#
Majority categories

##
Michael Anton Hoefnagel

We introduce the notion of a majority category - the categorical
counterpart of varieties of universal algebras admitting a majority term.
This notion can be thought to capture properties of the category of
lattices, in a way that parallels how Mal'tsev categories capture
properties of the category of groups. Among algebraic majority categories
are the categories of lattices, Boolean algebras and Heyting algebras.
Many geometric categories such as the category of topological spaces,
metric spaces, ordered sets, any topos, etc., are comajority categories
(i.e.~their duals are majority categories), and we show that, under mild
assumptions, the only categories which are both majority and comajority,
are the preorders. Mal'tsev majority categories provide an alternative
generalization of arithmetical categories to protoarithmetical categories
in the sense of Bourn. We show that every Mal'tsev majority category is
protoarithmetical, provide a counter-example for the converse implication,
and show that in the Barr-exact context, the converse implication also
holds. We can then conclude that a category is arithmetical if and only if
it is a Barr-exact Mal'tsev majority category, recovering in the varietal
context a well known result of Pixley.

Keywords:
arithmetical category, internal relation, Mal'tsev category, majority
category, majority algebra, majority term, lattice, protoarithmetical
category

2010 MSC:
18A05, 18C99, 08B05, 06B20, 18B35, 18B10, 18B30

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 10, pp 249-268.

Published 2019-04-05.

http://www.tac.mta.ca/tac/volumes/34/10/34-10.pdf

TAC Home