# Category (mathematics)

A category consists of a collection of objects with morphisms between them. A morphism $$f$$ goes from one object, say $$X$$, to another, say $$Y$$, and is drawn as an arrow from $$X$$ to $$Y$$. Note that $$X$$ may equal $$Y$$ (in which case $$f$$ is referred to as an endomorphism). The object $$X$$ is called the source or domain of $$f$$ and $$Y$$ is called the target or codomain of $$f$$, though note that $$f$$ itself need not be a function and $$X$$ and $$Y$$ need not be sets. This is written as $$f: X \rightarrow Y$$.

These morphisms must satisfy three conditions:

1. Composition: For any two morphisms $$f: X \rightarrow Y$$ and $$g: Y \rightarrow Z$$, there exists a morphism $$X \rightarrow Z$$, written as $$g \circ f$$ or simply $$gf$$.

2. Associativity: For any morphisms $$f: X \rightarrow Y$$, $$g: Y \rightarrow Z$$ and $$h:Z \rightarrow W$$ composition is associative, i.e., $$h(gf) = (hg)f$$.

3. Identity: For any object $$X$$, there is a (unique) morphism, $$1_X : X \rightarrow X$$ which, when composed with another morphism, leaves it unchanged. I.e., given $$f:W \rightarrow X$$ and $$g:X \rightarrow Y$$ it holds that: $$1_X f = f$$ and $$g 1_X = g$$.

Parents:

• Category theory

How mathematical objects are related to others in the same category.