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John Wiegley (Feb 10 2023 at 01:31):Hello, I was wondering if the following construction goes by another form/name:
Given categories C, D and E, and functors S : C -> D and T : E -> D, take the subcategory of comma <S, T> such that:
forall x : object of <S, T>, x.mor = id
That is, for every object in the comma category, whose components are (x : C, y : E, f : S(x) -> T(y)), the arrow f is always the identity.
John
Jonas Frey (Feb 10 2023 at 04:26):That sounds like the strict 2-pullback to me!
This concept is known simply as "2-pullback" in the 2-category literature, but since the nlab uses this term for what was/is traditionally known as "pseudopullback" (that's what you get when you ask the 's in your construction to be isos) , it's now better to always use the prefix "strict" if you want the "up to identity" rather than "up to iso" notion.
Or am I misunderstanding your question?
John Baez (Feb 10 2023 at 05:10):Is this the pullback in the category Cat of
John Baez (Feb 10 2023 at 05:10):?
Jonas Frey (Feb 10 2023 at 19:43):John Baez said:
Is this the pullback in the category Cat of
$C \stackrel{S}{\longrightarrow} D \stackrel{T}{\longleftarrow} E $
Yes.
Strict 2-pullbacks in the 2-category of categories coincide with ordinary pullbacks in the 1-category of categories.
John Wiegley (Feb 10 2023 at 20:55):Jonas Frey said:
That sounds like the strict 2-pullback to me!
Thank you! A pullback sounds exactly right for what's going on here, I just hadn't studied 2-pullbacks yet.