Category of topological spaces

Short description: Category whose objects are topological spaces and whose morphisms are continuous maps

In mathematics, the category of topological spaces, often denoted $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$, 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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ and of properties of topological spaces using the techniques of category theory is known as categorical topology.

N.B. Some authors use the name $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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

$U:\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}\to \mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$

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

$D:\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}\to \mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$

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

$I:\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}\to \mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$

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 $\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of $\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$ into $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$.

$\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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.

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

Limits and colimits

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

Specifically, if $F$ is a diagram in $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ and $(L,\phi :L\to F)$ is a limit of $UF$ in $\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$, the corresponding limit of $F$ in $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ is obtained by placing the initial topology on $(L,\phi :L\to F)$. Dually, colimits in $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ are obtained by placing the final topology on the corresponding colimits in $\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$.

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

Examples of limits and colimits in $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ include:

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

Other properties

• The monomorphisms in $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$, and in particular the category is not preadditive.
• $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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 $\mathbf{𝐂}\mathbf{𝐆}\mathbf{𝐇}\mathbf{𝐚}\mathbf{𝐮}\mathbf{𝐬}$ or the category of compactly generated weak Hausdorff spaces. However, $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ 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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$_• is a coslice category over $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$.
• The homotopy category $\mathbf{𝐡}\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ has topological spaces for objects and homotopy equivalence classes of continuous maps for morphisms. This is a quotient category of $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$. One can likewise form the pointed homotopy category $\mathbf{𝐡}\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$_•.
• $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ contains the important category $\mathbf{𝐇}\mathbf{𝐚}\mathbf{𝐮}\mathbf{𝐬}$ 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.
• $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ contains the full subcategory $\mathbf{𝐂}\mathbf{𝐆}\mathbf{𝐇}\mathbf{𝐚}\mathbf{𝐮}\mathbf{𝐬}$ 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 $\mathbf{𝐂}\mathbf{𝐆}\mathbf{𝐇}\mathbf{𝐚}\mathbf{𝐮}\mathbf{𝐬}$ a particularly convenient category of topological spaces that is often used in place of $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$.
• The forgetful functor to $\mathbf{𝐒}\mathbf{𝐞}\mathbf{𝐭}$ 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 $\mathbf{𝐋}\mathbf{𝐨}\mathbf{𝐜}$ 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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}$ with $\mathrm{\infty }\mathbf{𝐆}\mathbf{𝐫}\mathbf{𝐩}\mathbf{𝐝}$, 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 vector spaces
• Category of measurable spaces

References

• 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). "An initiation into convergence theory". in Mynard, Frédéric; Pearl, Elliott. Beyond Topology. Contemporary Mathematics. 486. pp. 115–162. doi:10.1090/conm/486/09509. ISBN 9780821842799. http://dolecki.perso.math.cnrs.fr/init_IX07.pdf. Retrieved 14 January 2021. 
• Dolecki, Szymon; Mynard, Frédéric (2014). "A unified theory of function spaces and hyperspaces: local properties". Houston J. Math. 40 (1): 285–318. http://dolecki.perso.math.cnrs.fr/18dolecki.pdf. 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.