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Bernd Losert (Feb 10 2024 at 17:29):Let (A, U) and (B, V) be two concrete categories over a category X. One can define a subcategory C of A × B whose objects (a, b) satisfy Ua = Vb and whose morphisms (f, g) satisfy Uf = Vg. If I am not mistaken, C is just the pullback of U and V. Does anyone have any reference with some theory about these kinds of categories?
Patrick Nicodemus (Feb 10 2024 at 18:29):Your description is very abstract, which of course we like as category theorists, but abstraction also means excluding information to focus on the (presumed) salient details of the problem. Maybe premature but my initial reaction is that not much can be said other than what follows from general properties of the pullbacks.
Do you have any questions in mind? For example, many classes of morphisms are stable under pullback, so if you have a conjecture of the form "if U has property P, then the canonical functor C -> B also has property P" then we could discuss this.
Reid Barton (Feb 10 2024 at 19:30):A minor comment: In general, imposing an equality of objects "Ua = Vb" isn't very sensible unless at least one of the functors U, V is an isofibration, which typically for concrete categories they always are (maybe it's part of your definition of concrete category?)
Bernd Losert (Feb 10 2024 at 19:56):Do you have any questions in mind?
Yes. Let W : C → X be the obvious forgetful functor. What can we say about W if both U and V are e.g. topological or monadic or solid? What about the case where e.g. U is monadic and V is topological. If A' is a reflective subcategory of A, is there are corresponding reflective subcategory in C?
John Baez (Feb 10 2024 at 20:07):Reid Barton said:
A minor comment: In general, imposing an equality of objects "Ua = Vb" isn't very sensible unless at least one of the functors U, V is an isofibration, which typically for concrete categories they always are (maybe it's part of your definition of concrete category?)
And a followup minor comment: instead of using an equation here I would by default specify an isomorphism. Thus, instead of using the pullback of U and V I would use their [[2-pullback]]. There are various flavors of 2-pullback but I usually use the iso-comma object. But when at least one of U or V is an isofibration this is equivalent to the pullback.
Bernd Losert (Feb 13 2024 at 00:31):Reid Barton said:
A minor comment: In general, imposing an equality of objects "Ua = Vb" isn't very sensible unless at least one of the functors U, V is an isofibration, which typically for concrete categories they always are (maybe it's part of your definition of concrete category?)
It is sensible though. Suppose U : Top → Set and V : PreOrd → Set. Then the pullback of U and V gives me the category of preordered topological spaces, which is a category that I need to deal with in my research.
Reid Barton (Feb 13 2024 at 01:34):Yes, because in this case both of your functors are isofibrations.
Bernd Losert (Feb 13 2024 at 01:40):Not sure what isofibrations are, but they are both topological and the pullback will also end up being topological.
Mike Shulman (Feb 13 2024 at 02:24):A [[topological concrete category]] (in the strict sense) is always an [[isofibration]].
Bernd Losert (Feb 13 2024 at 11:58):Ah, so "isofibration" is what is called "transportable" in the topological categories literature.