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nLab > Latest Changes: Grothendieck construction

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2010
    • (edited Feb 17th 2010)

    added to Grothendieck construction a section Adjoints to the Grothendieck construction

    There I talk about the left adjoint to the Grothendieck construction the way it is traditionally written in the literature, and then make a remark on how one can look at this from a slightly different perspective, which then is the perspective that seamlessly leads over to Lurie's realization of the (oo,1)-Grothendieck construction.

    There is a CLAIM there which is maybe not entirely obvious, but straightforward to check. I'll provide the proof later.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeFeb 18th 2010

    Grothendieck construction works more generally: for lax contravariant functors and colax covaraint functors, not only the pseudofunctors. This directedness aspect would be nice to emphasise.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2010

    Okay, but then let's put this in different sections.

    So what kind of fibration do lax functors correspond to under the construction?

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeFeb 18th 2010
    • (edited Feb 18th 2010)

    One of the Thomason's papers works in this generality, but I do not remember the conclusion. However the composition is well defined so the formal construction works whatever is obtained. One can not exchange lax and colax though.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 19th 2010

    @Urs: Since the Grothendieck construction is an equivalence, its inverse equivalence is both a left and a right adjoint to it. But it seems weird to me to talk about adjoints more specifically. Are you thinking of it as acting only on strict functors?

    Regarding lax functors, the most general statement is that an arbitrary functor X-->C corresponds to a lax functor C^op --> Prof. If all precartesian liftings exist, then the corresponding lax functor lands in Cat, while if precartesian arrows are closed under composition (hence are cartesian, and the functor is a fibration) then it is a pseudofunctor.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2010

    Mike,

    the section on adjoints is supposed to discuss the adjoint pair between all of Cat/C and [C^op,Cat], where it's not an equivalence.

    But I see that I didn't say this clearly, will try to imrpove on this.

    And, yes, internally I was thinking of strict functors.

    The point of this comment was meant as indicating how the construction described at (infinity,1)-Grothendieck construction does indeed reduce to the ordinary construction.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2010

    Oh, and thanks for the info on [C^op,Prof]. We should add these generalizations to the entry. At least point out a reference.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2010

    expanded the Idea-section at Grothendieck construction

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 11th 2010
    • (edited Mar 11th 2010)

    started writing out the details of the proof for that description of the left adjoint of the Grothendieck construction in terms of that cone construction. See the new section In terms of the cone construction.

    This requires more polishing notably to wards the end, which I fill in later when I am less tired. But I think the main ingredients are there.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2010

    added to Grothendieck construction a Properties-section with a formal statement of the equivalence induced by the construction.

    (I know this overlaps with stuff at Grothendieck fibration but it deserves to be stated here, too.)

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2010

    I further expanded Grothendieck construction, trying to bring it closer to something like a self-contained exposition.

    • CommentRowNumber12.
    • CommentAuthorFinnLawler
    • CommentTimeDec 23rd 2011

    At Urs’s suggestion, I’ve expanded on the lax colimit property of the Grothendieck construction here.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2011

    Thanks!

    • CommentRowNumber14.
    • CommentAuthoradeelkh
    • CommentTimeApr 8th 2014

    I added the reference

    However there is already the older reference

    which supposedly does the same thing. I don’t have time to look into the difference at the moment, but feel free to add a remark if you know.

    • CommentRowNumber15.
    • CommentAuthorTim_Porter
    • CommentTimeApr 8th 2014
    • (edited Apr 8th 2014)

    I fixed two dead links. (People change universities and old links do not continue to work! Perhaps Barr and Wells needs to be secured somewhere as the old URL failed. I replaced it by Mike Barr’s copy)

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2014

    I suppose this should be linked at (infinity,1)-Grothendieck construction (too).

    • CommentRowNumber17.
    • CommentAuthorKarol Szumiło
    • CommentTimeAug 13th 2014

    I have a question about what Mike wrote in #5 that functors with enough weakly cartesian morphisms (not necessarily closed under composition) correspond to lax 22-functors into small categories. It seems to me that this will only work if units are actually preserved on the nose. Without that even the existence of the Grothendieck construction (more precisely, the unitality of composition) seems problematic. Is that right?

