In the mathematical field of category theory, the **product** of two categories *C* and *D*, denoted *C* × *D* and called a **product category**, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.^{ [1] }

The product category *C* × *D* has:

- as objects:
- pairs of objects (
*A*,*B*), where*A*is an object of*C*and*B*of*D*;

- pairs of objects (
- as arrows from (
*A*_{1},*B*_{1}) to (*A*_{2},*B*_{2}):- pairs of arrows (
*f*,*g*), where*f*:*A*_{1}→*A*_{2}is an arrow of*C*and*g*:*B*_{1}→*B*_{2}is an arrow of*D*;

- pairs of arrows (
- as composition, component-wise composition from the contributing categories:
- (
*f*_{2},*g*_{2}) o (*f*_{1},*g*_{1}) = (*f*_{2}o*f*_{1},*g*_{2}o*g*_{1});

- (
- as identities, pairs of identities from the contributing categories:
- 1
_{(A, B)}= (1_{A}, 1_{B}).

- 1

For small categories, this is the same as the action on objects of the categorical product in the category ** Cat **. A functor whose domain is a product category is known as a bifunctor. An important example is the Hom functor, which has the product of the opposite of some category with the original category as domain:

- Hom :
*C*^{op}×*C*→**Set**.

Just as the binary Cartesian product is readily generalized to an *n*-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of *n* categories. The product operation on categories is commutative and associative, up to isomorphism, and so this generalization brings nothing new from a theoretical point of view.

**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*. A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Informally, category theory is a general theory of functions.

In mathematics, specifically category theory, a **functor** is a map between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

In category theory, a branch of mathematics, a **universal property** is an important property which is satisfied by a **universal morphism**. Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category. Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.

In category theory, a branch of mathematics, a **natural transformation** provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, a **category** is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.

In mathematics, specifically category theory, **adjunction** is a relationship that two functors may have. Two functors that stand in this relationship are known as **adjoint functors**, one being the **left adjoint** and the other the **right adjoint**. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.

In mathematics, specifically in category theory, a **preadditive category** is another name for an **Ab-category**, i.e., a category that is enriched over the category of abelian groups, **Ab**. That is, an **Ab-category****C** is a category such that every hom-set Hom(*A*,*B*) in **C** has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas:

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. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms associated with every pair of objects is replaced by an object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint.

In category theory, a category is **Cartesian closed** if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.

In category theory, the **product** of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

In mathematics, a **monoidal category** is a category **C** equipped with a bifunctor

In category theory, a branch of mathematics, a **monad** is an endofunctor, together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

In category theory, an abstract branch of mathematics, an **equivalence of categories** is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics. Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned. In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors. Functor categories are of interest for two main reasons:

In mathematics, particularly category theory, a **representable functor** is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

**Fibred categories** are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which *inverse images* of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space *X* to another topological space *Y* is associated the pullback functor taking bundles on *Y* to bundles on *X*. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

In mathematics, specifically in category theory, an **exponential object** or **map object** is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an **exponential law**.

This is a glossary of properties and concepts in category theory in mathematics.

In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called **hom-functors** and have numerous applications in category theory and other branches of mathematics.

In mathematics, the category **Rel** has the class of sets as objects and binary relations as morphisms.

- Definition 1.6.5 in Borceux, Francis (1994).
*Handbook of categorical algebra*. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Volume 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1. - Product category in
*nLab* - Mac Lane, Saunders (1978).
*Categories for the Working Mathematician*(Second ed.). New York, NY: Springer New York. pp. 49–51. ISBN 1441931236. OCLC 851741862.

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