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Stream: theory: category theory

Topic: non-split fibration

Daniel Teixeira (Oct 21 2024 at 21:49):

How can I see that the codomain fibration $\text{cod}:\mathcal C^{[1]}\to\mathcal C$ in general is not a split fibration? I'm missing something in my calculations, because to me it seems that lifting $f:x\to y$ to $f\circ 1_x = 1_y\circ f$ is functorial.

Kevin Carlson (Oct 21 2024 at 22:01):

A splitting of a fibration is a choice of Cartesian lifting for every pair of an arrow $f$ in the base and an object in the fiber over the codomain of $f$; your proposed splitting ignores the second half of this information. You may be confusing a split fibratjon with a fibration that, as a functor, is a split epimorphism. Certainly the codomain fibration is a split epi and you’ve shown why here.

Daniel Teixeira (Oct 21 2024 at 22:04):

ah, I see, given I'm not choosing the cartesian lift for $(f:x\to y,y'\in \text{cod}^{-1}(y))$, for that I could for instance take a pullback. But then the whole construction is only pseudofunctorial

Daniel Teixeira (Oct 21 2024 at 22:05):

do you have any non-trivial example of split fibration to keep in mind?
I guess one could skeletalize $\mathcal C$ in this case

Vincent Moreau (Oct 21 2024 at 22:08):

You can take a look at the example given by Streicher in Fibered Categories a la Jean Benabou, see Warning 1 under Definition 3.1. The fibration given is the surjective group homomorphism $\mathbf{Z} \to \mathbf{Z}/2\mathbf{Z}$, which is a Grothendieck fibration if we see the groups as one-object categories.

Daniel Teixeira (Oct 21 2024 at 22:08):

that fibration is not split

Vincent Moreau (Oct 21 2024 at 22:08):

This is another example of a NON-split fibration. On the other hand, if you want a split example, you can take the Fam fibration.

James Deikun (Oct 21 2024 at 22:10):

Or the domain fibration.

Daniel Teixeira (Oct 21 2024 at 22:12):

Vincent Moreau said:

This is an example of a NON-split fibration. On the other hand, if you want a split example, you can take the Fam fibration.

This is fibred in sets, right? I would consider this example trivial: every such fibration is split. For the codomain fibration, this would correspond to skeletalizing the category.

Vincent Moreau (Oct 21 2024 at 22:17):

If your question is if the fibers are discrete, then it is not the case as soon as the category on which we apply the Fam construction has a non-identity endomorphism. If we call such a morphism $u : X \to X$, then the morphism $(\operatorname{Id} : \{*\} \to \{*\}, u)$ lives in the fiber of $\{*\}$. Or perhaps I don't understand your question.

Vincent Moreau (Oct 21 2024 at 22:19):

The fiber over a set $I$ is the category $\mathbf{C}^I$.

Daniel Teixeira (Oct 21 2024 at 22:21):

oh, okay, I'm not familiar with this I guess I just misunderstood the construction at the 1lab

Vincent Moreau (Oct 21 2024 at 22:25):

As a reference, there is Categorical Logic and Type Theory by Jacobs, where the family fibration is described in section 1.2.