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

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Coalgebras have been proposed as formal basis for the semantics of objects in the sense of object-oriented programming. This paper shows that this semantics provides a smooth interpretation for subtyping, a central notion in object-oriented programming. We show that different characterisations of behavioural subtyping found in the literature can conveniently be expressed in coalgebraic terms. We also investigate the subtle difference between behavioural subtyping and refinement.

A duality between $\lambda $-ary varieties and $\lambda $-ary algebraic theories is proved as a direct generalization of the finitary case studied by the first author, F.W. Lawvere and J. Rosick’y. We also prove that for every uncountable cardinal $\lambda $, whenever $\lambda $-small products commute with $\mathcal{D}$-colimits in $\text{Set}$, then $\mathcal{D}$ must be a $\lambda $-filtered category. We nevertheless introduce the concept of $\lambda $-sifted colimits so that morphisms between $\lambda $-ary varieties (defined to be $\lambda $-ary, regular right adjoints) are precisely the functors...

A functional representation of the hyperspace monad, based on the semilattice structure of function space, is constructed.

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...