Editing Coproduct (section)

== Definition ==
Let <math>C</math> be a [[Category (mathematics)|category]] and let <math>X_1</math> and <math>X_2</math> be objects of <math>C.</math> An object is called the coproduct of <math>X_1</math> and <math>X_2,</math> written <math>X_1 \sqcup X_2,</math> or <math>X_1 \oplus X_2,</math> or sometimes simply <math>X_1 + X_2,</math> if there exist morphisms <math>i_1 : X_1 \to X_1 \sqcup X_2</math> and <math>i_2 : X_2 \to X_1 \sqcup X_2</math> that satisfies the following [[universal property]]: for any object <math>Y</math> and any morphisms <math>f_1 : X_1 \to Y</math> and <math>f_2 : X_2 \to Y,</math> there exists a unique morphism <math>f : X_1 \sqcup X_2 \to Y</math> such that <math>f_1 = f \circ i_1</math> and <math>f_2 = f \circ i_2.</math> That is, the following diagram [[Commutative diagram|commutes]]:

[[Image:Coproduct-03.svg|280px|center]]

The unique arrow <math>f</math> making this diagram commute may be denoted <math>f_1 \sqcup f_2,</math> <math>f_1 \oplus f_2,</math> <math>f_1 + f_2,</math> or <math>\left[f_1, f_2\right].</math> The morphisms <math>i_1</math> and <math>i_2</math> are called {{em|[[canonical injection]]s}}, although they need not be [[Injective function|injections]] or even [[Monomorphism|monic]].

The definition of a coproduct can be extended to an arbitrary [[Indexed family|family]] of objects indexed by a set <math>J.</math> The coproduct of the family <math>\left\{ X_j : j \in J \right\}</math> is an object <math>X</math> together with a collection of [[morphism]]s <math>i_j : X_j \to X</math> such that, for any object <math>Y</math> and any collection of morphisms <math>f_j : X_j \to Y</math> there exists a unique morphism <math>f : X \to Y</math> such that <math>f_j = f \circ i_j.</math> That is, the following diagram [[Commutative diagram|commutes]] for each <math>j \in J</math>:

[[Image:Coproduct-01.svg|160px|center]]

The coproduct <math>X</math> of the family <math>\left\{ X_j \right\}</math> is often denoted <math>\coprod_{j\in J}X_j</math> or <math>\bigoplus_{j \in J} X_j.</math>

Sometimes the morphism <math>f : X \to Y</math> may be denoted <math>\coprod_{j \in J} f_j</math> to indicate its dependence on the individual <math>f_j</math>s.