Editing Algebraic structure (section)

== Category theory ==

[[Category theory]] is another tool for studying algebraic structures (see, for example, Mac Lane 1998). A category is a collection of ''objects'' with associated ''morphisms.'' Every algebraic structure has its own notion of [[homomorphism]], namely any [[function (mathematics)|function]] compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a [[category (mathematics)|category]]. For example, the [[category of groups]] has all [[Group (mathematics)|groups]] as objects and all [[group homomorphism]]s as morphisms. This [[concrete category]] may be seen as a [[category of sets]] with added category-theoretic structure. Likewise, the category of [[topological group]]s (whose morphisms are the continuous group homomorphisms) is a [[category of topological spaces]] with extra structure. A [[forgetful functor]] between categories of algebraic structures "forgets" a part of a structure.

There are various concepts in category theory that try to capture the algebraic character of a context, for instance

* [[algebraic category]]
* [[essentially algebraic category]]
* [[presentable category]]
* [[locally presentable category]]
* [[Monad (category theory)|monadic]] functors and categories
* [[universal property]].