Zulip Chat Archive

Stream: condensed mathematics

Topic: simplex category variant


view this post on Zulip Johan Commelin (Mar 19 2021 at 09:36):

Do people ever work with the "simplex category" that only considers the maps [n][m][n] \to [m] for nmn \le m? Does it have a name? Because that seems to be all we need for this project.

view this post on Zulip Peter Scholze (Mar 19 2021 at 09:37):

semisimplicial objects?

view this post on Zulip Peter Scholze (Mar 19 2021 at 09:38):

I guess the "semisimplex category"? Let me look it up...

view this post on Zulip Peter Scholze (Mar 19 2021 at 09:40):

On the nLab https://ncatlab.org/nlab/show/semi-simplicial+object it did not get a special name

view this post on Zulip Johan Commelin (Mar 19 2021 at 09:43):

aha, thanks

view this post on Zulip Adam Topaz (Mar 19 2021 at 13:08):

I think semisimplicial is indeed the name that the HoTT folks gave such objects

view this post on Zulip Peter Scholze (Mar 19 2021 at 13:09):

semisimplicial is completely standard, I'm just not sure whether the index category has a name of its own

view this post on Zulip Adam Topaz (Mar 19 2021 at 13:09):

Oh I see

view this post on Zulip Adam Topaz (Mar 19 2021 at 13:10):

Semisimplex category? :smile:

view this post on Zulip Adam Topaz (Mar 19 2021 at 13:10):

Ha you already suggested that!

view this post on Zulip Adam Topaz (Mar 19 2021 at 13:13):

Maybe simplex_category_coface is a more descriptive name?

view this post on Zulip Adam Topaz (Mar 19 2021 at 13:14):

Or coface_category?

view this post on Zulip Johan Commelin (Mar 19 2021 at 13:19):

Anyway, I don't think that right now it makes sense to duplicate all the simplicial API to this new category.

view this post on Zulip Johan Commelin (Mar 19 2021 at 13:20):

Also, I don't know enough about 8.19 to see whether we use the degeneracy maps there. Probably we do.

view this post on Zulip Johan Commelin (Mar 19 2021 at 13:20):

For the construction of the double_complex used in the proof of 9.5, we only need the face maps.

view this post on Zulip Peter Scholze (Mar 19 2021 at 13:22):

Yes, the "alternating face map" complex doesn't use degeneracies. But yes, I think they are used in 8.19; I think the notion of a hypercover is really about simplicial objects, not semisimplicial ones

view this post on Zulip Johan Commelin (Mar 19 2021 at 13:26):

So then we'll just battle through the functoriality of compositions of not-all-face-maps.


Last updated: May 09 2021 at 23:10 UTC