论文标题
$ \ infty $ -properads
The equifibered approach to $\infty$-properads
论文作者
论文摘要
我们定义了$ \ infty $ - properads的概念，该概念通过允许具有多个输出的操作来概括$ \ infty $ -operads。专门针对每个操作具有单个输出的情况，可以在$ \ infty $ -operads上提供一个简单的新观点，但同时额外的通用性允许诸如Bordism类别之类的示例。我们还通过将它们与Hackney-Robertson-Yau的Segal $ \ infty $ -properads进行比较，将我们的$ \ infty $ properads解释为Segal Properades在一类图表上。将这两种方法结合起来提供了一种灵活的工具，用于与具有多个输入和输出的操作进行更高的代数。至关重要的是，这允许在$ \ infty $ properad上定义代数，以便例如，拓扑字段理论是Bordism $ \ infty $ -properad上的代数。本文的关键要素是在$ e_ \ infty $ - monoids之间的概念的概念，这是自由地图的表现良好的概括。我们还使用它来证明有关免费$ e_ \ infty $ monoids的事实，例如，在沿任意地图的回调下，免费的$ e_ \ infty $ monoids被关闭。
We define a notion of $\infty$-properads that generalises $\infty$-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on $\infty$-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our $\infty$-properads as Segal presheaves on a category of graphs by comparing them to the Segal $\infty$-properads of Hackney-Robertson-Yau. Combining these two approaches yields a flexible tool for doing higher algebra with operations that have multiple inputs and outputs. Crucially, this allows for a definition of algebras over an $\infty$-properad such that, for example, topological field theories are algebras over the bordism $\infty$-properad. The key ingredient to this paper is the notion of an equifibered map between $E_\infty$-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free $E_\infty$-monoids, for example that free $E_\infty$-monoids are closed under pullbacks along arbitrary maps.
