Classifying Vectoids and Generalisations of Operads

Abstract

A new generalisation of the notion of space, called "vectoid", is suggested in this work. Basic definitions, examples and properties are presented, as well as a construction of direct product of vectoids. Proofs of more complicated properties not used later are just sketched. Classifying vectoids of simplest algebraic structures, such as objects, algebras and coalgebras, are studied in some detail afterwards. Apart from giving interesting examples of vectoids not coming from spaces known before (such as ringed topoi), monoids in the endomorphism categories of these classifying vectoids turn out to provide a systematic approach to construction of different versions of the notion of an operad, as well as its generalisations, unknown before.