# Category theory

Category theory is a branch of pure mathematics concerned with the notion of structure, which is formalized by categories^{[1]}.

A category consists of a space of abstract objects and a set of arrows or morphisms for each pair of objects. Additionally, each object must have an identity morphism that loops back to itself, and there is a composition operator that allows for concatenating any two consecutive morphisms into a single morphism. This composition operator is associative and has the identity morphisms as its identity elements, that is, any morphism composed with the identity morphism of any of its two endpoints gives back the same morphism^{[2]}.

This may seem like an arbitrary concept, but it provides an extremely powerful framework for unifying many different areas of mathematics. A very intuitive category is the category of sets, where each object is a set and each morphism is a function between two sets. Identity morphisms are given by the identity function, which maps every element of a set to itself, and composition of morphisms is the composition of functions, which satisfies the associativity and identity element properties.

## Applications[edit]

Category theory is consistent with structuralism. From the perspective of category theory, objects are abstract and must be understood by their relations with other objects, unlike in set theory, where sets are defined by their elements.

This framework may also be useful for the formalization of Hegelian dialectics. The concept of *adjoint modality* formalizes a unity of opposites^{[3]}, but an explanation here would require the introduction of too many concepts.

There is a notion of *internal logic*^{[4]}, in which a category with a certain structure can give the semantics for a certain syntactic system of inference. This would be analogous to Hegel's objective logic and subjective logic distinction^{[5]}.

Category theory is also useful in functional programming, where certain structures can provide generalizations of many useful design patterns.

In Jacques Lacan's psychoanalytic theory, signifiers are meaningless in themselves, and only acquire meaning by their relations with other signifiers.