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Stream: learning: questions

Topic: Currying characterizes exponentials?

Nick Smith (Jul 22 2021 at 04:54):

The nLab page on closed monoidal categories says "Currying can be read as a characterization of the internal hom". I want to check I'm understanding this statement correctly, because it seems pretty important. Let me ask a question:

If I have a category, and a canonical way to compose every pair of morphisms $f: S \to X$ and $g: X \times Y \to T$ in the category to get a morphism $h: S \times Y \to T$, is that equivalent to saying I have a Cartesian closed category? i.e. is the concept of an "internal hom" fully captured by this property? Or phrased another way: if I have "partial composition" of morphisms as above, can I ignore the perspective that my category has exponential objects? Or by doing so, would I be ignoring other interesting properties of a Cartesian closed category?

Robin Piedeleu (Jul 22 2021 at 09:04):

I've probably misunderstood your question, but in a monoidal (not necessarily closed) category you already have a canonical way of composing $f : S \rightarrow X$ and $g : X\otimes Y \rightarrow T$ to get $h : S \otimes Y \rightarrow T$. It's just $h = g \circ (f \otimes id_Y)$, isn't it?

Nick Smith (Jul 22 2021 at 12:47):

Oh, I guess that works! I was thinking of an alternative process where you take the equivalent g, namely $g': X \to T^Y$, and then compose. Since the style of composition we're discussing doesn't depend on Cartesian closure after all, it obviously does not capture what it means to be Cartesian closed. I will have to think about this more and modify my question.

James Wood (Jul 22 2021 at 13:00):

The paragraph before says “This natural isomorphism [Hom(a⊗b,c)≃Hom(a,[b,c])] is called currying.”, so I think the remark just means that an internal hom is anything satisfying that natural isomorphism. The adjunction definition unfolds to exactly this.