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Stream: learning: questions

Topic: Kan is to groupoid as sSet is to... what?

Tim Hosgood (Jun 14 2023 at 14:45):

here's a question that I feel I should already know the answer to, but don't: Kan complexes are the $(\infty,1)$-analogue of groupoids; quasi-categories are the $(\infty,1)$-analogue of categories; simplicial sets are the $(\infty,1)$-analgoue of... what? just sets?

Tim Hosgood (Jun 14 2023 at 14:47):

maybe somewhat more concretely, in the $(\infty,1)$-setting we have fully faithful functors $\mathsf{Kan}\hookrightarrow\mathsf{Quasicat}\hookrightarrow\mathsf{sSet}$, and these inclusions have some adjoints — which category should appear in $\mathsf{Grpd}\hookrightarrow\mathsf{Cat}\hookrightarrow\mathsf{?}$ so that analogous adjoints exist (if possible)

Mike Shulman (Jun 14 2023 at 14:51):

Probably the closest analogue would be 2-truncated simplicial sets: presheaves on the full subcategory of $\Delta$ containing only [0], [1], [2].

Tim Hosgood (Jun 14 2023 at 14:56):

do these things turn up in many places? the thing that surprised me the most when I thought about this question was that, despite how great Kan complexes and quasi-categories are, simplicial sets themselves are still very interesting to study and there's a great deal of research concerning them — groupoids and categories are both very well studied, but I can't think of anywhere where I've read about 2-truncated simplicial sets before

Mike Shulman (Jun 14 2023 at 15:05):

The first application of simplicial sets I can think of that isn't just an application of Kan complexes or quasi-categories involves using them to index diagrams to take limits and colimits of, e.g. "bar constructions". An analogue of this does appear in category theory under the name (co) [[descent object]].

Mike Shulman (Jun 14 2023 at 15:05):

And, of course, the 1-truncated version is just reflexive graphs and (co)equalizers, and those appear all over the place.

Mike Shulman (Jun 14 2023 at 15:06):

Did you have other examples in mind?

Tim Hosgood (Jun 14 2023 at 16:03):

oh that's a nice point of view I hadn't thought about

Tim Hosgood (Jun 14 2023 at 16:03):

for me, simplicial sets are things that you do homotopy theory with, and the homotopy theory of simplicial presheaves is everywhere in the sort of work that i do

Tim Hosgood (Jun 14 2023 at 16:04):

I think I could say that 1-truncated simplicial sets naturally arise in this setting (in that you can just look at the 1-truncation of the Čech nerve for most classical purposes), but not 2-truncated

Tim Hosgood (Jun 14 2023 at 16:05):

I suppose the other thing is the two model structures on simplicial sets that give you quasi-categories or Kan complexes as your fibrant objects, but I don't know if (1-)groupoids and (1-) categories arise in some similar way from a construction on 2-truncated simplicial sets

Mike Shulman (Jun 14 2023 at 16:20):

When you say "simplicial sets are things that you do homotopy theory with", you're really treating simplicial sets as presentations of Kan complexes. So in that sense, the analogue in 1-category theory would just be working with groupoids.

Mike Shulman (Jun 14 2023 at 16:21):

1-category theory is pleasant enough that in most cases we don't need a notion of "presentation" for our 1-groupoids and can work with them directly, so for the most part you don't see an analogue of simplicial sets containing Kan complexes there.

Tim Hosgood (Jun 14 2023 at 16:31):

ok, I think you've sort of drilled down to what my real question should have been, which is "if simplicial sets are a context in which we can describe $(\infty,1)$-groupoids/-categories, then what is the analogous context for 1-groupoids/-categories?"

Tim Hosgood (Jun 14 2023 at 16:32):

in which case your answer of 2-truncated simplicial sets makes sense, because when we talk about categories and groupoids we talk about objects, morphisms, and commutativity of diagrams

Tim Hosgood (Jun 14 2023 at 16:32):

at least, that's how i'm understanding your answer!

Tim Hosgood (Jun 14 2023 at 16:33):

Tim Hosgood said:

I suppose the other thing is the two model structures on simplicial sets that give you quasi-categories or Kan complexes as your fibrant objects, but I don't know if (1-)groupoids and (1-) categories arise in some similar way from a construction on 2-truncated simplicial sets

is this something that still holds true in the 1-dimensional world?

Mike Shulman (Jun 14 2023 at 17:14):

Probably. You might have to use 3-truncated simplicial sets to get associativity. I don't remember if anyone has written down the details.