By J. P. May
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Additional info for A Concise Course in Algebraic Topology
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.