Category of topological spaces

In mathematics, the category of topological spaces, often denoted $\mathbf {Top}$, 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.

Some authors use the name $\mathbf {Top}$ for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak Hausdorff spaces.

As a concrete category

Like many categories, the category $\mathbf {Top}$ 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

$U:\mathbf {Top} \to \mathbf {Set}$

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

$D:\mathbf {Set} \to \mathbf {Top}$

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

$I:\mathbf {Set} \to \mathbf {Top}$

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 $\mathbf {Set}$). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of $\mathbf {Set}$ into $\mathbf {Top}$.

$\mathbf {Top}$ 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.

$\mathbf {Top}$ is the model of what is called a topological category. These categories are characterized by the fact that every structured source $(X\to UA_{i})_{I}$ has a unique initial lift $(A\to A_{i})_{I}$. In $\mathbf {Top}$ the initial lift is obtained by placing the initial topology on the source. Topological categories have many properties in common with $\mathbf {Top}$ (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).

Limits and colimits

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

Specifically, if $F$ is a diagram in $\mathbf {Top}$ and $(L,\varphi :L\to F)$ is a limit of $UF$ in $\mathbf {Set}$, the corresponding limit of $F$ in $\mathbf {Top}$ is obtained by placing the initial topology on $(L,\varphi :L\to F)$. Dually, colimits in $\mathbf {Top}$ are obtained by placing the final topology on the corresponding colimits in $\mathbf {Set}$.

Unlike many algebraic categories, the forgetful functor $U:\mathbf {Top} \to \mathbf {Set}$ does not create or reflect limits since there will typically be non-universal cones in $\mathbf {Top}$ covering universal cones in $\mathbf {Set}$.

Examples of limits and colimits in $\mathbf {Top}$ include:

• The empty set (considered as a topological space) is the initial object of $\mathbf {Top}$; any singleton topological space is a terminal object. There are thus no zero objects in $\mathbf {Top}$.
• The product in $\mathbf {Top}$ 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 $\mathbf {Top}$.

Other properties

• The monomorphisms in $\mathbf {Top}$ 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 $\mathbf {Top}$ 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 $\mathbf {Top}$, and in particular the category is not preadditive.
• $\mathbf {Top}$ 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 $\mathbf {CGHaus}$ or the category of compactly generated weak Hausdorff spaces. However, $\mathbf {Top}$ is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of convergence spaces.

Relationships to other categories

• The category of pointed topological spaces $\mathbf {Top}$_• is a coslice category over $\mathbf {Top}$.
• The homotopy category $\mathbf {hTop}$ has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of $\mathbf {Top}$. One can likewise form the pointed homotopy category $\mathbf {hTop}$_•.
• $\mathbf {Top}$ contains the important category $\mathbf {Haus}$ 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.
• $\mathbf {Top}$ contains the full subcategory $\mathbf {CGHaus}$ 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 $\mathbf {CGHaus}$ a particularly convenient category of topological spaces that is often used in place of $\mathbf {Top}$.
• The forgetful functor to $\mathbf {Set}$ 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 $\mathbf {Loc}$ 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.
• The homotopy hypothesis relates $\mathbf {Top}$ with $\mathbf {\infty Grpd}$, the category of ∞-groupoids. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo weak homotopy equivalence.

See also

• Category of groups – Category whose objects are groups and whose morphisms are group homomorphisms
• Category of metric spaces – Category whose objects are metric spaces and whose morphisms are metric maps
• Category of sets – Category whose objects are sets and whose morphisms are functions
• Category of topological spaces with base point – Topological space with a distinguished pointPages displaying short descriptions of redirect targets
• Category of topological vector spaces
• Category of measurable spaces
• Categorical topology

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories Archived 2015-04-21 at the Wayback Machine (4.2MB PDF). Originally publ. John Wiley & Sons. ISBN 0-471-60922-6. (now free on-line edition).
• Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
• Dolecki, Szymon (2009). "An initiation into convergence theory" (PDF). In Mynard, Frédéric; Pearl, Elliott (eds.). Beyond Topology. Contemporary Mathematics. Vol. 486. pp. 115–162. doi:10.1090/conm/486/09509. ISBN 9780821842799. 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.