Abstract

Abstracting the framework common to most flavors of functor calculus, one can define a calculus on a category M equipped with a distinguished class of weak equivalences to be a functor that associates to each object x of M a tower of objects in M that are increasingly good approximations to x, in some well defined, Taylor-type sense.  This definition dualizes in an obvious sense, giving rise to the notion of a cocalculus.   Such (co)calculi can be applied, for example, to testing whether morphisms in M are weak equivalences.In this talk, after making the definition above precise, I will describe machines for creating (co)calculi on functor categories Fun (C,M), naturally in both the source C and the target M. The naturality of this construction makes it possible to compare both different types calculi on the same functor category, as well as the same type of calculus on different functor categories.  I will briefly sketch a few examples. The key mechanism in the calculus machine is the natural construction of a comonad on a functor category Fun (D,M) from a cubical family of commuting localizations of D, and dually for the cocalculus machine.  (Joint work with Brenda Johnson and with Kristine Bauer, Robyn Brooks, Julie Rasmusen, and Bridget Schreiner.)