Editing Pushout (category theory) (section)

==Examples of pushouts==
Here are some examples of pushouts in familiar [[category (mathematics)|categories]]. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, though there may be other ways to construct it, they are all equivalent.

*Suppose that ''X'', ''Y'', and ''Z'' as above are [[set (mathematics)|sets]], and that ''f''&nbsp;:&nbsp;''Z''&nbsp;&rarr;&nbsp;''X'' and ''g''&nbsp;:&nbsp;''Z''&nbsp;&rarr;&nbsp;''Y'' are set functions. The pushout of ''f'' and ''g'' is the [[disjoint union]] of ''X'' and ''Y'', where elements sharing a common [[preimage]] (in ''Z'') are identified, together with the morphisms ''i''<sub>1</sub>, ''i''<sub>2</sub>  from ''X'' and ''Y'', i.e. <math>P = (X \sqcup Y)/\!\sim</math> where ''~'' is the [[Equivalence_relation#Comparing_equivalence_relations|finest equivalence relation]] (cf. also [[Closure_(mathematics)#Binary_relation_closures|this]]) such that ''f''(''z'')&nbsp;~&nbsp;''g''(''z'') for all ''z'' in ''Z''. In particular, if ''X'' and ''Y'' are [[subset]]s of some larger set ''W'' and ''Z'' is their [[intersection (set theory)|intersection]], with ''f'' and ''g'' the inclusion maps of ''Z'' into ''X'' and ''Y'', then the pushout can be canonically identified with the [[union (set theory)|union]] <math> X \cup Y \subseteq W</math>.
**A specific case of this is the cograph of a function.  If <math>f \colon X \to Y</math> is a function, then the '''cograph''' of a function is the pushout of {{mvar|f}} along the identity function of {{mvar|X}}. In elementary terms, the cograph is the quotient of <math>X \sqcup Y</math> by the equivalence relation generated by identifying <math>x \in X \subseteq X \sqcup Y</math> with <math>f(x) \in Y \subseteq X \sqcup Y</math>. A function may be recovered by its cograph because each equivalence class in <math>X \sqcup Y</math> contains precisely one element of {{mvar|Y}}. Cographs are dual to graphs of functions since the graph may be defined as the pullback of {{mvar|f}} along the identity of {{mvar|Y}}.<ref>Riehl, ''Category Theory in Context'', p. xii</ref><ref>{{cite web | url=https://math.stackexchange.com/questions/1350657/does-the-concept-of-cograph-of-a-function-have-natural-generalisations-exten | title=Does the concept of "cograph of a function" have natural generalisations / Extensions? }}</ref>
*The construction of [[adjunction space]]s is an example of pushouts in the [[category of topological spaces]]. More precisely, if ''Z'' is a [[subspace topology|subspace]] of ''Y'' and ''g'' : ''Z'' &rarr; ''Y'' is the [[inclusion map]] we can "glue" ''Y'' to another space ''X'' along ''Z'' using an "attaching map" ''f'' : ''Z'' &rarr; ''X''. The result is the adjunction space <math>X \cup_{f} Y</math>, which is just the pushout of ''f'' and ''g''.  More generally, all identification spaces may be regarded as pushouts in this way.
*A special case of the above is the [[wedge sum]] or one-point union; here we take ''X'' and ''Y'' to be [[pointed space]]s and ''Z'' the one-point space. Then the pushout is <math>X \vee Y</math>, the space obtained by gluing the basepoint of ''X'' to the basepoint of ''Y''.
*In the [[category of abelian groups]], pushouts can be thought of as "[[direct sum of abelian groups|direct sum]] with gluing" in the same way we think of adjunction spaces as "[[disjoint union topology|disjoint union]] with gluing". The [[zero group]] is a [[subgroup]] of every [[group (mathematics)|group]], so for any [[abelian group]]s ''A'' and ''B'', we have [[homomorphism]]s <math>f : 0 \to A</math> and <math>g : 0 \to B</math>. The pushout of these maps is the direct sum of ''A'' and ''B''. Generalizing to the case where ''f'' and ''g'' are arbitrary homomorphisms from a common domain ''Z'', one obtains for the pushout a [[quotient group]] of the direct sum; namely, we [[Modulo (jargon)|mod out]] by the subgroup consisting of pairs (''f''(''z''),&nbsp;−''g''(''z'')). Thus we have "glued" along the images of ''Z'' under ''f'' and ''g''. A similar approach yields the pushout in the [[category of modules|category of ''R''-modules]] for any [[Ring (mathematics)|ring]] ''R''.
*In the [[category of groups]], the pushout is called the [[free product with amalgamation]]. It shows up in the [[Seifert–van Kampen theorem]] of [[algebraic topology]] (see below).
*In '''CRing''', the category of [[commutative rings]] (a [[full subcategory]] of the [[category of rings]]), the pushout is given by the [[tensor product]] of rings <math>A \otimes_{C} B</math> with the morphisms <math>g': A \rightarrow A \otimes_{C} B</math> and <math>f': B \rightarrow A \otimes_{C} B</math> that satisfy <math> f' \circ g = g' \circ f </math>. In fact, since the pushout is the [[colimit]] of a [[Span (category theory)|span]] and the [[Pullback (category theory)|pullback]] is the limit of a [[Span (category theory)|cospan]], we can think of the tensor product of rings and the [[Pullback (category theory)|fibered product of rings]] (see the examples section) as dual notions to each other. In particular, let ''A'', ''B'', and ''C'' be objects (commutative rings with identity) in '''CRing''' and let ''f'' : ''C'' &rarr; ''A'' and ''g'' : ''C'' &rarr; ''B'' be morphisms ([[ring homomorphism]]s) in '''CRing'''. Then the tensor product is:
::<math>A \otimes_{C} B = \left\{\sum_{i \in I} (a_i,b_i) \; \big| \; a_i \in A, b_i \in B \right\} \Bigg/ \bigg\langle (f(c)a,b) - (a,g(c)b) \; \big| \; a \in A, b \in B, c \in C \bigg\rangle </math>
*See [[Free product of associative algebras]] for the case of non-commutative rings.
*In the multiplicative [[monoid]] of positive [[integer]]s <math>\mathbf{Z}_+</math>, considered as a category with one object, the pushout of two positive integers ''m'' and ''n'' is just the pair <math>\left(\frac{\operatorname{lcm}(m,n)}{m}, \frac{\operatorname{lcm}(m,n)}{n}\right)</math>, where the numerators are both the [[least common multiple]] of ''m'' and ''n''. Note that the same pair is also the pullback.