Classifying Vectoids and Generalisations of Operads

TL;DR

This paper introduces 'vectoids' as a new generalized space concept, explores their properties, and demonstrates how they systematically lead to new forms of operads and their generalizations.

Contribution

It proposes vectoids as a novel generalization of spaces, develops their foundational theory, and links them to the construction of new operad-like structures.

Findings

Vectoids generalize traditional spaces and include new examples.

Classifying vectoids relate to various algebraic structures.

Endomorphism monoids of classifying vectoids yield new operad generalizations.

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.