Category of topological spaces

In mathematics, the category of topological spaces, often denoted ${\displaystyle \mathbf {Top} }$, 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. The study of ${\displaystyle \mathbf {Top} }$ and of properties of topological spaces using the techniques of category theory is known as categorical topology.

N.B. Some authors use the name ${\displaystyle \mathbf {Top} }$ 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.

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

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

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 ${\displaystyle U}$ has both a left adjoint

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

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

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

which equips a given set with the indiscrete topology. Both of these functors are, in fact, right inverses to ${\displaystyle U}$ (meaning that ${\displaystyle UD}$ and ${\displaystyle UI}$ are equal to the identity functor on ${\displaystyle \mathbf {Set} }$). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of ${\displaystyle \mathbf {Set} }$ into ${\displaystyle \mathbf {Top} }$.

${\displaystyle \mathbf {Top} }$ is also fiber-complete meaning that the category of all topologies on a given set ${\displaystyle X}$ (called the fiber of ${\displaystyle U}$ above ${\displaystyle X}$) forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on ${\displaystyle X}$, while the least element is the indiscrete topology.

${\displaystyle \mathbf {Top} }$ 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 ${\displaystyle \mathbf {Top} }$ the initial lift is obtained by placing the initial topology on the source. Topological categories have many properties in common with ${\displaystyle \mathbf {Top} }$ (such as fiber-completeness, discrete and indiscrete functors, and unique lifting of limits).

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

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

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

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

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

• The monomorphisms in ${\displaystyle \mathbf {Top} }$ 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 ${\displaystyle \mathbf {Top} }$ 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 ${\displaystyle \mathbf {Top} }$, and in particular the category is not preadditive.
• ${\displaystyle \mathbf {Top} }$ 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 ${\displaystyle \mathbf {CGHaus} }$ or the category of compactly generated weak Hausdorff spaces. However, ${\displaystyle \mathbf {Top} }$ is contained in the exponential category of pseudotopologies, which is itself a subcategory of the (also exponential) category of convergence spaces.^[1]

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

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