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Higher isotropy

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J. Funk, P. Hofstra, and S. Khan

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.

http://www.tac.mta.ca/tac/volumes/33/20/33-20.pdf

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