You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Hi,
I'm wondering what sort of profunctors and form an adjunction internal to . By this I mean that there are natural transformations and satisfying the triangle identities.
nLab has a enigmatic remark about "semifunctors" in this situation but I'm not sure how a semifunctor would specify a profunctor.
In my own thinking, so far I have concluded that for all , and must be quotients of representables, but that's all I have so far.
Here is the argument for the last claim:
For , denote .
Consider one of the triangle identities: .
Take any and run it through this identity:
By the identity, we must have . So then every is the result of a -action of the single element , which only depends on . Thus there is a surjective transformation , where is the codomain of , sending to .
A similar argument can be made for .
Talking about semifunctors is just a cute trick for talking about functors to the Cauchy completion. If is a semifunctor, then the image of each identity is not necessarily an identity, but it is idempotent, and thus induces an actual functor into the Cauchy completion of . And oppositely.
So the Cauchy completion sends all idempotents to identity? I thought it just makes all idempotents split.
But how does any of this give a profunctor?
I didn't mean that the composite is a functor, only that from you can define a functor , by sending an object to the splitting of the idempotent .
Once you have such a functor, you get a profunctor in the usual way, and then restrict to .
Since has split idempotents, functors are equivalent to functors , so the restriction doesn't lose any information.