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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 $(\int F) \to C$?
added publication data to:
Hurkyl, I believe I’ve seen that.
added pointer to:
added pointer to:
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?
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 $P(I)$.
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 $Cat/C \to [C^{op}, Cat]$.
@jesuslop I think you’re right that must be a typo. He was probably halfway thinking of the contravariant Grothendieck construction.
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.)
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.)
added pointer also to:
added (here) the example of the arrow category
$\mathcal{S}^{I} \;\;\; \simeq \;\;\; \int_{B \in \mathcal{S}} \mathcal{S}_{/B}$and (here) the example of the category of “retractive spaces” $\mathcal{S}_{\mathcal{R}}$
$\mathcal{S}_{\mathcal{R}} \;\;\; \simeq \;\;\; \int_{B \in \mathcal{S}} \big( \mathcal{S}_{/\mathcal{B}} \big)^{\ast/}$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?
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…
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 $\mathcal{D}$ [a pseudofunctor] from the GF $\big(\int_{\mathcal{C}} \mathcal{D}, \pi_1\big)$ [its associated Grothenieck fibration]
This seems to directly contradict the first and constituting fact about the Grothendieck construction (here).
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.
Did you see that Cisinski et al. seem to claim (here) to have sorted this out?
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?
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 $\infty Cat$-valued functors.
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.
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?
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.)
added pointer to what I suppose is the original reference:
added the observation (here) that
for a pseudofunctor
$\mathbf{D}^{op} \overset{C_{(-)}}{\longrightarrow} Cat$and an adjunction
$\mathcal{D} \underoverset {\underset{R}{\longleftarrow}} {\overset{L}{\longrightarrow}} {\;\; \bot \;\;} \mathbf{D}$there is an induced covering adjunction between the Grothendieck constructions
$\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)$1 to 77 of 77