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It's well known that we can exhibit the comma category as a particular type of 2-limit in Cat. When working with 2-categories, there is a naïve comma object given by by boosting up the ordinary diagram for a comma category, but this fails to have many of the useful formal properties we might expect.

The answer is that we should consider appropriate 2-subcategories of the 2-comma category, the Grothendieck fibrations (or opposite Grothendieck fibrations depending on the application). However, at least as far as I can tell, these 2-categories $Cat\downarrow_{Fib} C$ and $Cat\downarrow_{OpFib} C$ do not have obvious universal constructions.

Is the situation hopeless, or is there a tricky way to realize these objects as 2 or 3-limits?

Please note: I am aware that there is an equivalence between these guys and the relevant pseudofunctor categories, but that is not what I'm looking for. That would be, for my purposes, begging the question.

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This is a (mixed-fog) idea: you have the (large) 2-category $\mathscr{C}$ of categories as 0-morphisms (objects) and fibrations as 1-morphisms and natural transformations as 2-morphisms. I seem that $\mathscr{C}$ is a representable 2-category (i.e. it has the comma's of any objects by itself) this is because the canonical proiection associated to a comma constructuion is a fibration. Then with pullback's you can make also (more general) comma object..and these are lax limits too See "Fibrations and Yoneda's lemma in a 2-category" (Lnm 420) – Buschi Sergio Feb 25 '11 at 20:32
Have you considered using the Gray monoidal structure instead of the cartesian one? – David Roberts Feb 25 '11 at 21:43
up vote 5 down vote accepted

I'm not sure whether you're working in Cat or 2-Cat; I'll assume the latter.

If B is an object of a finitely (2-)complete 2-category K, then the 2-category Fib(B) of fibrations over B is monadic over K/B, where 'fibration' is meant in Street's sense (see nLab or Street's Fibrations in bicategories in the Cahiers or the paper cited by Buschi Sergio above). The monad takes $f \colon E \to B$ to the projection $B/f \to B$. So Fib(B) has the universal property of an Eilenberg--Moore object, i.e. it's a lax limit.

So if K is 2-Cat, then as long as your definition of fibrations matches this one, you have your universal property.

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This is a good answer, but I wonder: can the monad on $K/B$ in question also be determined from $B\in K$ by limit constructions? – Mike Shulman Mar 4 '11 at 6:43
Interesting question... I suppose it depends on what you mean by 'determined from'. Certainly B/f is a limit, and you get a functor $K/B \to K$ together with a transformation to the constant functor at B, which uniquely determines an endofunctor of K/B. On the other hand, as you know from Street's paper, the monad structure ultimately comes from the fact that $\mathbf{2}$ is a monoid (in two different ways, lax- and colax-idempotent) in the simplex 2-category, and I don't really see where limits fit in there. Do you have any more specific ideas about it? – Finn Lawler Mar 5 '11 at 21:04
What I meant was, is the monad on K/B determined by a 2-categorical limit construction in Cat? – Mike Shulman Mar 15 '11 at 21:47
Offhand, I don't see how, because you need to use the limits in K, but 2-Cat (while complete) contains non-complete 2-categories, so limits there, I'd guess, can't produce limits in the 2-categories involved. Does that make any sense? – Finn Lawler Mar 16 '11 at 15:16

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