What is loop multiplication anyhow?

Abstract

In the beginning we are taught that multiplication of pointed loops is just the first followed by the second. Later we learn that the cobar construction on the chains of the base is a model for the chains on the loop space, that is, this cobar construction is chain equivalent to the chains on the loop space. The cobar construction is a certain tensor algebra and has a natural multiplication of tensors. This multiplication of tensors models the multiplication induced by the multiplication of loops. But, because of the simplicity of the definition of loop multiplication, this modeling is not obvious. We interpret loop multiplication so that this modeling becomes a clear consequence of the naturality of Eilenberg–Moore methods applied to multiple or iterated pullbacks. In contrast, we observe that, if we require that it be invariant under homological equivalence of differential coalgebras, there is no natural modeling of the comultiplication in the loop space. But, in rational homotopy theory, results of Milnor–Moore and Quillen show that there is a natural modeling of the Hopf algebra structure of the loop space.