Are all subcategories of locally finitely presentable categories that are closed under limits and $lambda$-filtered colimits also locally presentable? For full subcategories the answer is affirmative. Makkai and Pitts proved that in the case $lambda=
aleph_0$ the answer is affirmative also for all iso-full subcategories, emph{i.thinspace e.}, those containing with every pair of objects all isomorphisms between them. We discuss a possible generalization of this from $aleph_0$ to an arbitrary $lambda$.
Higher-order recursion schemes are recursive equations defining new operations from given ones called terminals. Every such recursion scheme is proved to have a least interpreted semantics in every Scotts model of lambda-calculus in which the termina
ls are interpreted as continuous operations. For the uninterpreted semantics based on infinite lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in lambda-calculus by an endofunctor Hlambda and they explained simultaneous substitution of lambda-terms by proving that the presheaf of lambda-terms is an initial Hlambda-monoid. Here we work with the presheaf of rational infinite lambda-terms and prove that this is an initial iterative Hlambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid.
