This is a glossary of properties and concepts in category theory in mathematics. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

A category A is said to be:

• small provided that the object class is a set (i.e., not proper class); otherwise large.
  • Cat is a category of all of small categories.
• locally small provided that the morphisms between every pair of objects A and B form a set.
• quasicategory provided that objects in A may not form a class and morphisms between objects A and B may not form a set.
  • CAT is a quasicategory of all categories.
• isomorphic to a category B provided that there exists an isomorphism between them.
• equivalent to a category B provided that there exists an equivalence between them.
• concrete provided that there exists a faithful functor from A to Set; e.g., Vec, Grp and Top.
• discrete provided that each morphism is the identity morphism.
• thin category provided that there is at most one morphism between objects A and B.
• a subcategory of a category B provided that there exists an inclusion functor from A to B.
• a full subcategory of a category B provided that the inclusion functor is full.

In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ... In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ... VEC is a TLA which can refer to several different things: The Victorian Electoral Commission. ... GRP may refer to: Gastrin Releasing Peptide Gibraltar Reform Party Glass-reinforced plastic Gross Rating Point in television GRP Records Grusin Rosen Production Gentoo Reference Platform (pre-compiled packages for Gentoo Linux) .GRP file extention; used for program groups for Program Manager, and also in an unrelated code for data... A top with sides marked in Braille A top, or spinning top, is a childrens toy that can be spun on an axis, balancing on a point. ... In category theory, a discrete category is a category whose only morphisms are the identity morphisms. ...

A morphism f in a category is said to be:

• an epimorphism provided that g = h whenever . In other words, f is the dual of a monomorphism.
• an identity provided that f maps an object A to A and for any morphisms g with domain A and h with codomain A, and .
• an inverse to a morphism g if is defined and is equal to the identity morphism on the domain of f, and is defined and equal to the identity morphism on the codomain of g. The inverse of g is unique and is denoted by f ^-1
• an isomorphism provided that there exists an inverse of f.
• a monomorphism provided that g = h whenever . In other words, f is the dual of an epimorphism.

In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... // Computer programming In object-oriented programming, object identity is a mechanism for distinguishing different objects from each other. ... Inverse typically means the opposite of something. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...

A functor F is said to be:

• a constant provided that F maps every object in a category to the same object A and every morphism to the identity on A.
• faithful provided that F is injective when restricted to each hom-set.
• full provided that F is surjective when restricted to each hom-set.
• isomorphism-dense (sometimes called essentially surjective) provided that for every B there exists an A such that F(A) is isomorphic to B.
• an equivalence provided that F is faithful, full and isomorphism-dense.
• reflect identities provided that if F(k) is an identity then k is an identity as well.

In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

An object A a category is said to be:

• isomorphic to an object B provided that there is an isomorphism between A and B.
• initial provided that there is exactly one morphism from A to each object B.
• terminal provided that there is exactly one morphism from each object B to A.