Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: Locally small fibrations

James Deikun (Nov 17 2025 at 13:53):

Locally small fibrations have a rather difficult definition. In keeping with the principle that if something in fibrations looks difficult it's because it has an instance or two of fiberwise opposite inlined, the condition that the fiberwise opposite is locally small is simple: that the inclusion of the wide subcategory of cartesian morphisms in the total category is a left adjoint.

This condition is well-motivated too: it means that small-generated copresheaves on the fibered category are themselves small. So the fibration itself being locally small means that small-generated presheaves are themselves small. This rephrasing makes it apparent how the condition, unlike for ordinary categories, is not self-dual, and that you often want both versions.

EDIT: I misremembered something I proved some time ago and made some misunderstandings based on that. See starting at #theory: category theory > Locally small fibrations @ 💬 for the real story.

Morgan Rogers (he/him) (Nov 17 2025 at 14:31):

@James Deikun was this supposed to be a reply in another topic?

James Deikun (Nov 17 2025 at 14:36):

No, it's just something I observed that I thought should be more widely known. I don't know whether it's in the literature or not but it seems unlikely it is widely known or I wouldn't have had to (re?)discover it.

Nathanael Arkor (Nov 17 2025 at 14:58):

Sounds worth adding to the nLab page on [[locally small fibration]]!

Mike Shulman (Nov 17 2025 at 17:20):

That doesn't look equivalent to the definition. And I'm pretty sure that local smallness of fibrations is indeed self-dual (relative to the fiberwise opposite).

James Deikun (Nov 17 2025 at 17:20):

Hm. I did notice the version of the definition on the nLab looks self-dual (unlike the version I was working against).

James Deikun (Nov 17 2025 at 17:21):

(Which might have been self-dual, but didn't look self-dual particularly.)

Mike Shulman (Nov 17 2025 at 17:29):

What version is that?

James Deikun (Nov 17 2025 at 17:29):

It's the version that used to be on the Stacks Project, which I can no longer find since it's been reorganized.

James Deikun (Nov 17 2025 at 17:36):

I think it was the same as the one in https://arxiv.org/abs/1801.02927

James Deikun (Nov 17 2025 at 17:41):

I came up with this while proving that the fundamental fibration is locally small if and only if the base is locally Cartesian closed.

Mike Shulman (Nov 17 2025 at 17:56):

Those should be equivalent.

James Deikun (Nov 17 2025 at 17:57):

Yes, they should be. But one is manifestly self-dual and the other is covertly self-dual.

James Deikun (Nov 17 2025 at 18:07):

(I clearly need to update the OP, but I'm holding off until it's clear what I need to update it to.)

James Deikun (Nov 17 2025 at 18:28):

I'm starting to think that I somehow smuggled in an assumption of global smallness. Although, that wouldn't make sense in the original context.

James Deikun (Nov 17 2025 at 19:44):

Aha! Found my notes. The problem was I misremembered the statement. The actual statement was: For each object $X$ of the total category, the functor $\mathcal{E}^\mathrm{cart}/X \xrightarrow{\mathrm{dom}} \mathcal{E}^\mathrm{cart} \xrightarrow{i} \mathcal E$ is a left adjoint. Only in the globally small case, where $\mathcal E$ itself has a terminal object, does it reduce to a single functor being an adjoint, because the $X$ for which this is a left adjoint form a sieve.

James Deikun (Nov 17 2025 at 20:01):

These functors sort of approximate the inclusion of the Cartesian arrows from below, so they're saying that functor is almost a left adjoint. Apparently this is still self-dual even though it's even less obvious than the Benabou-via-Streicher definition makes it look.

James Deikun (Nov 17 2025 at 20:05):

The connection to the definition on the nLab is more obvious. We have:

A Grothendieck fibration $p: C \to E$ is called locally small if, for every pair $A,B \in C$, there exists an object of $E_{pA \times pB}$, $(x,y) : I \to pA \times pB$, and a morphism $f: x^*A \to y^*B \in C_I$, which is terminal, in the sense that given another such datum $(J,z,w,g)$, there is a unique map $u: J \to I$ so that $x u = z, y u = w$, and the coherence isomorphisms identify $u^*f$ with $g$. (This is Elephant B.1.3.12).

James Deikun (Nov 17 2025 at 20:11):

Such a datum is equivalently up to unique iso, thanks to Cartesian lifting and factorization, a span $(J, \beta, b)$ where \beta is a Cartesian lift of $z$ and $b$ is the composition of a Cartesian lift of $w$ with $g$. You can get the original datum back from one of these and the categories of them are equivalent.

James Deikun (Nov 17 2025 at 20:17):

However, the category of such spans is equivalently also the comma category $L \downarrow B$ where $L : E^\mathrm{cart}/A \to E$ is the kind of functor from above. And if those commas have terminal objects for all $B$, then those functors have right adjoints.

James Deikun (Nov 18 2025 at 01:26):

Oh, and as for the interpretation: on globally small fibrations it's the same, the idea that copresheaves/presheaves are small when small-generated, but on other fibrations, it means that (co)presheaves restricted to some particular family of objects are small when the whole (co)presheaf is small-generated from within that family. In other words, that copresheaves/presheaves on globally small full subcategories are small when small-generated.