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeAug 13th 2014

    I think that’s right, it should be normal lax functors.

    • CommentRowNumber19.
    • CommentAuthorKarol Szumiło
    • CommentTimeAug 14th 2014

    OK, thanks! Do you know any source that discusses the Grothendieck construction in this generality?

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 14th 2014
    • (edited Aug 14th 2014)

    I’m sure there’s something related to it in the work that Bénabou is promising to write about ;-)

    • CommentRowNumber21.
    • CommentAuthorZhen Lin
    • CommentTimeAug 14th 2014

    Speaking of Grothendieck constructions, I wonder if any statement of the following kind is true / known:

    Let ℬ\mathcal{B} be a category and let F:𝔼′→𝔼F : \mathbb{E}' \to \mathbb{E} be a ℬ\mathcal{B}-indexed functor. If the components of FF are Grothendieck fibrations (+ possibly other conditions), then ∫F:∫𝔼′→∫𝔼\int F : \int \mathbb{E}' \to \int \mathbb{E} is also a Grothendieck fibration.

    In particular, for the case where 𝔼\mathbb{E} is the terminal ℬ\mathcal{B}-indexed category, this would recover the fact that the canonical projection is a Grothendieck fibration.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTimeAug 14th 2014

    @Karol: I can’t think of one off the top of my head.

    @Zhen: You may be looking for this theorem.

    • CommentRowNumber23.
    • CommentAuthorZhen Lin
    • CommentTimeAug 14th 2014

    That looks good, thanks! What I was thinking about was actually the two-sided Grothendieck construction 𝔼*𝔽\mathbb{E} \ast \mathbb{F}, which is a Grothendieck fibration over ∫𝔽\int \mathbb{F} and a Grothendieck opfibration over ∫𝔼\int \mathbb{E}.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeOct 17th 2017

    I added to Grothendieck construction the fact that it preserves local smallness, which came up at this thread.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeApr 12th 2018

    Add reference to Beardsley-Wong enriched Grothendieck construction.

    diff, v53, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2018
    • (edited Apr 12th 2018)

    I made “enriched” and “Jonathan Beardsley” become hyperlinks (here)

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 15th 2018

    Thanks, fixed now.

    I’d go ahead with any such changes yourself. There are plenty of them to be found. If you’re in any doubt, you can just fill in the changes box to alert nForum users.

    diff, v55, current

    • CommentRowNumber28.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 15th 2018
    • (edited Sep 15th 2018)

    Comment #27 is referring to this comment.

  29. The page says that the Grothendieck construction functor has both a left and right adjoint, and then goes on to describe the left adjoint while saying that “much of the traditional literature discusses (just) the right adjoint”. What is the right adjoint and where can I go to read about it?

    • CommentRowNumber30.
    • CommentAuthorAli Caglayan
    • CommentTimeFeb 15th 2019
    • CommentRowNumber31.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 16th 2019

    I don’t know what “traditional literature” the page is referring to, but the definition of a right adjoint valued in a presheaf category can be deduced by mapping out of the images of representables under its left adjoint. That is, if RR is this left adjoint, then by adjointness R(D)(c)R(D)(c) must be the hom-category (Cat/C)(∫(C(−,c)),D)(Cat/C)(\int( C(-,c)) , D) (at least if the adjunction is CatCat-enriched, which I assume it is).

    • CommentRowNumber32.
    • CommentAuthorGuest
    • CommentTimeApr 20th 2019
    Sorry if I made any mistakes, but how could Hom(c,v) be c/p? I think K(p) mentioned in the cone construction is (2,1)-category, so its hom-space is groupoid. But the comma category c/p usually has non-invertible morphisms... Am I right?

  33. created links for “Grothendieck construction for monoidal categories” and “Christina Vasilakopoulou”

    Joe M

    diff, v58, current

    • CommentRowNumber34.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 18th 2020

    Added a description of the right adjoint to the Grothendieck construction.

    diff, v61, current

    • CommentRowNumber35.
    • CommentAuthorEric
    • CommentTimeSep 14th 2020
    • (edited Sep 14th 2020)

    Hi there (a voice from the past here) :)

    Worlds are colliding and category theory is starting to pervade my universe of computational physics / finance and I’m trying to get a handle on Grothendieck construction.

