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Stream: theory: category theory

Topic: 2-functors preserving discrete fibrations

Matteo Capucci (he/him) (May 30 2025 at 14:57):

Consider a discrete fibration $p:E \to B$ in a 2-category $\cal K$, and a 2-functor $F:\cal K \to \cal H$.
I'm interested in conditions so that $Fp$ is again a discrete fibration.

Here's a possible one: if $F$ has a left 2-adjoint, then one can always factor a lifting problem for $Fp$ through the unit, and thus solve it in $\cal K$.
Alternatively, fibrations in a 2-category are algebras for the arrow monad. So preserving Cat-powers and pullbacks might be enough. This is implies by being left 2-adjoint.

Mike Shulman (May 30 2025 at 15:02):

Yes -- being a discrete fibration can be expressed by diagrams that use only finite (2-)limits, so it is preserved by any finite-limit-preserving functor.

Matteo Capucci (he/him) (May 30 2025 at 15:02):

Thanks!