A unified framework for generalized multicategories

By Geoff S. H. Cruttwell, Michael Shulman. Theory and Applications of Categories, 2009

95 citations recorded by the source corpus.

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the "lax algebras" or "Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7f2e753c2e4da9334dcd6ee2adae7badc4136b8f

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Citations in the corpus

Double Categories of Relations (2021, unknown): “Conventions and notation for the most part are adopted from those in [Cruttwell & Shulman, 2010] and [Schultz, 2015].”
∞-Categories for the Working Mathematician (2018, background): “6, which asserts that ∞-categories, functors, modules, and module maps in any ∞-cosmos define a virtual equipment, in the sense of Cruttwell and Shulman [15].”
The higher Morita category of n–algebras (2014, background): “4], generalized non-symmetric ∞-operads are an∞-categorical analogue of the fc-multicategoriesof Leinster [28] (also called virtual double categoriesin [12]), which are a common generalization of double categories and multicategories.”
Monoidal topology : a categorical approach to order, metric, and topology (2014, background): “The double-categorical presentation of lax extensions presented in 1.13 appears in [Cruttwell and Shulman, 2010] following a suggestion by Paré.”