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KZ-monadic categories and their logic

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Jiri Adamek and Lurdes Sousa

Given an order-enriched category, it is known that all its KZ-monadic
subcategories can be described by Kan-injectivity with respect to a
collection of morphisms. We prove the analogous result for Kan-injectivity
with respect to a collection H of commutative squares. A square is called
a Kan-injective consequence of H if by adding it to H Kan-injectivity is
not changed.
We present a sound logic for Kan-injectivity consequences and prove that
in ``reasonable" categories (such as $\Pos$ or $\Top_0$)
it is also complete for every set H of squares.

Keywords:
order-enriched category, Kan-injectivity, KZ-monad, Kan-injectivity logic,
locally ranked category

2010 MSC:
18C20, 18B35, 18D20, 54B30, 06B35,06D22, 18A15

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 10, pp 338-379.

Published 2017-02-27.

http://www.tac.mta.ca/tac/volumes/32/10/32-10.pdf

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