Advanced oil well drilling engineering: Handbook and by Bill J Mitchell

By Bill J Mitchell

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Bn A1 A2 . . An fn f1 f2 B1 ... 7: Arrow in B A consists of α ∈ βn and fi ∈ HomA (Ai , Bπ(α)(i) ), where π : βn Σn is the natural projection. 1). The tensor product is defined on objects and morphisms by block sum. Bm = (bn,m , idA1 , . . , idAn , idB1 , . . , idBm ). 6 with identity morphisms on the strings. Let (A, , c) be a braided strict monoidal category. By a result of Joyal and Street (in [15]), there exists a unique strict monoidal functor T : B A −→ A such that the following triangle commutes and T (c ) = c, for c the braiding of B A : i GB hh hh h id hh 4 A hh A 1 1 T 1  A.

3. Let P be a ribbon operad. A P-algebra structure on a space X is a morphism of ribbon operads φ : P −→ EndX . As in the symmetric case, the last definition is equivalent to requesting that there exist maps θj : P(j) × X j −→ X satisfying unit and associativity conditions (exactly the same as before) and such that θj (cr , x1 , . . , xj ) = θj (c, xπ(r)−1 (1) , . . , xπ(r)−1 (k) ), for all r ∈ Rβj . 4. The ribbon braid groups give rise to the following ribbon operad : Rβ = {Rβj }j∈N , with the action of Rβj given by right multiplication and with the operad maps defined by γ(r, s1 , .

May also showed that, conversely, Bn applied to an n-fold loop space Ωn Y provides an equivalent delooping (Bn Ωn Y Y ). n In what follows, we consider the behaviour of Ω and Bn with respect to G-actions and provide a recognition principle for algebras over Dn Let Dn connected Dn G −Topgl , Dn G. G −Top0 and G −Top∗n be the categories of grouplike, G-algebras and n-connected pointed G-spaces respectively. Those three categories are closed model categories with weak homotopy equivalences as weak equivalences [31].

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