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I see that Connes and Consani in the article listed there define a $\Gamma$-set as a pointed functor from $Fin_{\ast}$ to $Set_{\ast}$ rather than functors to $Set$ as we have here.
The latter would seem to sit better with Gamma-space.
The category of Γ-sets in this sense is no longer a topos
Why is it not a topos? Set_* is a slice topos, presheaves valued in a topos again form a topos.
Rather a co-slice, no?
Re #5: I confused slice and co-slice, I guess.
But this does raise a question: what kind of category does the co-slice category of a topos form?
But this does raise a question: what kind of category does the co-slice category of a topos form?
in this case it is not just a coslice but one from the global point (terminal object) which makes it have a sub object classifier I think. If that is worth anything.
There is a comment by Vladimir Sotirov in MO:a/4765697 on pointed sets having a subobject classifier but not being an elementary topos.
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