    As an initial step, I’m trying to understand the “universal Cat bundle” Cat *,l→CatCat_{*,l}\to Cat. The page refers to Cat *,lCat_{*,l} as a “slice”. Another good reference:

    • Section 2.2 (page 8) of http://math.jhu.edu/~dfuente6/notes/monoid.pdf

    also refers to this as a “slice”, but when I look at the definition, it looks like a coslice / under category */Cat*/Cat. What am I missing?

    • CommentRowNumber36.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 14th 2020
    • (edited Sep 14th 2020)

    It definitely is a coslice. Or, if you like, a slice under rather than a slice over.

    • CommentRowNumber37.
    • CommentAuthorEric
    • CommentTimeSep 14th 2020

    Thank you David :)

    • CommentRowNumber38.
    • CommentAuthorvarkor
    • CommentTimeSep 30th 2020

    Add conditions for the Grothendieck construction to have limits and colimits.

    diff, v63, current

    • CommentRowNumber39.
    • CommentAuthorHurkyl
    • CommentTimeOct 8th 2020
    • (edited Oct 8th 2020)

    I wanted to verify something. If we restrict to the case of strict functors between 1-categories, is it correct that the Grothendieck constructions are given by

    ∫:Cat C→Cat:F↦C •/⊗ CF \int : \Cat^C \to \Cat : F \mapsto C_{\bullet/} \otimes_{C} F ∫:Cat C op→Cat:F↦F⊗ CC /• \int : Cat^{C^{op}} \to \Cat : F \mapsto F \otimes_C C_{/\bullet}

    where C •/:C op→CatC_{\bullet/} : C^{op} \to \Cat and C /•:C→CatC_{/\bullet} : C \to \Cat are the coslice and slice constructions whose action on arrows is the pre/post composition functors, ⊗ C\otimes_C is the functor tensor product (induced by the ordinary product on CatCat), and both come equipped with a natural projection (∫F)→C(\int F) \to C?

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2020

    added publication data to:

    diff, v64, current

    • CommentRowNumber41.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 30th 2020

    Hurkyl, I believe I’ve seen that.

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2021

    added pointer to:

    diff, v67, current

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeDec 27th 2021

    added pointer to:

    • Francis Borceux, Section 8.3 of: [[Handbook of Categorical Algebra, Vol. 2: Categories and Structures Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

    But I was really looking for a source which would admit that the Grothendieck construction is generally the lax slice of representables over the given pre-stack.

    Now I see the statement is made explicit at the end of MO:a/387371. Is there a more citable reference that would say it this way?

    diff, v69, current

    • CommentRowNumber44.
    • CommentAuthorjesuslop
    • CommentTimeDec 27th 2021
    • (edited Dec 27th 2021)

    As for Borceaux 8.3, I think there can be a glitch there that I would like to confirm.

    He says, pg. 388 (ed. CUP 1994):

    Given a pseudo-functor P:ℰ⟶P: \mathcal{E} \longrightarrow Cat, we shall thus construct a fibration G:𝒢⟶ℰG: \mathcal{G} \longrightarrow \mathcal{E} whose fibre at I∈ℰI \in \mathcal{E} is precisely the category P(I)P(I) :

    • an object of 𝒢\mathcal{G} is a pair (I,X)(I, X) where I∈ℰI \in \mathcal{E} and X∈P(I)X \in P(I) are respectively objects of ℰ\mathcal{E} and P(I)P(I);
    • an arrow (J,Y)⟶(I,X)(J, Y) \longrightarrow(I, X) in 𝒢\mathcal{G} is a pair (α,f)(\alpha, f) where α:J⟶I\alpha: J \longrightarrow I and f:Y⟶P(α)(X)f: Y \longrightarrow P(\alpha)(X) are respectively arrows of ℰ\mathcal{E} and P(J)P(J);
    • G:𝒢⟶ℰG: \mathcal{G} \longrightarrow \mathcal{E} is just the first component functor, thus G(I,X)=IG(I, X)=I and G(α,f)=αG(\alpha, f)=\alpha.

    The problem is with f:Y→P(α)(X)f:Y \to P(\alpha)(X).

