Tested functor calculi

Functor calculus refers to a family of homotopy-theoretic frameworks that extend the ideas of differential calculus into categorical settings. Existing forms of functor calculus have proved remarkably useful, with applications ranging from algebraic $K$-theory to a variety of geometric problems. Their ubiquity suggests that it is worthwhile to search for new versions of functor calculus.

Recent work of Bandklayder, Bergner, Griffiths, Johnson, and Santhanam examines the homotopy theory encoded in the degree $n$ approximations  of Goodwillie calculus and discrete calculus. In each case, they identify (or in the case of discrete calculus, construct) a model structure that captures the relevant homotopical data and is controlled by a prescribed collection of test morphisms. By exploiting these test morphisms, they show that the resulting model category is cofibrantly generated, giving an explicit description of the generating acyclic cofibrations.

In this talk, I will describe the observation that these model structures can always be realised as left Bousfield localizations with respect to the corresponding sets of test morphisms. This viewpoint naturally suggests defining new calculi directly from chosen test morphisms. I will explain that such a ``tested'' calculus exists provided a certain technical condition on the test morphisms is satisfied, and then show how this condition can be reformulated in far more familiar terms, namely, that a degree $n$ functor automatically satisfies the conditions of being degree $n+1$. 

(The latter aspect of this talk is joint work-in-progress with Julie Bergner, Brenda Johnson, Rhiannon Griffiths and Rekha Santhanam.)