Weak factorization and transfer systems

Transfer systems are discrete objects that encode the homotopy theory of N∞ operads, i.e., the operads whose algebras are homotopy commutative monoids with a class of equivariant transfer (or norm) maps. They have a rich combinatorial structure defined in terms of the subgroup lattice of the group of equivariance, G. Indeed, if G is a cyclic p-group, there are Catalan-many transfer systems that assemble into the Tamari lattice (i.e., associahedron). In this talk, I will show that when G is finite Abelian, transfer systems are in natural bijection with weak factorization systems on the poset category of subgroups of G. This leads to a novel involution on the lattice of transfer systems, generalizing an observation of Balchin–Bearup–Pech–Roitzheim for cyclic groups of squarefree order. I will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg–Hill saturation conjecture.

This is joint work with Angélica Osorno and a team of Reed College undergraduates: Evan Franchere, Usman Hafeez, Peter Marcus, Weihang Qin, and Riley Waugh (the Electronic Collaborative Mathematics Research Group, or eCMRG).