topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
The following is the classical discussion of universal covering spaces in point-set topology.
(essential uniqueness of simply connected covering spaces)
Let be a topological space which is
Then if are two covering spaces over , , which are both path-connected and simply connected, then they are isomorphic as covering spaces.
Since both and are simply connected, the assumption of the lifting theorem for covering spaces is satisfied (this prop.). This says that there are horizontal continuous function making the following diagrams commute:
and that these are unique once we specify the image of a single point, which we may freely do (in the given fiber).
So if we pick any point and with and with and specify that and then uniqueness applied to the composites implies and .
(universal covering space)
Let be a topological space which is
Then a path-connected and simply connected covering space, is called the universal covering space of . This is well-defined, if it exists, up to isomorphism, by prop. .
(universal covering space reconstructed from free and transitive fundamental group representation)
Let be a topological space which is well-connected in that it is
By this prop. the covering space is connected and simply connected precisely if its monodromy representation is free and transitive. By the fundamental theorem of covering spaces every permutation representation of the fundamental groupoid arises as the monodromy of some covering space. Hence it remains to see that a free and transitive representation of exists. Let be any point, then is such a representation.
This is a functorial construction of a universal covering spaces
where denotes the full subcategory of pointed topological spaces on the well-connected spaces and is the subcategory of of pointed maps of spaces with objects the covering space maps.
Specifically, if is a space with basepoint , we define to be the space whose points are homotopy classes of paths in starting at , with the projection projecting to the endpoint of a path. We can equip this set with a topology coming from so that it becomes a universal covering space as above. As described at covering space, under the correspondence between covering spaces and -actions, the space corresponds to the “regular representation” of .
This regular representation can be seen to arise by taking a category of elements in the same way that the regular representation of a group is gotten by taking its action on itself: we can see that the universal covering groupoid in the slice category (see the universal covering -groupoid below) is just the category of elements of the action of on itself, and can be topologized in a natural way by lifting the topology on along the canonical projection ; decategorifying this yields .
We describe now how the universal cover construction may be understood from the nPOV, using a fragment of cohesive homotopy theory.
The basic idea is that the universal cover of a space is the homotopy fiber of the canonical morphism
from to its fundamental groupoid, which exists if both objects are regarded as “topological ∞-groupoids”. We may think of this as a precise way of the intuitive idea of “forcing to become trivial in a universal way”.
(cohesive higher toposes of topological groupoids)
Let be some ∞-cohesive site of spaces and consider
the corresponding cohesive (∞,1)-topos over this site. This is the (∞,1)-category of “topological ∞-groupoids” modeled on .
A canonical choice relating to the traditional discussion would be a small full subcategory of Top on locally contractible topological spaces, in which case the objects of might be called “locally contractible topological ∞-groupoids”.
A more restrictive choice would be CartSp, in which case the objects of might be called Euclidean-topological ∞-groupoids.
In fact for the discussion of just universal covering spaces as opposed to the higher stages in the Whitehead tower it would be sufficient and more natural to take the full subcategory of locally simply connected topological spaces and consider just the (2,1)-category of stacks over this site.
But the following discussion is completely formal and applies globally to all such realizations. Namely all we need is that
Assume that
is a cohesive (n,1)-topos
over the given base (n,1)-topos of n-groupoids.
such that its shape modality preserves homotopy fiber products over discrete objects (objects in the essential image of ).
We write
for the induced shape modality and
for its unit morphism ona given object .
Assumption is satisfied for the case of -toposes over an ∞-cohesive site and for -toposes by an -cohesive site as in example : by this prop..
Now consider any object, for instance a topological space regarded as a 0-truncated topological infinity-groupoid.
Assume, just for ease of discussion, that is geometrically connected in that the 0-truncation of its shape is contractible:
Using the axiom of choice in the base topos we choose a point
The universal cover of is the homotopy pullback in
The universal property of is immediate from the abstract setup:
is simply connected, in that
This is because is discrete by construction, and hence so is . So by assumption applying to the above square yields another homotopy pullback of the form
Now the long exact sequence of homotopy groups (in ) applied to this homotopy fiber sequence is of the form
which implies that and hence is indeed trivial.
Let be any other object of such that , then there is a morphism
This is because the naturality of the shape unit and of the truncation unit gives a homotopy commuting square of the form
and thus a cone over the diagram which defines via its universal property.
This shows that is the universal cover on abstract grounds.
We may also check explicitly that is given as the space of homotopy classes of paths in from the given basepoint. To that end we use that, at least over the site CartSp, the shape of is represented by the topological path ∞-groupoid. See at shape via cohesive path ∞-groupoid.
The following is old material that deserves to be harmonized a bit more with the above stuff.
Let be a suitably well behaved pointed space. The universal cover of is (equivalent to) the homotopy fiber of in the (∞,1)-category of topological ∞-groupoids.
In other words, the principal ∞-bundle classified by the cocycle is the universal cover : we have a homotopy pullback square
Urs Schreiber: may need polishing.
We place ourselves in the context of topological ∞-groupoids and regard both the space as well as its fundamental ∞-groupoid and its truncation to the fundamental groupoid as objects in there.
