On full and faithful Kan extensions (Q1273637)

This paper substantially extends the conceptual apparatus of universal (alg-universal, strongly universal) categories as by \textit{A. Pultr} and \textit{V. Trnková} [``Combinatorial, algebraic and topological representations of groups, semigroups and categories'', North-Holland, Amsterdam (1980; Zbl 0418.18004)]. Most of the embeddings of concrete categories (over \({\mathcal S}ets\)) constructed in universality theory are Kan extensions of their restrictions to some small left adequate subcategory of their domains; this provides the title and a pervasive theme of the paper. It is not practical to explain fully any of the eight results labelled `theorem', but the gist of theorem 5.11 is that given a concrete category \(U:{\mathcal K}\to{\mathcal S}ets\) with a factorization system \(({\mathcal E, \mathcal M})\) that \(U\) respects, and a suitable left adequate subcategory \(\mathcal A\) of \(\mathcal K\) and strong embedding \(F\) of \(\mathcal A\) in a variety of algebras \(\mathcal V\), and mild cardinality conditions, then the Kan \(\mathcal M\)-extension (defined in section 2 of this paper) of \(F\) over \(\mathcal K\) is a strong embedding.