We present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy. This homotopy theory takes the form of a Quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs. This new category of undirected graphs, denoted $\mathbf{U}$, plays a similar role for modular operads that the dendroidal category $\Omega$ plays for operads. We carefully study properties of $\mathbf{U}$, including the existence of certain factorization systems. Related structures, such as cyclic operads and stable modular operads, can be similarly treated using categories derived from $\mathbf{U}$.

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更高模块化操作数的图形类别

我们提出了一个弱版本的模块化操作数的同伦理论，其组合和收缩仅被定义为同伦。这种同伦理论采用 Quillen 模型结构的形式，该结构是针对某一类无向图的单纯预层集合。这种新的无向图类别，表示为 $\mathbf{U}$，对于模操作数起着类似于树状类别 $\Omega$ 在操作数中的作用。我们仔细研究了 $\mathbf{U}$ 的性质，包括某些分解系统的存在。相关结构，例如循环操作数和稳定模操作数，可以使用从 $\mathbf{U}$ 派生的类别进行类似处理。