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Stream: theory: category theory

Topic: Codomain fibration of a topos as a higher sheaf

Fernando Yamauti (Apr 10 2024 at 16:27):

Hi. Maybe I'm somewhat confused with the distinct definitions used in the literature. Still in Prop. 3.1 of this nlab page, it's claimed that the usual codomain fibration of a 1-topos $\mathcal{X}_{/(-)}$ is a (Cat-valued) 2-sheaf with respect to the canonical topology. On the other hand, it's well known that a (n, 1)-topos cannot define a continuous functor $\mathcal{X}^{op} \to Cat_{(n, 1)}$ unless we take the (n-2)-truncated objects.

Now, is there a nice way of seeing how these new non-invertible 2-cells recover descent? Also, is the same phenomenon expected to happen for higher n (say, for instance, n= (3, 2) or (3, 3))?

Reid Barton (Apr 11 2024 at 17:01):

The failure or success(?) of descent doesn't depend on the non-invertible 2-cells, just the invertible ones.
More specifically, the codomain fibration corresponds to the pseudofunctor that sends an object $X \in \mathcal{X}$ to the slice $\mathcal{X}/X$. Inside the category $\mathcal{X}/X$, there is the groupoid $(\mathcal{X}/X)^{\simeq}$ of objects and isomorphisms. This guy also satisfies descent, i.e., it is a stack of groupoids on $\mathcal{X}$.

Reid Barton (Apr 11 2024 at 17:03):

But inside $(\mathcal{X}/X)^{\simeq}$ we could also just look at the set of objects, and this doesn't form a stack (or sheaf, in this case).

Fernando Yamauti (Apr 11 2024 at 19:00):

Reid Barton said:

The failure or success(?) of descent doesn't depend on the non-invertible 2-cells, just the invertible ones.
More specifically, the codomain fibration corresponds to the pseudofunctor that sends an object $X \in \mathcal{X}$ to the slice $\mathcal{X}/X$. Inside the category $\mathcal{X}/X$, there is the groupoid $(\mathcal{X}/X)^{\simeq}$ of objects and isomorphisms. This guy also satisfies descent, i.e., it is a stack of groupoids on $\mathcal{X}$.

Hmm...I'm confused. Isn't requiring all colimits to be Van Kampen the same thing as requiring descent of the codomain fibration? If so, that cannot be true, say, for $Set_{/(-)} \colon Set^{op}\to Cat_{1}$, where $Cat_1$ is seen as a (2, 1)-category. On the other side, from the result I've initially mentioned, that should be true when one takes the aforementioned functor to take values, instead, in $Cat_1$ as a (2, 2)-category. No?

Reid Barton (Apr 11 2024 at 20:26):

Fernando Yamauti said:

Isn't requiring all colimits to be Van Kampen the same thing as requiring descent of the codomain fibration?

No I don't think so, or at least it's not what I meant by descent (or stack), which is just about van Kampen-ness of some specific "Cech nerve" diagrams. E.g. if $Y \to X$ is a surjection then a set over $X$ is the same (in the sense of an equivalence of categories, or an equivalence of groupoids) as a set over $Y$, together with an isomorphism between the two pullbacks to $Y \times_X Y$, that satisfy a cocycle condition.

Reid Barton (Apr 11 2024 at 20:30):

I am also frequently confused about the relationship between these conditions though.
I think maybe one way of looking at this situation is that the inclusion of $\mathcal{X}$ into stacks of categories (or groupoids) on $\mathcal{X}$ doesn't preserve all colimits (only the van Kampen ones), and that is why the codomain fibration, even though it is "representable" by a stack of categories (or groupoids), doesn't send all colimits in $\mathcal{X}$ to limits of categories (or groupoids).

Fernando Yamauti (Apr 11 2024 at 20:55):

Reid Barton said:

Fernando Yamauti said:

Isn't requiring all colimits to be Van Kampen the same thing as requiring descent of the codomain fibration?

No I don't think so, or at least it's not what I meant by descent (or stack), which is just about van Kampen-ness of some specific "Cech nerve" diagrams. E.g. if $Y \to X$ is a surjection then a set over $X$ is the same (in the sense of an equivalence of categories, or an equivalence of groupoids) as a set over $Y$, together with an isomorphism between the two pullbacks to $Y \times_X Y$, that satisfy a cocycle condition.

Sure. I've always meant Čech descent. Still, even if the sheaf condition on the canonical topology doesn't mean continuity (for some reason, I still think it does, though, but I need more time to think), in $Set$, the coequaliser with only $1$'s and identities is not Van Kampen, so something still not quite right. Also notice that in the nlab page (and also in the paper where the result is claimed), they are only requiring morphisms between the restrictions of the sections to the the double intersections, not isomorphisms as you've required above (i.e, a limit in a (2,2)-category as opposed to a (2,1)-category).

Fernando Yamauti (Apr 13 2024 at 03:06):

Ok. I guess I impulsively replied and I've spilled a bunch of bullshit. So let me just clean it partially. Certainly the coequaliser with 1's is not Van Kampen, but the one with three 1's is so. Also, no definition of descent leave the morphisms on intersections $Y \times_X Y$ free, what can happen is that they might be only invertible up to a 2-cell.

I guess what was confusing me was my memory of Lurie's proof of Prop.1.3.1.7 in SAG (p.117) claiming that the $\mathcal{C}$-sheaf condition with respect to the canonical topology coincides with continuity for $\infty$-topoi when $\mathcal{C}$ is complete. Everything seems to follow formally and $n = \infty$ doesn't seem to be explicitly required.