What is a coproduct in an enriched category?

Question

Does anybody know how to define coproducts in enriched categories ? For example I was wondering if they could be thought of as some kind of weighted colimits. I am particularly interested in if/how coproducts are defined in categories enriched over weighted sets (that is, additively weighted categories).

Answer 1

If your base of enrichment has a $\rm copy$ morphism, there is a functor

$$ \Delta : {\cal A} \to {\cal A} \boxtimes {\cal A} $$

where the codomain is using the monoidal structure on the 2-category ${\cal V}\text{-Cat}$ inherited from $\cal V$. $\Delta$ "copies" objects and morphisms, in the obvious sense (you need $\cal V$ to have a $\rm copy$ map in order to define $\Delta_{AB} : {\cal A}(A,A') \to {\cal A}(A,A') \otimes {\cal A}(A,A')$).

Definition. An "enriched coproduct" is a left adjoint for $\Delta$.

One can probably argue differently and use a certain discrete weight, but I can't say from the top of my head if these definitions are equivalent:

Let $\cal I[2]$ be the discrete $\cal V$-category on a two-element set: $\hom(x,y)$ is the monoidal unit if $x=y$, and the initial object otherwise (I am assuming the base of enrichment has all limits and colimits needed to state everything I need, plus symmetry of course). The binary coproduct of two objects $X,Y\in\cal A$ is a colimit for a functor $F : {\cal I[2]} \to \cal A$ which is defined on objects sending $0\in [2]$ to $X$, and $1\in [2]$ to $Y$, weighted by the functor constant... at the monoidal unit? At the terminal object?

Answer 2

Let $\mathcal{C}$ be a $\mathcal{V}$-enriched category and let $A$ and $B$ be objects in $\mathcal{C}$. A $\mathcal{V}$-enriched coproduct $A + B$ is an object in $\mathcal{C}$ together with morphisms $j : I \to \mathcal{C} (A, A + B)$ and $k : I \to \mathcal{C} (B, A + B)$ in $\mathcal{V}$ such that, for each object $C$ in $\mathcal{C}$, each object $V$ in $\mathcal{V}$, and each pair of morphisms $f : V \to \mathcal{C} (A, C)$ and $g : V \to \mathcal{C} (B, C)$ in $\mathcal{V}$, there is a unique morphism $[f, g] : V \to \mathcal{C} (A + B, C)$ such that the following diagrams commute: $$\require{AMScd} \begin{CD} V \otimes I @>{[f, g] \otimes j}>> \mathcal{C}(A + B, C) \otimes \mathcal{C} (A, A + B) \\ @V{\cong}VV @VV{\circ}V \\ V @>>{f}> \mathcal{C} (A, C) \end{CD}$$ $$\begin{CD} V \otimes I @>{[f, g] \otimes k}>> \mathcal{C}(A + B, C) \otimes \mathcal{C} (B, A + B) \\ @V{\cong}VV @VV{\circ}V \\ V @>>{g}> \mathcal{C} (B, C) \end{CD}$$

If $\mathcal{V}$ has binary (cartesian!) products, this can be more succinctly expressed as saying that the coproduct is a $\mathcal{V}$-enriched representation of the $\mathcal{V}$-enriched functor $\mathcal{C} (A, -) \times \mathcal{C} (B, -)$. (Expanding the definitions here actually yields many further conditions that I have omitted from the above because they are automatically satisfied.) If $\mathcal{V}$ has enough limits and colimits, this can be expressed as a special case of $\mathcal{V}$-enriched conical colimits, which are in turn special cases of $\mathcal{V}$-enriched weighted colimits. The details are in Kelly's textbook.