It seems to be a well-known fact that there is a ``one-to-one correspondence'' between prestacks and fibered categories. Here a prestack (called a pseudo-functor in SGA1) means a contravariant lax functor F on a small category taking values in the 2-category of small categories in which the structure natural transformation

FfoFg -> F(gof)

is invertible.

For example, Vistoli says in this note that "the theory of fibered categories is equivalent to the theory of pseudo-functors" at the end of section 3.1.

Is this "equivalence" an equivalence of 2-categories? If so, where can I find a proof?

  • $\begingroup$ Do you really mean "lax functor", so that there is only a noninvertible morphism Ff o Fg -> F(g o f)? $\endgroup$ – Reid Barton Oct 28 '09 at 20:02
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    $\begingroup$ Ugh, in that paper before Def. 3.10 is the sentence "What we get instead of a functor is what is called a pseudo-functor, or, in a more modern terminology, a lax 2-functor." But I think in current usage those two terms mean very different things! See ncatlab.org/nlab/show/lax+functor $\endgroup$ – Reid Barton Oct 28 '09 at 20:11
  • $\begingroup$ Thanks. What I meant was "lax functor" in which FfoFg -> F(gof) is invertible. I will edit the question. $\endgroup$ – Dai Tamaki Oct 28 '09 at 20:16
  • $\begingroup$ Lax and pseudo is usually (also in "modern" terminology) meant to be distinct: pseudo means that all higher coherences are invertible cells, while for lax they can be any cells. So a pseudo-functor (the same thing as a weak 2-functor from a 1-category to a bicategory) respects composition up to a 2-isomorphism, which a lax functor which respect it only up to an arbitrary 2-cell. $\endgroup$ – Urs Schreiber Oct 29 '09 at 9:02
  • $\begingroup$ According to the definitions is Vistoli's notes, not every pseudofunctor (or fibered category) is a prestack. It needs to satisfy descent conditions for morphisms, $\endgroup$ – Andrea Ferretti Feb 2 '10 at 21:30

I don't have a reference right now, but I hope this answer is useful. If nothing else, perhaps you could comment on why this doesn't answer your question.

A pseudofunctor is exactly the same thing as a fibered category with a choice of cleavage (a cleavage is a choice of cartesian arrow over every morphism in the base category with given target in the fiber). That is, there is an isomorphism between the (2-)category of pseudofunctors and the (2-)category of fibered categories with cleavage (where the morphisms don't have to respect the cleavage).

By the axiom of choice, every fibered category has a cleavage, and any two choices of cleavage are canonically isomorphic (via the identity functor; remember that the functor need not respect the cleavage). So the category of fibered categories with cleavage is equivalent to the category of fibered categories, and this is an equivalence in the usual 1-categorical sense. That is, you have two functors (the forget-cleavage and choose-cleavage functors) whose compositions are naturally isomorphic to the the identity. I don't think you need to use any kind of 3-morphism even though you're dealing with 2-categories.

  • $\begingroup$ I took "lax functor" to mean that it comes equipped with a natural isomorphism F-oF- and F(-o-). $\endgroup$ – Anton Geraschenko Oct 28 '09 at 20:14
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    $\begingroup$ Thanks. Vistoli's note and your argument combined answered my question. $\endgroup$ – Dai Tamaki Oct 28 '09 at 20:34
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    $\begingroup$ I don't think that your statement that the 2-categories of (I'll call them) pseudofunctors and fibered categories with cleavage are isomorphic is literally true. That would mean for instance that passing from a fibered category with cleavage to a pseudofunctor and back would give you the same fibered category. I don't think you can (or want to) achieve this even in the lower-categorical situation of "fibered sets". $\endgroup$ – Reid Barton Oct 28 '09 at 20:47
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    $\begingroup$ I believe what is true is that the 2-categories of pseudofunctors and cloven fibrations are "strictly 2-equivalent". That means you have strict 2-functors in either direction whose composites are strictly 2-naturally isomorphic to the identity. This is the usual notion of equivalence of for enriched categories specialized to enrichment over Cat. $\endgroup$ – Mike Shulman Oct 28 '09 at 22:47
  • $\begingroup$ @Mike: Yes, that is exactly true. $\endgroup$ – Harry Gindi May 24 '10 at 16:29

This is called the Grothendieck construction. At that link there are further links to the full statement.

The full statement is that the (oo,1)-category of (oo,1)-functors from C^op to ooCat is (oo,1)-equivalent to that of Cartesian fibraitons of (oo,1)-cats over C.

  • $\begingroup$ Ah, the infinity-categorical statement is so nice. $\endgroup$ – Kevin H. Lin Oct 29 '09 at 14:30
  • $\begingroup$ This is not what I intended to ask, but seems interesting. $\endgroup$ – Dai Tamaki Oct 29 '09 at 19:55
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    $\begingroup$ Well, this is the generalization of what you intended to ask. What you asked sits in there as the special case where all the (oo,1)-categories in question are 1-truncated -- i.e. are categories. But, I agree, if there were any justice in the world somebody would have written into that nLab entry a nice discussion of the ordinary Grothendieck construction already. Anyone reading this here should feel challanged to do so! So that next time when the answer comes upwe can we can point the person asking to a nice summed up archived answer. $\endgroup$ – Urs Schreiber Oct 29 '09 at 20:44

What he means is, in this context, since you are considering psuedofunctors into the 2-category of groupoids where all 2-cells are invertible, lax functors are the same as pseudofunctors. The more modern terminology for pseudofunctor is "weak functor". And yes, Vistoli's definition of prestack is such a weak functor which is separated with respect to whatever Grothendieck topology you have floating around- although, I think this is confusing; I call such weak functors prestacks as well to keep them in good analogy with presheaves. Anyhow, there is an equivalence between the 2-category of categories fibred in groupoids over C and the 2-category of contravariant weak functors from C to groupoids, where the later 2-category has weak natural transformations as arrows (so each naturality square is fixed by a 2-cell (necessarily invertible in this context)), and so-called "modifications" as 2-cells. I am not sure of a reference for this equivalence, but, you should be able to spell this out yourself :-).

  • $\begingroup$ This can be found in, for example, Sketches of an Elephant. $\endgroup$ – David Carchedi Mar 24 '10 at 0:05
  • $\begingroup$ Are you asserting that all fibered categories are fibered in groupoids? The target of the pseudofunctor is the strict 2-category Cat. $\endgroup$ – Harry Gindi May 25 '10 at 6:22
  • $\begingroup$ No, but the question had the word "stack" in it, and for most people, this means "stack of groupoids". $\endgroup$ – David Carchedi May 25 '10 at 11:41

In "FIbred and COfibred CAtegories, John Walter Gray, La Jolla, 1965" at pag. 32-33 there is a short tractation (no proofs, but are elementary, may be tedious), about your question. In short:

FIbration (by Clivage) correspond to pseudo-functors (see http://ncatlab.org/nlab/show/lax+natural+transformation) and cartesian functors (clivage preserving functors) are identified to pseudo-natural-trasformation,

I think that: 1) more in general functors between fibration (commutative functorial traiangle) are identified by colax-natural-transformation.

2) By naturality and funtoriality of inverse images follow that the natural trasformatins between cartesian functors are identified by modification between the associate pseudo-natural-transformation.

Excuse for my bad ENglish.

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    $\begingroup$ Your English is quite good, really!. The Shift key on your keyboard, on the other hand, seems to be getting stuck :) $\endgroup$ – Mariano Suárez-Álvarez May 24 '10 at 20:12

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