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Stream: theory: category theory

Topic: representable promonoids and how to find them

fosco (May 26 2020 at 22:12):

Assume that a category of presheaves $[A,Set]$ is monoidal with respect to a convolution product. Then one can recover on $A$ a promonoidal structure by convoluting (convolving?) representables; I think I need a "representability criterion" on the promonoidal multiplication $P : A \times A \rightsquigarrow A$ ensuring that $P(x,y;z)=A(x\otimes y, z)$ for a certain monoidal structure on $A$. Is there such a thing? (I am not 100% clueless on this question, but I would like to know if there's a better approach than the one I have in mind)

sarahzrf (May 26 2020 at 22:15):

convolving, indeed

sarahzrf (May 26 2020 at 22:16):

when you say "a convolution product", what do you mean exactly if not the one defined in terms of a promonoidal structure on $A$ to begin with?

sarahzrf (May 26 2020 at 22:16):

just a tensor product?

John Baez (May 27 2020 at 00:04):

@fosco - are you asking when a promonoidal structure on $A$ (i.e. a monoidal structure on $[A, \mathsf{Set}]$) arises from a monoidal structure on $A$ via Day convolution?

John Baez (May 27 2020 at 00:05):

Do you want an answer different from the obvious answer, namely "if and only if the tensor product of representables is representable?"

fosco (May 29 2020 at 09:21):

Yes, I would like a condition to check on the promonoidal structure that allows me to say "oh, look, $P(a,b;c) \cong \hom(a\otimes b,c)$!")