I was surprised by Thomas's previous post, because I knew that if E has set-indexed copowers then Fam(E) can be identified with the glueing of Delta, and is thus a topos. (I haven't seen Pieter Hofstra's thesis, so I wasn't aware that he had made a different claim.) In fact set-indexed copowers in E (a slightly weaker condition than cocompleteness, cf. A2.1.7 in the Elephant) is necessary as well as sufficient for Fam(E) to be a topos. Here's a proof: Let me write objects of Fam(E) in the form (I, (A_i | i \in I)) where I is a set and the A_i are objects of E. Noting that the forgetful functor sending (I,(A_i)) to I is represented by the object (1,(0)) where 0 is the initial object of E, it's easy to see that if Fam(E) is cartesian closed then objects of the form (1,(A)) form an exponential ideal, i.e. any exponential (1,(A))^(I,(B_i)) is of the form (1,(C)). In particular, if the exponential (1,(A))^(I,(1 | i \in I)) exists, it is of the form (1,(C)) where C is an I-fold power of A in E. So E has arbitrary set-indexed powers; but E^op is monadic over E, so it also has set-indexed powers, i.e. E has set-indexed copowers. Peter Johnstone On Fri, 21 Apr 2017, Thomas Streicher wrote:
Let E be a topos then Fam(E) -> Set is certainly a fibered topos but by Th.6.2.3 of Pieter Hofstra's Thesis Fam(E) is a topos iff E is an atomic category (in the sense of Johnstone's 1977 book on Topos Theory, exercise 12 on p. 257). But in atomic categories all morphisms are epic and thus Fam(E) is a topos only if E is trivial.
Alas, there is a flaw in Pieter's Th.6.2.3 (which certainly is not crucial for the main results of his otherwise very nice Thesis). Actually, it can be seen quite easily: if E is a cocomplete topos then Fam(E) is equivalent to the glueing of Delta : Set -> E which is known to be a topos.
So it seems to be open to characterize those toposes E for which Fam(E) is a topos. In particular, I don't know the answer for E the free topos (with nno) or a realizability topos. In the latter case we know that glueing of Nabla (right adjoint to Gamma) is a topos but it's different from Fam(E).
I'd be grateful about any suggestions even for these particular cases!
Thomas
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