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Stream: theory: category theory

Topic: Slices vs families

Brendan Murphy (Apr 08 2024 at 19:09):

We work in a material set theory. Let $X$ be a set. It's well know that the categories $\mathsf{Set}/X$ and $\prod_{x \in X} \mathsf{Set}$ are equivalent. Are they isomorphic? I would think not, but I'm not sure how to prove it

Brendan Murphy (Apr 08 2024 at 19:36):

Maybe you can use the fact that the projection to $\mathsf{Set}$ is determined uniquely up to isomorphism by having a left exact left adjoint?

Kevin Carlson (Arlin) (Apr 08 2024 at 21:01):

At least there shouldn't be any such isomorphism that's natural in $X$--I think that captures the basic issue that $\mathsf{Set}^X$ can have the same fiber over different $x$ es but $\mathsf{Set}^X$ can't.

John Baez (Apr 08 2024 at 21:17):

There's a typo in your comment, Arlin.

Mike Shulman (Apr 09 2024 at 03:36):

They're isomorphic if $X=\emptyset$ or $X=1$... (-:O