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 : Top → Set
to the
category of sets which assigns to each topological space the underlying set and to each continuous map the underlying
function.
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 the model of what is called a
topological category. These categories are characterized by the fact that every
structured source has a unique
initial lift. 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, φ : 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, φ : 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.
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.
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
denseimages 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
localesLoc 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.
Category of groups – category in mathematicsPages displaying wikidata descriptions as a fallback
Category of metric spaces – mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphismsPages displaying wikidata descriptions as a fallback
Category of sets – Category in mathematics where the objects are sets
Herrlich, Horst: Categorical topology 1971–1981. In: General Topology and its Relations to Modern Analysis and Algebra 5, Heldermann Verlag 1983, pp. 279–383.