By Vaisman

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2 with p(y) = (xl, x2) and re(y) = x. Let cL~'L~(ul, u2) : m *Sign'l+~2(L1 | L2) ~ + p*(Sign'~(L1) [] $ign'2( L2) ) be the unique isomorphism equal to IdLi| associativity condition (cf. (c)' above). over y. These isomorphisms satisfy the Let us define the brading local system Sign by Sign~ = S i g n ' ( i ) , r u2). Here 1 denotes the standard vector space k. 10. E x a m p l e . Standard local systems J , Z. Let /2 C Y+, let 7r : J ---+ I be an unfolding of u, n = card(J), rr: ~)J~ - - + ~),o the corresponding projection.

For each ~ E Y+, set r 9c8. 4 | = (M | y)" = %-+o((M []H)"). Here g/z_~0 : AJ(D"(2)) > Ad(D ~) denotes the functor of nearby cycles for the function D"(2) ----+ D sending (z; (tj)) to z. 2" The factorization isomorphisms r induce the factorization isomorphisms between the sheaves (3,4 | H ) " . This defines a factorizable sheaf Ad | iV'. One sees at once t h a t this construction is functorial; thus it defines a functor of tensor product | : 5-8 x ~ 8 ~ 5-8. The subcategory ) r 8 C he8 is stable under the tensor product.

6. 2. For ~ E Y+, let us consider the corresponding (relative over A 1) configuration scheme f " : Q~A~ -----+ AI" For the brevity we will omit the subscript /A~ indicating that we are dealing with the relative version of configuration spaces. We denote by Q"" (resp. Q,O) the subspace of configurations with the points distinct from z l , z 2 (resp. also pairwise distinct). We set ' Q ' ~ = Q~~ etc. 1). For L, E Y+ and #1, #2 E Xe such that #1 + #2 - L, = 2pc, let J21,~2 denote the local system over 'Q"~ which is the inverse image of the local system ~;~,,~ over P"(2) ~ Let J2~',,~ denote the perverse sheaf over 'Q~'(C) which is the middle extension of J2~,~[dim Q"].