A version of Graeme Segal’s perspective on classifying spaces A group G acting on a set X determines a category GX with object set X and GX(x, y) = {g : gx = y}. For example G∗ is the category whose single object has endomorphism monoid G. Its classifying space is denoted BG. The action of G on itself by left translation provides a category GG that is “unicursal” – it’s nonempty and has exactly one morphism between any two objects – so its classifying space, denoted EG, is contractible. The group G acts from the left on GG and hence on EG. The map EG ↓ BG induced by the map of G-sets G → ∗ is a principal G-bundle. These constructions can also be made internally to the category Top. EG is still contractible, and (at least if G is a Lie group) EG ↓ BG continues to be a principle G-bundle. ˇ A map π : Y → X deﬁnes the “descent category” C(π) with objects Y , morphisms ˇ Y ×X Y , and structure morphisms the evident maps. There’s a natural functor : C(π) → cX, where cX denotes the “constant category” with objects and morphisms both given by X. The nerve N Č(π) is the “Čech simplicial object” associated to π. It is equipped with a simplicial map to N cX, which is the constant simplicial object with value X. In particular let U be a cover of the space X, and let a Y = U U ∈U with its evident projection to X. Write Č(U) in this case, and write xU,V for the morphism determined by x ∈ U ∩ V . An n-simplex in N Č(U) is then a sequence of n + 1 elements of U together with a point in their intersection. Suppose we are given a principal G-bundle p : P ↓ X, and a trivializing open cover U equipped with trivializations (equivariant ﬁberwise isomorphisms) tU : p−1 U → U × G. These data determine a functor θ : Č(U) → G∗ . To decide where to send xU,V , look at t−1 t U V (U ∩ V ) × G −→ p−1 (U ∩ V ) −→(U ∩ V ) × G. This equivariant map is determined by where (y, 1) is sent, for each y ∈ U ∩ V ; write γU,V : U ∩ V → G for this map. Then the functor is deﬁned by sending xU,V 7→ γU,V (x) . Passing to classifying spaces, we have BČ(U) θ / BG X Proposition. (1) θ∗ EG ∼ = ∗ P . (2) If P is numerable, is a homotopy equivalence.

A Version of Graeme Segal’s Perspective on Classifying Spaces

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