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# Strict 2-category

In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. 

## Algebraic theory

Informally in mathematical logic, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables.

## Bicategory

In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism.

## Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

## Category of small categories

In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

## Charles Ehresmann

Charles Ehresmann (19 April 1905 – 22 September 1979) was a French mathematician who worked in differential topology and category theory.

## Enriched category

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category.

## Functor

In mathematics, a functor is a map between categories.

## Higher category theory

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.

## Initial and terminal objects

In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

## Jonathan Mock Beck

Jonathan Mock Beck (aka Jon Beck; 11 November 1935 &ndash; 11 March 2006, Somerville, Massachusetts) was an American mathematician, who worked on category theory and algebraic topology.

## Monoidal category

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.

## Morphism

In mathematics, a morphism is a structure-preserving map from one mathematical structure to another one of the same type.

Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity.

## Presheaf (category theory)

In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf.

## Product category

In the mathematical field of category theory, the product of two categories C and D, denoted and called a product category, is an extension of the concept of the Cartesian product of two sets.

## Topos

In mathematics, a topos (plural topoi or, or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site).

## William Lawvere

Francis William Lawvere (born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.

## References

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