Category theory

. . . la mathématique est l’art de donner le même nom à des choses différentes.

. . . mathematics is the art of giving the same name to different things.

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Henri Poincaré [cite:@riehl2017category]

Categories are a generalisation of both monoids and preordered sets.[cite:@aamek1990abstract]

Notation

Composition of morphisms is typically written as you'd expect, with $(f\circ g)x=f\left(g\left(x\right)\right)$.

Alternatively, one may write in "diagrammatic order," with morphisms given in the order in which they are 'applied.' This either looks like $(fg)x=g(fx)$ or $(f;g)x=g(fx)$.[cite:@riehl2017category]

See also

• Functors

• Examples of categories