Operads, clones, and distributive laws
Pierre-Louis Curien (PPS, INRIA Paris - Rocquencourt)
TL;DR
This paper explores the categorical foundations of operads, clones, and distributive laws by analyzing their construction through monads on categories and profunctors, unifying various approaches.
Contribution
It introduces a unified framework connecting non-symmetric operads, symmetric operads, and clones via monads on Cat and profunctors, clarifying their relationships.
Findings
Operads and clones arise from specific monads on Cat.
The profunctor approach unifies previous categorical analyses.
Connections between different operad theories are clarified.
Abstract
We show how non-symmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo-)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day's tensor products, or via analytical functors) fit with the profunctor approach.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models