Direct sum of categories Is there a notion of a direct sum of two categories?
E.g. let $A$ be an algebra, $\text{Rep} A$ the category of finite dimensional left $A$-modules and $\text{Vect}$ the category of finite dimensional vector spaces (of a fixed field, the same as for $\text{Rep} A$).
What would $\text{Rep} A\oplus \text{Vect}$ mean?
 A: There's only one reasonable notion of direct sum of additive categories, like the module categories in your example. $A\oplus B$ has objects the pairs $(a,b)$ and morphisms $(A\oplus B)((a,b),(a',b'))=A(a,a')\oplus B(b,b')$. This has the universal property of a product in the category of additive categories, but unfortunately not of a coproduct in general. There aren't even natural inclusions $A, B\to A\oplus B$ unless $A$ and $B$ have zero objects. In that case you have additive fully faithful embeddings $a\mapsto (a,0_B)$ etc, and every object $(a,b)$ becomes the biproduct $(a,0_B)\oplus (0_A,b)$. This says that the inclusions of and projections on $A$ and $ B$ satisfy a categorified biproduct identity 
$$i_Ap_A\oplus i_Bp_B\cong \mathrm{id}_{A\oplus B}$$
It's not much work now to show that we have canonical equivalences, even isomorphisms, of categories $\mathrm{Add}(C,A\oplus B)\cong \mathrm{Add}(C,A)\oplus \mathrm{Add}(C,B)$ for any $C$. If $C$ has biproducts, then we also get the dual identity $\mathrm{Add}(A\oplus B,C)\cong \mathrm{Add}(A,C)\oplus\mathrm{Add}(B,C)$. So $A\oplus B$ satisfies the universal properties of the biproduct in the category, or really, the 2-category, of additive categories with biproducts. The latter then becomes the canonical example of an "additive 2-category."
