This paper is about an invariant of small categories called \emph{isotropy}. Every small category C has associated with it a presheaf of groups on C, called its isotropy group, which in a sense solves the problem of making the assignment C |-> Aut(C) functorial. Consequently, every category has a canonical congruence that annihilates the isotropy; however, it turns out that the resulting quotient may itself have non-trivial isotropy. This phenomenon, which we term higher order isotropy, is the subject of our investigation. We show that with each category C we may associate a sequence of groups called its higher isotropy groups, and that these give rise to a sequence of quotients of C. This sequence leads us to an ordinal invariant for small categories, which we call isotropy rank: the isotropy rank of a small category is the ordinal at which the sequence of quotients stabilizes. Our main results state that each small category has a well-defined isotropy rank, and moreover, that for each small ordinal one may construct a small category with precisely that rank. It happens that isotropy rank of a small category is an instance of the same concept for Grothendieck toposes, for which corresponding statements hold. Most of the technical work in the paper is concerned with the development of tools that allow us to compute (higher) isotropy groups of categories in terms of those of certain suitable subcategories.
Keywords: utomorphism groups, algebraic invariants of categories, toposes
2010 MSC: 08A35, 18A23, 18A32, 18B25, 18D35
Theory and Applications of Categories, Vol. 33, 2018, No. 20, pp 537-582.
Published 2018-06-03.
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