    I think Wikipedia and nLab agree that it should be f:P(α)(Y)→Xf:P(\alpha)(Y) \to X living in P(I)P(I).

    • CommentRowNumber45.
    • CommentAuthorHurkyl
    • CommentTimeDec 27th 2021
    • (edited Dec 27th 2021)

    Changed the phrasing in the introduction to “adjoints to the Grothendieck construction”. Or maybe fixed a glitch, I’m not certain what the intent was.

    The previous phrasing made it sound like an equivalence is induced by restricting to Fib(C)Fib(C), but I don’t think that’s accurate since Fib(C)Fib(C) is not a full subcategory of Cat/CCat/C. So while the adjunction factors through the adjoint equivalence [C op,Cat]→Fib(C)[C^{op}, Cat] \to Fib(C), the equivalence Fib(C)→[C op,Cat]Fib(C) \to [C^{op}, Cat] is not the map you get by including Fib(C)⊆Cat/CFib(C) \subseteq Cat/C and applying Cat/C→[C op,Cat]Cat/C \to [C^{op}, Cat].

    diff, v70, current

    • CommentRowNumber46.
    • CommentAuthorMike Shulman
    • CommentTimeDec 28th 2021

    @jesuslop I think you’re right that must be a typo. He was probably halfway thinking of the contravariant Grothendieck construction.

    • CommentRowNumber47.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2021
    • (edited Dec 30th 2021)

    back to #43:

    For what it’s worth, I see that the statement in question (identifying the Grothendieck construction with the restricted slicing of pre-stacks) is made explicit in:

    (It’s not a big deal to see that this is the case, but it’s of some importance, which is why I was hunting for canonical citations.)

    • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2022

    added pointer to:

    (this should eventually go into an entry titled “enriched Grothendieck construction” – currently that term is redirecting to the entry Grothendieck construction, here.)

    diff, v72, current

    • CommentRowNumber49.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2022

    added pointer also to:

    • Liang Ze Wong, The Grothendieck Construction in Enriched, Internal and ∞-Category Theory, PhD thesis, Univ. Washington (2019) [pdf]

    diff, v72, current

    • CommentRowNumber50.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2023

    I have touched formatting and hyperlinking in the first few paragraphs of the Definition-section (here), in an attempt to beautify it a little more (still much room left, though)

    diff, v74, current

    • CommentRowNumber51.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2023

    have replaced the illustrating diagram by a tikzcd-rendering (here)

    diff, v74, current

    • CommentRowNumber52.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2023
    • (edited Apr 1st 2023)

    added (here) the example of the arrow category

    𝒮 I≃∫ B∈𝒮𝒮 /B \mathcal{S}^{I} \;\;\; \simeq \;\;\; \int_{B \in \mathcal{S}} \mathcal{S}_{/B}

    and (here) the example of the category of “retractive spaces” 𝒮 ℛ\mathcal{S}_{\mathcal{R}}

    𝒮 ℛ≃∫ B∈𝒮(𝒮 /ℬ) */ \mathcal{S}_{\mathcal{R}} \;\;\; \simeq \;\;\; \int_{B \in \mathcal{S}} \big( \mathcal{S}_{/\mathcal{B}} \big)^{\ast/}

    diff, v75, current

    • CommentRowNumber53.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2023

    added (here) a tikzed schema diagram also for the contravariant construction

    diff, v75, current

    • CommentRowNumber54.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2023

    added (here) the example of tangent categories

    diff, v75, current

    • CommentRowNumber55.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023

    Cite “Grothendieck Construction Considered Harmful”

    diff, v76, current

    • CommentRowNumber56.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2023

    You say in the entry:

    A call to think twice before using a Grothendieck construction

    This sounds mysterious. Briefly scanning over the note, I am getting the impression that what you have in mind are issues you ran into when considering categorical semantics of some kind of directed type theory?

    diff, v77, current

    • CommentRowNumber57.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023
    Correct, but I think my remarks apply more generally. Basically, the Grothendieck construction generalizes the ∑-type from sets/types to categories. The main point in my note is that it is just one out of a zoo of ∑-like constructions for category-like objects, and I believe too often people pick the Grothendieck construction because they are unaware of all the alternatives.
    • CommentRowNumber58.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2023

    Sounds interesting. How about adding a comment to this effect to the entry, so that the mystery is resolved a little.

