Moment categories and operads

Clemens Berger

A moment category is endowed with a distinguished set of split idempotents, called moments, which can be transported along morphisms. Equivalently, a moment category is a category with an active/inert factorisation system fulfilling two simple axioms. These axioms imply that the moments of a fixed object form a monoid, actually a left regular band. Each locally finite unital moment category defines a specific type of operad which records the combinatorics of partitioning moments into elementary ones. In this way the notions of symmetric, non-symmetric and n-operad correspond to unital moment structures on Γ, Δ and Θ_n respectively. There is an analog of the plus construction of Baez-Dolan taking a unital moment category C to a unital hypermoment category C^+. Under this construction, C-operads get identified with C^+-monoids, i.e. presheaves on C^+ satisfying strict Segal conditions. We show that the plus construction of Segal's category Γ embeds into the dendroidal category Ω of Moerdijk-Weiss.

Keywords: Moment category; Operad; Active/inert factorisation system; Strict Segal condition; Plus construction; Monadicity

2020 MSC: 18A32, 18M60; 18M85, 18N70

Theory and Applications of Categories, Vol. 38, 2022, No. 39, pp 1485-1537.

Published 2022-12-14.

TAC Home