Geodesic
Moment categories and operads
Theory and applications of categories, Tome 38 (2022), pp. 1485-1537

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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.

Publié le :
Classification : 18A32, 18M60, 18M85, 18N70
Keywords: Moment category, Operad, Active/inert factorisation system, Strict Segal condition, Plus construction, Monadicity 
@article{TAC_2022_38_a38,
     author = {Clemens Berger},
     title = {Moment categories and operads},
     journal = {Theory and applications of categories},
     pages = {1485--1537},
     publisher = {mathdoc},
     volume = {38},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2022_38_a38/}
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Clemens Berger. Moment categories and operads. Theory and applications of categories, Tome 38 (2022), pp. 1485-1537. http://geodesic.mathdoc.fr/item/TAC_2022_38_a38/