TITLE: Groupoids as models of symmetric simplicial sets
QUESTION [6 upvotes]: 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.

REPLY [7 votes]: 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.