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Stream: theory: category theory

Topic: companion as "teleporting"

Christian Williams (Jan 11 2023 at 19:55):

A functor $f:A\to B$ induces a "companion" profunctor $B(f,1):A\to B$ (yes, we may have opposite conventions) whose values are $B(f,1)(a,b) = B(fa,b)$. I think of this as "teleporting" $a$ into $B$ and looking outward; the companion allows $a$ to be in "superposition" in both $A$ and $B$ at the same time.

Christian Williams (Jan 11 2023 at 19:55):

But in the collage of $B(f,1)$, the morphism $a\to fa$ is not literally an identity. I think I want $a\to fa$ to be an identity, for the Fib does not have companions issue I've discussed in another stream.

Christian Williams (Jan 11 2023 at 19:55):

In topos theory, the "union" of two subobjects is formed by first forming their pullback, and then pushing out, to identify what they have in common. Doing that here, for the inclusion of $A$ and $B$ into the collage, would identify $a$ with $fa$. This seems intuitively nice, and I want to use this construction. Does anyone know anything about this? Thanks.

Christian Williams (Jan 11 2023 at 22:10):

Okay, what I said is not quite right; the pullback-pushout doesn't identify $a:A$ with $fa:B$, because they include on opposite sides of the collage. A modified version of the construction may still work, but it looks like it's going in the wrong direction for my research question. Anyway, just curious if anyone has thoughts on this topic, so I'll leave it here.