The canonical morphism hence given by the inclusion of constant paths may be regarded as a cocycle for a -principal ∞-bundle, respectively for a -principal bundle.
Let be the set of connected components of , regarded as a topological -groupoid, and choose any section of the projection .
Then the -principal -bundle classified by the cocycle is its homotopy fiber, i.e. the homotopy pullback
We think of this topological -groupoid as the universal covering -groupoid of . To break this down, we check that its decategorification gives the ordinary universal covering space:
for this we compute the homotopy pullback
where we assume to be connected with chosen baspoint just to shorten the exposition a little. By the laws of homotopy pullbacks in general and homotopy fibers in particular, we may compute this as the ordinary pullback of a weakly equivalent diagram, where the point is resolved to the universal -principal bundle
(More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the global model structure on simplicial presheaves on Top. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in -prestacks. Then by left exactness of -stackification, the image of that in -stacks is still a homotopy pullback. )
The topological groupoid has as objects homotopy classes rel endpoints of paths in starting at and as morphisms it has commuting triangles
in . The topology on this can be deduced from thinking of this as the pullback
in simplicial presheaves on Top. Unwinding what this means we find that the open sets in are those where the endpoint evaluation produces an open set in .
Now it is immediate to read off the homotopy pullback as the ordinary pullback
Since is categorically discrete, this simply produces the space of objects of over the points of , which is just the space of all paths in starting at with the projection being endpoint evaluation.
This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in
On the other hand, we can view a space as the little -topos of -sheaves on . If is locally connected and locally simply connected in the “coverings” sense, then is locally 1-connected.
In fact, for the construction of the universal cover we require only the (2,1)-topos of sheaves (stacks) of groupoids on , so we will work in that context because it is simpler. The construction can be adapted, however, to produce a “universal cover” of any locally 1-connected -topos.
Let be any (2,1)-topos which is locally 1-connected. This means that in the unique global sections geometric morphism , the functor has a left adjoint , which is automatically -indexed. The fundamental groupoid of is defined to be , where is the terminal object of .
As discussed here in the -case, the construction of is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends ). Thus we have a geometric morphism (where we regard as the (2,1)-topos ).
Suppose, for simplicity, that is connected. Then is also connected, and so we have an essentially unique functor . We define the universal cover of to be the pullback (2,1)-topos:
(where of course denotes the terminal (2,1)-topos ).
Now observe that is a local homeomorphism of toposes, since we have . Since local homeomorphisms of toposes are stable under pullback, is also a local homeomorphism, i.e. there exists an object and an equivalence over . Moreover, it is not hard to see that can be identified with the pullback
in . Note that the bottom map is applied to the unique map , while the right-hand map is the unit of the adjunction .
In order to see that this is a sensible definition, we first observe that is itself locally 1-connected (since it is etale over ). Moreover, it is actually 1-connected, which is equivalent to saying that . This is because the “Frobenius reciprocity” condition for the adjunction (which is equivalent to saying that is -indexed) applied to the defining pullback of implies that we also have a pullback
which clearly implies that .
Thus, is a connected and simply connected space with a local homeomorphism to , but is it a covering space? In other words, is it locally trivial? Since we have supposed that is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects such that . In particular, we have a 1-connected object and a regular 1-epic? .
We claim that if is any 1-connected object of , then is trivialized (or split) over , in that is equivalent, over , to for some . For pulling back the defining pullback to , we obtain
But , so to give a map over is the same as to give a map in . But , since is 1-connected, and is connected, so there is only one such morphism. Therefore, the two maps in the pullback above are in fact the same, and in particular both are the pullback to of the map . Thus, is equivalent to , where is the loop object of , i.e. what we might call the fundamental group of the connected (2,1)-topos .
Therefore, since is trivialized over any 1-connected object, and is generated by 1-connected objects, is locally trivial. Moreover, since is a discrete object of , so is . Thus, if we specialize all this to the case of (2,1)-sheaves on a topological space, then we conclude that is an honest 1-sheaf on which, when regarded as a local homeomorphism over , is locally trivial (hence a covering space), connected, and 1-connected—i.e. a universal cover of .
(real line is universal covering of circle)
Let
be the real line with its Euclidean metric topology;
be the circle with its subspace topology induced from the Euclidean plane.
Consider the continuous function
This exhibits the universal covering space (def. ) of the circle.
Let be any point. It is clear that we have a homeomorphism of the form
and hence a homeomorphism of the form
Now for two distinct point in , their complements constitute an open cover
and so this exhibits as being covering spaces.
Now
is path-connected and locally path connected (this example);
is simply connected (this example).
Therefore exhibits as a universal covering space of , by def. .
See also the references at covering space.
A detailed treatment is available in
Original discussion of universal covering spaces in terms of equivalence classes of based paths:
Formulation in cohesive homotopy theory
and explicitly so in cohesive homotopy type theory:
Discussion of equivariant universal covers in the generality of equivariant homotopy theory:
Last revised on November 18, 2023 at 05:24:17. See the history of this page for a list of all contributions to it.