Editing Category of topological spaces (section)

==As a concrete category==

Like many categories, the category '''Top''' is a [[concrete category]], meaning its objects are [[Set (mathematics)|sets]] with additional structure (i.e. topologies) and its morphisms are [[function (mathematics)|function]]s preserving this structure. There is a natural [[forgetful functor]]

{{block indent|''U'' : '''Top''' &rarr; '''Set'''}}

to the [[category of sets]] which assigns to each topological space the underlying set and to each continuous map the underlying [[function (mathematics)|function]].

The forgetful functor ''U'' has both a [[left adjoint]]

{{block indent|''D'' : '''Set''' &rarr; '''Top'''}}

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

{{block indent|''I'' : '''Set''' &rarr; '''Top'''}}

which equips a given set with the [[indiscrete topology]]. Both of these functors are, in fact, [[Inverse function#Left and right inverses|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 embedding]]s of '''Set''' into '''Top'''.

'''Top''' is also ''fiber-complete'' meaning that the [[lattice of topologies|category of all topologies]] on a given set ''X'' (called the ''[[fiber (mathematics)|fiber]]'' of ''U'' above ''X'') forms a [[complete lattice]] when ordered by [[set inclusion|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]] <math>(X \to UA_i)_I</math> has a unique [[initial lift]] <math>( A \to A_i)_I</math>. 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).