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Stream: learning: questions

Topic: Reference regarding Top and related categories

Bernd Losert (Dec 30 2022 at 15:55):

Does anyone know of a reference that contains results about Top and related categories. For example, I'm looking for results about the forgetful functors OrdTop -> Top, G-Top -> Top, G-Top -> G-Set, TopGrp -> Top, G-OrdTop -> G-Top, etc.

Graham Manuell (Jan 01 2023 at 12:53):

Maybe Abstract and Concrete Categories - The Joy of Cats is somewhat relevant?

Bernd Losert (Jan 01 2023 at 18:56):

Yes. Unfortunately, they only talk about TopGrp -> Top a little. There is no mention of any of the other functors.

Morgan Rogers (he/him) (Jan 02 2023 at 08:52):

What kind of result are you looking for? The second and fourth functors you mention are monadic, but the third is not (for the others I would need to check); is that the kind of statement you're looking for? Or some more general properties like faithfulness?

Bernd Losert (Jan 02 2023 at 18:04):

Yeah, I would like to know whether they are monadic, topological, what kinds limits and colimits are created, lifted, reflected and so on.

Bernd Losert (Jan 02 2023 at 18:05):

I am writing a research paper, so it would be nice if I had a reference that I can quote.

Graham Manuell (Jan 03 2023 at 14:30):

What do you mean by OrdTop? Does the order relation have to be closed or not? If it must be closed it won't be topological over Top, but it should be topological over Haus. Otherwise, I think it will be topological over Top. In any case, all the functors you mention should be semitopological / [[solid functors]]. I don't know a reference though.

Bernd Losert (Jan 03 2023 at 17:38):

OrdTop is just Top where the objects are partially ordered. The order does not have to be closed.

Bernd Losert (Jan 03 2023 at 17:41):

That semitopological stuff is interesting. Thanks for mentioning it.

Bernd Losert (Jan 03 2023 at 23:59):

Oh, this solid stuff is in Joy of Cats. Hmm.. How does knowing that G-OrdTop -> G-Top is solid tell me how to e.g. construct colimits in G-OrdTop from colimits in G-Top.

Reid Barton (Jan 04 2023 at 08:47):

I think for this question you don't need much; if $F_1 : C_1 \to D$ and $F_2 : C_2 \to D$ are colimit-preserving functors (with all the categories cocomplete) then colimits in $C_3 = C_1 \times_D C_2$ are computed componentwise. (Here $C_3$ is the pseudopullback, but since your functors are isofibrations you can also use the strict pullback.)

Graham Manuell (Jan 05 2023 at 14:57):

The nlab page on solid functors explains how to lift colimits.