Lurdes Sousa

Well-Pointed Coalgebras

For endofunctors of varieties preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. The initial algebra consists of all well-pointed coalgebras that are well-founded in the sense of Osius and Taylor. And initial algebras are precisely the final well-founded coalgebras. Finally, the initial iterative algebra consists of all finite well-pointed coalgebras. Numerous examples are discussed e.g. automata, graphs, and labeled transition systems.

The Orthogonal Subcategory Problem and the Small Object Argument

A classical result of P. Freyd and M. Kelly states that in “good” categories, the Orthogonal Subcategory Problem has a positive solution for all classes

On Final Coalgebras of Power-Set Functors and Saturated Trees To George Janelidze on the Occasion of His Sixtieth Birthday

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Power-Set Functors and Saturated Trees.

of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on

On Kan-injectivity of Locales and Spaces

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A Formula for Codensity Monads and Density Comonads

, the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular

On reflective subcategories of varieties

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KAN INJECTIVITY IN ORDER-ENRICHED CATEGORIES

-injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal λ every member of the class is either λ-presentable or an epimorphism. Both the Orthogonal Subcategory Problem and the Small Object Argument are valid for quasi-presentable classes. Surprisingly, in locally ranked categories (used previously to generalize Quillen’s result), this is no longer true: we present a class

Order-preserving reflectors and injectivity

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A logic of implications in algebra and coalgebra

of morphisms, all but one being epimorphisms, such that the orthogonality subcategory