Timeline for Self-enrichments of rigid monoidal categories

6 events

when toggle format	what		by	license	comment
Feb 9, 2021 at 14:24	comment	added	Tim Campion		@TimCromby There's a notational ambiguity. I'll write $Hom(X,Y)$ for the set of morphisms and $[X,Y]$ for the internal hom. As Fernando says, one defines $[X,Y] = X^\ast \otimes Y$. The idea to keep in mind is that if you already had the closed monoidal structure, then you would have defined $X^\ast = [X,I]$. You would have a map $X^\ast \otimes Y = [X,I] \otimes [I,Y] \to [X,Y]$ internalizing the composition operation $Hom(X,I) \times Hom(I,Y)\to Hom(X,Y)$, and using rigidity you'd have shown this map to be an isomorphism. When setting up the closed structure, you run that picture backwards
Feb 9, 2021 at 12:06	comment	added	Jake Wetlock		@Fernando: How does one relate the set $Hom(X,Y)$ with $V^* \otimes Y$?
Feb 9, 2021 at 11:57	history	edited	Jake Wetlock	CC BY-SA 4.0	added 4 characters in body
Feb 9, 2021 at 11:56	comment	added	Jake Wetlock		@Fernando: I have added right thanks!
Feb 9, 2021 at 10:34	comment	added	Fernando Muro		$Hom(X,Y)=X^{*}\otimes Y$. BTW you need symmetry in the first line, or that would just be right closed.
Feb 9, 2021 at 10:04	history	asked	Jake Wetlock	CC BY-SA 4.0