    I gather instead of saying

    A call to think twice before using a Grothendieck construction

    it might be appropriate to say

    For a discussion of constructions closely related to but different from the Grothendieck construction, see…

    • CommentRowNumber59.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2023

    Looking closer at your note now.

    You claim to see two “flaws” of the Grothendieck construction, “Flaw 1.9” (p. 4) and “Flaw 1.10” (p. 5).

    It seems that “flaw 1.9” comes down to the feeling that

    the GC and the twisted GC are equally panthodic, yet the twisted GC is rarely used, if ever. This should indicate that something is off.

    But just a few lines above you note that what you call the “twisted GC” is “an instance of the GC.” This seems to render the previous “flaw” a moot point by your own account, no?

    \,

    In “flaw 1.10” you claim that

    it is still impossible to reconstruct 𝒟\mathcal{D} [a pseudofunctor] from the GF (∫ 𝒞𝒟,π 1)\big(\int_{\mathcal{C}} \mathcal{D}, \pi_1\big) [its associated Grothenieck fibration]

    This seems to directly contradict the first and constituting fact about the Grothendieck construction (here).

    • CommentRowNumber60.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023
    • (edited Apr 17th 2023)
    Regarding flaw 1.9: both points that I'm making are certainly connected, but the conclusion in any case will be that with the same input data, one can form two instances of the Grothendieck construction. So when in need of a ∑-type for categories, even if one decides to use a Grothendieck construction, this is a point to stop and reflect which one you need.

    Regarding flaw 1.10: I state "Indeed, in the GF, we can recognize cartesian arrows, which are arrows that can serve as mappings of the functorial action of D. However, these mappings are not unique on the nose." They are of course unique up to isomorphism, and so the pseudofunctor can be reinstated up to 2-equivalence (?), but not up to equality. The computational content of the original pseudofunctor's action on 1-cells has been discarded and new computational content for the reinstated pseudofunctor needs to be derived from the computational content of the proof of being a Grothendieck fibration (if that property has been stated in such a way that it contains computational content), (edit: or from the axiom of choice), which may not lead to the same outcome on the nose.

    I will rephrase the remark.
    • CommentRowNumber61.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023

    Rephrase exposition for reference to “Grothendieck Construction Considered Harmful”.

    diff, v79, current

    • CommentRowNumber62.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023
    • (edited Apr 17th 2023)

    Well I’m not sure how this would work in constructive mathematics, now that I think of it. Could you rebuild the pseudofunctor without defining a Grothendieck fibration in such a way that it is actually a cloven one? This page suggests that without cleavage, you can merely build a pseudofunctor to the 2-category of categories, anafunctors, and ananatural transformations. Without cleavage and with choice, you can rebuild the pseudofunctor, but its computational content may differ from the original one when considered on the nose.

    • CommentRowNumber63.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2023
    • (edited Apr 17th 2023)

    Did you see that Cisinski et al. seem to claim (here) to have sorted this out?

    • CommentRowNumber64.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023
    • (edited Apr 17th 2023)
    I am aware of their work, but I think many of my remarks are especially relevant when there is directedness at more than a single level. I have clarified this on the nlab page.
    (Assuming "this" refers to (infty, 1)-directed type theory at large.)
    • CommentRowNumber65.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2023

    Then maybe best to move the paragraph out of the references-section at the ordinary Grothendieck construction (on which your claim seems at best weak to me) into the bulk of directed type theory, or maybe into a new entry (n,r)-Grothendieck construction?

    • CommentRowNumber66.
    • CommentAuthoranuyts
    • CommentTimeApr 17th 2023
    Makes sense, I'll do something along those lines.
    • CommentRowNumber67.
    • CommentAuthoranuyts
    • CommentTimeApr 18th 2023

    Make the note at “A Vision for Natural Type Theory” shorter again, and put a paragraph under “generalizations”.

    diff, v81, current

    • CommentRowNumber68.
    • CommentAuthorHurkyl
    • CommentTimeApr 18th 2023
    • (edited Apr 18th 2023)

    Regarding flaw 1.9, if I recall correctly, my experience has been in the other direction: that the majority of usage is for the Grothendieck construction for contravariant functors, although not to the point I would consider the covariant Grothendieck construction’s usage to be rare. E.g. it’s invoked to establish equivalences between Grothendieck fibrations and contravariant Cat-valued pseudofunctors. Or Lurie’s exposition for the straightening/unstraightening construction for the equivalence between cartesian fibrations of ∞\infty-categories and ∞Cat\infty Cat-valued functors.

