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CT: Composition

In category theory, composition is the joining together of two morphisms to create a new morphism.

Two morphisms f and g are composable if fs arrow ends where gs starts. i.e. f : A \rightarrow B and g : B \rightarrow C would make f and g composable.

Composition is described using the composition operator \circ, pronounced after. So f composed with g as described above would be written g \circ f, or g after f when spoken aloud.

\begin{xy} \xymatrix{ A \ar@{->}[r]^{f} \ar@/_/@{->}[rr]_{g \circ f} & B \ar@{->}[r]^{g} & C } \end{xy}

This concept is very closely related to composition in functional programming. This is because it's the same concept, where the category in question is the category of types.