Linked to the page finite set in the comments about what finite sets are meant for $[FinSet, Set]$.

]]>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 !

]]>Added more concrete construction of the pointed set classifier.

Steve Vickers

]]>- Added remarks on “finite” in FinSet.
- Added remarks on object classifier as generalized space of “sets”

Steve Vickers

]]>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*.

Spelled out here the argument for why $PSh(FinSet^{op})$ is the classifying topos for objects by pointing to this fact.

(Just for completeness.)

Also added the remark that similarly $PSh((FinSet_\ast)^{op})$ is the classifying topos for pointed objects.

]]>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.

Nice page, thanks.

]]>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…

I added to your remark a related point of view, and linked to some notes of mine.

]]>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.