Category of topological spaces

Template:Short description 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.

Some authors use the name $𝐓𝐨𝐩$ 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 $𝐓𝐨𝐩$ 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

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

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

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 $(X\to U{A}_i{)}_I$ has a unique initial lift $(A\to A_i{)}_I$. 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 $U:𝐓𝐨𝐩\to 𝐒𝐞𝐭$ 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 $(L,\phi :L\to F)$ is a limit of $UF$ in $𝐒𝐞𝐭$, the corresponding limit of $F$ in $𝐓𝐨𝐩$ is obtained by placing the initial topology on $(L,\phi :L\to F)$. Dually, colimits in $𝐓𝐨𝐩$ are obtained by placing the final topology on the corresponding colimits in $𝐒𝐞𝐭$.

Unlike many algebraic categories, the forgetful functor $U:𝐓𝐨𝐩\to 𝐒𝐞𝐭$ 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 $𝐂𝐆𝐇𝐚𝐮𝐬$ or the category of compactly generated weak Hausdorff spaces. However, $𝐓𝐨𝐩$ 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 $𝐓𝐨𝐩$_• 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.
• The homotopy hypothesis relates $𝐓𝐨𝐩$ with $\infty 𝐆𝐫𝐩𝐝$, the category of ∞-groupoids. The conjecture states that ∞-groupoids are equivalent to topological spaces modulo weak homotopy equivalence.

See also

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• Category of measurable spaces
• Categorical topology

References

• Adámek, Jiří, Herrlich, Horst, & Strecker, George E.; (1990). Abstract and Concrete Categories Template:Webarchive (4.2MB PDF). Originally publ. John Wiley & Sons. Template:ISBN. (now free on-line edition).
• Template:Dolecki Mynard Convergence Foundations Of Topology
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• 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.