Homotopy-coherent algebra via Segal conditions

Abstract

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", bywhich we mean an $\infty$-category equipped with a factorization system and a collection of "elementary" objects. Examples of structures that occur as such "Segal $\mathcal{O}$-spaces" for an algebraic pattern $\mathcal{O}$ include $\infty$-categories, $(\infty,n)$-categories, $\infty$-operads, $\infty$-properads, and algebras for an $\infty$-operad in spaces. In the first part of this paper we set up a general frameworkn for algebraic patterns and their Segal objects, including conditions under which the latter are preserved by left and right Kan extensions. In particular, we obtain necessary and sufficent conditions on a pattern $\mathcal{O}$ for free Segal $\mathcal{O}$-spaces to be described by an explicit colimit formula, in which case we say that $\mathcal{O}$ is "extendable". In the second part of the paper we explore the relationship between extendable algebraic patterns and polynomial monads, by which we mean cartesian monads on presheaf $\infty$-categories that are accessible and preserve weakly contractible limits. We first show that the free Segal $\mathcal{O}$-space monad for an extendable pattern $\mathcal{O}$ is always polynomial. Next, we prove an $\infty$-categorical version of Weber's Nerve Theorem for polynomial monads, and use this to define a canonical extendable pattern from any polynomial monad, whose Segal spaces are equivalent to the algebras of the monad. These constructions yield functors between polynomial monads and extendable algebraic patterns, and we show that these exhibit full subcategories of "saturated" algebraic patterns and "complete" polynomial monads as localizations, and moreover restrict to an equivalence between the $\infty$-categories of saturated patterns and complete polynomial monads.