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Stream: learning: questions

Topic: Self-enrichment without symmetry

Amar Hadzihasanovic (Feb 12 2021 at 10:42):

A symmetric monoidal closed, locally small category is “self-enriched” as in Section 1.6 of Kelly's book. I've found this referenced, for example in this comment by Mike Shulman, as “the symmetric case”, and in general it doesn't look like symmetry plays an essential role, except in a monoidal biclosed category there's a choice to be made between left and right homs. Does anyone know of a source where the non-symmetric case is spelled out?

Nathanael Arkor (Feb 12 2021 at 11:33):

This appears in Eilenberg–Kelly's Closed categories as Theorem 5.2, in the context of closed categories (one can then consider when the closed structure is representable, admitting a tensor product, which appears later in the paper).

Todd Trimble (Feb 12 2021 at 16:23):

Basic Concepts in Enriched Category Theory.

Mike Shulman (Feb 12 2021 at 22:08):

Note that whether "left" or "right" homs are the correct choice may depend on whether you write composition in an enriched category as $C(y,z)\otimes C(x,y) \to C(x,z)$ or as $C(x,y)\otimes C(y,z) \to C(x,z)$. Plus, of course, deciding which hom is "left" or "right" may depend on whether you write your internal-hom adjunction as $V(x\otimes y,z) \cong V(x,y\multimap z)$ or $V(x\otimes y,z) \cong V(y,x\multimap z)$.