Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an “algebraic pattern”, by which we mean an ∞-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 ∞-categories, $(\infty ,n)$-categories, ∞-operads (including symmetric, non-symmetric, cyclic, and modular ones), ∞-properads, and algebras for a (symmetric) ∞-operad in spaces.

In the first part of this paper we set up a general framework for algebraic patterns and their associated Segal objects, including conditions under which the latter are preserved by left and right Kan extensions. In particular, we obtain necessary and sufficient 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 ∞-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 ∞-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 ∞-categories of saturated patterns and complete polynomial monads.

可以用由“代数模式”确定的Segal型极限条件来描述许多同伦同调的代数结构，在这种情况下，我们指的是配备有因子分解系统和“基本”对象集合的∞类。诸如“ Segal”之类的结构的示例$\mathcal{Ø}$-空格”代表代数模式 $\mathcal{Ø}$ 包括∞类， $（\infty ，ñ）$-类，∞-算子（包括对称的，非对称的，循环的和模的算子），∞-properads和空间中的（对称）∞-算子的代数。

在本文的第一部分中，我们为代数模式及其相关的Segal对象建立了通用框架，包括通过左和右Kan扩展保留后者的条件。特别是，我们在模式上获得了必要和充分的条件$\mathcal{Ø}$ 免费西格尔 $\mathcal{Ø}$-空间由一个明确的colimit公式描述，在这种情况下，我们说 $\mathcal{Ø}$ 是“可扩展的”。

在本文的第二部分中，我们探讨了可扩展代数模式与多项式单子态之间的关系，通过这种关系我们指的是前捆∞类上的笛卡尔单子态，它们可以访问并保持弱可收缩的极限。我们首先展示免费的Segal$\mathcal{Ø}$-space monad可扩展模式 $\mathcal{Ø}$总是多项式。接下来，我们证明了用于多项式Monad的Weber神经定理的∞分类版本，并使用它来定义任何多项式Monad的正则可扩展模式，其Segal空间等于该monad的代数。这些构造在多项式单子和可扩展的代数模式之间产生函子，并且我们证明了它们表现出“饱和”代数模式和“完整”多项式单子的完整子类别作为局部化，而且还限制了饱和模式和∞类别之间的等价性。完整的多项式单子。