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

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.
• CommentTimeApr 8th 2014

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 $2$-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 $\mathcal{B}$ be a category and let $F : \mathbb{E}' \to \mathbb{E}$ be a $\mathcal{B}$-indexed functor. If the components of $F$ are Grothendieck fibrations (+ possibly other conditions), then $\int F : \int \mathbb{E}' \to \int \mathbb{E}$ is also a Grothendieck fibration.

In particular, for the case where $\mathbb{E}$ is the terminal $\mathcal{B}$-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 $\mathbb{E} \ast \mathbb{F}$, which is a Grothendieck fibration over $\int \mathbb{F}$ and a Grothendieck opfibration over $\int \mathbb{E}$.

• 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.

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

• 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.

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

Comment #27 is referring to this comment.

1. 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 $R$ is this left adjoint, then by adjointness $R(D)(c)$ must be the hom-category $(Cat/C)(\int( C(-,c)) , D)$ (at least if the adjunction is $Cat$-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?
2. created links for “Grothendieck construction for monoidal categories” and “Christina Vasilakopoulou”

Joe M

• CommentRowNumber34.
• CommentAuthorDmitri Pavlov
• CommentTimeMar 18th 2020

• 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” $Cat_{*,l}\to Cat$. The page refers to $Cat_{*,l}$ 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 $*/Cat$. 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.

• 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

$\int : \Cat^C \to \Cat : F \mapsto C_{\bullet/} \otimes_{C} F$ $\int : Cat^{C^{op}} \to \Cat : F \mapsto F \otimes_C C_{/\bullet}$

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

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeDec 30th 2020

• CommentRowNumber41.
• CommentAuthorTodd_Trimble
• CommentTimeDec 30th 2020

Hurkyl, I believe I’ve seen that.

• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeAug 28th 2021

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeDec 27th 2021

• 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?

• 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 $P: \mathcal{E} \longrightarrow$ Cat, we shall thus construct a fibration $G: \mathcal{G} \longrightarrow \mathcal{E}$ whose fibre at $I \in \mathcal{E}$ is precisely the category $P(I)$ :

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

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

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

• 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 $Fib(C)$, but I don’t think that’s accurate since $Fib(C)$ is not a full subcategory of $Cat/C$. So while the adjunction factors through the adjoint equivalence $[C^{op}, Cat] \to Fib(C)$, the equivalence $Fib(C) \to [C^{op}, Cat]$ is not the map you get by including $Fib(C) \subseteq Cat/C$ and applying $Cat/C \to [C^{op}, Cat]$.

• 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.)