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.
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 $X$ obtained by taking the disjoint union $A + B$ and identifying $a \in A$ with $b \in B$ if there exists $x \in C$ such that $f(x) = a$ and $g(x) = b$ (and all identifications that follow to keep equality an equivalence relation).
This construction comes up, for example, when $C$ is the intersection of the sets $A$ and $B$, and $f$ and $g$ are the obvious inclusions. Then the pushout is just the union of $A$ and $B$.
Note that there are maps $i_A : A \to X$, $i_B : B \to X$ such that $i_A(a) = [a]$ and $i_B(b) = [b]$ 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 $h: X \to Y$ such that
and
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 $x$ is also called the pushout. It has the universal property already described above in the special case of the category $Set$.
Other terms: $x$ is a cofibred coproduct of $a$ and $b$, or (especially in algebraic categories when $f$ and $g$ are monomorphisms) a free product of $a$ and $b$ with $c$ amalgamated, or more simply an amalgamation (or amalgam) of $a$ and $b$.
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 $\{f_i: c \to a_i\}_{i \in I}$. Thus an ordinary pushout is the case where $I$ has cardinality $2$.
Note that the concept of pushout is dual to the concept of pullback: that is, a pushout in $C$ is the same as a pullback in $C^{op}$.
See pullback for more details.
(pushouts as coequalizers)
If coproducts exist in some category, then the pushout
is equivalently the coequalizer
of the two morphisms induced by $f$ and $g$ into the coproduct of $b$ with $c$.
(pushouts preserves epimorphisms and isomorphisms)
Pushouts preserve epimorphisms and isomorphisms:
If
is a pushout square in some category then:
if $g$ is a epimorphism then $f_\ast g$ is an epimorphism;
if $g$ is an isomorphism then $f_\ast g$ is an isomorphism.
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.
pushout of strong monomorphism in quasitopos
Suppose that $(\mathrm{T},\mathcal{C})$ is either
Suppose that
is a commutative diagram in $\mathcal{C}$ such that
Then
See at quasitopos this lemma. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are regular ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in
where $\chi_i$ is the classifying map of $i$) and therefore strong.
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