By J. P. May

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**Example text**

It is universal if p(π(E , e)) = {e}. Clearly a covering space is regular or universal if and only if its associated covering of fundamental groupoids is regular or universal. A covering of groupoids is universal if and only if π(B, b) acts freely on Fb , and then Fb is isomorphic to π(B, b) as a π(B, b)-set. Specializing to covering spaces, this sharpens our earlier claim that the elements of Fb and π1 (B, b) are in bijective correspondence. 5. The classification of coverings of groupoids Fix a small connected groupoid B throughout this section and the next.

Show that the space of G-maps G/H −→ X is naturally homeomorphic to X H . In particular, O(G/H, G/K) ∼ = (G/K)H . 8. THE CONSTRUCTION OF COVERINGS OF SPACES 33 6. Let X be a G-space. Show that passage to fixed point spaces, G/H −→ X H , is the object function of a contravariant functor X (−) : O(G) −→ U . CHAPTER 4 Graphs We define graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space. 1. The definition of graphs We give the definition in a form that will later make it clear that a graph is exactly a one-dimensional CW complex.

For (ii), there is a map g : e −→ e′ since E is connected. Conjugation by g gives a homomorphism π(E , e) −→ π(E , e′ ) that maps under p to conjugation of π(B, b) by its element p(g). For (iii), the surjectivity of p on St(e) gives that any f ∈ π(B, b) is of the form p(g) for some g ∈ St(e). If e′ is the target of g, then p(π(E , e′ )) is the conjugate of p(π(E , e)) by f . The fibers Fb of a covering of groupoids are related by translation functions. Definition. Let p : E −→ B be a covering of groupoids.