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Background

Given any function $$ f: A \times B \to C, \ \text{or} \ f \in \mathrm{Hom}(A \times B, C) $$ we can "curry" the function by a sort of reverse evaluation: $$ \mathrm{curry}: \mathrm{Hom}(A \times B, C) \to \mathrm{Hom}(A, \mathrm{Hom}(B,C)), $$ or $$ \ \mathrm{curry}(f): A \to (B \to C) $$ with $$ (\mathrm{curry}(f))(a) = f(a,\cdot). $$

My Question

Can we define a mapping $$ *: \mathrm{Hom}(A \times B, C) \to \mathrm{Hom}(\mathrm{Hom}(A,B),C) $$ with $$ *(f): (A \to B) \to C $$ and is there a general name for this? Also, can we define some mapping $$ \mathrm{Hom}(A, \mathrm{Hom}(B,C)) \to \mathrm{Hom}(\mathrm{Hom}(A,B),C)) $$ which somehow preserves structure nicely?

Apologies for how simple-minded and unmotivated this question seems. It's just a result of messing about with things. This also may not be related to category theory so please feel free to remove the tag if so. Any recommended reading material is also greatly appreciated.

Andrew Whelan
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    You might want to read this: https://en.wikipedia.org/wiki/Exponential_object – xavierm02 Feb 10 '17 at 12:48
  • Also: https://en.wikipedia.org/wiki/Hom_functor#Internal_Hom_functor – xavierm02 Feb 10 '17 at 12:51
  • I'm not a category theorist but my feeling is that the well-behaved constructions like your $\operatorname{curry}$ that I have seen are built up from composition and evaluation, so if there is such a construction one can probably find it. Example: there is a map $\operatorname{Hom}(A\times B,C)\to \operatorname{Hom}(A\times \operatorname{Hom}(A,B), C)$ that maps $\phi \in \operatorname{Hom}(A\times B, C)$ to $(a,g)\mapsto [\operatorname{curry}(\phi)(a)\circ g] (a) = \phi(a,g(a))$, where $a\in A$ and $g\in \operatorname{Hom}(A,B)$ – Ben Blum-Smith Feb 10 '17 at 13:14

1 Answers1

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No.

First off, categorically this can't be a natural transformation in $A$ since the source, in both examples, is contravariant with respect to $A$ and the target is covariant in $A$. This suggests that any such function would need to care about what $A$ was.

Sure enough, if we set $A$ and $C$ to $\emptyset$ we get $\text{Hom}(\emptyset,\emptyset)\to \text{Hom}(1,\emptyset) \cong \text{Hom}(\text{Hom}(\emptyset,B),\emptyset)$ where $1$ is some singleton set. But $\text{Hom}(\emptyset,\emptyset) \cong 1$ while $\text{Hom}(1,\emptyset) = \emptyset$. There are no functions $1 \to \emptyset$, and so there can be no family of functions $\text{Hom}(A\times B,C)\to\text{Hom}(\text{Hom}(A,B),C)$ parameterized by arbitrary sets $A$, $B$, and $C$.

Derek Elkins left SE
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