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Bernd Losert (Feb 21 2024 at 11:17):According to exercise 27D.(d) of "The Joy of Cats", a concretely cartesian closed topological construct (A, U) with function spaces is isomorphic to Set. Thus, it seems like categories like Conv (of convergence spaces and continuous functions) are isomorphic to Set? Could this exercise be a mistake?
Morgan Rogers (he/him) (Feb 21 2024 at 14:10):I'm not familiar enough with the terms in Joy of Cats to know what everything means in that question. What's the definition of "topological construct"? And does "concretely" modify "cartesian closed" or is is just referring to a category that is both concrete and cartesian closed?
Bernd Losert (Feb 21 2024 at 14:11):A topogical construct is a pair (A, U) where A is category and U : A → Set is a topological functor.
Bernd Losert (Feb 21 2024 at 14:12):"concretely cartesian closed" means that the functor U : A → Set preserves cartesian closedness.
Bernd Losert (Feb 21 2024 at 14:16):For cartesian closed constructs, "has function spaces" basically means that constant functions in Set are always A-morphisms.
Morgan Rogers (he/him) (Feb 21 2024 at 14:18):Does that mean that if factors through the terminal object then it is in the image of ?
Bernd Losert (Feb 21 2024 at 14:22):For a cartesian closed topological construct, it says:
(A, U) has function spaces if and only if A × T* → A is an isomorphism for each A-object A.
Here, T is the terminal object and T* is the terminal object equipped with the discrete structure.
Morgan Rogers (he/him) (Feb 21 2024 at 14:29):Is Conv cartesian closed?
Morgan Rogers (he/him) (Feb 21 2024 at 14:30):(It's not clear to me if this answers my question hahaha)
Morgan Rogers (he/him) (Feb 21 2024 at 14:47):At any rate, I think the argument that the exercise is looking for is to use cartesian closedness to somehow show that must be full, which means that it's full and faithful and hence a reflective and coreflective subcategory of Set, which limits the possibilities to Set itself and the degenerate category with a single object. It does seem strange to me that the exercise would say "isomorphic to Set", since we surely only get an equivalence here, but that's not so problematic.
Bernd Losert (Feb 21 2024 at 15:02):Yes, Conv is cartesian closed.
Bernd Losert (Feb 21 2024 at 15:03):And constant functions are always continuous in Conv.
Bernd Losert (Feb 21 2024 at 15:07):I am almost certain that it is also concretely cartesian closed too.
James Deikun (Feb 21 2024 at 15:23):If is a Cartesian closed functor, then all functions on the underlying sets of convergence spaces are maps of convergence spaces. This seems unlikely to me!
Mike Shulman (Feb 21 2024 at 15:37):Regarding "isomorphic", probably their notion of topological construct is also an [[amnestic functor]].
Morgan Rogers (he/him) (Feb 21 2024 at 15:48):Alright, I gather that Joy of Cats is rather more concerned with grounding everything in sets than the broader category theory community is. That's fine, I guess.
Bernd Losert (Feb 21 2024 at 16:04):James Deikun said:
If is a Cartesian closed functor, then all functions on the underlying sets of convergence spaces are maps of convergence spaces. This seems unlikely to me!
So you are saying that if C(A, B) is the exponentional object of all continuous functions from A to B, then U(C(A,B)) is not ?
Bernd Losert (Feb 21 2024 at 16:06):Ah, that makes sense actually. There are plenty of discontinuous functions after all.
Bernd Losert (Feb 21 2024 at 16:13):Mike Shulman said:
Regarding "isomorphic", probably their notion of topological construct is also an [[amnestic functor]].
Yes, topological functors are amnestic.
Bernd Losert (Feb 21 2024 at 16:15):Morgan Rogers (he/him) said:
Alright, I gather that Joy of Cats is rather more concerned with grounding everything in sets than the broader category theory community is. That's fine, I guess.
Yes. The foundations of Joy of Cats relies on sets, classes and "conglomerates".
Mike Shulman (Feb 21 2024 at 16:16):Bernd Losert said:
Yes, topological functors are amnestic.
(According to Joy of Cats.)
Bernd Losert (Mar 07 2024 at 00:12):Suppose you have two cartesian-closed categories (A, U) an (B, V) over some base category X that is also cartesian-closed. Under what conditions is the pullback or pseudopullback of U and V also cartesian closed?
I am particularly interested in the case where (A, U) and (B, V) are well-fibered topological constructs. Acording to Joy of Cats, it's enough to prove that for any object m in the pullback, the functor m × - preserves colimits. Here's my attempt at proving this: Let c be a colimit in the pullback category C. Then Wc is also a colimit in X = Set, where W is the forgetful functor C → Set. Since Set is cartesian closed, Wm × Wc ≅ W(m × c) is a colimit in Set, and since W is topological, we can lift it to a colimit c' in C. Now I somehow need to argue that c' and m × c are isomorphic, but I can't think of any argument here.