24th of June, 2021. Part of the Topos Institute Colloquium.
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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. Such calculi could 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 a machine for creating calculi on functor categories Fun (C,M) that is natural in both the source C and the target M. Our calculi arise by comparison of the source category C with a tower of test categories, equipped with cubical structure of progressively higher dimension, giving rise to sequences of resolutions of functors from C to M, built from comonads derived from the cubical structure on the test categories. The stages of the towers of functors that we obtain measure how far the functor we are analyzing deviates from being a coalgebra over each of these comonads. The naturality of this construction makes it possible to compare both different types calculi on the same functor category, arising from different towers of test categories, and the same type of calculus on different functor categories, given by a fixed tower of test categories.
(Joint work with Brenda Johnson)
