Neurology Mathematics Ideas66.pdf - Together with formula(3.14 this implies that for any \u03ba \u2208 2Z we have \u03ba \u2212\u03ba 0 0 0 0 0 M \u22a0 I0 \u22a0 N \u2243 M \u22a0

Neurology Mathematics Ideas66.pdf - Together with...

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Together with formula (3.14), this implies that for any κ 2 Z we have M κ I 0 κ N ≃ M 0 κ I κ 0 0 N ≃ M 0 I 0 0 N ≃ M 0 N . (5.13) Example 5.7. The equivalences (5.11) lead to distinguished objects in certain gluing categories: Let A be a finite tensor category and M an A -bimodule. Via the Eilenberg-Watts equivalences (3.39) the gluing categories for the defect one-manifolds I ր κ ( M ) := κ M I κ M I ւ κ ( M ) := κ M M I κ I տ κ ( M ) := κ M M I κ I ց κ ( M ) := κ M I κ M (5.14) are canonically equivalent to the category L ex A , A ( M , M ) of bimodule endofunctors: we have T( I ր κ ( M )) = M 1 M κ I κ 1 (3.54) ≃ L ex A , A ( M , M κ I κ ) (5.11) ≃ L ex A , A ( M , M ) and T( I ւ κ ( M )) = M κ I κ 1 M 1 (3.8) I κ κ M 1 M 1 M 1 M 1 (3.54) ≃ L ex A , A ( M , M ) , (5.15) and similarly for the other two gluing circles. The so obtained endofunctor categories have the identity functor as a distinguished object. Remark 5.8. The pre-images of the identity functor under the equivalences (5.15) constitute distinguished objects in the respective gluing categories. Using the precise form of the equiva-
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