Category of Topological Spaces
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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. 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 categories with topological manifolds or with compactly generated spaces as objects and continuous maps as morphisms.

## As a concrete category

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 : TopSet

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 : SetTop

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

I : SetTop

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 ${\displaystyle (X\to UA_{i})_{I}}$ has a unique initial lift ${\displaystyle (A\to A_{i})_{I}}$. 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, ? : 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, ? : 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:

## Relationships to other categories

• The category of pointed topological spaces Topo 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 hTopo.
• 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.

## 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, Ji?í, 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).