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I think the definition of the Grothendieck construction was wrong. The explicit definition was right, but the description in terms of a generalized universal bundle didn’t work out to that, if by “the category of pointed categories” was meant for the functors to preserve the points, which is the usual meaning of a category of pointed objects. I corrected this by using the lax slice. Since while I was writing it I got confused with all the op’s, I decided that the reader might have similar trouble, so I changed it to do the covariant version first and then the contravariant.
’the category of pointed categories’ should have weakly pointed functors as 1-arrows, and as you say is the lax slice $*\downarrow Cat$. I think the definition was lost in translation somewhere between my brain and (probably) Urs’ fingers :) (and this was a few years ago since that discussion!)
Redacted
As discussed at generalized universal bundle (I think) the “generalized universal bundle” over $C$ is the pullback
$\array{ P_* C &\to& * \\ \downarrow && \downarrow \\ C^I &\stackrel{d_1}{\to}& C } \,,$for the given point $* \to C$ and the relevant path object $C^I$.
For $C = Cat$ the point is the terminal category and $C^I$ is the 1-categorical internal hom $[\Delta[1], Cat]$, the category whose objects are functors and whose morphisms are filled commuting diagrams.
This gives the correct construction, and this is what is meant.
I am thinking that eventually there should be a way to make precise the sense in which this kind of construction produces “comma object-weak pullbacks” in analogy to how the analogous construction for undirected interval objects gives homotopy pullbacks.
I don’t know what a “filled commuting diagram” is; if “filled” means that it contains a 2-cell, then it doesn’t commute. Squares containing a 2-cell is what you need as the morphisms in order to get the right thing (i.e. the lax slice) as the “generalized universal bundle,” and that version of “$[\Delta[1],Cat]$” is only the internal hom relative to the lax version of the Gray tensor product on $2Cat$. It’s certainly not the internal hom for any closed structure on $Cat$.
Commuting up to a 2-isomorphism, yes. And I mean the 2-categorical internal hom: 2-functors, pseudonatural transformations and modifications.
So this diagram
$\array{ && * \\ & \swarrow &\swArrow& \searrow \\ C_x &&\to&& C_y }$that describes what the Grothendieck construction does over a morphism is the (single) component of a pseudonatural transformation between two 2-functors $\Delta[1] \to Cat$ where after restriction along $d_1 : \Delta[0] \to \Delta[1]$ everything is required to be constant on the terminal category.
No, the point is that it can’t commute up to a 2-isomorphism; you have to allow the transformation sitting in the square to be non-invertible. In other words, you need to use lax natural transformations. If it were a pseudonatural transformation, then in that diagram you just drew, the 2-morphism would be invertible – but in the Grothendieck construction a morphism in the total category from $a$ to $b$ over $f:x\to y$ is a map $f_!(a)\to b$ which is not necessarily invertible.
No, the point is that it can’t commute up to a 2-isomorphism; you have to allow the transformation sitting in the square to be non-invertible.
Right of course. Or else it works for the groupoid version.
Okay, now looking back at the discussion I clearly didn’t give very concentrated replies here, but it seems to me that this notwithstanding, it is kind of noteworthy that the Grothendieck construction is forming this limit here
$\array{ \int F &\to&&\to& * \\ \downarrow &&&& \downarrow \\ && [\Delta[1], Cat^{op}] &\to& Cat^{op} \\ \downarrow && \downarrow \\ C &\to& Cat^{op} }$for a suitable hom-obect $[\Delta[1],Cat^{op}]$.
Speaking just about the lax (co)slice for the top-right pullback hides a bit this nice structure.
With the suitable notion of “weak pullback” we should just be saying that the Grothendieck construction is the “weak pullback” of the point.
Maybe. I prefer thinking about comma objects (including slice categories) as weighted limits in their own right, rather than always constructing them as a double pullback, but I can see that that other perspective can also be helpful. It also makes me uneasy when we say “weak” to include “lax” because some people use “weak” to mean specifically “pseudo.”
…although perhaps that is being too persnickety. I do like the “generalized universal bundle” point of view, although as this discussion shows there are some subtleties to beware of when you start having noninvertible higher cells around.
as this discussion shows there are some subtleties to beware of when you start having noninvertible higher cells around.
Yes, indeed, I took the message home here that the discussion of this on the Lab, for the special case at hand but also for the general situation at generalized universal bundle, needs to be improved. I’ll try to look into this when I have more leisure.
This week and next one I am attending conferences and need to focus my energy a little bit elsewhere, though.
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