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Stream: theory: category theory

Topic: simplicial sets and sheaves

sarahzrf (Apr 08 2020 at 21:07):

is there some kind of relationship between being a kan complex and being a sheaf? they both have a kind of "gluing is possible" flavor

sarahzrf (Apr 08 2020 at 21:07):

somewhat parallel-ly: are there any interesting coverages you can put on the simplex category?

sarahzrf (Apr 08 2020 at 21:09):

if you look at it in its presentation as the category of finite linear graphs, it seems tempting to me to imagine that you could get some category-adjacent notion out of "sheaf with respect to an appropriate coverage"

Reid Barton (Apr 08 2020 at 21:10):

Being a Kan complex is a right lifting property, being a sheaf is a right orthogonality property. You can also express a right orthogonality property as two right lifting properties: one saying that the lift exists and one saying that any two lifts are equal.

sarahzrf (Apr 08 2020 at 21:11):

:o

sarahzrf (Apr 08 2020 at 21:11):

i never learned my orthogonality stuff :sob:

Reid Barton (Apr 08 2020 at 21:15):

Categories (viewed as simplicial sets via the nerve construction) are also characterized by being right orthogonal to the "spine into simplex" maps $\Delta^1 \amalg_{\Delta^0} \cdots \amalg_{\Delta^0} \Delta^1 \to \Delta^n$, which I think is what you're asking about, but even though these maps do look like the sort of maps you would get out of a coverage, they can't really be one because otherwise the category of categories would be a topos, which it isn't!

sarahzrf (Apr 08 2020 at 21:15):

hmm, i guess if you put a coverage on the simplex category that lets you cover any of its objects by a bunch of 1-simplices, then all of your data is already present in F([1])

Reid Barton (Apr 08 2020 at 21:15):

Probably it is a good exercise to find out exactly what goes wrong here.

sarahzrf (Apr 08 2020 at 21:15):

:thinking:

sarahzrf (Apr 08 2020 at 21:16):

i guess geometrically speaking those arent covers anyway

sarahzrf (Apr 08 2020 at 21:16):

lol

sarahzrf (Apr 08 2020 at 21:16):

i was thinking too hard about the graph picture

Reid Barton (Apr 08 2020 at 21:17):

I mean, I think this graph picture and the idea that if you glue n arrows to m arrows you get (n+m) arrows is very important. In higher dimensions you would want to glue pasting diagrams together.

Reid Barton (Apr 08 2020 at 21:17):

It just isn't a coverage specifically because one of the axioms fails

sarahzrf (Apr 08 2020 at 21:18):

oh i didnt mean those would be the only things in the coverage

sarahzrf (Apr 08 2020 at 21:18):

just that any coverage including them would make sheaves be determined by just one object

sarahzrf (Apr 08 2020 at 21:19):

or are you saying that no coverage can include them

Reid Barton (Apr 08 2020 at 21:19):

I think the smallest coverage which includes them is rather boring

sarahzrf (Apr 08 2020 at 21:19):

what is it?

sarahzrf (Apr 08 2020 at 21:20):

i am playing with forces i do not understand here :ghost:

Reid Barton (Apr 08 2020 at 21:28):

If you "cover" $\Delta^2$ by its two short edges and then pull back along the long edge $\Delta^1 \to \Delta^2$, you get the two inclusions of the endpoints as a "cover" of $\Delta^1$.

sarahzrf (Apr 08 2020 at 21:32):

oh, yikes

sarahzrf (Apr 08 2020 at 21:33):

i have yet to properly internalize the axiom for a coverage :sob:

sarahzrf (Apr 08 2020 at 21:33):

i'm sure i've read it in full at least once...

Joachim Kock (Apr 08 2020 at 22:08):

$\Delta$ does not in itself have a good Gronthendieck topology, but a significant subcategory has, and does the job. Namely, $\Delta$ has an active-inert factorisation system: the active maps are the endpoint-preserving maps; the inert maps are the distance-preserving maps. The subcategory $\Delta_{\text{in}}$ of inert maps has a bona fide Grothendieck topology, where the covers are the surjective families -- surjective meaning both on edges and vertices. Every element $[n]$ has a canonical cover by $[1] \amalg_{[0]} \cdots \amalg_{[0]} [1]$, so a sheaf on $\Delta_{\text{in}}$ is the same thing as a presheaf on the elementary objects, the full subcategory of $\Delta_{\text{in}}$ spanned by $[0]$ and $[1]$. That's the same thing as a directed graph.

