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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.

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    $\begingroup$ 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. $\endgroup$ Jul 28 '20 at 23:23
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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.

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    $\begingroup$ 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. $\endgroup$ Jul 29 '20 at 2:13

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