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Reid Barton (May 05 2020 at 17:54):The introduction to the Elephant contains a number of different descriptions of what a topos is (or is like), including:
(v) ‘A topos is a totally cocomplete object in the meta-2-category CART of cartesian (i.e. , finitely complete) categories’
This sounds like a definition I would like if I knew what it meant, but I failed to find anything in the book about this (quite possibly because I didn't look hard enough), or for that matter anywhere else. Does anyone know what this sentence is supposed to mean?
vikraman (May 05 2020 at 18:15):I think totally cocomplete means that yoneda has a left adjoint.
John Baez (May 05 2020 at 18:16):The only phrase I don't understand is "totally cocomplete". I think I know how to internalize the concepts of limits and colimits to a sufficiently nice 2-category: @Joe Moeller wrote up a bunch of stuff about that.
Reid Barton (May 05 2020 at 18:17):Oh, like https://ncatlab.org/nlab/show/total+category.
John Baez (May 05 2020 at 18:18):I do not, however, understand how the subobject classifier is supposed to come out of this sentence. Is this supposed to be a characterization of, or just a fact about, topoi?
Reid Barton (May 05 2020 at 18:18):I think it is a characterization, but I don't know quite what it means yet.
sarahzrf (May 05 2020 at 18:18):arent grothendieck topoi characterized by some odd cocompleteness properties that you might not expect to produce a subobject classifier
Reid Barton (May 05 2020 at 18:19):If I interpret it as meaning that the left adjoint to Yoneda preserves finite limits, then it sounds like a plausible characterization. Let me think about that.
Reid Barton (May 05 2020 at 18:20):(Also, in case it's not obvious, I think Grothendieck topos is what's meant here.)
John Baez (May 05 2020 at 18:21):Okay, now I'm guessing this is Street's characterization of Grothendieck topoi, just because that contains the word "lex total".
Nathanael Arkor (May 05 2020 at 18:21):Reid Barton said:
If I interpret it as meaning that the left adjoint to Yoneda preserves finite limits, then it sounds like a plausible characterization. Let me think about that.
I think this is true, and is essentially the topic of Street's paper Notions of topos.
Oscar Cunningham (May 05 2020 at 18:25):John Baez said:
I do not, however, understand how the subobject classifier is supposed to come out of this sentence. Is this supposed to be a characterization of, or just a fact about, topoi?
'Total' implies every limit preserving functor is representable. So to show there was a subobject classifier you would have to show the subobject functor sent colimits to limits.
Mike Shulman (May 05 2020 at 18:34):Yes, this is a reference to Street's characterization. As stated at the nLab page John linked to, it's a characterization of Grothendieck topoi, so Street's proof goes by checking Giraud's axioms. One can then apply the standard theorem that a Grothendieck topos is an elementary topos. I don't know whether one could prove more directly that any lex-total category is a elementary topos.
Mike Shulman (May 05 2020 at 18:35):I think the introduction to the Elephant is a bit misleading in that some of its phrases are descriptions of Grothendieck topoi while some of them are descriptions of elementary topoi, and IIRC nothing is said about which is which.
Reid Barton (May 05 2020 at 18:36):Thanks everyone--I think this suffices for what I was originally interested in, but I also wonder whether there is actually a sensible notion of "totally cocomplete object of " for suitable 2-categories
Reid Barton (May 05 2020 at 18:38):Or whether I am just supposed to take the Yoneda embedding as a fixed construction, and then when it makes sense as a morphism of , to ask for a left adjoint to it as a morphism of .
Mike Shulman (May 05 2020 at 18:41):Not in an arbitrary 2-category, I think. But it makes sense in any proarrow equipment or Yoneda structure, and I believe LEX can be given those structures.