Algebraically-topological systems and attachments

The paper continues the study of the authors on relationships between \emph{topological systems} of S.~Vickers and \emph{attachments} of C.~Guido. We extend topological systems to \emph{algebraically-topological systems}. A particular instance of the latter, called \emph{attachment system}, incorporates the notion of attachment, thus, making it categorically redundant in mathematics. We show that attachment systems are equipped with an internal topology, which is similar to the topology induced by locales. In particular, we provide an attachment system analogue of the well-known categorical equivalence between sober topological spaces and spatial locales.