Is $\operatorname{Hom}(A\oplus B, C)=\operatorname{Hom}(A,C)\oplus\operatorname{Hom}(B,C)$? Where can I find general rules for $\operatorname{Hom}$?

Question

Question is in the title. More specific, where can I find general calculation rules for $\operatorname{Hom},\otimes,\oplus$? I need them not very often, but if I need these rules, I can't find anything. Especially those things that seem simple, but are not trivial, are difficult for me.

In this case I want to calculate $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}\oplus\mathbb{Z},\mathbb{C}^*)$ ($\mathbb{C}^*$ is the unit group). If I would have $\mathbb{Z}^2$ it would be easy and we would have $\operatorname{Hom}(\mathbb{Z}^2,\mathbb{C}^*)=(\mathbb{C}^*)^2$ right? So is the result from the question just $\mathbb{C}^*\oplus\mathbb{C}^*$? Can I simplify this somehow? What exactly is the difference to $(\mathbb{C}^*)^2$?

Answer 1

This is true. The most common lemma that is used is that right adjoints preserve limits (if they are contravariant, then they send colimits to limits, and we still say that they are limit preserving - we always name things by the target).

Here, contravariant Hom is right adjoint, and so sends coproducts to products. It so happens that this category has biproducts, which are simulataneously products and coproducts. So it preserves them.

A text on category theory should have all of the necessary rules on Hom and tensor, except perhaps several interesting ones on how these relate to the finitely generated condition.

Here is a list of the four most common rules, which apply in a category with limits and colimits:

1. $\text{Hom}( \text{colim } X_i, Y) \cong \text{lim } \text{Hom}(X_i, Y)$.

2. In the case where the colimit is coproduct and the limit is product, we get $\text{Hom}(X \amalg Y, Z) \cong \text{Hom}(X, Z) \prod \text{Hom}(Y, Z)$

3. $\text{Hom}( X, \text{lim } Y_i) \cong \text{lim } \text{Hom}(X, Y_i)$.

4. The case where the limit is product above.

These four rules will give you most things you ever need. Further common results arise when something is required of one of the objects. Usually such situations reduce to one of the rules above, e.g.

5. $\text{Hom}_R(L, M ) \otimes S \stackrel{\cong}{\rightarrow} \text{Hom}_{R \otimes S} (L \otimes S, M \otimes S)$ The post here has sufficient conditions and a proof.

The post I linked to has links to other common ones, which might seem pretty fancy to a newcomer. The idea is just like you're thinking: bundle up the common results so that whenever you're working in XYZ category, you know the basics without having to think very much.