Pushout (category theory)

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms "f" : "Z" → "X" and "g" : "Z" → "Y" with a common domain. The pushout consists of an object "P" along with two morphisms "X" → "P" and "Y" → "P" that complete a commutative square with the two given morphisms "f" and "g". In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are formula_1 and formula_2.
The pushout is the categorical dual of the pullback.
Universal property.
Explicitly, the pushout of the morphisms "f" and "g" consists of an object "P" and two morphisms "i"1 : "X" → "P" and "i"2 : "Y" → "P" such that the diagram
commutes and such that ("P", "i"1, "i"2) is universal with respect to this diagram. That is, for any other such set ("Q", "j"1, "j"2) for which the following diagram commutes, there must exist a unique "u" : "P" → "Q" also making the diagram commute:
As with all universal constructions, the pushout, if it exists, is unique up to a unique isomorphism.
Examples of pushouts.
Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, though there may be other ways to construct it, they are all equivalent.
Construction via coproducts and coequalizers.
Pushouts are equivalent to coproducts and coequalizers (if there is an initial object) in the sense that:
All of the above examples may be regarded as special cases of the following very general construction, which works in any category "C" satisfying:
In this setup, we obtain the pushout of morphisms "f" : "Z" → "X" and "g" : "Z" → "Y" by first forming the coproduct of the targets "X" and "Y". We then have two morphisms from "Z" to this coproduct. We can either go from "Z" to "X" via "f", then include into the coproduct, or we can go from "Z" to "Y" via "g", then include. The pushout of "f" and "g" is the coequalizer of these new maps.
Application: the Seifert–van Kampen theorem.
The Seifert–van Kampen theorem answers the following question. Suppose we have a path-connected space "X", covered by path-connected open subspaces "A" and "B" whose intersection "D" is also path-connected. (Assume also that the basepoint * lies in the intersection of "A" and "B".) If we know the fundamental groups of "A", "B", and their intersection "D", can we recover the fundamental group of "X"? The answer is yes, provided we also know the induced homomorphisms
formula_16
and
formula_17
The theorem then says that the fundamental group of "X" is the pushout of these two induced maps. Of course, "X" is the pushout of the two inclusion maps of "D" into "A" and "B". Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when "D" is simply connected, since then both homomorphisms above have trivial domain. Indeed this is the case, since then the pushout (of groups) reduces to the free product, which is the coproduct in the category of groups. In a most general case we will be speaking of a free product with amalgamation.
There is a detailed exposition of this, in a slightly more general setting (covering groupoids) in the book by J. P. May listed in the references.