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Kan injectivity in order-enriched categories

Published online by Cambridge University Press:  02 December 2014

JIŘÍ ADÁMEK ,
LURDES SOUSA  and
JIŘÍ VELEBIL
JIŘÍ ADÁMEK
Affiliation:
Institute of Theoretical Computer Science, Technical University of Braunschweig, Braunschweig, Germany Email: adamek@iti.cs.tu-bs.de
LURDES SOUSA
Affiliation:
Polytechnic Institute of Viseu and Centre for Mathematics of the University of Coimbra, Coimbra, Portugal Email: sousa@estv.ipv.pt
JIŘÍ VELEBIL
Affiliation:
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czech Republic Email: velebil@math.feld.cvut.cz
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Abstract

Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1.

For every class $\mathcal{H}$ of morphisms, we study the subcategory of all objects that are Kan-injective with respect to $\mathcal{H}$ and all morphisms preserving Kan extensions. For categories such as Top0 and Pos, we prove that whenever $\mathcal{H}$ is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock–Zöberlein monad. However, this does not generalise to proper classes, and we present a class of continuous mappings in Top0 for which Kan-injectivity does not yield a monadic category.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 25 , Issue 1 , January 2015 , pp. 6 - 45
Copyright
Copyright © Cambridge University Press 2014 

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