Talk:Hom functor

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[edit] Notation

Personally, I find the notation

Hom(A,–)

awkward and hard to understand; I would much prefer the "standard" notation

λx.Hom(A,x)

which makes it clear that the free variable is the splat in the second place. The lambda notation is particularly handy in proofs, especially in Hom-type proofs, because it explicltly names the bound variable. However, I can't say I've seen the lambda notation much in category-theory type books, so I don't want to just jam it into the article, as I'm afraid that might cause trouble. FWIW, the lambda notation is more common in discussions of cartesian closed categories, as that is where it comes to the fore. linas (talk) 05:54, 23 November 2007 (UTC)

As far as I know using lambda calculus notation is unheard of in treatments of category theory given by mathematicians. Also, it is hard to see what advantage is conferred by naming the "variable" except for maybe a psychological one to people who are used to doing that. I grew up on and use the (–) notation without encountering any cognitive dissonance. Also, these are not just blanks for "variables" (objects) anyway: they have to accept morphisms as well. In this way Hom(A, –) is not just a single function but a pair of functions, so the lambda calc notation is potentially also misleading. - 129.100.75.90 (talk) 21:40, 28 February 2008 (UTC)

[edit] Currying

The WP article currying has the particularly elegent statement:

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product f \colon (X \times Y) \to Z and the morphisms to an exponential object g \colon X \to Z^Y . In other words, currying is the statement that product and Hom are adjoint functors; this is the key property of being a Cartesian closed category.

I'd like to copy it over to this article; I think its true, except that I have not personally verified that Hom and product are really adjoint (I don't quite have the depth to competently do so), and so got cold feet performing this copy. linas (talk) 06:10, 23 November 2007 (UTC)