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I'm looking for the right jargon and notation to represent the following situation:

Let's say you have two functions, one that maps values from $A$ to $B$ and one that maps values from $C$ to $D$.

  • How do I denote the set of functions that maps from one set to another set?
  • Does a set of functions (in this context) have a standard name?

If I have a function that maps one set of functions to another set of functions, then it seems like there has to be at least two additional functions that map the domain of the first set to the domain of the second, and a second function that maps the range of the first set to the range of the second. These two additional mappings could be injective, surjective or bijective.

  • Do these two functions (domain to domain and range to range) have a standard name?
  • What $\LaTeX$ arrows should be used to denote the different cases (injective, surjective and bijective)?

Thanks!

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The wikipedia page for injective and surjective functions mention that sometimes the arrows $f:X↣Y$ and $f:X↠Y$ are used for an injective and surjective function, respectively. Of course, there's also the occasional $\hookrightarrow$ for the canonical injection. I do not have a link for all of the common TeX arrows though. –  Tyler Jun 1 '11 at 22:20
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2 Answers

up vote 2 down vote accepted

In category theory it's standard to use the notation $\text{Hom}(A, B)$ or $\text{Mor}(A, B)$ (which stand for "homomorphism" and "morphism" respectively) or, if you want to be completely precise, $\text{Hom}_{\text{Set}}(A, B)$ (which specifies that we are in the category of sets) to refer to the set of functions between two sets $A$ and $B$ (or more generally to the set of morphisms between two objects in a category).

I'm not sure I understand your second question. Giving a function $f : \text{Hom}(A, B) \to \text{Hom}(C, D)$ does not entail giving either a function $A \to C$ or a function $B \to D$.

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It's been a while since I've looked at CT, but I'll have to go dust off some books. I see now your reasoning about not requiring two functions with respect to question two. –  lewellen Jun 2 '11 at 3:18
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Have a look at Wikipedia's article on Function space.

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