2014
DOI: 10.1017/s0960129514000024
|View full text |Cite
|
Sign up to set email alerts
|

Kan injectivity in order-enriched categories

Abstract: Abstract. Continuous lattices were characterised by Martín Escardó as precisely the objects that are Kan-injective w.r.t. a certain class of morphisms. We study Kan-injectivity in general categories enriched in posets. An example: ω-CPO's are precisely the posets that are Kan-injective w.r.t. the embeddings ω ֒→ ω + 1 and 0 ֒→ 1.For every class H of morphisms we study the subcategory of all objects Kan-injective w.r.t. H and all morphisms preserving Kan-extensions. For categories such as Top 0 and Pos we prove… Show more

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1
1

Citation Types

1
27
0

Year Published

2015
2015
2022
2022

Publication Types

Select...
5

Relationship

3
2

Authors

Journals

citations
Cited by 10 publications
(28 citation statements)
references
References 32 publications
(91 reference statements)
1
27
0
Order By: Relevance
“…In [8] we showed that KZ-monadic subcategories are precisely the KZ-reflective categories closed under left adjoint retractions (i.e., the equality gx = yf between morphisms of X with f in A and x and y both left adjoint retractions implies that g also belongs to A). In [1] we proved that in well-behaved categories, namely in locally ranked ones, every Kan-injective subcategory LInj(H) with H a set is indeed a KZ-monadic subcategory. When A is KZ-reflective in X, with F : X → A the corresponding reflector functor, A LInj is precisely the class of all morphisms f of X such that Ff is a left adjoint section in A, that is, there is a morphism (Fh) * in A with (Fh) * Fh = id and Fh(Fh) * ≤ id ( [8]).…”
Section: Kz-reflective Subcategoriesmentioning
confidence: 95%
See 3 more Smart Citations
“…In [8] we showed that KZ-monadic subcategories are precisely the KZ-reflective categories closed under left adjoint retractions (i.e., the equality gx = yf between morphisms of X with f in A and x and y both left adjoint retractions implies that g also belongs to A). In [1] we proved that in well-behaved categories, namely in locally ranked ones, every Kan-injective subcategory LInj(H) with H a set is indeed a KZ-monadic subcategory. When A is KZ-reflective in X, with F : X → A the corresponding reflector functor, A LInj is precisely the class of all morphisms f of X such that Ff is a left adjoint section in A, that is, there is a morphism (Fh) * in A with (Fh) * Fh = id and Fh(Fh) * ≤ id ( [8]).…”
Section: Kz-reflective Subcategoriesmentioning
confidence: 95%
“…In this section, we recall the notions of Kan-injectivity and KZ-reflective subcategory, and some of their properties, which are presented in [8] and [1].…”
Section: Preliminariesmentioning
confidence: 99%
See 2 more Smart Citations
“…An ordered category is thus a special form of 2category, and thus the well-developed theory of 2-categories (see, e.g., [29]) can be applied. To site just a couple of examples where the 2-categorical machinery works very well for particular order-enriched categories we mention [30,31,32], which involves a translation of a 2-categorical notion to a condition on a monad known as the Kock-Zöberlein condition, and [33] in the area of ordered universal algebra. However, as noted generally already in [34], the standard 2-categorical constructions yield the 'wrong' results in certain ordered categories arising in computer science.…”
Section: Order Extensionsmentioning
confidence: 99%