#
Double power monad preserving adjunctions are Frobenius

##
Christopher Townsend

We give a direct proof that between two toposes, F and
E, bounded over a base topos S, adjunctions
L -| R: Loc_F -> Loc_E over Loc_S are
Frobenius if and only if R commutes with the double power locale monad
and finite coproducts. The proof uses only certain categorical properties
of the category of locales, Loc. This implies that between
categories axiomatized to behave like categories of locales, it does not
make a difference whether maps are defined as structure preserving
adjunctions (i.e. those that commute with the double power monads) or
Frobenius adjunctions.

Keywords:
topos, locale, geometric morphism, Frobenius reciprocity, power monad

2010 MSC:
06D22, 18D35, 18B40, 22A22

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 17, pp 476-491.

Published 2018-05-28.

http://www.tac.mta.ca/tac/volumes/33/17/33-17.pdf

TAC Home