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Stream: theory: category theory

Topic: iterated fibrations

Matteo Capucci (he/him) (Mar 29 2023 at 14:04):

Given $p:E \to B$ and $q:F \to B$ [[Grothendieck fibrations]], a morphism $\phi : p \to q$ is a fibration between them (in the sense of [[fibration in a 2-category]]) iff $\phi$ makes the evident triangle commute and $\phi : E \to F$ is a Grothendieck fibration itself. That's a well-known theorem proven, for instance in this paper by Hermida.
In particular this theorem tells us that we don't need to check wheter $\phi$ preserves $p$-cartesian arrows. If it is a fibration, then the following commutes automatically (for any $f : b \to b'$ in $B$):
image.png

Matteo Capucci (he/him) (Mar 29 2023 at 14:04):

On the other hand, each $\phi_b$ is itself a fibration, so one might wonder whether the above square witnesses $f^*$ being a map of fibrations. In other words, does $f^*$ preserve $\phi$-cartesian arrows?

Mike Shulman (Mar 29 2023 at 15:59):

Yes, this should be true, if you construct the action of $f^*$ on arrows by factoring through cartesian arrows and then use the fact that cartesian arrows are cancelable on one side.

Matteo Capucci (he/him) (Mar 29 2023 at 20:06):

Thanks Mike! Though I don't think I understand what you mean by 'cartesian arrows are cancelable on one side'.

Mike Shulman (Mar 29 2023 at 20:22):

If $g\circ f$ and $g$ are cartesian, then $f$ is cartesian.