nLab
pushout

Context

Category theory

Infinity-limits

Contents

Idea

A pushout is an ubiquitous construction in category theory providing a 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:

C f g A B \array{ &&&& C &&&& \\ & && f \swarrow & & \searrow g && \\ && A &&&& B }

the ‘pushout’ of this diagram is the set XX obtained by taking the disjoint union A+BA + B and identifying aAa \in A with bBb \in B if there exists xCx \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:AXi_A : A \to X, i B:BXi_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:

C f g A B i A i B X \array{ &&&& C &&&& \\ & && f \swarrow & & \searrow g && \\ && A &&&& B \\ & && {}_{i_A}\searrow & & \swarrow_{i_B} && \\ &&&& X &&&& }

In fact, the pushout is the universal solution to finding a commutative square like this. In other words, given any commutative square

C f g A B j A j B Y \array{ &&&& C &&&& \\ & && f \swarrow & & \searrow g && \\ && A &&&& B \\ & && {}_{j_A}\searrow & & \swarrow_{j_B} && \\ &&&& Y &&&& }

there is a unique function h:XYh: 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:

c f g a b \array{ &&&& c &&&& \\ & && f \swarrow & & \searrow g && \\ && a &&&& b }

Such a diagram is called a span. If the colimit exists, we obtain a commutative square

c f g a b i a i b x \array{ &&&& c &&&& \\ & && f \swarrow & & \searrow g && \\ && a &&&& b \\ & && {}_{i_a}\searrow & & \swarrow_{i_b} && \\ &&&& x &&&& }

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, 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:ca i} iI\{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

General

In a quasitopos

Proposition

pushout of strong monomorphism in quasitopos

Suppose that (T,𝒞)(\mathrm{T},\mathcal{C}) is either

Suppose that

O 0,1 O 1,1 m h O 0,0 O 1,0\array{ O_{0,1} & \to & O_{1,1} \\ \downarrow m &&\downarrow h \\ O_{0,0} & \to & O_{1,0} }

is a commutative diagram in 𝒞\mathcal{C} such that

    • mm is T\mathrm{T} in 𝒞\mathcal{C}
    • the diagram is a pushout in 𝒞\mathcal{C}

Then

    • hh is T\mathrm{T} in 𝒞\mathcal{C}
    • the diagram is a pullback in 𝒞\mathcal{C}

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

1 ! t B χ i Ω\array{ & & 1 \\ & \mathllap{!} \nearrow & \downarrow \mathrlap{t} \\ B & \underset{\chi_i}{\to} & \Omega }

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

Examples

Revised on June 19, 2017 03:26:28 by Urs Schreiber (46.183.103.17)