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Stream: learning: questions

Topic: Partial application in a monoidal category

view this post on Zulip Bruno Gavranović (Jan 19 2021 at 13:11):

If I'm in a concrete monoidal category and have a map f:A⊗B→Cf :A \otimes B \to C, can I say that for every a:A a:A there is a map f(a,−):B→C,f(a, -):B \to C, or do I need some extra structure? I'm thinking I might need Cartesian, or monoidal closed

view this post on Zulip Jules Hedges (Jan 19 2021 at 13:12):

Definitely monoidal closed, I think

view this post on Zulip Jules Hedges (Jan 19 2021 at 13:14):

If you just have a plain monoidal category and a state a:I→Aa : I \to A then you can build B≅I⊗B→A⊗B→CB \cong I \otimes B \to A \otimes B \to C

view this post on Zulip Bruno Gavranović (Jan 19 2021 at 13:14):

Note that I'm not trying to say that for every a:Aa:A there is an object f(a,−):[A,B]f(a, -):[A, B], but rather to say there's just a morphism f(a,−):A→Bf(a, -):A \to B (but maybe you already knew that)

view this post on Zulip Bruno Gavranović (Jan 19 2021 at 13:15):

Jules Hedges said:

If you just have a plain monoidal category and a state a:I→Aa : I \to A then you can build B≅I⊗B→A⊗B→CB \cong I \otimes B \to A \otimes B \to C

Ah yes, I think that's what I needed