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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2010
    • (edited Feb 17th 2010)

    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.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 17th 2010

    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).

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeFeb 17th 2010

    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.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 17th 2010

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2010
    • (edited Feb 17th 2010)

    really useful to motivate algebraic stacks.

    Now where did algebraic stacks come from here? Do you mean it is useful to motivate stacks ?

    • CommentRowNumber6.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 17th 2010

    I'm writing an expository thing up about algebraic stacks, so that's what first came to mind, but yeah, I meant stacks.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2010
    • (edited Feb 18th 2010)

    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.

    • CommentRowNumber8.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 18th 2010
    • (edited Feb 18th 2010)

    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.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2010

    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.

    • CommentRowNumber10.
    • CommentAuthorHarry Gindi
    • CommentTimeFeb 18th 2010

    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.).

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2010
    • (edited Feb 18th 2010)

    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.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2010

    added two more references to Grothendieck fibration: Joyal's CatLab entry and Vistoli's notes.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeSep 10th 2011

    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).

  1. 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.

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 6th 2014

    I thought the following argument works. Suppose ϕ:ee\phi:e'\to e is weakly cartesian for p:EBp:E\to B. Since pp is a fibration, there is a cartesian arrow ψ:ee\psi:e''\to e with p(ψ)=p(ϕ)p(\psi) = p(\phi). Then the universal properties of ϕ\phi and ψ\psi give inverse isomorphisms eee'\to e'' and eee''\to e' lying over the identity. Thus, since ϕ\phi is isomorphic to a cartesian arrow, it is also cartesian.

  2. Yes, I think that works!

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2014

    Please be so kind to add a remark on this to the entry, so that the next reader stumbling over this will know.

  3. 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.)

    • CommentRowNumber19.
    • CommentAuthorJon Beardsley
    • CommentTimeApr 13th 2017

    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.

    • CommentRowNumber20.
    • CommentAuthorMike Shulman
    • CommentTimeApr 15th 2017

    Thanks, I agree; I added more links and made the terminology align.

    • CommentRowNumber21.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 17th 2020
    • (edited Mar 17th 2020)

    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.

    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 18th 2020

    Added a paragraph explaining Grothendieck fibrations versus presheaves of categories.

    diff, v84, current

    • CommentRowNumber23.
    • CommentAuthorHurkyl
    • CommentTimeJan 13th 2021

    Add a bit more language to the “idea” section to ensure the reader doesn’t start off with the mistaken assumption that Fib(B)Fib(B) is a full subcategory of Cat/BCat/B.

    diff, v87, current

    • CommentRowNumber24.
    • CommentAuthorSam Staton
    • CommentTimeApr 3rd 2021

    elaboration of the properties section – limits and cartesian closed fibrations. Hope I didn’t make a mistake.

    diff, v88, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2021

    added pointer to:

    diff, v90, current

    • CommentRowNumber26.
    • CommentAuthorvarkor
    • CommentTimeNov 8th 2021

    Mention “total category” and “base category” terminology.

    diff, v91, current

  4. I previously added an example (vector bundles) and forgot to fill out this box!

    Emilio Verdooren

    diff, v93, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeJan 22nd 2023

    added pointer to:

    diff, v94, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2023

    have hyperlinked “fibered limit” in the paragraph here (in order to create that entry now)

    diff, v96, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023

    added pointer to:

    diff, v98, current

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeApr 27th 2023
    • (edited Apr 27th 2023)

    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

    • Jean Bénabou, Fibrations petites et localement petites, C. R. Acad. Sci. Paris 281 Série A (1975) 897-900 [gallica]

    diff, v99, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2023
    • (edited May 5th 2023)

    have tried to brush-up the list of references (publication data, order of and comments around citations)

    diff, v102, current

    • CommentRowNumber33.
    • CommentAuthormaxsnew
    • CommentTimeJun 3rd 2023

    Add the simple fibration as an example

    diff, v103, current

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2023

    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.)

    diff, v104, current

    • CommentRowNumber35.
    • CommentAuthormaxsnew
    • CommentTimeJun 3rd 2023

    hyperlink the environment comonad

    diff, v105, current

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2023
    • (edited Jun 4th 2023)

    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.

    • CommentRowNumber37.
    • CommentAuthormattecapu
    • CommentTimeOct 18th 2023

    remove redirection to ’morphism of fibration’, put it in fibred functor

    diff, v106, current

    • CommentRowNumber38.
    • CommentAuthormattecapu
    • CommentTimeOct 18th 2023
    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeOct 27th 2023

    added pointer to:

    diff, v107, current

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2023
    • (edited Nov 8th 2023)

    haved polished the Definition-section (here) by adding formatting, changing to more suggestive symbols, and adding a TikZ-diagram for the definition of Cartesian morphisms

    diff, v109, current

    • CommentRowNumber41.
    • CommentAuthorvarkor
    • CommentTimeDec 5th 2023

    Add a reference to factorisation systems.

    diff, v110, current

    • CommentRowNumber42.
    • CommentAuthorvarkor
    • CommentTimeDec 28th 2023

    Added a cross-reference to foliated category.

    diff, v111, current

    • CommentRowNumber43.
    • CommentAuthorvarkor
    • CommentTimeDec 28th 2023

    Added two more references for the connection between fibrations and factorisation systems.

    diff, v112, current

    • CommentRowNumber44.
    • CommentAuthorvarkor
    • CommentTimeJan 3rd 2024

    Mentioned that fibrations can be characterised as algebras for a lax-idempotent pseudomonad.

    diff, v113, current

  5. The previous notation was too confusing.

    Tomas Jakl

    diff, v115, current

    • CommentRowNumber46.
    • CommentAuthormaxsnew
    • CommentTimeMar 8th 2024

    Cite Jacobs for the term “simple fibration”.

    (Sorry about the delay, Urs, I missed your later comments)

    diff, v116, current