Symmetric monoidal closed objects

Posted in Uncategorized by Mike Stay on 2012 April 12

I conjecture that there’s a compact closed bicategory Th(SMCC) such that the 2-category hom(Th(SMCC), Prof) of

  • sylleptic monoidal functors (of bicategories),
  • braided monoidal transformations and
  • monoidal modifications

is equivalent as a 2-category to the 2-category SMCC of

  • symmetric monoidal closed categories,
  • braided monoidal closed functors, and
  • monoidal closed natural isomorphisms.

If we model Th(SMCC) in the compact closed bicategory 3Cob2, then we get

where I didn’t draw

  • the right unitor
  • the pentagon equation for the associator
  • the triangle equations for the unitors
  • the hexagon equations for the braiding
  • the yanking for internal hom
  • a,b,c,l,r composed with their formal inverses equals the identity

but I think they’re pretty obvious given this other stuff.


3 Responses

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  1. David said, on 2012 April 12 at 5:09 pm

    The last six really are quite trivially solved.

  2. Mike Stay said, on 2012 May 8 at 8:34 am

    I got the adjunctions backwards. They should map from the cap+cup to the cylinder, from nothing to the sphere, from the pair of cylinders to the X, and from the O to the cylinder.

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