Lexic.us

Definition of Pushout

1. Noun. (category theory) Given a pair of morphisms $f:Z\backslash rightarrow\; X$ and $g:\; Z\backslash rightarrow\; Y$ with a common domain, ''Z'', their '''pushout''' is a pair of morphisms $i\_1:X\backslash rightarrow\; P$ and $i\_2:Y\backslash rightarrow\; P$ as well as their common codomain, ''P'', such that the equation $i\_1\; \backslash circ\; f\; =\; i\_2\; \backslash circ\; g$ is satisfied, and for which there is the ''universal property'' that for any other object ''Q'' for which there are also morphisms $j\_1:\; X\backslash rightarrow\; Q$ and $j\_2:\; Y\backslash rightarrow\; Q$; there is a unique morphism $u:\; P\backslash rightarrow\; Q$ such that $u\; \backslash circ\; i\_1\; =\; j\_1$ and $u\; \backslash circ\; i\_2\; =\; j\_2$. ¹

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