Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: concrete simplicial sets

Zhen Lin Low (Feb 14 2022 at 23:53):

Recall that a concrete simplicial set is a simplicial set $X$ such that the unit $X \to \textrm{cosk}_0 (X)$ is a monomorphism. Let $\mathbf{C}$ be the category of concrete simplicial sets. It is a reflective full subcategory of $\textbf{sSet}$ and contains the standard simplices $\Delta^n$, so it is locally finitely presentable. It is even a quasitopos. If I'm not mistaken, the inclusion $\mathbf{C} \hookrightarrow \textbf{sSet}$ induces an equivalence of homotopy categories, but the reflector does not. Nonetheless, is there a model structure on $\mathbf{C}$ with the same weak equivalences and a Quillen equivalence with $\textbf{sSet}$?

Reid Barton (Feb 15 2022 at 00:30):

Interesting question. Maybe it would be a good idea to try to transfer the model structure using the sd-Ex adjunction somehow.

Zhen Lin Low (Feb 15 2022 at 01:00):

That was my first thought, actually. It doesn't work for a stupid reason: the barycentric subdivision of a general simplicial set may fail to be concrete. Then I thought of using the barycentric subdivision of the nerve of the category of simplices but that is also non-concrete, I think. There has to be some subdivision functor that works though...

Zhen Lin Low (Feb 15 2022 at 15:59):

It seems fat geometric realisation is homotopy equivalent to geometric realisation. Given that, maybe taking the double barycentric subdivision of the "underlying" semisimplicial set works? In general, the barycentric subdivision of any simplicial set with free degeneracies is the nerve of a category and the barycentric subdivision of the nerve of a category is the nerve of a poset.