Anna Frascella - Department of Mathematics E. De Giorgi", University of Salento, P. O. Box ۱۹۳, ۷۳۱۰۰ Lecce, Italy
Cosimo Guido - Department of Mathematics E. De Giorgi", University of Salento, P. O. Box ۱۹۳, ۷۳۱۰۰ Lecce, Italy
Sergey A. Solovyov - Department of Mathematics, University of Latvia, Zellu iela ۸, LV-۱۰۰۲ Riga, Latvia and Institute of Mathematics and Computer Science, University of Latvia, Raina bulvaris ۲۹, LV-۱۴۵۹ Riga, Latvia

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.

Algebraically-topological system, Attachment system, Categorically-algebraic topology, Dual attachment pair, Localic algebra, Localification of systems, (Variety-based) pointless topology, Spatialization of systems, Topological theory morphism, Variety