Algebras over infinity-operads

We develop a notion of an algebra over an infinity-operad with values in infinity-categories which is completely intrinsic to the formalism of dendroidal sets. Its definition involves the notion of a coCartesian fibration of dendroidal sets and extends Lurie’s definition of a coCartesian fibration of simplicial sets. We show how, for a dendroidal set X, the coCartesian fibrations over X fit together to form an infinity-category coCart(X). Using a generalization of the Grothendieck construction, we prove that coCart(X) is equivalent to the infinity-category of algebras in infinity-categories over the simplicial operad associated to X. This equivalence can be restricted to give an equivalence between algebras taking values in infinity-groupoids (or equivalently, spaces) and the infinity-category of so-called left fibrations over X.

💡 Research Summary

The paper develops a fully intrinsic definition of an algebra over an ∞‑operad with values in ∞‑categories, using only the language of dendroidal sets. The central construction is a notion of coCartesian fibration of dendroidal sets, which generalizes Lurie’s coCartesian fibrations of simplicial sets. For any dendroidal set X, the collection of coCartesian fibrations over X assembles into an ∞‑category denoted coCart(X).

The authors first recall the classical Grothendieck construction for categories and extend it to operads, defining a “Grothendieck construction for operads” that sends a weak algebra over an operad to a cofibred operad. They then introduce a left adjoint to this construction and show that, after suitable restriction of the codomain, the construction yields an equivalence.

In the second part they study left fibrations and topological algebras (algebras valued in ∞‑groupoids). They prove that left fibrations over a dendroidal set correspond precisely to algebras taking values in spaces, providing an operadic analogue of Lurie’s equivalence between left fibrations and space‑valued functors.

The core technical development appears in Sections 3 and 4, where coCartesian corollas and coCartesian fibrations are defined. The authors prove basic stability properties, show how to mark dendroidal sets, and construct a combinatorial model structure (the Cisinski‑Moerdijk model structure) whose fibrant objects are exactly the ∞‑operads. Normalization of dendroidal sets and the notion of normal monomorphisms are used to control cofibrations.

Section 4 builds the ∞‑category coCart(X) by taking the homotopy‑coherent nerve of the full simplicial subcategory of a marked model category consisting of fibrant‑cofibrant objects over X. This yields a robust ∞‑categorical environment for coCartesian fibrations.

Section 5 contains the main theorem (Theorem 5.2): there is an equivalence of ∞‑categories
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