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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.
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.
Okay, but then let's put this in different sections.
So what kind of fibration do lax functors correspond to under the construction?
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.
@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.
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.
Oh, and thanks for the info on [C^op,Prof]. We should add these generalizations to the entry. At least point out a reference.
expanded the Idea-section at Grothendieck construction
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.
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.)
I further expanded Grothendieck construction, trying to bring it closer to something like a self-contained exposition.
At Urs’s suggestion, I’ve expanded on the lax colimit property of the Grothendieck construction here.
Thanks!
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.
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)
I suppose this should be linked at (infinity,1)-Grothendieck construction (too).
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?
I think that’s right, it should be normal lax functors.
OK, thanks! Do you know any source that discusses the Grothendieck construction in this generality?
I’m sure there’s something related to it in the work that Bénabou is promising to write about ;-)
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.
@Karol: I can’t think of one off the top of my head.
@Zhen: You may be looking for this theorem.
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 $\int \mathbb{E}$.
I added to Grothendieck construction the fact that it preserves local smallness, which came up at this thread.
Add reference to Beardsley-Wong enriched Grothendieck construction.
I made “enriched” and “Jonathan Beardsley” become hyperlinks (here)
Comment #27 is referring to this comment.
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?
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).
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:
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?
It definitely is a coslice. Or, if you like, a slice under rather than a slice over.
Thank you David :)
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 $(\int F) \to C$?
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