By Dieudonne J.
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The previous lemmas give some feeling for the structure of O(G) and lead to the following alternative description. Lemma. The category O(G) is isomorphic to the category G whose objects are the subgroups of G and whose morphisms are the distinct subconjugacy relations γ −1 Hγ ⊂ K for γ ∈ G. If we regard G as a category with a single object, then a (left) action of G on a set S is the same thing as a covariant functor G −→ S . ) If B is a small groupoid, it is therefore natural to think of a covariant functor T : B −→ S as a generalization of a group action.
Again, while it suffices to think in terms of locally contractible spaces, appropriate generality demands a weaker hypothesis. We say that a space B is semi-locally simply connected if every point b ∈ B has a neighborhood U such that π1 (U, b) −→ π1 (B, b) is the trivial homomorphism. Theorem. If B is connected, locally path connected, and semi-locally simply connected, then B has a universal cover. Proof. Fix a basepoint b ∈ B. We turn the properties of paths that must hold in a universal cover into a construction.
Let wU be the category of weak Hausdorff spaces. We have the functor k : wU −→ U , and we have the forgetful functor j : U −→ wU , which embeds U as a full subcategory of wU . Clearly U (X, kY ) ∼ = wU (jX, Y ) for X ∈ U and Y ∈ wU since the identity map kY −→ Y is continuous and continuity of maps defined on compactly generated spaces is compactly determined. Thus k is right adjoint to j. We can construct colimits and limits of spaces by performing these constructions on sets: they inherit topologies that give them the universal properties of colimits and limits in the classical category of spaces.