Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I made the former entry "fibered category" instead a redirect to Grothendieck fibration. It didn't contain any addition information and was just mixing up links. I also made category fibered in groupoids redirect to Grothendieck fibration
I also edited the "Idea"-section at Grothendieck fibration slightly.
That big query box there ought to be eventually removed, and the important information established in the discussion filled into a proper subsection in its own right.
So what's a less evil idea, a Grothendieck fibration or a pseudofunctor? It seems like the grothendieck fibration is the "strictification" of a pseudofunctor (strictification as in how all bicategories are equivalent to strict 2-categories, not as in taking equivalence classes).
The fact that the equivalence of 2-categories is provided in one direction by a functor which can be understood as the strictification is a notion about a particular equivalence of 2-functors, not about the intrinsic properties of the definition of members in one of the 2-categories. True strictification is of course the composition and is called the first Street's construction on a lax functor (his paper Two constructions on a lax functor in Cahiers, now online). Fibration is much better having a property instead of non/motivated particular choice of the structure; but I have no arguments at which level would these be evil. I think the concept of a pseudofunctor is not evil as it has the inner cells where needed, but the particular choice of a pseudofunctor should be evil as a chocie of a member of the sub-2-category of all pseudofunctors corresponding to the same fibered category. But Toby and Mike will know much better this time.
That's true, it's just that the notion of a pseudofunctor as a presheaf of categories (or groupoids) seems really useful to motivate algebraic stacks.
really useful to motivate algebraic stacks.
Now where did algebraic stacks come from here? Do you mean it is useful to motivate stacks ?
I'm writing an expository thing up about algebraic stacks, so that's what first came to mind, but yeah, I meant stacks.
Okay, I see.
This is related to the general observation that I keep making, that there is a certain school of thinking in which happily all sorts of abstract category theory are used -- except that whenever a choice of site has to be made, alwyays only the algebraic site is considered. I find this curious. I know how it came about historically, but it's funny how people stick to it.
Toen has lecture notes on a generalization of algebraic stacks to general "geometric contexts", about which I might write something up on nLab. The thing is, defining a "generic" stack is probably only a quarter of the work required to define an algebraic, or more generally, geometric stack. Here are the notes if you haven't seen them before.
There is a stub entry geometric stack that is badly in need of attention. You'd do me a grand favor if you could put some items from your "expository thing" in there.
I'll first have to write up something about geometric contexts (tbh I haven't gotten to geometric stacks yet. I'm in the process of reading those notes.).
good, whatever you have, put it in that entry. For instance you already gave me a useful reference here on the forum. You know that it is forbidden to mention anything useful here on the forum without making sure that it survived on the wiki in some form! ;-)
So at least archive the reference to Toen's lecture notes in the entry on geometric stacks.
added two more references to Grothendieck fibration: Joyal's CatLab entry and Vistoli's notes.
I added a section “Properties” to Grothendieck fibration, as a place to put remarks about various things one can lift along a fibration (limits, colimits, factorizations).
I have a small technical question here. It it true (as the article says) that “In a fibration, every weakly cartesian arrow is cartesian”? If so, can someone explain how to demonstrate this? I can see how to show that in an opfibration, every weakly cartesian arrow is cartesian, and dually, in a fibration that every weakly opcarteesian arrow is opcartesian, but I can’t see how to show the quoted statement.
I thought the following argument works. Suppose is weakly cartesian for . Since is a fibration, there is a cartesian arrow with . Then the universal properties of and give inverse isomorphisms and lying over the identity. Thus, since is isomorphic to a cartesian arrow, it is also cartesian.
Yes, I think that works!
Please be so kind to add a remark on this to the entry, so that the next reader stumbling over this will know.
Alright, I added a summary of Mike’s hint that I think would have been enough to get me by. (I also explicitly introduced the terminology “cartesian lifting of f to e”, which I hope is okay with you all.)
made “weakly cartesian” link to prefibered category since these seem like they should either be the same thing or are closely related. Would be interested to know if I’m wrong.
Thanks, I agree; I added more links and made the terminology align.
Sharon Hollander’s thesis constructs a Quillen equivalence between three model categories: Grothendieck fibrations in groupoids, pseudofunctors valued in groupoids, and presheaves valued in groupoids.
Is there a written reference for the same statement with groupoids replaced by categories?
There are in fact two ways to rectify a Grothendieck fibration: the left adjoint “adds formal pullbacks” and the right adjoint “chooses pullbacks along all possible morphisms”. Ideally, both would be discussed in such a source.
