2019/05/08 by Gal, Adam, Gal, Elena
#Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.03376
The notion of a higher Segal space was introduced by Dyckerhoff and Kapranov as a general framework for studying higher associativity inherent in a wide range of mathematical objects. In the present work we formalize the connection between this notion and the notion of \mathbbA_∞-algebra. We introduce the notion of a "d-lax \mathbbA_∞-algebra object" which generalizes the notion of an \mathbbA_∞-algebra object. We describe a construction that assigns to a simplicial object S_\bullet in a category \mathscrS a datum of higher associators. We show that this datum defines a d-lax \mathbbA_∞-algebra object in the category of correspondences in \mathscrS precisely when S_\bullet is a (d+1)-Segal object. More concretely we prove that for n≥ d the "n-dimensional associator" is invertible. The so called "upper" and "lower" d-Segal conditions which originally come from the geometry of polytopes appear naturally in our construction as the two conditions which together imply the invertibility of the d-dimensional associator. A corollary is that for d=2, our construction defines an \mathbbA_∞-algebra in the (∞,1)-category of correspondences in \mathscrS with the 2-Segal conditions implying invertibility of all associativity data.