In an earlier work, Batanin has shown that an important class of definitions of higher categories could be apprehended together simply as monads over globular sets. This allowed him to generalize the notion of polygraph, initially introduced by Street and Burroni for strict categories, to all algebraic globular higher categories. In this work, we refine this perspective and introduce new constructions and properties for this class of higher categories. In particular, we define the notion of cellular extension and its associated free construction, from which we obtain another definition of polygraphs and the adjunction between globular algebras and polygraphs. We moreover introduce two criteria allowing one to use most of the constructions of this article without having to describe explicitly the underlying globular monad.
Keywords: globular algebras, truncable monads, string diagrams, Eilenberg--Moore construction, formal theory of monads
2020 MSC: 18C15,18C20,18N30
Theory and Applications of Categories, Vol. 45, 2026, No. 44, pp 1810-1903.
Published 2026-08-10.
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