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I have splitt off from classifying topos an entry classifying topos for the theory of objects and added the statement about the relation to finitary monads.
I added to your remark a related point of view, and linked to some notes of mine.
Ah, thanks, that’s nice!!
We should add some kind of remark concerning $[\mathbb{P}^{op}, Set]$ also as an Example at monoidal topos. And I guess we should still add a pointer to an explanation of $\mathbb{P}$ to the entry. I can’t right now, though, am in a rush here…
Nice page, thanks.
I have added a stub for permutation category, just for completeness. In the course of this I noticed that we already have braid category! I have now cross-linked that a bit more such as to make it easier to find.
added also at classifying topos for the theory of objects remarks on the $\infty$-case:
Similarly
$PSh(FinSet_\ast^{op})$ is the classifying topos for pointed objects.
write $Fin\infty Grpd$ for the full sub-(∞,1)-category on ∞Grpd which is generated under finite (∞,1)-colimits from the point $\ast$ (HA, def. 1.4.2.8), then the (∞,1)-presheaf (∞,1)-topos $PSh_\infty(Fin\infty Grpd^{op})$ is the classifying (∞,1)-topos for objects;
write $Fin\infty Grpd_\ast$ for pointed finite $\infty$-groupoids in this sense, then $PSh_\infty((Fin\infty Grpd_\ast)^{op})$ is the classifying $(\infty,1)$-topos for pointed objects. See also at spectrum object via excisive functors.
Fixed some typos and highlighted (hopefully in a correct way) the role of the category of elements in this. It would be nice if one could bring the description of the generic object in section 2 notationally and conceptually in line with the description given in the current section 3 ! Thanks anyway for polishing the entries on ’geometric’ logics and adding clarifications !
Linked to the page finite set in the comments about what finite sets are meant for $[FinSet, Set]$.
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