2022/11/04 by Barkan, Shaul, Steinebrunner, Jan · 1 citation
#18M85 #18N70 #55P99 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.02576
We define a notion of ∞-properads that generalises ∞-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on ∞-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our ∞-properads as Segal presheaves on a category of graphs by comparing them to the Segal ∞-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 ∞-properad such that, for example, topological field theories are algebras over the bordism ∞-properad. The key ingredient to this paper is the notion of an equifibered map between E_∞-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free E_∞-monoids, for example that free E_∞-monoids are closed under pullbacks along arbitrary maps.