Finitary monads on globular sets and notions of computad they generate

Michael Batanin – 21 May 1997

The pasting operation in the theory of strict n-categories is well understood, but the corresponding operation in weak categories is not. This is my attempt to initiate this investigation. I prove that every finitary monad A on the category of globular sets generates an appropriate category of n-computads (for every integer n). If we denote by A(n) the "n-skeleton" of A (this is a monad on the category of n-globular sets) then we have an forgetful functor from the category of A(n)-algebras to the category of n-computads; moreover, this functor has a left adjoint (the construction of the free A(n)-algebra generated by an n-computad is interesting by itself as it uses machinery applicable in many different situations). The counit of this adjunction may be considered as a generalized pasting operation for A(n)-algebras. As examples I consider the following monads on globular sets: The monad D_s whose algebras are omega-categories. It generates the ordinary notion of n-computad and the usual pasting operation in n-categories. The monad B on 2-globular sets the algebras of which are bicategories. The corresponding notion of 2-computad is considered. The pasting theorem in this case asserts that the result of pasting of a 2-dimensional diagram depends only on a bracketing of 1-dimensional skeleton of it (so this is an another proof of a Dominic Verity result). The monad G on 3-globular sets whose algebras are Gray-categories. The corresponding notion of 3-computad is Gray-computad that M.McIntyre and T.Trimble used in their calculus of progressive 3D-diagrams.