In mathematics, a **complete category** is a category in which all small limits exist. That is, a category *C* is complete if every diagram *F* : *J* → *C* (where *J* is small) has a limit in *C*. Dually, a **cocomplete category** is one in which all small colimits exist. A **bicomplete category** is a category which is both complete and cocomplete.

The existence of *all* limits (even when *J* is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.

A weaker form of completeness is that of finite completeness. A category is **finitely complete** if all finite limits exists (i.e. limits of diagrams indexed by a finite category *J*). Dually, a category is **finitely cocomplete** if all finite colimits exist.

It follows from the existence theorem for limits that a category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) products. Since equalizers may be constructed from pullbacks and binary products (consider the pullback of (*f*, *g*) along the diagonal Δ), a category is complete if and only if it has pullbacks and products.

Dually, a category is cocomplete if and only if it has coequalizers and all (small) coproducts, or, equivalently, pushouts and coproducts.

Finite completeness can be characterized in several ways. For a category *C*, the following are all equivalent:

*C*is finitely complete,*C*has equalizers and all finite products,*C*has equalizers, binary products, and a terminal object,*C*has pullbacks and a terminal object.

The dual statements are also equivalent.

A small category *C* is complete if and only if it is cocomplete.^{ [1] } A small complete category is necessarily thin.

A posetal category vacuously has all equalizers and coequalizers, whence it is (finitely) complete if and only if it has all (finite) products, and dually for cocompleteness. Without the finiteness restriction a posetal category with all products is automatically cocomplete, and dually, by a theorem about complete lattices.

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- The following categories are bicomplete:
**Set**, the category of sets**Top**, the category of topological spaces**Grp**, the category of groups**Ab**, the category of abelian groups**Ring**, the category of rings, the category of vector spaces over a field*K*-Vect*K*, the category of modules over a commutative ring*R*-Mod*R***CmptH**, the category of all compact Hausdorff spaces**Cat**, the category of all small categories**Whl**, the category of wheels**sSet**, the category of simplicial sets^{ [2] }

- The following categories are finitely complete and finitely cocomplete but neither complete nor cocomplete:
- The category of finite sets
- The category of finite abelian groups
- The category of finite-dimensional vector spaces

- Any (pre)abelian category is finitely complete and finitely cocomplete.
- The category of complete lattices is complete but not cocomplete.
- The category of metric spaces,
**Met**, is finitely complete but has neither binary coproducts nor infinite products. - The category of fields,
**Field**, is neither finitely complete nor finitely cocomplete. - A poset, considered as a small category, is complete (and cocomplete) if and only if it is a complete lattice.
- The partially ordered class of all ordinal numbers is cocomplete but not complete (since it has no terminal object).
- A group, considered as a category with a single object, is complete if and only if it is trivial. A nontrivial group has pullbacks and pushouts, but not products, coproducts, equalizers, coequalizers, terminal objects, or initial objects.

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, an **abelian category** is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, **Ab**. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very *stable* categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

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, a **pre-abelian category** is an additive category that has all kernels and cokernels.

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, an **equalizer** is a set of arguments where two or more functions have equal values. An equalizer is the solution set of an equation. In certain contexts, a **difference kernel** is the equalizer of exactly two functions.

In category theory, the **coproduct**, or **categorical sum**, is a 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.

The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of *objects* and *arrows*, where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

In category theory, a **coequalizer** is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer.

In mathematics, the **category of topological spaces**, often denoted **Top**, is the category whose objects are topological spaces and whose morphisms are continuous maps. 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 category theory, a branch of mathematics, a **pushout** is the colimit of a diagram consisting of two morphisms *f* : *Z* → *X* and *g* : *Z* → *Y* with a common domain. The pushout consists of an object *P* along with two morphisms *X* → *P* and *Y* → *P* that complete a commutative square with the two given morphisms *f* and *g*. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are and .

In category theory, a branch of mathematics, a **pullback** is the limit of a diagram consisting of two morphisms *f* : *X* → *Z* and *g* : *Y* → *Z* with a common codomain. The pullback is often written

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

In category theory, a branch of mathematics, the **diagonal functor** is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects *within* the category : a product is a universal arrow from to . The arrow comprises the projection maps.

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.

In mathematics, a **Grothendieck category** is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

In mathematics, specifically category theory, a **posetal category**, or **thin category**, is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class. As suggested by the name, the further requirement that the category be skeletal is often assumed for the definition of "posetal"; in the case of a category that is posetal, being skeletal is equivalent to the requirement that the only isomorphisms are the identity morphisms, equivalently that the preordered class satisfies antisymmetry and hence, if a set, is a poset.

In mathematics, **compact objects**, also referred to as **finitely presented objects**, or **objects of finite presentation**, are objects in a category satisfying a certain finiteness condition.

- ↑ Abstract and Concrete Categories, Jiří Adámek, Horst Herrlich, and George E. Strecker, theorem 12.7, page 213
- ↑ Riehl, Emily (2014).
*Categorical Homotopy Theory*. New York: Cambridge University Press. p. 32. ISBN 9781139960083. OCLC 881162803.

- Adámek, Jiří; Horst Herrlich; George E. Strecker (1990).
*Abstract and Concrete Categories*(PDF). John Wiley & Sons. ISBN 0-471-60922-6. - Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics**5**((2nd ed.) ed.). Springer. ISBN 0-387-98403-8.

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