The oo-Categorical Eckmann-Hilton Argument

The classical "Eckmann-Hilton argument" states that given a set with two unital binary operations that satisfy the interchange law, the two operations must coincide and moreover, this operation is associative and commutative. If we assume that both binary operations where associative to begin with, this result says that two interchanging commutative monoid structures on a set must coincide and be commutative. In this form, the Eckmann-Hilton argument has a higher homotopical generalization in terms of the "additivity theorem". Namely, the Boardman-Vogt tensor product of the operads E_n and E_m is E_(n+m). In joint work with Tomer Schlank we give a (different) generalization of the non-associative Eckmann-Hilton argument in terms of a lower bound on the connectivity of the spaces of n-ary operations of  the Boardman-Vogt tensor product of any two reduced oo-operads P and Q in terms of the connectivity of P and Q. In this talk, I will give a quick introduction to oo-operads and the Boardman-Vogt tensor product, state the main results and, if time permits, sketch the proof.