Pushout

The pushout (also cofiber product , cocartesian square , fiber sum , amalgamated sum ) is a term from the mathematical subfield of category theory . It is the dual construction for pullback .

Pushout of modules

Let and two homomorphisms between modules over a ring . If one sets , the pushout of and is defined as ${\displaystyle \alpha _{1}:X\rightarrow X_{1}}$ ${\displaystyle \alpha _{2}:X\rightarrow X_{2}}$ ${\displaystyle R}$ ${\displaystyle Q:=\{(\alpha _{1}(x),\alpha _{2}(x)):\,x\in X\}\subset X_{1}\oplus X_{2}}$ ${\displaystyle \alpha _{1}}$ ${\displaystyle \alpha _{2}}$

${\displaystyle P:=(X_{1}\oplus X_{2})/Q}$ with the homomorphisms

${\displaystyle \varphi _{1}:X_{1}\rightarrow P,\,\varphi _{1}(x_{1}):=(x_{1},0)+Q}$ and

${\displaystyle \varphi _{2}:X_{2}\rightarrow P,\,\varphi _{2}(x_{2}):=(0,-x_{2})+Q}$

One can show that and that has the following universal property : ${\displaystyle \varphi _{1}\circ \alpha _{1}=\varphi _{2}\circ \alpha _{2}}$ ${\displaystyle P,\varphi _{1},\varphi _{2}}$

If there is any module with homomorphisms and such that , then there is exactly one homomorphism with and . ${\displaystyle Y}$ ${\displaystyle R}$ ${\displaystyle \psi _{1}:X_{1}\rightarrow Y}$ ${\displaystyle \psi _{2}:X_{2}\rightarrow Y}$ ${\displaystyle \psi _{1}\circ \alpha _{1}=\psi _{2}\circ \alpha _{2}}$ ${\displaystyle \rho :P\rightarrow Y}$ ${\displaystyle \psi _{1}=\rho \circ \varphi _{1}}$ ${\displaystyle \psi _{2}=\rho \circ \varphi _{2}}$

Pushout in categories

Motivated by the above example, the pushout is defined in any category as follows.

Let there be and two morphisms of one category. A pair of morphisms in this category is called a pushout of if: ${\displaystyle \alpha _{1}:X\rightarrow X_{1}}$ ${\displaystyle \alpha _{2}:X\rightarrow X_{2}}$ ${\displaystyle (\varphi _{1},\varphi _{2})}$ ${\displaystyle \varphi _{i}:X_{i}\rightarrow P}$ ${\displaystyle (\alpha _{1},\alpha _{2})}$

• ${\displaystyle \varphi _{1}\circ \alpha _{1}=\varphi _{2}\circ \alpha _{2}}$
• If there is a pair of morphisms with , then there is exactly one morphism with and . ${\displaystyle (\psi _{1},\psi _{2})}$ ${\displaystyle \psi _{i}:X_{i}\rightarrow Y}$ ${\displaystyle \psi _{1}\circ \alpha _{1}=\psi _{2}\circ \alpha _{2}}$ ${\displaystyle \rho :P\rightarrow Y}$ ${\displaystyle \psi _{1}=\rho \circ \varphi _{1}}$ ${\displaystyle \psi _{2}=\rho \circ \varphi _{2}}$

Sometimes you just call the object a pushout, meaning that there are morphisms that meet the above definition. The diagram too ${\displaystyle P}$ ${\displaystyle \varphi _{i}:X_{i}\rightarrow P}$

${\displaystyle {\begin{array}{ccc}X&{\xrightarrow {\alpha _{1}}}&X_{1}\\\downarrow _{\alpha _{2}}&&\downarrow _{\varphi _{1}}\\X_{2}&{\xrightarrow {\varphi _{2}}}&P\end{array}}}$

is sometimes referred to as a pushout. The notation is analogous to the pullback . ${\displaystyle P=X_{1}\sqcup _{X}X_{2}}$

Examples

• Every pullback in a category is a pushout in the dual category , because obviously the pushout is exactly the concept that is dual to the pullback. ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}^{op}}$
• In an Abelian category , the pushout is closed

${\displaystyle {\begin{array}{ccc}X&{\xrightarrow {\alpha _{1}}}&X_{1}\\\downarrow _{0}&&\\0&&\end{array}}}$

like the coke of . ${\displaystyle \alpha _{1}}$

• If the zero object is an additive category with the above designation , the pushout is equal to the direct sum . ${\displaystyle X}$ ${\displaystyle X_{1}\oplus X_{2}}$
• The introductory example shows that there are always pushouts in the category of modules. ${\displaystyle R}$
• There is always a pushout in the category of groups. With the above designations, this is equal to the free product modulo that generated by the normal divider with the natural images. This construction occurs in Seifert-van Kampen's theorem . ${\displaystyle X_{1}*X_{2}}$ ${\displaystyle \{\alpha _{1}(x)\alpha _{2}(x)^{-1}:\,x\in X\}}$ ${\displaystyle N}$ ${\displaystyle \varphi _{i}:X_{i}\rightarrow X_{1}*X_{2}\rightarrow X_{1}*X_{2}/N}$
• In the category of commutative rings with one element, the pushout with the above designations is equal to the tensor product provided with the one and the multiplication determined by the one . ${\displaystyle X_{1}\otimes _{X}X_{2}}$ ${\displaystyle 1\otimes 1}$ ${\displaystyle (a\otimes b)\cdot (c\otimes d):=(a\cdot c)\otimes (b\cdot d)}$
• In the category of sets this is the pushout , where the equivalence relation generated by is on the disjoint union . ${\displaystyle (X_{1}\sqcup X_{2})/{\sim }}$ ${\displaystyle \sim }$ ${\displaystyle \{(\alpha _{1}(x),\alpha _{2}(x)):x\in X\}}$ ${\displaystyle X:=X_{1}\sqcup X_{2}}$
• Pushouts of topological spaces can be described in a similar way. These play a role in adhesive structures.

Individual evidence

1. ↑ Louis D. Tarmin: Lineare Algebra, Modules 2 , Book X Verlag (April 2008), ISBN 3-9346-7151-9 , sentence 4.158.3
2. ^ Peter Hilton: Lectures in Homological Algebra , American Mathematical Society (2005), ISBN 0-8218-3872-5 , definition 4.1
3. ^ Joseph J. Rotman: An Introduction to the Theory of Groups . Springer, Graduate Texts in Mathematics, 1995, ISBN 0-3879-4285-8 , Theorem 11.58