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| In [[category theory]], a branch of [[mathematics]], a '''pushout''' (also called a '''fibered coproduct''' or '''fibered sum''' or '''cocartesian square''' or '''amalgamed sum''') is the [[colimit]] of a [[diagram (category theory)|diagram]] consisting of two [[morphism]]s ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' with a common [[Domain of a function|domain]]: it is the colimit of the [[Span (category theory)|span]] <math>X \leftarrow Z \rightarrow Y</math>.
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| The pushout is the [[dual (category theory)|categorical dual]] of the [[pullback (category theory)|pullback]].
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| ==Universal property==
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| Explicitly, the pushout of the morphisms ''f'' and ''g'' consists of an object ''P'' and two morphisms ''i''<sub>1</sub> : ''X'' → ''P'' and ''i''<sub>2</sub> : ''Y'' → ''P'' for which the following diagram [[commutative diagram|commutes]]:
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| :[[Image:Categorical pushout.svg|125px]]
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| Moreover, the pushout (''P'', ''i''<sub>1</sub>, ''i''<sub>2</sub>) must be [[universal property|universal]] with respect to this diagram. That is, for any other such set (''Q'', ''j''<sub>1</sub>, ''j''<sub>2</sub>) for which the following diagram commutes, there must exist a unique ''u'' : ''P'' → ''Q'' also making the diagram commute:
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| :[[Image:Categorical pushout (expanded).svg|225px]]
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| As with all universal constructions, the pushout, if it exists, is unique up to a unique [[isomorphism]].
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| ==Examples of pushouts==
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| Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, there may be other ways to construct it, but they are all equivalent.
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| 1. Suppose that ''X'', ''Y'', and ''Z'' as above are [[Set (mathematics)|sets]], and that ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''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 certain morphisms from ''X'' and ''Y''.
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| 2. 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'' → ''Y'' is the [[inclusion map]] we can "glue" ''Y'' to another space ''X'' along ''Z'' using an "attaching map" ''f'' : ''Z'' → ''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.
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| 3. 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''.
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| 4. In the category of [[abelian group]]s, 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]] with gluing". The zero group is a subgroup of every group, so for any abelian groups ''A'' and ''B'', we have [[homomorphisms]]
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| :''f'' : 0 → ''A'' | |
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| and
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| :''g'' : 0 → ''B''.
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| 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''),−''g''(''z'')). Thus we have "glued" along the images of ''Z'' under ''f'' and ''g''. A similar trick yields the pushout in the category of ''R''-[[Module (mathematics)|module]]s for any [[Ring (mathematics)|ring]] ''R''.
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| 5. 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).
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| 6. 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. In particular, let ''A'', ''B'', and ''C'' be objects (commutative rings with identity) in '''CRing''' and let ''f'' : ''C'' → ''A'' and ''g'' : ''C'' → ''B'' be morphisms (ring [[homomorphisms]]) in '''CRing'''. Then the tensor product,
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| :<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>
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| 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> defines the pushout in '''CRing'''. Since the pushout is the [[colimit]] of a span and the [[pullback]] is the limit of a 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.
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| ==Properties==
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| *Whenever ''A''∪<sub>''C''</sub>''B'' and ''B''∪<sub>''C''</sub>''A'' exist, there is an isomorphism ''A''∪<sub>''C''</sub>''B'' ≅ ''B''∪<sub>''C''</sub>''A''.
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| *Whenever the pushout ''A''∪<sub>''A''</sub>''B'' exists, there is an isomorphism ''B'' ≅ ''A''∪<sub>''A''</sub>''B'' (this follows from the universal property of the pushout).
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| ==Construction via coproducts and coequalizers==
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| Pushouts are equivalent to coproducts, and coequalizers (if there is an initial object) in the sense that:
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| * Coproducts are a pushout from the initial object, and the coequalizer of ''f'', ''g'' : ''X'' → ''Y'' is the pushout of [''f'', ''g''] and [1<sub>''X''</sub>, 1<sub>''X''</sub>], so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
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| * Pushouts can be constructed from coproducts and coequalizers, as described below (the pushout is the coequalizer of the maps to the coproduct).
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| All of the above examples may be regarded as special cases of the following very general construction, which works in any category ''C'' satisfying:
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| * For any objects ''A'' and ''B'' of ''C'', their [[coproduct]] exists in ''C'';
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| * For any morphisms ''j'' and ''k'' of ''C'' with the same domain and target, the [[coequalizer]] of ''j'' and ''k'' exists in ''C''.
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| In this setup, we obtain the pushout of morphisms ''f'' : ''Z'' → ''X'' and ''g'' : ''Z'' → ''Y'' by first forming the coproduct of the targets ''X'' and ''Y''. We then have two morphisms from ''Z'' to this coproduct. We can either go from ''Z'' to ''X'' via ''f'', then include into the coproduct, or we can go from ''Z'' to ''Y'' via ''g'', then include. The pushout of ''f'' and ''g'' is the coequalizer of these new maps.
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| ==Application: The Seifert–van Kampen theorem==
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| Returning to topology, the [[Seifert–van Kampen theorem]] answers the following question. Suppose we have a path-connected space ''X'', covered by path-connected open subspaces ''A'' and ''B'' whose intersection is also path-connected. (Assume also that the basepoint * lies in the intersection of ''A'' and ''B''.) If we know the [[fundamental group]]s of ''A'', ''B'', and their intersection ''D'', can we recover the fundamental group of ''X''? The answer is yes, provided we also know the induced homomorphisms
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| <math>\pi_1(D,*) \to \pi_1(A,*)</math>
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| and
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| <math>\pi_1(D,*) \to \pi_1(B,*).</math>
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| The theorem then says that the fundamental group of ''X'' is the pushout of these two induced maps. Of course, ''X'' is the pushout of the two inclusion maps of ''D'' into ''A'' and ''B''. Thus we may interpret the theorem as confirming that the fundamental group functor preserves pushouts of inclusions. We might expect this to be simplest when ''D'' is [[simply connected]], since then both homomorphisms above have trivial domain. Indeed this is the case, since then the pushout (of groups) reduces to the [[free product]], which is the coproduct in the category of groups. In a most general case we will be speaking of a [[free product with amalgamation]].
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| There is a detailed exposition of this, in a slightly more general setting ([[covering space|covering]] [[groupoid]]s) in the book by J. P. May listed in the references.
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| ==References==
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| *[[J.P. May|May, J. P.]] ''A concise course in algebraic topology.'' University of Chicago Press, 1999.
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| *:An introduction to categorical approaches to algebraic topology: the focus is on the algebra, and assumes a topological background.
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| == External links ==
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| *[http://nlab.mathforge.org/nlab/show/pushout pushout in nLab ]
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| [[Category:Limits (category theory)]]
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