# Category of topological spaces

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**.

N.B. Some authors use the name **Top** for the category with topological manifolds as objects and continuous maps as morphisms.

## As a concrete category

Like many categories, the category **Top** is a concrete category (also known as a *construct*), 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**.

The construct **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.

The construct **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).

## Limits and colimits

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**.

## Other properties

- 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**.

## Relationships to other categories

- 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 topological spaces with the Hausdorff property 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.

## References

- 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.
- 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).