Year: 2014
Type: article
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 asTop0andPos, 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 inTop0for which Kan-injectivity does not yield a monadic category.
Authors: JIŘÍ ADÁMEK, LURDES SOUSA, JIŘÍ VELEBIL
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2147115247