Talk:Hom functor

Notation

Personally, I find the notation

Hom(A,–)

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

$\lambda x.{\mbox{Hom}}(A,x)$

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)

The problem with the “placeholder” notation (e.g. $A\times -$) is that it may be ambiguous in complex terms, i.e. $F(A\times -)$ may be $\lambda B.F(A\times B)$ or $F(\lambda B.A\times B)$. This is true for any $n$-ary functor for $n>1$. --Beroal (talk) 14:52, 19 July 2011 (UTC)

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)

Fixed. linas (talk) 13:45, 23 August 2012 (UTC)

Commutation of Hom with limits and colimits

That's what I was looking for here... I think it should be said. Thank you by advance for the contributor who will do it !! —Preceding unsigned comment added by 129.20.38.150 (talk) 17:20, 26 November 2009 (UTC)

I added a single sentence that now says this. It still doesn't explain it in any intuitive manner. Sorry, maybe later... linas (talk) 14:31, 23 August 2012 (UTC)

analogy with pushforward and pullback two different categories

From a beginner's perspective, the covariant and the contravariant versions of the functor have much similarity with the more familiar notions of pushforward and pullback respectively. Will it be technically incorrect/misleading to include an explanation with the analogy? For example, I was thinking of something like the following to explain Hom(-,B).

Fix an element B from the category C (for example, for the category of manifolds, B can be any particular manifold, say the 2-sphere). Then, for every other element X in the category (every other manifold), there can exist functions f:X → B (e.g. immersions of X in B). Given a particular X, we define Hom(X, B) to be the set of all such functions from X to B (i.e., in this example, f∈Hom(X, B)). Thus we construct a new category, C_h, where each element are such spaces of functions (i.e. Hom(X, B)∈C_h). Thus, corresponding to every element X,Y,... in the original category C, there are elements Hom(X, B),Hom(Y, B),... in C_h, each of which represent the set of functions from X,Y,... to B.

Now consider a map φ:X → Y, for X,Y in C. Then, for a given function f_Y: Y → B such that f_Y ∈ Hom(Y, B), one can construct f_X : X → B such that f_X=f_Y ο φ. This is simply the pullback of f_Y by φ. In terms of the Hom functor, it defines a morphism Hom(Y, B)→Hom(X, B). We write f_X=Hom(f_Y,B)

Also, is it true that B has to be in the same category as X or Y? For example, in Universal coefficient theorem for cohomology this does not seem to be true. That means the first and the second elements of Hom(-,-) can be from different categories. This is not clear from present statement of the formal definition.

- Subh83 (talk | contribs) 23:48, 16 November 2011 (UTC)

I think this is confusing, as it forces the reader to think about un-natural things, like 'what the heck is a manifold?' and 'what the heck is a 2-sphere?', which have nothing to do with anything if you are a computer programmer studying Hom in lambda calculus and currying. Note also, that the word 'pullback' has a rather different defintion in category theory: the pullback (category theory) is a certain kind of diagram (category theory), it is a limit (category theory) and it should be noted that the internal Hom preserves limits, and thus preserves pullbacks... so using the word 'pullback' in this article would lead to considerable confusion.linas (talk) 13:54, 23 August 2012 (UTC)

Illustration

I have created a simple illustration for the covariant functor, hoping it will help people to grasp the concept more easily, as well as help in remembering the definition of the functor. Below is the figure along with description. It is possible to create a similar illustration for the contravariant functor. Feedback will be appreciated before I place it in the article.

Illustration of $Hom(A,-)$ functor: Fix $A\in {\mathcal {C}}$. Then there are sets of morphisms $Hom(A,X)=\{\alpha _{1},\alpha _{2},\cdots \}$, $Hom(A,Y)=\{\beta _{1},\beta _{2},\cdots \}$, etc. Also, for $f:X\rightarrow Y$, we define a map $Hom(A,f):Hom(A,X)\rightarrow Hom(A,Y)$, i.e. $Hom(A,f):\{\alpha _{1},\alpha _{2},\cdots \}\rightarrow \{\beta _{1},\beta _{2},\cdots \}$ by $\alpha \mapsto f\circ \alpha$ (illustrated by the dashed arrow). Thus $Hom(A,-)$ is a covariant functor.

- Subh83 (talk | contribs) 20:09, 1 March 2012 (UTC)

Two quick remarks: the Hom functor is a functor into Set, and this diagram does not show Set anywhere. So I don't think it actually illustrates the idea. The other remark: typically images are inserted into articles as thumbnails, so you need to be sure the diagram still looks good and is readable when it is much much smaller.linas (talk) 03:22, 23 August 2012 (UTC)

Why Homsets and not Homclases.

Is this restriction necessary, other than for the applicability of Yoneda's Lemma?

Since one of the aims of category theory seems to be to make everything as general as possible

Why not define Hom(A,–) : C → Class (category of classes) — Preceding unsigned comment added by 86.27.207.97 (talk) 23:17, 22 February 2013 (UTC)

Commutative diagram

In the section Formal definition it is written: " For any pair of morphisms f:B->B' and h:A'->A " "Both paths send g:A->B to f°g°h:A'->B'." And there is a diagram showing this, but I think in the diagram h is the written h^-1 because h should go from A to A' Is there a typo or am I missing something?

Thank you 5.171.96.12 (talk) 15:54, 28 November 2022 (UTC)