Complete category

Last updated

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 : JC (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.

Contents

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.

Theorems

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:

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.

Examples and nonexamples

Related Research Articles

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 IX.

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 : ZX and g : ZY with a common domain. The pushout consists of an object P along with two morphisms XP and YP 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.

Category of rings category in mathematics

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.

References

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

Further reading