Recall the types of set mappings from the section on defining mathematical structures. It can be useful to generalize both sets and mappings, which takes us from set theory to **category theory**. Category theory eliminates any dependence upon elements, referring only to classes of generic **objects**. The **class** (AKA collection) of objects \({\textrm{ob}(C)}\) of a class \({C}\) sometimes may be defined as sets with a certain structure, but in category theory they are left completely abstract, with the following definitions built upon them:

**Morphisms**: a set \({\textrm{mor}(X,Y)}\) (also denoted \({\textrm{hom}(X,Y)}\) or \({C(X,Y)}\)) of morphisms between \({X}\) and \({Y}\) is defined for every \({X,Y\in \textrm{ob}(C)}\); every \({\textrm{mor}(X,X)}\) includes an identity \({\mathbf{1}_{X}}\)**Composition**: an operator \({\circ}\) is defined between morphisms that is distributive and respects the identity, i.e. for morphisms \({m\colon X\to Y}\) and \({n\colon Y\to Z}\) we have \({n\circ m\colon X\to Z}\) with \({(n\circ m)\circ l=n\circ(m\circ l)}\) and \({m\circ\mathbf{1}_{X}=m=\mathbf{1}_{Y}\circ m}\)

A **category** \({C}\) then consists of a class of objects \({\textrm{ob}(C)}\), a collection of sets of morphisms \({\textrm{mor}(X,Y)}\) between these objects, and a morphism composition operator. It is helpful in understanding these definitions to consider their application to sets and mappings. In this case, a class of objects would consist of sets along with a structure; morphisms would be mappings between these objects; and the category would consist of the class and the mappings.