The equifibered approach to $\infty$-properads

Abstract

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.

92 pages, 4 figures. v3: Revised in response to report. Removed appendix as it was no longer needed - a version of it will be made available on the website of the second author. To appear in Advances in Mathematics