Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
The notion of coproduct is a generalization to arbitrary categories of the notion of disjoint union in the category Set.
For a category and two objects, their coproduct is an object in 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 such that we have a commuting diagram
This morphism is called the copairing of and . The morphisms and are called coprojections or sometimes “injections” or “inclusions”, although in general they may not be monomorphisms.
Notation. The coproduct is also denoted or , especially when it is disjoint (or if your fonts don't include ‘’). The copairing is also denoted or (when possible) given vertically: .
A coproduct is thus the colimit over the diagram that consists of just two objects.
More generally, for any set and a collection of objects in indexed by , their coproduct is an object
equipped with maps
such that this is universal among all objects with maps from the .
In Set, the coproduct of a family of sets is the disjoint union of sets.
This makes the coproduct a categorification of the operation of addition of natural numbers and more generally of cardinal numbers: for two finite sets and the cardinality operation, we have
In Top, the coproduct of a family of spaces is the space whose set of points is and whose open subspaces are of the form (the internal disjoint union) where each is an open subspace of . This is typical of topological concrete categories.
In Grp, the coproduct is the free product, whose underlying set is not a disjoint union. This is typical of algebraic categories.
In Ab, in Vect, the coproduct is the subobject of the product consisting of those tuples of elements for which only finitely many are not 0.
In Cat, the coproduct of a family of categories is the category with
In Grpd, the coproduct follows Cat rather than Grp. This is typical of oidifications: the coproduct becomes a disjoint union again.
A coproduct in is the same as a product in the opposite category .
When they exist, coproducts are unique up to unique canonical isomorphism, so we often say “the coproduct.”
A coproduct indexed by the empty set is an initial object in .
Revised on June 8, 2015 13:22:54
by Mike Shulman