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
A pushout is an ubiquitous construction in category theory providing a a colimit for the diagram . It is dual to the notion of a pullback.
In the category Set a ‘pushout’ is a quotient of the disjoint union of two sets. Given a diagram of sets and functions like this:
the ‘pushout’ of this diagram is the set obtained by taking the disjoint union and identifying with if there exists such that and (and all identifications that follow to keep equality an equivalence relation).
This construction comes up, for example, when is the intersection of the sets and , and and are the obvious inclusions. Then the pushout is just the union of and .
Note that there are maps , such that and respectively. These maps make this square commute:
In fact, the pushout is the universal solution to finding a commutative square like this. In other words, given any commutative square
there is a unique function such that
Since this universal property expresses the concept of pushout purely arrow-theoretically, we can formulate it in any category. It is, in fact, a simple special case of a colimit.
A pushout is a colimit of a diagram like this:
Such a diagram is called a span. If the colimit exists, we obtain a commutative square
and the object is also called the pushout. It has the universal property already described above in the special case of the category .
Other terms: is a cofibred coproduct of and , or (especially in algebraic categories when and are monomorphisms) a free product of and with amalgamated, or more simply an amalgamation (or amalgam) of and .
The concept of pushout is a special case of the notion of wide pushout (compare wide pullback), where one takes the colimit of a diagram which consists of a set of arrows . Thus an ordinary pushout is the case where has cardinality .
Note that the concept of pushout is dual to the concept of pullback: that is, a pushout in is the same as a pullback in .
See pullback for more details.