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**.

- As a concrete category
- Limits and colimits
- Other properties
- Relationships to other categories
- See also
- Citations
- References

N.B. Some authors use the name **Top** for the categories with topological manifolds or with compactly generated spaces as objects and continuous maps as morphisms.

Like many categories, the category **Top** is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor

*U*:**Top**→**Set**

to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function.

The forgetful functor *U* has both a left adjoint

*D*:**Set**→**Top**

which equips a given set with the discrete topology, and a right adjoint

*I*:**Set**→**Top**

which equips a given set with the indiscrete topology. Both of these functors are, in fact, right inverses to *U* (meaning that *UD* and *UI* are equal to the identity functor on **Set**). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of **Set** into **Top**.

**Top** is also *fiber-complete* meaning that the category of all topologies on a given set *X* (called the * fiber * of *U* above *X*) forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on *X*, while the least element is the indiscrete topology.

**Top** is the model of what is called a topological category. These categories are characterized by the fact that every structured source has a unique initial lift . In **Top** the initial lift is obtained by placing the initial topology on the source. Topological categories have many properties in common with **Top** (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).

The category **Top** is both complete and cocomplete, which means that all small limits and colimits exist in **Top**. In fact, the forgetful functor *U* : **Top** → **Set** uniquely lifts both limits and colimits and preserves them as well. Therefore, (co)limits in **Top** are given by placing topologies on the corresponding (co)limits in **Set**.

Specifically, if *F* is a diagram in **Top** and (*L*, *φ* : *L* → *F*) is a limit of *UF* in **Set**, the corresponding limit of *F* in **Top** is obtained by placing the initial topology on (*L*, *φ* : *L* → *F*). Dually, colimits in **Top** are obtained by placing the final topology on the corresponding colimits in **Set**.

Unlike many *algebraic* categories, the forgetful functor *U* : **Top** → **Set** does not create or reflect limits since there will typically be non-universal cones in **Top** covering universal cones in **Set**.

Examples of limits and colimits in **Top** include:

- The empty set (considered as a topological space) is the initial object of
**Top**; any singleton topological space is a terminal object. There are thus no zero objects in**Top**. - The product in
**Top**is given by the product topology on the Cartesian product. The coproduct is given by the disjoint union of topological spaces. - The equalizer of a pair of morphisms is given by placing the subspace topology on the set-theoretic equalizer. Dually, the coequalizer is given by placing the quotient topology on the set-theoretic coequalizer.
- Direct limits and inverse limits are the set-theoretic limits with the final topology and initial topology respectively.
- Adjunction spaces are an example of pushouts in
**Top**.

- The monomorphisms in
**Top**are the injective continuous maps, the epimorphisms are the surjective continuous maps, and the isomorphisms are the homeomorphisms. - The extremal monomorphisms are (up to isomorphism) the subspace embeddings. In fact, in
**Top**all extremal monomorphisms happen to satisfy the stronger property of being regular. - The extremal epimorphisms are (essentially) the quotient maps. Every extremal epimorphism is regular.
- The split monomorphisms are (essentially) the inclusions of retracts into their ambient space.
- The split epimorphisms are (up to isomorphism) the continuous surjective maps of a space onto one of its retracts.
- There are no zero morphisms in
**Top**, and in particular the category is not preadditive. **Top**is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all spaces. When this feature is desired, one often restricts to the full subcategory of compactly generated Hausdorff spaces**CGHaus**. However,**Top**is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of convergence spaces.^{ [1] }

- The category of pointed topological spaces
**Top**_{•}is a coslice category over**Top**. - The homotopy category
**hTop**has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of**Top**. One can likewise form the pointed homotopy category**hTop**_{•}. **Top**contains the important category**Haus**of Hausdorff spaces as a full subcategory. The added structure of this subcategory allows for more epimorphisms: in fact, the epimorphisms in this subcategory are precisely those morphisms with dense images in their codomains, so that epimorphisms need not be surjective.**Top**contains the full subcategory**CGHaus**of compactly generated Hausdorff spaces, which has the important property of being a Cartesian closed category while still containing all of the typical spaces of interest. This makes**CGHaus**a particularly*convenient category of topological spaces*that is often used in place of**Top**.- The forgetful functor to
**Set**has both a left and a right adjoint, as described above in the concrete category section. - There is a functor to the category of locales
**Loc**sending a topological space to its locale of open sets. This functor has a right adjoint that sends each locale to its topological space of points. This adjunction restricts to an equivalence between the category of sober spaces and spatial locales.

- ↑ Dolecki 2009 , pp. 1-51

**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 category theory, a branch of mathematics, a **Grothendieck topology** is a structure on a category *C* that makes the objects of *C* act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a **site**.

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 prototypical 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 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 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 category theory, an **epimorphism** is a morphism *f* : *X* → *Y* that is right-cancellative in the sense that, for all objects *Z* and all morphisms *g*_{1}, *g*_{2}: *Y* → *Z*,

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, 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 mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label **Stone duality**, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics.

In mathematics, the category **Ab** has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in **Ab**.

In category theory, **Met** is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by Isbell (1964).

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.

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

In mathematics, particularly in homotopy theory, a **model category** is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes. The concept was introduced by Daniel G. Quillen (1967).

In mathematics, a full subcategory *A* of a category *B* is said to be **reflective** in *B* when the inclusion functor from *A* to *B* has a left adjoint. This adjoint is sometimes called a *reflector*, or *localization*. Dually, *A* is said to be **coreflective** in *B* when the inclusion functor has a right adjoint.

In mathematics, the **category of topological vector spaces** is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted **TVect** or **TVS**.

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, 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.

- Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990).
*Abstract and Concrete Categories*(4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition). - Dolecki, Szymon; Mynard, Frederic (2016).
*Convergence Foundations Of Topology*. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. - Dolecki, Szymon (2009). Mynard, Frédéric; Pearl, Elliott (eds.). "An initiation into convergence theory" (PDF).
*Beyond Topology*. Contemporary Mathematics Series A.M.S.**486**: 115–162. Retrieved 14 January 2021. - Dolecki, Szymon; Mynard, Frédéric (2014). "A unified theory of function spaces and hyperspaces: local properties" (PDF).
*Houston J. Math*.**40**(1): 285–318. Retrieved 14 January 2021. - Herrlich, Horst:
*Topologische Reflexionen und Coreflexionen*. Springer Lecture Notes in Mathematics 78 (1968). - Herrlich, Horst:
*Categorical topology 1971–1981*. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279–383. - Herrlich, Horst & Strecker, George E.: Categorical Topology – its origins, as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971. In: Handbook of the History of General Topology (eds. C.E.Aull & R. Lowen), Kluwer Acad. Publ. vol 1 (1997) pp. 255–341.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.