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Abstract: Given an algebraic theory which can be described by a possibly symmetricoperad $P$, we propose a definition of the \emph{weakening} or\emph{categorification} of the theory, in which equations that hold strictlyfor $P$-algebras hold only up to coherent isomorphism. This generalizes thetheories of monoidal categories and symmetric monoidal categories, and severalrelated notions defined in the literature. Using this definition, we generalizethe result that every monoidal category is monoidally equivalent to a strictmonoidal category, and show that the -strictification- functor has aninteresting universal property, being left adjoint to the forgetful functorfrom the category of strict $P$-categories to the category of weak$P$-categories. We further show that the categorification obtained isindependent of our choice of presentation for $P$, and extend some of ourresults to many-sorted theories, using multicategories.

Autor: M. R. Gould



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