# Various notions of (co)simplicial (pre)sheaves

###### 30th May, 2022

For the first time, I have released into the wild a preprint of which I am the sole author, and had no real supervision. This is a scary moment indeed — how do I know that I haven’t written complete made-up nonsense? It’s true that I talked with a couple of close colleagues about the results, and they nodded in vague agreement, but the responsibility of checking the actual formal details is all on me. Even worse, I wanted to include some results about something that I don’t really have any formal experience with. Anyway, I hope the resulting paper is at least mildly “good” (whatever that might mean). It’s called “Various notions of (co)simplicial (pre)sheaves”, and is now on the arXiv: 2205.15185.

# Where this paper came from

Ever since my PhD thesis, I’ve been thinking on-and-off about these
things called “sheaves on the Čech nerve”, which were introduced by
Green^{1}, and then later in a different way
by Toledo and Tong^{2}, in order to do some nice
homotopical construction in complex geometry. The vague idea is that a
sheaf \mathcal{F} on a space X can be split up into finer parts, with an
individual sheaf \mathcal{F}^p for each
simplicial level of the Čech nerve of (a cover of) X, i.e. instead of just describing what \mathcal{F} looks like on all of X, we describe what it looks like on all the
open sets in a cover of X (this is
\mathcal{F}^0), and then what it looks
like on each pairwise intersection U\cap
V of open sets from the cover (this is \mathcal{F}^1), and then what it looks like
on each triple-wise intersection U\cap V\cap
W of open sets from the cover (this is \mathcal{F}^2), and… and so on. One reason
for wanting to do this is that sometimes we want to endow our sheaf
\mathcal{F} with some extra structure
or data (such as a *Koszul
connection*, in the case of Green/Toledo–Tong) that might not
exist globally (as is indeed the case for holomorphic connections),
*but* that exists *locally* — decomposing our sheaf \mathcal{F} into different simplicial levels
over the Čech nerve lets us do exactly this!

Anyway, what always annoyed me about this story (however petty this
might be) was the terminology: a bunch of sheaves (\mathcal{F}^p)_{p\in\mathbb{N}} living over
a simplicial space (X_p)_{p\in\mathbb{N}} were called
*simplicial sheaves*, but they’re *not* simplicial objects
in the category of sheaves on X, and,
even worse, they don’t actually really even look very simplicial (the
morphisms all end up going the wrong way, making them look
*co*simplicial). So is there any relation between these things
(which I gave the uninventive but fairly descriptive name of *sheaves
on a simplicial space*) and actual simplicial sheaves? Since the
only reference that I know of which actually talks about these sheaves
on simplicial spaces is Green’s aforementioned thesis (and Toledo–Tong’s
aforementioned summary of it), I couldn’t find if anybody else actually
even cared about these objects, let alone cared enough to figure out if
they also deserved the name of “simplicial sheaves”.

# An unexpected application

↟When I finally got the motivation to sit down and write things down,
I realised that there was also another pretty large gap in what I knew
about these things: the second thing you normally learn about after
learning about sheaves is *sections* of a sheaf. So what is a
section of a sheaf on a simplicial space? In one of life’s little
coincidences, when I started thinking about this specific question
seriously, I was working in an office three floors above Vincent
Wang-Maścianica, who patiently helped me to understand the basics of
string diagrams as well as some other stuff that he was working on. The
reason this is a coincidence is because, one afternoon (to mildly
dramatise the story) he saw a diagram on my board and said “oh, I didn’t
realise you were studying X”, to which I replied, “I’m not, I’m thinking
about sections of sheaves on simplicial spaces”, to which we both
replied “oh… interesting”.

The upshot of the ensuing conversations with Vincent left me
reasonably convinced that I could find an application for sheaves on
simplicial spaces outside their one appearance in complex geometry: they
could be used to describe “stuff, stuff relating that stuff, and the
order in which those relations should evolve”. To give a more concrete
example of this, consider the following scenario: we have some objects
x_1,\ldots,x_n in a category that has a
nice notion of tensor product (e.g. a symmetric monoidal category), and
we want to understand what endomorphisms of x_1\otimes x_2\otimes \ldots\otimes x_n look
like. One way of getting such endomorphisms is to build them up from
smaller endomorphisms! For example, say we have objects A, B, C, and D,
and an endomorphism f\in\operatorname{End}(A\otimes B) and
another endomorphism g\in\operatorname{End}(B\otimes C\otimes D).
Then we can get an endomorphism on A\otimes
B\otimes C\otimes D by composing f and g,
after extending them by identity morphisms: (\mathrm{id}_A\otimes
g)\circ(f\otimes\mathrm{id}_C\otimes\mathrm{id}_D). But of course
we could have also composed these in the other order, doing g first and then f, or we could have even picked different
endomorphisms, or we could have had *more* endomorphisms on
tensor products of *different* subsets of our objects.

In the paper, I (timidly) make the argument that all of these choices
can be bundled up into one sheaf on a simplicial space, and then that a
single choice of all the data described above corresponds to a
*section* of this sheaf. There are some caveats in this
construction (for example, the method that I describe requires that we
fix the number and type of endomorphisms beforehand), but I think that
most of them could probably be done away with by somebody who actually
knows something about these very restrictive sorts of string diagrams
(e.g. if there are some “generating” shapes of these diagrams then you
could maybe consider building a simplicial space for each one and then
gluing them altogether with some limit-y process).

Really, the only reason that I actually include this construction at all (beyond wanting to make people in another field aware of these objects that I find so fascinating) is in the hope that somebody comes along and makes it much better! I think it could be extended to describe much more general string diagrams, with copies and deletes and whatnot, but this is not something that I have the knowledge to do on my own.

This application is, I think, not particularly useful, and might just
be an example of “if all you have is a hammer, then everything looks
like a nail”, or even just a tautology (“of *course* sections of
a sheaf on a simplicial space describe such things — they’re just
objects in the totalisation of the cosimplicial simplicial mapping space
from a cofibrant replacement of a point, and the simplicial nature is
exactly describing the notion of linear order”, or something like this,
says some hypothetical expert), but “hey ho”, thought I, “why not just
write it down anyway — maybe somebody else will also like to think about
such things from this point of view”.

# What’s actually in the paper

↟The paper is mostly a survey, but contains some proofs of what I’m calling “pre-folklore results” — results which I think would be folklore if I’d ever actually heard anybody else care about them!

The first two sections (after the introduction) define three (or
really four) notions of “simplicial sheaf” and describes how they relate
to one another; the next section recalls a technical construction that
shows how to understand this story in the specific setting of (locally)
ringed spaces via a lovely construction^{3} of
lax homotopy limits of model categories by Bergner; the final section is
split into two, and contains a summary of the application of sheaves on
simplicial spaces to coherent sheaves in complex geometry, and then this
semi-conjectural construction relating sections of a sheaf on a
simplicial space to string diagrams describing endomorphisms generated
by endomorphisms.

If anybody does actually read this and have any comments, corrections, criticisms, or complaints, then please do let me know — you can leave a comment here, or email me, or tweet at me, or whatever you like. Happy reading!