nLab pushout



Category theory

Limits and colimits



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 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:

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: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:

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: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.


In any category


(pushouts as coequalizers)

If coproducts exist in some category, then the pushout

is equivalently the coequalizer

of the two morphisms induced by ff and gg into the coproduct of bb with cc.


(pushouts preserves epimorphisms and isomorphisms)

Pushouts preserve epimorphisms and isomorphisms:


is a pushout square in some category then:

  1. if gg is a epimorphism then f *gf_\ast g is an epimorphism;

  2. if gg is an isomorphism then f *gf_\ast g is an isomorphism.


(pasting law for pushouts)

Consider a commuting diagram of the following shape in any category:

If the left square is a pushout, then the total rectangle is a pushout if and only if the right square is a pushout.


See the proof of the dual property for pullbacks.


The converse implication does not hold: it may happen that the outer and the right square are pushouts, but not the left square.


See the proof of the dual proposition for pullbacks.

In a quasitopos


pushout of strong monomorphism in quasitopos

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

Suppose that

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}


  • 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

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



A pushout of injections of Sets is called the union of the sets.


A pushout of groups in Grps is called their amalgamated free product


In topology, space attachments are pushouts in Top.


Early use of the terminology “pushout”:

Early use of the terminology “amalgamated sums”:

Textbook accounts:

Last revised on May 29, 2023 at 15:36:09. See the history of this page for a list of all contributions to it.