# When the homsets of a category are structured.

Example 1. Let $\cal M$ denote the category of magmas and functions, not necessarily structure-preserving. Then given objects $X$ and $Y$ of $\cal M$, the hom"set" $\mathrm{Hom}(X,Y)$ is probably best viewed as a magma as opposed to a set, with the pointwise operation inherited from $Y$. Furthermore we have a right (but not left) distributivity law. In particular, given objects $X, Y$ and $Z$ and arrows $f:X \rightarrow Y$ and $g,g' : Y \rightarrow Z,$

$$(g+g')\circ f = (g \circ f) + (g' \circ f).$$

Example 2. Now let $\cal N$ denote the category of medial magmas and magma homomorphisms. Then given objects $X$ and $Y$ of $\cal N$, the hom"set" $\mathrm{Hom}(X,Y)$ can be viewed as a medial magma in its own right. Furthermore, we have both left and right distributivity laws. In the sense that given objects $X, Y$ and $Z$ and arrows $f,f':X \rightarrow Y$ and $g,g' : Y \rightarrow Z,$

$$(g+g')\circ f = (g \circ f) + (g' \circ f)$$

$$g \circ (f + f') = (g \circ f) + (g \circ f').$$

• This is the notion of a category enriched over a monoidal category. "Enriched category" will work as a good search term; there's even a short chapter on them in Mac Lane. Dec 9 '13 at 11:18
• @MaliceVidrine, I've been looking over the wikipedia page dedicated to enriched categories for the past half hour or so before asking this question, but honestly I cannot see the connection. What have monoidal categories got to do with anything? Dec 9 '13 at 11:25
• @user18921 A monodical category have a operation $\otimes$ which can emulate the composition. Imagine you don't have such an $\otimes$ : what could it means to compose arrows (what even is an arrow when $\hom(a,b)$ is an abstract object of a category) in the enriched category ?
– Pece
Dec 9 '13 at 11:43
• @Pece, still not getting it. Why is $\mathrm{hom}(a,b)$ an abstract object of a category? For example, if $a$ and $b$ are magmas, well $\mathrm{hom}(a,b)$ should also denote some kind of magma. Magmas have elements, so that isn't very abstract. Dec 9 '13 at 11:49
• If you have a commutative algebraic theory, then the category of models has a canonical monoidal closed structure. Dec 9 '13 at 12:36

So in a general category, we can just view the $Hom$-objects as being sets, the monoidal operation being the Cartesian product, and the "unit" is any terminal object. It seems trivial to note, but the composition operation out of $Hom(A,B)\times Hom(B,C)$ is a morphism in the category of sets, while the Cartesian product is also a set.
This observation becomes less trivial when we move to, say, Abelian groups or categories as $Hom$-objects. What we have in, say, $\mathbf{Cat}$ is not just a category structure on the $Hom$ objects, but the composition operation that maps $B^A\times C^B$ to $C^A$ is itself functorial. The trivial category is obviously a unit for the product operation in $\mathbf{Cat}$, and functors from it can pick out the identity functor in any category $A^A$, which is the endofunctor of $A$ that acts as a unit in the composition operation.
That's the point about monoidal categories. There are lots of arbitrary structures you could add to a $Hom$-set, but not every category's composition operation will respect that structure in an appropriate way. Further, not every category has the right structure for its objects to be the $Hom$-objects of a category. If you diagram out how the composition operation works in $\mathbf{Set}$, then you can see it's exactly the kind of structure that a category's being monoidal makes possible. And enrichment is the condition that composition respects the additional $Hom$ structure in the appropriate way.