Top denotes the category whose objects are topological spaces and whose morphisms are continuous functions between them. Its isomorphisms are the homeomorphisms.
Often one considers (sometimes by default) subcategories of nice topological spaces such as compactly generated topological spaces, notably because these are cartesian closed. There other other convenient categories of topological spaces. With any one such choice understood, it is often useful to regard it as “the” category of topological spaces.
The homotopy category of given by its localization at the weak homotopy equivalences is the classical homotopy category Ho(Top). This is the central object of study in homotopy theory, see also at classical model structure on topological spaces. The simplicial localization of Top at the weak homotopy equivalences is the archetypical (∞,1)-category, equivalent to ∞Grpd (see at homotopy hypothesis).
We discuss universal constructions in Top, such as limits/colimits, etc. The following definition suggests that universal constructions be seen in the context of as a topological concrete category (see Proposition 4 below):
Let be a class of topological spaces, and let be a bare set. Then
For a set of functions out of , the initial topology is the topology on with the minimum collection of open subsets such that all are continuous.
For a set of functions into , the final topology is the topology on with the maximum collection of open subsets such that all are continuous.
For a single topological space, and a subset of its underlying set, then the initial topology , def. 1, is the subspace topology, making
a topological subspace inclusion.
Conversely, for an epimorphism, then the final topology on is the quotient topology.
Let be a small category and let be an -diagram in Top (a functor from to ), with components denoted , where and a topology on . Then:
The limit of exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. 1, for the functions which are the limiting cone components:
The colimit of exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology, def. 1 for the component maps of the colimiting cocone
(e.g. Bourbaki 71, section I.4)
The required universal property of is immediate: for
any cone over the diagram, then by construction there is a unique function of underlying sets making the required diagrams commute, and so all that is required is that this unique function is always continuous. But this is precisely what the initial topology ensures.
The case of the colimit is formally dual.
The limit over the empty diagram in is the point with its unique topology.
For a set of topological spaces, their coproduct is their disjoint union.
The equalizer of two continuous functions in is the equalizer of the underlying functions of sets
(hence the largets subset of on which both functions coincide) and equipped with the subspace topology, example 1.
The coequalizer of two continuous functions in is the coequalizer of the underlying functions of sets
(hence the quotient set by the equivalence relation generated by for all ) and equipped with the quotient topology, example 2.
two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted
(Here is also called the pushout of , or the cobase change of along .) If is an inclusion, one also write and calls this the attaching space.
By example 8 the pushout/attaching space is the quotient topological space
of the disjoint union of and subject to the equivalence relation which identifies a point in with a point in if they have the same pre-image in .
(graphics from Aguilar-Gitler-Prieto 02)
As an important special case of example 9, let
be the canonical inclusion of the standard (n-1)-sphere as the boundary of the standard n-disk (both regarded as topological spaces with their subspace topology as subspaces of the Cartesian space ).
Then the colimit in Top under the diagram, i.e. the pushout of along itself,
is the n-sphere :
(graphics from Ueno-Shiga-Morita 95)
for the forgetful functor that sends a topological space to its underlying set and which regards a continuous function as a plain function on the underlying sets.
Prop. 1 means in particular that:
(But it does not create or reflect them.)
Mono-/Epimorphisms, Quotients and Intersections
For proof see there.
- Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
An axiomatic desciption of along the lines of ETCS for Set is discussed in
- Dana Schlomiuk, An elementary theory of the category of topological space, Transactions of the AMS, volume 149 (1970)