On the category of directed graphs we have the free-category monad. The category $\Delta$ appears now as the Kleisli category of this monad restricted to $\Delta_{\text{in}}$. In other words, it is the category of free categories on linear graphs. Now consider the nerve functor from categories to simplicial sets. The nerve theorem says that a simplicial set is the nerve of a category if and only if its restriction to $\Delta_{\text{in}}$ is a sheaf.

The nice thing about this description and the whole set-up is that it generalises to many fancier structures, such as operads, properads, globular operads, and so on. This is Weber theory. It is very general, and depends only on some nice properties of the monad -- for example it is enough if it is a so-called local right adjoint monad. (In the abstract theory the active and inert maps are called generic and free, respectively: 'free' means 'in the image of the free part of the adjunction. In the category case, the free=inert maps in Delta are those that are graph maps. Generic maps are characterised by a universal property with respect to the monad.)

Recently, working in the context of $\infty$-categories, Chu and Haugseng have taken another viewpoint on this kind of calculus: they axiomatise the theory of operad-like structures by starting with the notion of active-inert factorisation system and a notion of elementary object, and build everything from there.

(PS: The conditions of the nerve theorems can be formulated as a sheaf condition, but since it is only a sheaf condition on a subcategory, sheaf theory does not seem to be so important for these structures. Usually the condition is rather regarded as a generalised Segal condition.)

Here are four references:

Blog post by Tom Leinster: How I learned to love the nerve construction,
https://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html

M. Weber: Familial 2-functors and parametric right adjoints, TAC 2007.

J. Kock: Polynomial functors and trees, IMRN 2011.

H. Chu, R. Haugseng: Homotopy-coherent algebra via Segal conditions, https://arxiv.org/abs/1907.03977

(Only a biased person would include reference 3 in this list.)

Matt Feller (Apr 09 2020 at 00:12):

I don't know much about sheaves and Grothendieck topologies myself, but there is a stackexchange post explaining why there are no interesting Grothendieck topologies on the simplex category: https://math.stackexchange.com/questions/1633230/a-grothendieck-topology-on-delta

John Baez (Apr 09 2020 at 00:14):

Neat. I'm too lazy to follow the details of the argument... so I lazily wonder if this is a special case of some more general theorem.

Morgan Rogers (he/him) (Apr 09 2020 at 09:02):

Both @Reid Barton and @Joachim Kock talk about lengths/distances in $[n]$-simplices. This seems strange to me, since the morphisms in the simplex category don't see distances; I could take my geometric realisation of $\Delta[n]$ to be regular, right? Is the "long edge" taken to be a particular one under some convention? Are the "distance-preserving maps" just the injective ones?

Amar Hadzihasanovic (Apr 09 2020 at 09:32):

They are thinking of the simplex category as the category of finite ordinals $\{0 < \ldots < n\}$ and order-preserving maps. The “distance” between $i$ and $j$ is $|j - i|$.

Amar Hadzihasanovic (Apr 09 2020 at 09:34):

So for example the distance preserving maps from $\Delta[1]$ to $\Delta[2]$ are $0 \mapsto 0, 1 \mapsto 1$ and $0 \mapsto 1$, $1 \mapsto 2$.

Morgan Rogers (he/him) (Apr 09 2020 at 09:34):

Oh of course! I keep slipping into thinking about these things geometrically rather than combinatorially :rolling_eyes: thakns!

Amar Hadzihasanovic (Apr 09 2020 at 09:37):

Another way of seeing the factorisation system would be to actually define $\Delta$ to be the full subcategory of $\mathrm{Cat}$ on the free categories generated by the finite linear graphs. The inert maps are exactly those that map generators to generators, ie the polygraphic maps...

Amar Hadzihasanovic (Apr 09 2020 at 09:40):

(Which is the same as “those that come from injective maps of graphs” btw)

Amar Hadzihasanovic (Apr 09 2020 at 09:42):

This is said in Joachim's post, but “as a theorem” whereas I find it easier to think of it as a definition -- it depends on which description of the categories at play you are more comfortable with!