Category theory




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


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.


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.


Revised on August 10, 2016 05:22:26 by Thomas Holder (