Homotopy limits of triples (Q1873749)

Let \(\mathbf E\) be a unitary ring spectrum, and define a functor \(E:{\mathcal S}_* \to {\mathcal S}_*\) from the category of pointed spaces to itself by the rule \(E:X\mapsto \Omega^\infty(\Sigma^\infty X\wedge {\mathbf E})\). If \(E\) is an \(S\)-algebra or a symmetric spectrum, then \(E\) is a triple on \({\mathcal S}_*\) and is also a continuous functor in the sense that the natural map \[ \text{ map}_*(X,Y) \to \text{ map}_*(EX,EY) \] of spaces is continuous. There is then a cosimplical resolution \(E^\bullet X\) and partial terms \(E^{\bullet \leq n} = {\text{ Tot}}_n(E^\bullet_{\text{ fib}} X)\), (where \(E^\bullet_{\text{ fib}} X\) comes with a Reedy fibrant approximation). The main result shows that \(E_n\) is a homotopy functor, and that the induced functor on the homotopy category is a triple.