Editing Monoid (section)

== Relation to category theory ==
{{Group-like structures}}
Monoids can be viewed as a special class of [[category theory|categories]]. Indeed, the axioms required of a monoid operation are exactly those required of [[morphism]] composition when restricted to the set of all morphisms whose source and target is a given object.{{sfn|ps=|Awodey|2006|p=10}} That is,
: ''A monoid is, essentially, the same thing as a category with a single object.''
More precisely, given a monoid {{math|(''M'', •)}}, one can construct a small category with only one object and whose morphisms are the elements of {{math|''M''}}. The composition of morphisms is given by the monoid operation&nbsp;{{math|•}}.

Likewise, monoid homomorphisms are just [[functor]]s between single object categories.{{sfn|ps=|Awodey|2006|p=10}}  So this construction gives an [[equivalence of categories|equivalence]] between the [[category of monoids|category of (small) monoids]] '''Mon''' and a full subcategory of the category of (small) categories '''Cat'''. Similarly, the [[category of groups]] is equivalent to another full subcategory of '''Cat'''.

In this sense, category theory can be thought of as an extension of the concept of a monoid. Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid.

Monoids, just like other algebraic structures, also form their own category, '''Mon''', whose objects are monoids and whose morphisms are monoid homomorphisms.{{sfn|ps=|Awodey|2006|p=10}}

There is also a notion of [[monoid (category theory)|monoid object]] which is an abstract definition of what is a monoid in a category. A monoid object in [[category of sets|'''Set''']] is just a monoid.