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Asad Saeeduddin (Mar 07 2022 at 15:34):Is there such a thing as a "cotopos", which has all finite colimits and copower objects? If so, what (if anything) is the use of it? Is it uninteresting as merely being a "category whose opposite is a topos"?
Joe Moeller (Mar 07 2022 at 15:36):Topo-logie by Joyal and Anel is essentially about this I believe.
Morgan Rogers (he/him) (Mar 07 2022 at 16:16):That's not really what Joyal and Anel talk about, actually (although I understand why you might think that). When they say "a topos is the dual of a logos", what they mean is a formal dual, in the same way that a locale is the formal dual of a frame: the data in each is formally the same, but we think of the arrows of the category as going the opposite way around.
Morgan Rogers (he/him) (Mar 07 2022 at 16:18):Asad Saeeduddin said:
Is there such a thing as a "cotopos", which has all finite colimits and copower objects? If so, what (if anything) is the use of it? Is it uninteresting as merely being a "category whose opposite is a topos"?
They are rarely talked about, but dualities can produce non-trivial presentations of them. For example, I can identify with the category of complete atomic Boolean algebras, so the latter form a cotopos.
Joe Moeller (Mar 08 2022 at 03:44):Oh right, I level slipped. They were thinking about the opposite of the category of topoi, not the opposite of a topos.
Matteo Capucci (he/him) (Mar 08 2022 at 10:24):There is some literature on 'complement topoi', which have dual properties to topoi (in particular, their internal logic is paraconsistent) https://ojs.victoria.ac.nz/ajl/article/view/1819
That said, it's more a different attitude on topoi than a completely different mathematical notion: every topos gives rise to a complement topos and viceversa.
Peter Arndt (Mar 08 2022 at 11:14):A "complement topos" is exactly the same thing as a topos. The author just chooses a different name to emphasize that he uses the topos properties in a different way than usual, to give semantics to a different logic than the usual intuitionistic logic.
Jon Sterling (Mar 08 2022 at 19:45):Wow that is confusing.
Naso (Mar 09 2022 at 02:22):does a cotopos have a quotient object classifier?
Fawzi Hreiki (Mar 09 2022 at 10:32):Yes, in the weak sense of quotient meaning epimorphism, not quotient by equivalence relation.
However, since every monomorphism in a topos is effective, every epi in a cotopos is effective. So in fact you do have this stronger notion of quotient being classified.
Mike Shulman (Mar 09 2022 at 15:35):You mean coclassified?
Fawzi Hreiki (Mar 09 2022 at 16:25):Yes. Classified by maps out of the classifier (just to be clear)
Peter Arndt (Apr 07 2022 at 16:49):I found the notice too late, but two days ago there was a talk about this:
Peter Arndt (Apr 07 2022 at 16:50):3:00 PM (Coimbra time): Zurab Janelidze (Stellenbosch Univ., South Africa)
Title: In search for an algebraically sound notion of a subobject in the dual of a topos
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Abstract: A noetherian form over a category is a faithful bifibration over it, whose fibres are lattices, and which satisfies certain self-dual axioms ensuring the validity of homomorphism theorems (as stated relative to the form) from abstract algebra, such as the isomorphism theorems and homological diagram lemmas. In all standard examples, such bifibration can be obtained as the bifibration of subobjects. However, the same bifibration of the category of sets or of its dual category, neither are noetherian forms. In this talk we describe a noetherian form over the category of sets, formulate its intrinsic self-dual properties, and show that any topos admits a noetherian form having these properties. This result is similar to, but not the same as the known theorem of Dominique Bourn that the dual of any topos is a protomodular category. The talk is based on a joint work in progress with Francois van Niekerk.