Algebraic theories by Wraith, Gavin

By Wraith, Gavin

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G. 4. , a natural factorization of the category CM×M that is associated to a Cartesian product of symplectic manifolds. However, our construction of the symplectic 2-category and the induced functor in Sect. 5 is such that the objects of CM×M are general Lagrangian submanifolds of M × M , not just split Lagrangians L × L ⊂ M × M arising from objects L ∈ ObjCM and L ∈ ObjCM in the categories associated to the factors of the Cartesian product. Homological algebra allows one to formulate a sense in which refined versions of these categories may be equivalent, CM×M ∼ CM ⊗ CM , but it would likely require significant restrictions on the geometry of the symplectic manifolds M, M .

1, functoriality then requires F ([Y ]) = Lι−Y , LY01 , LY12 , . . , LY(n−1)n , L(ι+Y )−1 to be given by the algebraic composition in Symp of the corresponding Lagrangian submanifolds. 1—but also allow for a diffeomorphism Ψ : Y → Z that intertwines ± boundary identifications, Ψ ◦ ι± Y = ιZ . The latter induces a Cerf decomposition Z = Ψ (Y01 ) ∪Ψ (Σ1 ) Ψ (Y12 ) ∪ . . ∪Ψ (Σn−1 ) Ψ (Y(n−1)n ) with Σi := Y(i−1)i ∩ Yi(i+1) ⊂ Y , whose value under F is F ([Z]) = = LΨ |∂ − Y ◦ι−Y , LΨ (Y01 ) , LΨ (Y12 ) , .

While step 1 fixes the functor F on all objects, steps 2 and 3 fix explicit Lagrangians F ([Y ]) = L Y only for simple morphisms Y as LZφ = Lφ for cylindrical cobordisms, LYα = Lα for 2-handle attachments, and LYα− = LαT for their adjoint 1-handle attachments. To determine the value of the functor F ([Y ]) = [L Y ] on a general cobordism Y ∈ Mor Bor2+1 (Σ, Σ ), we choose a Cerf decomposition Y = Y01 ∪Σ1 Y12 . . ∪Σk−1 Y(k−1)k into a composable chain of simple morphisms Yij ∈ Mor Bor2+1 (Σi , Σj ) from Σ0 = Σ to Σk = Σ .

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