I hope and pray you and your family are all safe and well.
I was wondering under what conditions a category C can be written as an exponential B^A (a category of contravariant functors interpreting a theory A into a background B). Reyes, Reyes, and Zolfaghari note (on p. 81 of their book: Generic Figures and their Glueings) that the answer to the above question is a theorem of Roos in SGA4, p. 415.
Would you be kind enough to direct me to an English version of Roos theorem.
If I may, I'd like to add to my earlier question the following:
If a category C has adequate (A) and discrete (B) subcategories, then objects of C can be represented as contravariant functors A --> B.
Please correct me if I'm mistaken. Also, is this related to the theorem of Roos in SGA4?
Roos' Theorem, as stated on page 415 of SGA4 (volume 1), says that the three following conditions on a topos E are equivalent :
i) The family of essential points of E is conservative. (NB : A point is called "essential" when its fiber functor not only has a right adjoint but also a left adjoint.)
ii) The full sub-category of E consisting in objects which are connected - non empty and projective is generating.
iii) E is a presheaf topos.
At the end of the volume, there is a reference to three notes (in French) of Roos at "Comptes Rendus de l'Académie des Sciences", with the general title "Distributivity of colimits with respect to limits in toposes". This reference is to CR 259 (August and September 1964) : p. 969-972, 1605-1608 and 1801-1804.