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In mathematics, in the field of category theory, a **discrete category** is a category whose only morphisms are the identity morphisms:

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

**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, particularly in category theory, a **morphism** is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

- hom
_{C}(*X*,*X*) = {id_{X}} for all objects*X* - hom
_{C}(*X*,*Y*) = ∅ for all objects*X*≠*Y*

Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set

- | hom
_{C}(*X*,*Y*) | is 1 when*X*=*Y*and 0 when*X*is not equal to*Y*.

Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.

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.

Any class of objects defines a discrete category when augmented with identity maps.

In set theory and its applications throughout mathematics, a **class** is a collection of sets that can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.

Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full.

In mathematics, specifically category theory, a **subcategory** of a category *C* is a category *S* whose objects are objects in *C* and whose morphisms are morphisms in *C* with the same identities and composition of morphisms. Intuitively, a subcategory of *C* is a category obtained from *C* by "removing" some of its objects and arrows.

The limit of any functor from a discrete category into another category is called a product, while the colimit is called a coproduct. Thus, for example, the discrete category with just two objects can be used as a diagram or diagonal functor to define a product or coproduct of two objects. Alternately, for a general category **C** and the discrete category **2**, one can consider the functor category **C**^{2}. The diagrams of **2** in this category are pairs of objects, and the limit of the diagram is the product.

In category theory, a branch of mathematics, the abstract notion of a **limit** captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a **colimit** generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits.

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, 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, the direct product of 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.

The functor from **Set** to **Cat** that sends a set to the corresponding discrete category is left adjoint to the functor sending a small category to its set of objects. (For the right adjoint, see indiscrete category.)

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.

An **indiscrete category** is a category *C* in which every hom-set *C*(*X*, *Y*) is a singleton. Every class *X* gives rise to an indiscrete category whose objects are the elements of *X* with exactly one morphism between any two objects. Any two nonempty indiscrete categories are equivalent to each other. The functor from **Set** to **Cat** that sends a set to the corresponding indiscrete category is right adjoint to the functor that sends a small category to its set of objects.

In category theory, a branch of mathematics, when we say that a construction satisfies a **universal property**, it means that this construction can be seen as an initial or terminal object of some other category. By "universal property" one may mean either a universal initial or terminal morphism. The nomination of initial/terminal morphisms comes from the fact that these two morphisms are respectively initial/terminal objects in their corresponding comma categories which has been made from the original ones. Intuitively, by *universal* we mean that the initial/terminal morphism(s) is (are) the "most general" constructions in that category with those properties.

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 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*.

In mathematics, specifically in category theory, an **additive category** is a preadditive category **C** admitting all finitary biproducts.

In category theory, a category is considered **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 **coproduct**, or **categorical sum**, is a category-theoretic construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the "least specific" object to which each object in the family admits a morphism. It is the category-theoretic dual notion to the categorical product, which means the definition is the same as the product but with all arrows reversed. Despite this seemingly innocuous change in the name and notation, coproducts can be and typically are dramatically different from products.

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

In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps or some other variant; for example, objects are often assumed to be compactly generated. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of **Top** and of properties of topological spaces using the techniques of category theory is known as **categorical topology**.

In mathematics, the category **Grp** has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

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.

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 category theory, a branch of mathematics, a **diagram** is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a *function* from a fixed index *set* to the class of *sets*. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a *functor* from a fixed index *category* to some *category*.

In mathematics, the **category of rings**, denoted by **Ring**, is the category whose objects are rings and whose morphisms are ring homomorphisms. Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.

In mathematics, a **topos** is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The **Grothendieck topoi** find applications in algebraic geometry; the more general **elementary topoi** are used in logic.

- Robert Goldblatt (1984).
*Topoi, the Categorial Analysis of Logic*(Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications, and available online at Robert Goldblatt's homepage.

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