pushout in nLab

Contents

Contents

Idea

A pushout is an ubiquitous construction in category theory providing a colimit for the diagram •←•→•\bullet\leftarrow\bullet\rightarrow\bullet. It is dual to the notion of a pullback.

Pushouts in SetSet

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 XX obtained by taking the disjoint union A+BA + B and identifying a∈Aa \in A with b∈Bb \in B if there exists x∈Cx \in C such that f(x)=af(x) = a and g(x)=bg(x) = b (and all identifications that follow to keep equality an equivalence relation).

This construction comes up, for example, when CC is the intersection of the sets AA and BB, and ff and gg are the obvious inclusions. Then the pushout is just the union of AA and BB.

Note that there are maps i A:A→Xi_A : A \to X, i B:B→Xi_B : B \to X such that i A(a)=[a]i_A(a) = [a] and i B(b)=[b]i_B(b) = [b] 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 h:X→Yh: X \to Y such that

hi A=j A h i_A = j_A

and

hi B=j B. h i_B = j_B .

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.

Definition

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 xx is also called the pushout. It has the universal property already described above in the special case of the category SetSet.

Other terms: xx is a cofibred coproduct of aa and bb, or (especially in algebraic categories when ff and gg are monomorphisms) a free product of aa and bb with cc amalgamated sum (Gabriel & Zisman (1967), p. 1) or more simply an amalgamation (or amalgam) of aa and bb.

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 {f i:c→a i} i∈I\{f_i: c \to a_i\}_{i \in I}. Thus an ordinary pushout is the case where II has cardinality 22.

Note that the concept of pushout is dual to the concept of pullback: that is, a pushout in CC is the same as a pullback in C opC^{op}.

See pullback for more details.

Properties

In any category

In a quasitopos

See at quasitopos this lemma. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms i:A→Bi: A \to B in a topos are regular (ii being the equalizer of the arrows χ i,t∘!:B→Ω\chi_i, t \circ !: B \to \Omega in

where χ i\chi_i is the classifying map of ii) and therefore strong.

Examples

• amalgamation property

• wide pushout

• coequalizer

• colimit

• pullback

• pushout-product

• homotopy pushout

References

Early use of the terminology “pushout”:

• Barry Mitchell, Section I.7 of: Theory of categories, Pure and Applied Mathematics 17, Academic Press (1965) [ISBN:978-0-12-499250-4]

Early use of the terminology “amalgamated sums”:

• Pierre Gabriel, Michel Zisman, p 1 of: Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer (1967) [doi:10.1007/978-3-642-85844-4, pdf]

Textbook accounts:

• Saunders MacLane, p. 65-66 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 2.5 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)