Michael Ching: Pro-operads and the classification of Taylor towers

Abstract.

In previous work, Greg Arone and I showed that Taylor towers of functors from based spaces or spectra to spectra are classified by modules over certain "pro-operads". Thomas Blom and I now extend that classification to spectrum-valued functors on any pointed compactly generated ∞-category C. In doing so, we associate a pro-operad to C, such that the inverse limit of that pro-operad consists of the derivatives of the identity functor on C. One twist is that what we call a pro-operad is perhaps not quite what you expect: it is not (necessarily) a cofiltered system of operads, but a slightly more general algebraic structure based on the notion of a closed, but not necessarily monoidal, ∞-category.