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Stream: theory: category theory

Topic: bifunctor spans->rels

Matteo Capucci (he/him) (Sep 04 2024 at 15:18):

At [[Rel]], it is said that there is a lax bifunctor $|-|:Span(C) \to Rel(C)$.
Is such a bifunctor actually strong? Afaiu it sends $A \leftarrow R \to B$ to $|R| = \{(a,b) \mid \exists r \in R(a,b).\top\} \subseteq A \times B$.
Then $|R| \odot |S| = \{(a,c) \mid \exists b \in B \exists r \in R(a,b) \land \exists s \in S(b,c)\}$ while $|R \odot S| = \{(a,c) \mid \exists k \in R\odot S(a,c)\} = \{(a,c) \mid \exists b \in B \exists r \in R(a,b) \land \exists s \in S(b,c)\}$.

Matteo Capucci (he/him) (Sep 04 2024 at 15:20):

That's a proof 'in Set' but there's also a diagrammatic proof involving the lifting property of ortoghonal factorization system of the regular category C, and the fact that mono+regular epi = iso

Matteo Capucci (he/him) (Sep 04 2024 at 15:21):

Am I missing something?

Mike Shulman (Sep 04 2024 at 20:15):

Looks right to me.

Kevin Carlson (Sep 04 2024 at 20:50):

FWIW Barr's embedding theorem should neatly justify the pointwise reasoning without worrying about the diagrammatic proof.

Mike Shulman (Sep 05 2024 at 00:31):

Or just the internal logic of a regular category.

Matteo Capucci (he/him) (Sep 05 2024 at 06:10):

Kevin Carlson said:

Barr's embedding theorem

ah cool, I didn't know about that

Matteo Capucci (he/him) (Sep 05 2024 at 06:10):

I'm gonna edit the nLab then