Coproduct (category theory)

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This needs fleshing out with things like notation, compared to coproduct and such

Note: see product and coproduct compared for a definition written in parallel with the product definition. This demonstrates how close the concepts are.

Definition

Given a pair of objects $A$ and $B$ in a category $\mathscr{C}$ a coproduct (of $A$ and $B$) is a:

• Wedge $\xymatrix{ A \ar[r]^{i_A} & S & B \ar[l]_{i_B} }$ (in $\mathscr{C}$) such that:
  • for any other wedge $\xymatrix{ A \ar[r]^{f_A} & X & B \ar[l]_{f_B} }$ in $\mathscr{C}$
    • there exists a unique arrow $S\mathop{\longrightarrow}^mX$ (called the mediating arrow) such that the following diagram commutes:

$\xymatrix{ A \ar[d]_{i_A} \ar[dr]^{f_A} & \\ S \ar[r]^{m} & X \\ B \ar[u]^{i_B} \ar[ur]_{f_B} &}$
Diagram of the coproduct of $A$ and $B$