对∞-properads的等纤维化方法
The equifibered approach to $\infty$-properads
AI总结：
本文提出一种广义的∞-properads概念，允许操作有多个输出，从而为∞-operads提供新视角，并通过等纤维化方法定义了Segal预示例，使拓扑场论成为bordism ∞-properad的代数。
摘要：
我们定义了一种广义的∞-properads，通过允许操作有多个输出来扩展∞-operads。专门考虑每个操作有单一输出的情况，为∞-operads提供了一个简单的新视角，同时额外的通用性允许了如边界范畴的例子。我们还通过将∞-properads解释为图的Segal预示例，比较它们与Hackney-Robertson-Yau的Segal ∞-properads。结合这两种方法，得到一个灵活的工具，用于具有多个输入和输出的操作的更高代数。关键在于等纤维化映射的概念，这是E_∞-monoids自由映射的合理推广。我们还利用此概念证明了关于自由E_∞-monoids的事实，例如自由E_∞-monoids在任意映射下的拉回下闭合。
英文摘要：
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.
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