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**Example text**

6). 9) f ∗ Y g∗ Z ∼ = (f × g)∗ (Y Z). 10) f! W g! X ∼ = (f × g)! (W X). Now take A = A , B = B and f = g. For ex-spaces Y and Z over B, f ∗ (Y ∧B Z) ∼ = f ∗ ∆∗B (Y Z) ∼ = (∆B ◦ f )∗ (Y Z). 9), ∼ ∆∗ (f × f )∗ (Y f ∗ Y ∧A f ∗ Z = A Z) ∼ = ((f × f ) ◦ ∆A )∗ (Y Z). The right sides are isomorphic since ∆B ◦ f = (f × f ) ◦ ∆A . Similarly, ∼ f! ∆∗ (f × id)∗ (Y X) ∼ f! (f ∗ Y ∧A X) = = f! ((f × id) ◦ ∆A )∗ (Y X), A while ∼ ∆∗ (id × f )! (Y X). Y ∧B f! 11. It is illuminating conceptually to go further and consider group actions from an external point of view.

The fiber Xb is a based Gb -space with Gb -fixed basepoint s(b), where Gb is the isotropy group of b. Recall from [105, II§1] the distinction between the category KG of G-spaces and nonequivariant maps and the category GK of G-spaces and equivariant maps; the former is enriched over GK , the latter over K . We have a similar dichotomy on the ex-space level. Here we have a conflict of notation with our notation for categories of ex-spaces, and we agree to let KG,B denote the category whose objects are the ex-G-spaces over B and whose morphisms are the maps of underlying ex-spaces over B, that is, the maps f : X −→ Y such that f ◦ s = t and q ◦ f = p.

1. The category GKG/H of ex-G-spaces over G/H is equivalent to the category HK∗ of based H-spaces. Proof. The equivalence sends an ex-G-space (Y, p, s) over G/H to the Hspace p−1 (eH) with basepoint the H-fixed point s(eH). Its inverse sends a based H-space X to the induced G-space G ×H X, with the evident structure maps. More formally, recall that there are “induction” and “coinduction” functors ι! and ι∗ from H-spaces to G-spaces that are left and right adjoint to the forgetful functor ι∗ that sends a G-space Y to Y regarded as an H-space.