A chain rule in the calculus of homotopy functors (Q1428381)

The authors state and prove the chain rule of their title. Given a weak homotopy endo-functor on simplicial sets, the derivative spectrum admits a right action by the loop group of the domain, and a left action by the loop group of the codomain. In forming a composite, the loop group of the intermediate set acts on both derivatives and the derivative of the composite is stably equivalent to the balanced smash product with respect to these two actions. Several hypotheses are required for this chain rule to hold, but the authors give a number examples of interest where all such hypotheses are satisfied.