On continuity of accessible functors (Q2674400)

The theory of locally presentable categories was initiated in [\textit{P. Gabriel} and \textit{F. Ulmer}, Lokal präsentierbare Kategorien. (Locally presentable categories). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0225.18004)]. Given a cardinal \(\alpha\), the author defines the category \(\boldsymbol{Lp}_{\alpha}\)\ of locally \(\alpha\)-presentable categories (as objects), functors preserving all small limits and all \(\alpha\)-filtered colimits (as morphisms) and natural transformations (as \(2\)-morphisms). This paper aims to give a characterization of the morphism out of a locally \(\alpha\)-presentable \ category \(\mathcal{K}\in\boldsymbol{Lp}_{\alpha}\) in terms of those preserving \(\gamma\)-small limits for some determined \(\gamma\). The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] establishes the main result (Theorem 2.9) claiming that for any locally \(\alpha\)-presentable \ category \(\mathcal{K}\) there exists a regular cardinal \(\gamma\)\ such that an \(\alpha\)-accessible \(F:\mathcal{K} \rightarrow\mathcal{L}\)\ with \(\mathcal{L}\)\ locally \(\alpha\)-presentable, is continuous iff it preserves all \(\gamma\)-small limits. This can be interpreted as a new adjoint functor theorem for \(\alpha\)-accessible functors out of a locally \(\alpha\)-presentable \ category. \item[\S 3] establishes an enriched version of the main result based on the notion of locally presentable \(\mathcal{V}\)-category [\textit{G. M. Kelly}, Cah. Topologie Géom. Différ. Catégoriques 23, 3--42 (1982; Zbl 0538.18006)]. An adjoint functor theorem specialized to the \(\alpha\)-accessible case is obtained (Theorem 3.4). It is also established (Theorem 3.7) that a small \(\mathcal{V}\)-category is accessible in the sense of [\textit{F. Borceux} et al., Theory Appl. Categ. 4, 47--72 (1998; Zbl 0981.18006)] iff it is Cauchy complete. \end{itemize}