I am only aware of a quasicategorical analog, as explained in the work of Joyal, Lurie, and Heuts-Moerdijk. But these sources do not talk about the middle category (the one with pseudofunctors), which, of course, can also be treated using available rectification tools. In any case, quasicategories are a bit of an overkill here.
added pointer to:
added pointer to:
added pointer to:
added the DOI-link for
and deleted the whimsical commentary around it; instead moved it way up the list to after the Grothendieck references
and added
I have added more hyperlinks, to this example and the following one.
But more substantially I’d urge again to provide more details or else a pointer to the literature. (I assume by the “environment comonad” you mean the one whose coproduct is induced from the diagonal map on the given object.)
I do think we need an independent reference for this usage and even then it deserve disambiguation. Because, the references that I am aware of use “environment” to refer not to the co-reader but for the reader monad, e.g.:
Also I think the term “indexed monad” needs some substantiation, and of course “simple fibration”.
It seems like you read about these things somewhere, before remarking on them on the nLab. So if you don’t want to spell out details yourself, just add a pointer to where you read about them, so that readers can see what is behind your paragraph.
remove redirection to ’morphism of fibration’, put it in fibred functor
Added links to fibred functor and fibred transformation.
added pointer to:
Added a cross-reference to foliated category.
added pointer to:
Removed an old query box:
The following discussion brings out some interesting points about the equivalence between fibrations and pseudofunctors.
+–{: .query} Sridhar Ramesh: I have a (possibly stupid) question about the nature of this equivalence. I assume the idea here is that moving from a cloven fibration to the corresponding pseudofunctor is in some sense “inverse” to carrying out the Grothendieck construction in the other direction. But, in trying to get a good intuition for the nuances of non-splittable fibrations, I seem to be stumbling upon just in what sense this is so. For example, consider the nontrivial group homomorphism from Z (integer addition) to Z_2 (integer addition modulo 2); this gives us a non-splittable fibration (and, for that matter, an opfibration), for which a cleavage can be readily selected. No matter what cleavage is selected, the corresponding (contravariant) pseudofunctor from Z_2 to Cat, it would appear to me, is the one which sends the unique object in Z_2 to the subcategory of Z containing only even integers (let us call this 2Z), and which sends both of Z_2’s morphisms to identity; thus, it is actually a genuine functor, and indeed, a “constant” functor. Applying the Grothendieck construction now, I would seem to get back the projection from Z_2 X 2Z onto Z_2. But can this really be equivalent to the fibration I started with? After all, Z and Z_2 X 2Z are very different groups. So either “equivalence” means something trickier here than I realize, or I keep making a mistake somewhere along the line. Either way, it’d be great if someone could help me see the light.
Mike Shulman: Good question! I think the missing subtlety is that a pseudofunctor is not uniquely determined by its action on objects and morphisms, even if its domain is a mere category (or a mere group); there are also natural coherence isomorphisms to take into account. For instance, if is the nonidentity element of , then , so even if acts by the identity on , a pseudofunctor also contains the additional data of a natural automorphism of the identity of , i.e. a (central) element of . If you start from , then depending on your cleavage your element can be anything that’s 2 mod 4, while if you start from , your element can be anything that’s 0 mod 4. Finally, there is a pseudonatural equivalence between two such pseudofunctors just when their corresponding elements differ by a multiple of 4, so you get exactly two equivalence classes of pseudofunctors, corresponding to the two groups and . Of course we are reproducing the classification of group extensions via group cohomology.
By the way, this sort of thing (by which I mean, the cohomology class that classifies some categorical structure arising as the trace of a coherence isomorphism) happens in lots of other places too. For instance, a 2-group is classified by a group , an abelian group , an action of on , and an element in . If you replace a 2-group by a skeletal one, then is the group of objects (which is strictly associative and unital, by skeletality), is the group of endomorphisms of the unit, and the action is defined by “whiskering”. The cohomology class comes from the associator isomorphism, which can (and often must) still be nontrivial even though the multiplication is “strictly associative” at the level of objects (by skeletality).
Toby: So the multiplication is strictly associative, but the -group itself is not a strict -group, since it uses a different associator than the identity. As in the example of the pseudofunctor from to , there is some additional structure here which is not trivial, even though it seems like it could be.
Sridhar Ramesh: Ah, of course, that’s what I was missing. Thanks, both of you; that clears it all up. =–
Re #48: It is a useful sort of example to consider, though, not only for the conceptual content and the link to cohomology, but also because it’s explicitly a (set-theoretically) small example. It just came in a vehicle which is deprecated today. I’ll see about adding it to the examples section, and then link back to the discussion here.
1 to 50 of 50