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Stream: theory: category theory

Topic: supply of comonoids

Matteo Capucci (he/him) (Oct 18 2021 at 09:59):

A Markov category is an affine monoidal category $(\mathbf C, 1, \otimes)$ such that every object $X$ has an assigned comonoid structure $(\mathsf{copy}_X, !_X)$
What kind of datum is this assignment? Is it a section of $\mathrm{Comon}(\mathbf C) \to \mathbf C$? Or would that imply naturality of $\mathsf{copy}$, which we reject in order to not degenerate into cartesianity?

Matteo Capucci (he/him) (Oct 18 2021 at 10:02):

(One would add monoidal to section in order to satisfy the rest of axioms for copy and delete in a Markov category)

fosco (Oct 18 2021 at 10:24):

What do you mean by "section"? A right-adjoint-right-inverse?

Matteo Capucci (he/him) (Oct 18 2021 at 10:52):

A right inverse

Matteo Capucci (he/him) (Oct 18 2021 at 10:53):

@Toby Smithe pointed me to this paper which seems to answer my question: https://arxiv.org/pdf/1908.02633.pdf

Matteo Capucci (he/him) (Oct 18 2021 at 10:53):

If we ask for functors then we get in trouble with naturality