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Stream: learning: questions

Topic: Walking arrow in a 2-category

Fawzi Hreiki (Feb 25 2021 at 22:16):

I don't know if this is a naive question, or if it even makes sense. In a 2-category, the comma objects $(f/g)$ determine the 2-morphisms in the category (this is just their UMP). In Cat, the natural transformations are determined by the walking arrow $2 = (0 \rightarrow 1)$ which is a join $(1 \star 1)$ and hence a particular kind of co-comma object $f\star g$. This is obviously reminiscent of the fact that $1$ is strong generator in Set. Is there anyway to make sense of this? For example, what implications does it have on the internal logic of the 2-category?

Mike Shulman (Feb 26 2021 at 02:08):

It's true that 1 is a strong generator in Cat in a 2-categorical sense, i.e. if $f:A\to B$ induces an equivalence of hom-categories $K(1,A) \to K(1,B)$ then $f$ is an equivalence.