The equifibered approach to $\mathrm{\infty }$-properads

We define a notion of $\mathrm{\infty }$-properads that generalises $\mathrm{\infty }$-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on $\mathrm{\infty }$-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our $\mathrm{\infty }$-properads as Segal presheaves on a category of graphs by comparing them to the Segal $\mathrm{\infty }$-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 $\mathrm{\infty }$-properad such that, for example, topological field theories are algebras over the bordism $\mathrm{\infty }$-properad. The key ingredient to this paper is the notion of an equifibered map between $E_{\mathrm{\infty }}$-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free $E_{\mathrm{\infty }}$-monoids, for example that free $E_{\mathrm{\infty }}$-monoids are closed under pullbacks along arbitrary maps.