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Stream: theory: category theory

Topic: algebra of a profunctor

Matteo Capucci (he/him) (Jul 04 2021 at 15:31):

Does anyone know where this concept is defined?

Matteo Capucci (he/him) (Jul 04 2021 at 15:31):

The nLab page doesn't specify any references

Timothy Porter (Jul 04 2021 at 15:56):

Why not ask on the nForum?

John Baez (Jul 04 2021 at 16:16):

Because those nLab people are so scary. :goblin:

John Baez (Jul 04 2021 at 16:20):

Seriously: you can look at the history of an nLab page, see who wrote something, and ask that person a question.

Mike Shulman (Jul 04 2021 at 17:59):

The creation and naming of the nLab page was discussed here.

Matteo Capucci (he/him) (Jul 04 2021 at 18:56):

I wanted to but I didn't find a thread for that page on the nForum

Matteo Capucci (he/him) (Jul 04 2021 at 18:58):

Mike Shulman said:

The creation and naming of the nLab page was discussed here.

Thanks, let's see if Ramesh replies

Matteo Capucci (he/him) (Jul 04 2021 at 18:59):

Reading the first message in that thread it seems like he came up with the definition, though

John Baez (Jul 04 2021 at 19:28):

Matteo Capucci (he/him) said:

I wanted to but I didn't find a thread for that page on the nForum

I see. You can always just make up a new nForum thread whose title is the title of some nLab page - and if doesn't already exist you should, since it just means someone forgot to.

dusko (Jul 06 2021 at 02:58):

Matteo Capucci (he/him) said:

Does anyone know where this concept is defined?

i either don't understand the question, or don't understand what is there to be defined. we define things to stipulate what we are talking about, when it might be this or that (as per aristotle). eg we define tomato as a fruit at school and a vegetable at home. but if we agree what is a profunctor and what is an algebra, then the profunctor deetermines a monad on the cocompletion of its domain, and a comonad on the completion on its codomain, and the monad determines the algebras. or are there are other things around profunctors that would like to be called algebras? maybe someone should define them as something else. vegetables?

fosco (Jul 06 2021 at 04:56):

dusko said:

Matteo Capucci (he/him) said:

Does anyone know where this concept is defined?

i either don't understand the question, or don't understand what is there to be defined. we define things to stipulate what we are talking about, when it might be this or that (as per aristotle). eg we define tomato as a fruit at school and a vegetable at home. but if we agree what is a profunctor and what is an algebra, then the profunctor deetermines a monad on the cocompletion of its domain, and a comonad on the completion on its codomain, and the monad determines the algebras. or are there are other things around profunctors that would like to be called algebras? maybe someone should define them as something else. vegetables?

The definition on that nLab page leaves me a bit confused; I prefer your approach: an endoprofunctor $$H : C^{op}\times C \to Set$ corresponds to a left adjoint $\hat H : [C^{op},Set] \to [C^{op},Set]$, and from the adjunction $(\hat H\dashv R)$ one gets a monad on $[C^{op},Set]$; an algebra for $H$ is an algebra for the monad $R\hat H$. The problem I have with this definition is that _every_ profunctor, not only an endoprofunctor, generates such a monad, whereas an "algebra" is usually defined as an object on which a monoid object is acting, being as an action map $M\otimes X \to X$ or, by transposition, as a map of monoids $M \to End(X)$.

Matteo Capucci (he/him) (Jul 06 2021 at 09:32):

Replying to @dusko, I'm trying to understand where the definition in the nLab comes from.
It doesn't seem to be what you and @fosco suggest, unless I'm mistaken. I think the concept they instantiate is that of algebra (or better, module) of an endomorphism in a bicategory (you can find such definition on the nlab too)

Nathanael Arkor (Jul 06 2021 at 11:56):

If the concept is the same as an algebra for an endomorphism, then you can find a reference in section 2.3.1 of Lenisa–Power–Watanabe's Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads.

Nathanael Arkor (Jul 06 2021 at 11:56):

You may have to unwind some definitions, though.

Matteo Capucci (he/him) (Jul 06 2021 at 13:00):

Thanks!