    • CommentRowNumber69.
    • CommentAuthorjonsterling
    • CommentTimeApr 18th 2023

    I think there is a lot to disagree with in the note under discussion, not only in the discussion of the “flaws” of the Grothendieck construction, but also in the discussion of self-classification (it is clear that not everything can be classified by a big version of itself! That does not mean it isn’t a wonderful thing when it is possible, and allusions to modes do not change this.). Everyone is certainly entitled to their opinion, and there are some interesting points made in there, but maybe for the nlab, I strongly think we should avoid taking on board such controversial viewpoints that seem to be deeply speculative.

    • CommentRowNumber70.
    • CommentAuthoranuyts
    • CommentTimeApr 18th 2023
    • (edited Apr 18th 2023)
    I'm fine either way, people can find it on my website. Or you could simply label it as controversial.
    • CommentRowNumber71.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    It may be unclear what it means to label a mathematical text as controversial.

    How about, instead, you first develop your note to the point that it contains a tangible proposition; and then we check it out to put it back on the nLab?

    • CommentRowNumber72.
    • CommentAuthoranuyts
    • CommentTimeApr 18th 2023
    I'm okay with that. However, I did add this part:
    https://ncatlab.org/nlab/show/Grothendieck+construction#other_like_constructions_for_categories
    which contains a tangible construction (rather than a proposition), which comes from the note. The necessity to cite one's sources would imply the removal of that section as well.
    • CommentRowNumber73.
    • CommentAuthoranuyts
    • CommentTimeApr 18th 2023
    I've rolled back to version 75. @Urs: there were two edits of yours in between. I believe these also concerned my added citation.
    • CommentRowNumber74.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2023

    In the section “(Co)Limits in a Grothendieck construction” (here) — which used to just state sufficient conditions for (co)limits to exist, I added an indication of how they are actually computed under these conditions.

    (May try to harmonize notation with the previous material tomorrow.)

    diff, v84, current

    • CommentRowNumber75.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2023

    added pointer to what I suppose is the original reference:

    • Alexander Grothendieck, §VI.8 of: Revêtements Étales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61 (SGA 1) , LNM 224 Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]

    diff, v86, current

    • CommentRowNumber76.
    • CommentAuthorUrs
    • CommentTimeMay 17th 2023

    In the section of (co)limits in a Grothendieck construction (here) I have made explicit the example of the Cartesian product (here), which comes out as the “external cartesian product” on the fiber categories.

    diff, v90, current

    • CommentRowNumber77.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2023
    • (edited May 31st 2023)

    added the observation (here) that

    for a pseudofunctor

    D op⟶C (−)Cat \mathbf{D}^{op} \overset{C_{(-)}}{\longrightarrow} Cat

    and an adjunction

    𝒟⊥⟵R⟶LD \mathcal{D} \underoverset {\underset{R}{\longleftarrow}} {\overset{L}{\longrightarrow}} {\;\; \bot \;\;} \mathbf{D}

    there is an induced covering adjunction between the Grothendieck constructions

    (∫x∈𝒟C L(x))⊥⟵R^⟶L^(∫x∈DC x) \Big( \underset {x \in \mathcal{D}} {\textstyle{\int}} C_{L(x)} \Big) \; \underoverset {\underset{\widehat R}{\longleftarrow}} {\overset{\widehat L}{\longrightarrow}} {\;\; \bot \;\;} \; \Big( \underset {\mathbf{x} \in \mathbf{D}} {\textstyle{\int}} C_{\mathbf{x}} \Big)

    diff, v92, current

  78. Added intuition for why the Grothendieck construction is an equivalence for groupoids.

    Owen Lynch

    diff, v97, current

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