### $\U0001d519$-Cat is locally presentable or locally bounded if $\U0001d519$ is so.

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A logic of orthogonality characterizes all “orthogonality consequences" of a given class $\Sigma $ of morphisms, i.e. those morphisms $s$ such that every object orthogonal to $\Sigma $ is also orthogonal to $s$. 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 $\Sigma $ of morphisms such that all members except a set are regular epimorphisms and (b) for all classes $\Sigma $, without...

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 $3$-element set is colimit-dense in ${\mathrm{\mathbf{S}\mathbf{e}\mathbf{t}}}^{\mathrm{op}}$, and spaces of countable dimension are colimit-dense in ${\mathrm{\mathbf{V}\mathbf{e}\mathbf{c}}}^{\mathrm{op}}$.

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 $\mathcal{S}$ of morphisms in a locally presentable category $\mathcal{C}$ of structures, the orthogonal class of objects is a small-orthogonality...

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 ${\aleph}_{1}$-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-${\aleph}_{0}$-generated objects in CMet, CPMet and CCMet, answering questions by Di Liberti and Rosický. Other results include...