Category

up:: 028 MOC Category Theory

A category $C$ is composed of:

1. A class of elements called objects, denoted $Ob(C)$
2. For each pair of objects $x,y\in Ob(C)$, a class of elements called morphisms between $x$ and $y$ $Hom(x,y)$ (called Hom-Set of morphisms from $x$ to $y$ ─ though it need not be a set)¹
3. For each pair of morphisms $f\in Hom(x,y),g\in Hom(y,z)$, a morphism $g\circ f\in Hom(x,z)$, called their composite
4. For each object $x$, an identity morphism $1_x\in Hom(x,x)$

The rules that apply to a category are:

1. Composition of morphism is associative: $(f\circ g)\circ h=f\circ (g\circ h)$
   
2. Identity morphism acts as two-sided composition identity: $\forall f\in Hom(x,x):1_x\circ f=f\circ 1_x=f$
   

Properties

• To every category $C$ there is an Opposite Category $C^{op}$ which is the same category but with all morphisms flipped

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References

• category in nLab
• Category (mathematics) - Wikipedia
• ABRAMSKY, Samson; TZEVELEKOS, Nikos. Introduction to categories and categorical logic. In: New Structures for Physics, p. 3-94, 2011.

Footnotes

1. The class of all morphisms in $C$ is the disjoint union of all $Hom(x,y)$, for all $x,y\in Ob(C)$. ↩