Pushout (category theory)
Encyclopedia
In category theory
, a branch of mathematics
, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamed sum) is the colimit of a diagram consisting of two morphism
s f : Z → X and g : Z → Y with a common domain: it is the colimit of the span
.
The pushout is the categorical dual
of the pullback
.
:
Moreover, the pushout (P, i1, i2) must be universal
with respect to this diagram. That is, for any other such set (Q, j1, j2) 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
.
1. Suppose that X and Y as above are sets. Then if we write Z for their intersection
, there are morphisms f : Z → X and g : Z → Y given by inclusion. The pushout of f and g is the union of X and Y together with the inclusion morphisms from X and Y.
2. The construction of adjunction space
s is an example of pushouts in the category of topological spaces
. More precisely, if Z is a subspace of Y and g : Z → Y is the inclusion map
we can "glue" Y to another space X along Z using an "attaching map" f : Z → X. The result is the adjunction space
which is just the pushout of f and g. More generally, all identification spaces may be regarded as pushouts in this way.
3. A special case of the above is the wedge sum
or one-point union; here we take X and Y to be pointed space
s and Z the one-point space. Then the pushout is
, the space obtained by gluing the basepoint of X to the basepoint of Y.
4. In the category of abelian group
s, pushouts can be thought of as "direct sum with gluing" in the same way we think of adjunction spaces as "disjoint union
with gluing". The zero group is a subgroup of every group, so for any abelian groups A and B, we have homomorphisms
and
The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group
of the direct sum; namely, we mod out
by the subgroup consisting of pairs (f(z),-g(z)). Thus we have "glued" along the images of Z under f and g. A similar trick yields the pushout in the category of R-module
s for any ring R.
5. In the category of groups
, the pushout is called the free product with amalgamation. It shows up in the Seifert-van Kampen theorem of algebraic topology (see below).
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.
s 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
and
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 groupoid
s) in the book by J. P. May listed in the references.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamed sum) is the colimit of a diagram consisting of two morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s f : Z → X and g : Z → Y with a common domain: it is the colimit of the span
Span (category theory)
A span, in category theory, is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks , spans can be considered as morphisms in a category of fractions....
.The pushout is the categorical dual
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
of the pullback
Pullback (category theory)
In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain; it is the limit of the cospan X \rightarrow Z \leftarrow Y...
.
Universal property
Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X → P and i2 : Y → P for which the following diagram commutesCommutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...
:
Moreover, the pushout (P, i1, i2) must be universal
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
with respect to this diagram. That is, for any other such set (Q, j1, j2) 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
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
.
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, there may be other ways to construct it, but they are all equivalent.1. Suppose that X and Y as above are sets. Then if we write Z for their intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
, there are morphisms f : Z → X and g : Z → Y given by inclusion. The pushout of f and g is the union of X and Y together with the inclusion morphisms from X and Y.
2. The construction of adjunction space
Adjunction space
In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be a continuous map...
s is an example of pushouts in the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
. More precisely, if Z is a subspace of Y and g : Z → Y is the inclusion map
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
we can "glue" Y to another space X along Z using an "attaching map" f : Z → X. The result is the adjunction space
which is just the pushout of f and g. More generally, all identification spaces may be regarded as pushouts in this way.3. A special case of the above is the wedge sum
Wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0 ∼ y0:X\vee Y = \;/ \sim,\,where ∼ is the...
or one-point union; here we take X and Y to be pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
s and Z the one-point space. Then the pushout is
, the space obtained by gluing the basepoint of X to the basepoint of Y.4. In the category of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s, pushouts can be thought of as "direct sum with gluing" in the same way we think of adjunction spaces as "disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
with gluing". The zero group is a subgroup of every group, so for any abelian groups A and B, we have homomorphisms
- f : 0 → A
and
- g : 0 → B.
The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of the direct sum; namely, we mod out
Modulo (jargon)
The word modulo is the Latin ablative of modulus which itself means "a small measure."It was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801...
by the subgroup consisting of pairs (f(z),-g(z)). Thus we have "glued" along the images of Z under f and g. A similar trick yields the pushout in the category of R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s for any ring R.
5. In the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
, the pushout is called the free product with amalgamation. It shows up in the Seifert-van Kampen theorem of algebraic topology (see below).
Properties
- Whenever A∪CB and B∪CA exist, there is an isomorphism A∪CB ≅ B∪CA.
- Whenever the pushout A∪AB exists, there is an isomorphism B ≅ A∪AB (this follows from the universal property of the pushout).
Construction via coproducts and coequalizers
Pushouts are equivalent to coproducts, and coequalizers (if there is an initial object) in the sense that:- Coproducts are a pushout from the initial object, and the coequalizer of f, g : X → Y is the pushout of [f, g] and [1X, 1X], so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
- Pushouts can be constructed from coproducts and coequalizers, as described below (the pushout is the coequalizer of the maps to the coproduct).
All of the above examples may be regarded as special cases of the following very general construction, which works in any category C satisfying:
- For any objects A and B of C, their coproductCoproductIn category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
exists in C; - For any morphisms j and k of C with the same domain and target, the coequalizerCoequalizerIn category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...
of j and k exists in C.
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
Returning to topology, 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 is also path-connected. (Assume also that the basepoint * lies in the intersection of A and B.) If we know the fundamental groupFundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s 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
and
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
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
, 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 groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
s) in the book by J. P. May listed in the references.
External links
- Interactive Web page which generates examples of pushouts in the category of finite sets. Written by Jocelyn Paine.