Category theory

Limits and colimits



The notion of coproduct is a generalization to arbitrary categories of the notion of disjoint union in the category Set.


For CC a category and x,yObj(C)x, y \in Obj(C) two objects, their coproduct is an object xyx \coprod y in CC equipped with two morphisms

such that this is universal with this property, meaning such that for any other object with maps like this

there exists a unique morphism (f,g):xyQ(f,g) : x \coprod y \to Q such that we have the following commuting diagram:

This morphism (f,g)(f,g) is called the copairing of ff and gg. The morphisms xxyx\to x\coprod y and yxyy\to x\coprod y are called coprojections or sometimes “injections” or “inclusions”, although in general they may not be monomorphisms.

Notation. The coproduct is also denoted a+ba+b or a⨿ba\amalg b, especially when it is disjoint (or aba \sqcup b if your fonts don't include ‘⨿\amalg’). The copairing is also denoted [f,g][f,g] or (when possible) given vertically: {fg}\left\{{f \atop g}\right\}.

A coproduct is thus the colimit over the diagram that consists of just two objects.

More generally, for SS any set and F:SCF : S \to C a collection of objects in CC indexed by SS, their coproduct is an object

sSF(s) \coprod_{s \in S} F(s)

equipped with maps

F(s) sSF(s) F(s) \to \coprod_{s \in S} F(s)

such that this is universal among all objects with maps from the F(s)F(s).




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