Coproduct
Definition

Let C {\displaystyle C} be a category and let X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} be objects of C. {\displaystyle C.} An object is called the coproduct of X 1 {\displaystyle X_{1}} and X 2, {\displaystyle X_{2},} written X 1 ⊔ X 2, {\displaystyle X_{1}\sqcup X_{2},} or X 1 ⊕ X 2, {\displaystyle X_{1}\oplus X_{2},} or sometimes simply X 1 + X 2, {\displaystyle X_{1}+X_{2},} if there exist morphisms i 1: X 1 → X 1 ⊔ X 2 {\displaystyle i_{1}:X_{1}\to X_{1}\sqcup X_{2}} and i 2: X 2 → X 1 ⊔ X 2 {\displaystyle i_{2}:X_{2}\to X_{1}\sqcup X_{2}} satisfying the following universal property: for any object Y {\displaystyle Y} and any morphisms f 1: X 1 → Y {\displaystyle f_{1}:X_{1}\to Y} and f 2: X 2 → Y, {\displaystyle f_{2}:X_{2}\to Y,} there exists a unique morphism f: X 1 ⊔ X 2 → Y {\displaystyle f:X_{1}\sqcup X_{2}\to Y} such that f 1 = f ∘ i 1 {\displaystyle f_{1}=f\circ i_{1}} and f 2 = f ∘ i 2. {\displaystyle f_{2}=f\circ i_{2}.} That is, the following diagram commutes:

The unique arrow f {\displaystyle f} making this diagram commute may be denoted f 1 ⊔ f 2, {\displaystyle f_{1}\sqcup f_{2},} f 1 ⊕ f 2, {\displaystyle f_{1}\oplus f_{2},} f 1 + f 2, {\displaystyle f_{1}+f_{2},} or. {\displaystyle \left.} The morphisms i 1 {\displaystyle i_{1}} and i 2 {\displaystyle i_{2}} are called canonical injections, although they need not be injections or even monic.

The definition of a coproduct can be extended to an arbitrary family of objects indexed by a set J. {\displaystyle J.} The coproduct of the family { X j: j ∈ J } {\displaystyle \left\{X_{j}:j\in J\right\}} is an object X {\displaystyle X} together with a collection of morphisms i j: X j → X {\displaystyle i_{j}:X_{j}\to X} such that, for any object Y {\displaystyle Y} and any collection of morphisms f j: X j → Y {\displaystyle f_{j}:X_{j}\to Y} there exists a unique morphism f: X → Y {\displaystyle f:X\to Y} such that f j = f ∘ i j. {\displaystyle f_{j}=f\circ i_{j}.} That is, the following diagram commutes for each j ∈ J {\displaystyle j\in J}:

The coproduct X {\displaystyle X} of the family { X j } {\displaystyle \left\{X_{j}\right\}} is often denoted ∐ j ∈ J X j {\displaystyle \coprod _{j\in J}X_{j}} or ⨁ j ∈ J X j. {\displaystyle \bigoplus _{j\in J}X_{j}.}

Sometimes the morphism f: X → Y {\displaystyle f:X\to Y} may be denoted ∐ j ∈ J f j {\displaystyle \coprod _{j\in J}f_{j}} to indicate its dependence on the individual f j {\displaystyle f_{j}} s.
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