Morita-types of Reedy categories

A homotopy-type is a set together with data that codifying how it is presented as a quotient of other sets; this additional data enables inductive inclusion-exclusion counting techniques. Sets are examples of homotopy-types, as are groups & groupoids. One way to organize a homotopy-type is as a simplicial set. The linearization of a homotopy-type is thusly a simplicial vector space. A classical theorem (Dold-Kan) states that a simplicial vector space can be recodified in two other, simpler, ways: as a chain complex of vector spaces; as a filtered vector space. I’ll start this talk by motivating the notion of a homotopy-type, and introducing simplicial objects as representations of the simplex category (akin to how an involution on a set is a representation of the cyclic group of order 2). I’ll cast the above classical theorem as an identification of the Marita-type of the simplex category, and its categorified Mobius inversion. I’ll state a vast generalization of this classical result, which identifies the Marita-type & Mobius inversion of essentially any category that is suitably filtered (ie, generalized Reedy). Explicating this identification amounts to identifying the left and right null spaces of transition matrices associated to directed graphs. I’ll list examples of such filtered categories, and work out some of these null spaces.