[Thesis] Higher limits via homotopical algebra
Published:
This PhD thesis focuses on describing the higher limits of a functor using homotopical tools rather than the classical methods of Homological Algebra. This work focuses on functors over a filtered category in which every endomorphism is an isomorphism (EI-category) and takes values in modules. Firstly, two model category structures are presented in this category of functor: one suitable for contravariant functors and another for covariant functors. In this context, higher limits and colimits are described through fibrant and cofibrant replacements, respectively. Then, based on the combinatorial properties of these EI-categories, an explicit construction of both replacements is provided. In addition, variations of these replacements are presented to adapt them to the problem of study: describing vanishing bounds and ranks for the higher limits.
In the case of partially ordered categories (posets for short), it is shown that pseudo-projective property is equivalent to cofibrant in the covariant functors category described in this work. A notion of Mackey functor for posets is also introduced, inspired by the classical notion of Mackey functor for orbit categories. In this case, it is proven that Mackey functors with an additional notion of quasi-unit are cofibrant; therefore, their higher colimits vanish in positive degrees.
Using the combinatorial structure of the replacement and the presented computation tools, explicit vanishing bounds for the higher limits are proven. Using different strategies, these are described based on the geometry of the poset, local bounds of higher limits, and filtrations from atomic functors.
Finally, the case of higher limits of functors indexed on CL-shellable posets is studied in detail. These posets have the homotopy type of a wedge sum of spheres of the same dimension, so the higher limits in strictly positive degrees of a constant functor are concentrated in a single degree. Motivated by this particular case, a sufficient property for a functor is abstracted, which guarantees that its higher limits vanish for dimensions lower than the length of the poset. As an example of application, the case of the family of n-linear forms functors in hyperplane arrangements is described.