so the variances are opposite (as with profunctors). But there's the nice fact that these things are _(co)-simplicial spaces_, and actually we don't end up in $\mathsf{Set}$ most of the time, but instead something smaller: there is always some sort of ring/algebra structure on the image (e.g. simplicial differential forms).

I suppose my question boils down to the following. We have two bits of extra structure* : the simplicial space structure, and the ring/algebra structure — does this give us any nice properties, or the ability to do some constructions that are not normally possible? As a bonus bit of niceness, there's also the fact that these things look like profunctors into the topological simplices, which are somehow extra nice (initial or terminal or something maybe) contractible spaces.

• I tried to avoid saying "structure", given the discussions on property vs. structure vs. stuff that I've yet to fully internalise, but just wrote it anyway.

Mike Shulman (Apr 15 2020 at 19:26):

Tim Hosgood said:

Then I'm interested in functors $F\colon X_\bullet\times\Delta^\bullet\to\mathsf{Set}$.

What does that mean? In what way is $X_\bullet\times\Delta^\bullet$ a category, so as to be the domain of a functor?

Tim Hosgood (Apr 15 2020 at 19:39):

maybe I'm stretching the similarities a bit thin here... I suppose I'm thinking of it as a functor from $\mathsf{Top}$, with $X$ variable

(since this whole thing was motivated mostly by "oh this looks a bit like a profunctor" and nothing more concrete, I wouldn't be surprised if this questions ends up being nonsensical)

Mike Shulman (Apr 15 2020 at 19:45):

$X_\bullet\times\Delta^\bullet$ is itself a functor from $\Delta^{\mathsf{op}}\times \Delta$ to $\mathsf{Top}$, so you can regard it as a sort of topological profunctor. Is that what you have in mind?

Tim Hosgood (Apr 15 2020 at 21:51):

that makes more sense, yes, and then I suppose I'm just composing it with something from $\mathsf{Top}$ to $\mathsf{Alg}_k$ or whatever

Tim Hosgood (Apr 15 2020 at 21:51):

so is there a theory of topological profunctors?

Reid Barton (Apr 15 2020 at 22:18):

Maybe you could spell out concretely the kind of object you have in mind? For example "for every n >= 0, and every ???, a set, together with for every morphism of $\Delta$, ..." or whatever.

Reid Barton (Apr 15 2020 at 22:18):

Then we could see if it can be repacked into something we recognize.

Reid Barton (Apr 15 2020 at 22:19):

Is it the same as your question about "simplicial sheaves"? I think I understood that one better.

Mike Shulman (Apr 15 2020 at 22:37):

There's a theory of enriched profunctors over any enriching base, which you can take to be Top.

Tim Hosgood (Apr 15 2020 at 22:45):

Reid Barton said:

Is it the same as your question about "simplicial sheaves"? I think I understood that one better.

it came up in the same area of research, but isn't really linked to it much beyond that

Tim Hosgood (Apr 15 2020 at 22:46):

Reid Barton said:

Maybe you could spell out concretely the kind of object you have in mind? For example "for every n >= 0, and every ???, a set, together with for every morphism of $\Delta$, ..." or whatever.

this seems like a good suggestion: I'll do that soon!

Tim Hosgood (Apr 16 2020 at 14:43):

Firstly, to make me not look like a crackpot: I am fully aware that what follows is very vague and meandering, and I'm speaking about things I don't really know much about, but am hoping that if I spell them out here then somebody will be able to say something more coherent :-)
Secondly, this will probably be a long post, so sorry about that.

Background. Let $X_\bullet$ be a simplicial complex manifold. Define a _simplicial differential $r$-form_ to be a family $\{\omega_p\}_{p\in\mathbb{N}}$ where each $\omega_p$ is a form on $X_p\times\Delta^p$ that is holomorphic on $X_p$ and smooth on $\Delta^p$ (n.b. this isn't really a very good/formal definition, but it can be made so, and hopefully gets the idea across well enough) such that, for all _coface_ maps $f_p^i\colon[p-1]\to[p]$, we have that $(X_\bullet f_p^i\times\mathrm{id})^*\omega_{p-1} = (\mathrm{id}\times f_p^i)^*\omega_p$ as sections of $\Omega^r(X_p\times\Delta^{p-1})$. We can define a differential and get a _simplicial de Rham complex_. There is also a quasi-isomorphism (_fibre integration_) that lets us recover the usual Čech-de Rham bicomplex.

Vague observations.

1. The condition that the $\omega_\bullet$ must satisfy is basically the same as the equivalence relation found in the definition of the fat geometric realisation. The fat geometric realisation computes the homotopy colimit, and a very hand-wavy reason for this is that the $\Delta^p$ are all contractible and thus somehow trivial in the context of topological spaces. (I think this can be formalised via framings maybe). In particular, $X_\bullet\times\Delta^\bullet\twoheadrightarrow\operatorname{hocolim}X_\bullet$
2. Analogously, $\operatorname{holim}X_\bullet\hookrightarrow\mathrm{Hom}(X_\bullet,k)\otimes\Omega_{\Delta^\bullet}^\star$, with the de Rham complex also being somehow trivial in the context of commutative (dg-)$k$-algebras: $\Omega_{\Delta^\bullet}^\star\simeq k[0]$.

Vague question. Is there some way of expressing the above in "neat" terms, maybe using profunctors, or maybe just by making this analogous algebraic statement somehow formally "dual" to the topological understanding (that uses the fat geometric realisation)?

Tim Hosgood (Apr 16 2020 at 14:44):

Reading back, I realise that there really isn't a very clear question here. Maybe somebody will be able to say something enlightening anyway!