# Groupoids as models of symmetric simplicial sets

In the Elephant, Peter Johnston remarks that internal categories may be regarded as simplicial objects that “preserve all limits that happen to exist in $$\Delta^{op}$$“ (I guess you might call this a flat functor). This is because the join in $$\Delta$$ is a limit.

Does a similar statement exist for the symmetric simplicial set/groupoid correspondence? It’s clear that the nerve of a groupoid yields a symmetric simplicial set, the question is then whether or not they correspond to flat functors on the site for symmetric simplicial sets.

• Note that there is already a notion of "flat functor" which is indeed quite close to what you are talking about (ncatlab.org/nlab/show/flat+functor), but these are not what you are talking about. Informally, a flat functor preserve all limits, "even these that do not exist". Here you just have a limit preserving functor. Jul 28 '20 at 23:23

You can definitely characterize groupoids as presheaves on $$Fin_+$$ preserving some colimtis (i.e. sending some colimits in $$Fin_+$$ to limits in Set). In fact Groupoids are the presheaf on $$Fin_+$$ that preserve the colimits comming from $$\Delta$$.

However, the Category $$Fin_+$$ has much more colimits than $$\Delta$$, in fact it all non-empty finite colimits. Groupoids do not preserves most of these colimits.

For example, in $$Fin_+$$ you have $$\{1\} \coprod \{1\} \simeq \{1,2\}$$ , but given a a groupoid $$G$$, $$G(\{1\})$$ is the set of objects, and $$G(\{1,2\})$$ is the set of arrow, so the isomorphisms $$G(\{1,2\}) \simeq G(\{1\}) \times G(\{1\})$$ means that your groupoid has a unique arrow between any two objects.

In fact, I claim that the colimits in $$Fin_+$$ that exists are exactly the finite non-empty colimits, and the category of presheaf preserving them is equivalent to the category of sets. In terms of the usual description of groupoids as presheaf on $$Fin_+$$, these corresponds to anti-discrete groupoids.

• Of course, so if you preserve products then you get the pair groupoid, whereas if you only preserve the necessary pullbacks your models are groupoids. Jul 29 '20 at 2:13