#
Covariant presheaves and subalgebras

##
Ulrich Höhle

For small involutive and integral quantaloids ${\cal Q}$ it is shown that
covariant presheaves on symmetric ${\cal Q}$-categories are equivalent to
certain subalgebras of a specific monad on the category of symmetric
${\cal Q}$-categories. This construction is related to a weakening of the
subobject classifier axiom which does not require the classification of
all subalgebras, but only guarantees that classifiable subalgebras are
uniquely classifiable. As an application the identification of closed left
ideals of non-commutative $C^*$-algebras with certain
"open", subalgebras of freely generated algebras is given.

Keywords:
Involutive quantale, involutive quantaloid, symmetric ${\cal
Q}$-category, covariant presheaf, monad of weak singletons,
classifiable
subalgebra, closed left ideal

2000 MSC:
06F07, 18C15, 18D20, 18F20

*Theory and Applications of Categories,*
Vol. 25, 2011,
No. 13, pp 342-367.

Published 2011-07-14.

http://www.tac.mta.ca/tac/volumes/25/13/25-13.dvi

http://www.tac.mta.ca/tac/volumes/25/13/25-13.ps

http://www.tac.mta.ca/tac/volumes/25/13/25-13.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/25/13/25-13.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/25/13/25-13.ps

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