Neurology Mathematics Ideas78.pdf - The relevant silent objects are obtained via a remarkable interplay between algebraic structures in finite tensor

Neurology Mathematics Ideas78.pdf - The relevant silent...

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The relevant silent objects are obtained via a remarkable interplay between algebraic struc- tures in finite tensor categories and the geometry of framings. The gluing categories for tadpole circles are canonically equivalent to (twisted) centers: T( Q A + ) = I 0 2 = Z ( A ) and T( Q A ) = I 0 2 = Z 4 ( A ) (5.37) (recall Definition 3.6 of the twisted center Z κ ). Thus in particular they contain canonical objects, namely the monoidal unit in Z ( A ) and the distinguished invertible object (see page 32) in Z 4 ( A ), respectively. We define the respective silent objects to be these two specific objects: ( Q A + ) := 1 ∈ Z ( A ) = T( Q A + ) and ( Q A ) := D A Z 4 ( A ) = T( Q A ) . (5.38) In the second step we consider an arbitrary fillable gluing circle or interval L . There exists (uniquely up to isomorphism) a defect surface D tad L which is a fillable disk (in the sense of Definition 5.10) such that L is its outer boundary and each of its inner boundary circles is a tadpole circle Q i
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