In category theory, a branch of mathematics, a pushout is a way to combine two maps that start from the same object. These maps are called f: Z → X and g: Z → Y. A pushout includes an object P and two maps from X and Y to P. These maps form a diagram where the paths from Z to X to P and from Z to Y to P match, creating a square that works consistently. The pushout is the most flexible way to complete this square. It is often written as P = X ⊔ Z Y or P = X + Z Y.

A pushout is the opposite of a pullback in category theory.

Universal property

A pushout of the morphisms f and g is made up of an object P and two morphisms, i₁ from X to P and i₂ from Y to P. These morphisms create a diagram where the paths from X to Y through P match the direct path from X to Y. The setup (P, i₁, i₂) is special because it is the best possible way to complete this diagram. This means that if there is another similar setup (Q, j₁, j₂) that also completes the diagram, there must be exactly one way to connect P to Q so that all paths in the diagram match.

If a pushout exists, it is the only one possible, except for changes that do not affect its structure.

Examples of pushouts

Here are examples of pushouts in familiar categories. In each case, we describe a way to construct an object that belongs to the same group of equivalent pushouts, even though other methods may exist.

• If X, Y, and Z are sets, and f: Z → X and g: Z → Y are functions, the pushout of f and g is formed by combining X and Y without overlapping elements. Then, elements that share a common origin in Z are grouped together. This creates a new set P, written as (X ⊔ Y)/∼, where ∼ is the strictest rule that matches f(z) with g(z) for every z in Z. If X and Y are subsets of a larger set W, and Z is their shared part, then the pushout becomes the union of X and Y within W. A specific example is the cograph of a function f: X → Y. The cograph is formed by combining X and Y and grouping elements x in X with their corresponding f(x) in Y. A function can be recovered from its cograph because each group in the combined set contains exactly one element from Y. Cographs are related to function graphs, which are defined as pullbacks instead of pushouts.

• In the category of topological spaces, creating adjunction spaces is an example of a pushout. If Z is a part of Y and g: Z → Y is the inclusion map, then we can attach Y to another space X along Z using a map f: Z → X. The result is the adjunction space X ∪ₐ Y, which is the pushout of f and g. Similarly, all identification spaces can be seen as pushouts.

• A simple case is the wedge sum, where X and Y are spaces with a single basepoint. Here, Z is the one-point space, and the pushout becomes X ∨ Y, the space formed by connecting the basepoint of X to the basepoint of Y.

• In the category of abelian groups, pushouts can be thought of as combining groups and connecting them where they overlap, similar to how adjunction spaces combine spaces. The zero group is a subgroup of every group, so for groups A and B, we have maps f: 0 → A and g: 0 → B. The pushout of these maps is the direct sum of A and B. For general maps f: Z → A and g: Z → B, the pushout is a group formed by combining A and B and grouping elements (f(z), -g(z)) for all z in Z. This process connects A and B along the images of Z under f and g. A similar method applies to R-modules for any ring R.

• In the category of groups, the pushout is called the free product with amalgamation. This concept appears in the Seifert–van Kampen theorem in algebraic topology.

• In CRing, the category of commutative rings, the pushout is the tensor product of rings A ⊗ₐ B, with maps g': A → A ⊗ₐ B and f': B → A ⊗ₐ B that satisfy f' ∘ g = g' ∘ f. Since pushouts are colimits of spans and pullbacks are limits of cospans, the tensor product of rings and the fibered product of rings are dual concepts. If A, B, and C are commutative rings with maps f: C → A and g: C → B, then the tensor product is defined as A ⊗ₐ B.

• For non-commutative rings, the pushout is described in the context of free products of associative algebras.

• In the multiplicative monoid of positive integers, considered as a category with one object, the pushout of two integers m and n is the pair (lcm(m, n)/m, lcm(m, n)/n), where lcm(m, n) is the least common multiple of m and n. This pair is also the pullback.

Properties

• If the pushout A ⊔ C B exists, then the pushout B ⊔ C A also exists, and these two pushouts are naturally isomorphic, meaning A ⊔ C B ≅ B ⊔ C A.
• In an abelian category, all pushouts exist, and they preserve cokernels in the following way: if (P, i₁, i₂) is the pushout of maps f: Z → X and g: Z → Y, then the natural map from coker(f) to coker(i₂) is an isomorphism, and the natural map from coker(g) to coker(i₁) is also an isomorphism.
• There is a natural isomorphism (A ⊔ C B) ⊔ B D ≅ A ⊔ C D. Specifically, this means: if maps f: C → A, g: C → B, and h: B → D are given, and the pushout of f and g is defined by maps i: A → P and j: B → P, and the pushout of j and h is defined by maps k: P → Q and l: D → Q, then the pushout of f and h ∘ g is defined by maps k ∘ i: A → Q and l: D → Q.

Construction via coproducts and coequalizers

Pushouts are the same as coproducts and coequalizers (if there is an initial object) in the following way:

• Coproducts are a type of pushout that starts with the initial object. The coequalizer of two functions f and g from X to Y is the same as the pushout of [f, g] and [1X, 1X]. This means that if a category has pushouts and an initial object, it also has coequalizers and coproducts.
• Pushouts can be made using coproducts and coequalizers. To create a pushout of two morphisms f: Z → X and g: Z → Y, first form the coproduct of X and Y. Then, there are two paths from Z to this coproduct: one through f and one through g. The pushout is the coequalizer of these two paths.

All of the above examples are special cases of a general method that works in any category C where:
– For any two objects A and B in C, their coproduct exists in C.
– For any two morphisms j and k in C that share the same domain and target, their coequalizer exists in C.

In this setup, the pushout of f: Z → X and g: Z → Y is created by first forming the coproduct of X and Y. Then, two morphisms from Z to this coproduct are defined: one using f and one using g. The pushout is the coequalizer of these two morphisms.

Application: the Seifert–van Kampen theorem

The Seifert–van Kampen theorem helps answer this question: If a space X is made up of two smaller spaces A and B, both of which are path-connected, and their overlapping area A ∩ B is also path-connected, can we find the fundamental group of X if we know the fundamental groups of A, B, and A ∩ B? The answer is yes, but only if we also know how the fundamental group of A ∩ B connects to the fundamental groups of A and B. These connections are described by special functions called induced homomorphisms. The theorem then states that the fundamental group of X is the result of combining these two homomorphisms in a specific way, called a pushout. In simpler terms, X can be thought of as the pushout of the ways A ∩ B is included in A and B. This means the fundamental group of X is built from the fundamental groups of A and B, linked together by the fundamental group of A ∩ B. When A ∩ B is simply connected (meaning it has no loops that cannot be shrunk to a point), the process becomes simpler because the homomorphisms from A ∩ B to A and B involve no complex structures. In this case, the pushout reduces to a free product, which is a basic way to combine groups. In more general cases, the process involves a free product with amalgamation, which is a more complex form of combining groups. A more detailed explanation of this idea, using a broader concept called covering groupoids, can be found in the book by J. P. May listed in the references.