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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 $\phi:e'\to e$ is weakly cartesian for $p:E\to B$. Since $p$ is a fibration, there is a cartesian arrow $\psi:e''\to e$ with $p(\psi) = p(\phi)$. Then the universal properties of $\phi$ and $\psi$ give inverse isomorphisms $e'\to e''$ and $e''\to e'$ lying over the identity. Thus, since $\phi$ 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.
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