# “Distributivity” of unary operations

Motivating example: given an algebraic object, there are many useful subobjects which, seen as operations on the category of objects, follow a distributivity-like law over things like products. That's not very clearly-worded, so as an example consider the units of a ring: (AxB)* = A* x B*. There are plenty of other examples. I don't know much category theory at all - maybe category theory isn't the right tag for this - but this strikes me as the sort of thing someone must have studied. Can anyone recommend material for me to read in this sort of direction? (Sorry if that's a bit vague.)

Speaking of tags, I was tempted to include ac.commutative-algebra but I'm not convinced that commutativity is actually an important property here.

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I think you're looking for preservation of limits (products in your example). I didn't check it, but I see in Wikipedia (en.wikipedia.org/wiki/Adjoint_functors) that the "group of units" functor has a left adjoint, and therefore it preserves all limits, including binary products. – user2734 May 1 '10 at 16:42

Let me elaborate on unknown (google)'s comment. If $\text{Rng}$ denotes the category of unital rings and $\text{Grp}$ denotes the category of groups, then it turns out that "group of units" is a functor $\text{Rng} \to \text{Grp}$. In fact, it's representable; the group of units of a ring $R$ can be identified with $\text{Hom}(\mathbb{Z}[x, x^{-1}], R)$ (the set of all ring homomorphisms from $\mathbb{Z}[x, x^{-1}]$ to $R$), where the group structure on the Hom-set comes from a Hopf algebra structure on $\mathbb{Z}[x, x^{-1}]$. And covariant representable functors preserve limits, including binary products. (Actually, whether this argument is valid depends on the answer to this question which I posted yesterday.)
(A closely related functor is $\text{Hom}(\mathbb{Z}[x], R)$, which is none other than the forgetful functor $\text{Rng} \to \text{Set}$ which sends a ring to its set of elements. The fact that this functor preserves limits is precisely why the product of rings can be constructed using a product of the underlying sets. In more complicated categories where the obvious forgetful functor doesn't preserve limits, this isn't necessarily true.)
Closely related to representability (although I've never been clear on the precise relationship) is the fact that "group of units" has a left adjoint $\text{Grp} \to \text{Rng}$ which constructs the group ring of a group. And functors which have left adjoints preserve limits.
@Qiaochu Yuan: Thanks for making it an answer! One connection between representable functors and adjoint functors is that if a functor $G$ has a left adjoint then it is surely representable: Saying that a functor $G$ is representable is like saying that there is a universal arrow from a one-object set $1$ to $G$ (Prop. 3.2.2, p. 60 in Mac Lane), and for this we can take the unit $\eta_1\colon 1\to G(F1)$ (with $F$ the left adjoint of $G$). For a condition for the opposite direction, see Ex. 1, p. 131 of Mac Lane. – user2734 May 1 '10 at 18:43
... I meant $G:C\to \mathbf{Set}$ – user2734 May 1 '10 at 18:46