This work presents a detailed analysis of the combinatorics of modular operads. These are operad-like structures that admit a contraction operation as well as an operadic multiplication. Their combinatorics are governed by graphs that admit cycles, and are known for their complexity. In 2011, Joyal and Kock introduced a powerful graphical formalism for modular operads. This paper extends that work. A monad for modular operads is constructed and a corresponding nerve theorem is proved, using Weber's abstract nerve theory, in the terms originally stated by Joyal and Kock. This is achieved using a distributive law that sheds new light on the combinatorics of modular operads.

这项工作详细分析了模块化操作数的组合。这些是类似操作数的结构，允许收缩操作以及操作数乘法。它们的组合由允许循环的图控制，并以其复杂性而闻名。2011 年，Joyal 和 Kock 为模块化操作数引入了强大的图形形式。本文扩展了这项工作。使用 Weber 的抽象神经理论，按照最初由 Joyal 和 Kock 陈述的术语，构造了用于模运算的单子并证明了相应的神经定理。这是使用分配定律实现的，该定律为模块化操作数的组合提供了新的思路。