-Cat is locally presentable or locally bounded if is so.
-purity and orthogonality.
A characterization of locally -presentable categories
A convenient category for directed homotopy.
A duality relative to a limit doctrine.
A logic of injectivity.
A logic of orthogonality
A logic of orthogonality characterizes all “orthogonality consequences" of a given class of morphisms, i.e. those morphisms such that every object orthogonal to is also orthogonal to . A simple four-rule deduction system is formulated which is sound in every cocomplete category. In locally presentable categories we prove that the deduction system is also complete (a) for all classes of morphisms such that all members except a set are regular epimorphisms and (b) for all classes , without...
Algebraically closed and existentially closed substructures in categorical context.
Categorical domain theory: Scott topology, powercategories, coherent categories.
Colimit-dense subcategories
Among cocomplete categories, the locally presentable ones can be defined as those with a strong generator consisting of presentable objects. Assuming Vopěnka’s Principle, we prove that a cocomplete category is locally presentable if and only if it has a colimit dense subcategory and a generator consisting of presentable objects. We further show that a -element set is colimit-dense in , and spaces of countable dimension are colimit-dense in .
Definable orthogonality classes in accessible categories are small
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class of morphisms in a locally presentable category of structures, the orthogonal class of objects is a small-orthogonality...
Dense morphisms of monads.
Esquisses inductives et presque inductives
Finitely presentable morphisms in exact sequences.
Introduction to coalgebra.
Metric enrichment, finite generation, and the path coreflection
We prove a number of results involving categories enriched over CMet, the category of complete metric spaces with possibly infinite distances. The category CPMet of path complete metric spaces is locally -presentable, closed monoidal, and coreflective in CMet. We also prove that the category CCMet of convex complete metric spaces is not closed monoidal and characterize the isometry--generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include...
More on injectivity in locally presentable categories.
More on orthogonality in locally presentable categories
Multi-bimodels
-permutable locally finitely presentable categories.