simplex category variant

Stream: condensed mathematics

[Johan Commelin]

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

[Peter Scholze]

semisimplicial objects?

[Peter Scholze]

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

[Peter Scholze]

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

[Johan Commelin]

aha, thanks

[Adam Topaz]

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

[Peter Scholze]

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

[Adam Topaz]

Oh I see

[Adam Topaz]

Semisimplex category? :smile:

[Adam Topaz]

Ha you already suggested that!

[Adam Topaz]

Maybe simplex_category_coface is a more descriptive name?

[Adam Topaz]

Or coface_category?

[Johan Commelin]

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

[Johan Commelin]

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

[Johan Commelin]

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

[Peter Scholze]

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

[Johan Commelin]

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