Quantum families of maps

TL;DR

This survey introduces quantum families of maps within non-commutative topology, discussing their properties, existence, and quantum semigroup structures, with classical analogies and examples.

Contribution

It provides a foundational overview of quantum families of maps, highlighting their properties, existence, and semigroup structures in a non-commutative setting.

Findings

Quantum families can possess special properties.

Quantum spaces are endowed with quantum semigroup structures.

Many classical analogies and examples are provided.

Abstract

In this survey article we give basic introduction to the theory of quantum families of maps. We begin with a general look at non-commutative (or "quantum") topology. Then we formulate all our results in this language. Existence of quantum families possessing special properties is discussed and we show that these quantum spaces are canonically endowed with quantum semigroup structures. Classical analogy is emphasized at each step and many examples are described.