Of course, the case I'm really interested in is where we start with a well-pointed topos with nno and recover at least a boolean topos with nno satisfying *IAC*.
From this we can use stack semantics to get a model of ETCS.
Best, David Roberts On 5 January 2013 01:52, Ohad Kammar <ohad.kammar@ed.ac.uk> wrote:
Dear Andrej,
The following formulation of your question has negative answers:
Let Cat be the category of small categories and functors between them, and SCat be its full sub-category of small categories in which all epis split. Let U : SCat -> Cat be the full inclusion.
U does not have a left adjoint, nor a right adjoint.
Proof:
Let C be the following category: It has 3 objects: 0 - an initial object A and B.
Apart from the identities and the initial maps, C has the following 3 morphisms: a parallel pair f, g : A -> B an endomorphism h : B -> B
The non-trivial compositions are given by:
x o h = x, for x = f, g, and h.
Note that C is in SCat, as it has no trivial epimorphisms: the non-trivial arrow into A equalises f and g, which are different, and the non-trivial arrows into B equalise h and id.
Let F : C -> C be the endofunctor that swaps f with g.
If U had a left adjoint, then SCat was a full reflective subcategory of Cat, it was complete. Consider the equaliser of F, Id : C->C. Whatever it is in SCat, this coequaliser cannot be preserved by U, as the equaliser in Cat is given by dropping f and g. This resulting subcategory has an epi, the initial arrow into A, that doesn't split.
Similarly, if U had a right adjoint, then SCat was cocomplete. Then the coequaliser of F, Id : C->C in Cat would be C with g dropped (=identified with f). But then the initial map into A becomes epi, without a section. Thus this colimit is not in SCat, and U doesn't preserve it.
The same construction actually lies within the category Cat_ac that Richard described (with specified sections), as C has no non-trivial sections. Thus the same proof applies to his formulation, and his forgetful functor also isn't a right nor a left adjoint.
Ohad.
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