Collections are omnipresent within models: collections of references can represent relations between objects, and collections of values can represent object attributes. Consequently, manipulating models often consists of performing operations on collections. For example, transformations create target collections from given source collections. Similarly, constraint evaluations perform computation on collections. Recent research works focus on making such transformations or constraint evaluations active (i.e. incremental, or live). However, they propose their own solutions to the issue by the introduction of specific languages and/or systems. This paper proposes a mathematical formalism, centered on collections and independent of languages and systems, that describes how the implementation of standard operations on collections can be made active. The formalism also introduces a reversed active assignment dedicated to bidirectional operations. A case study illustrates how to use the formalism and its Active Kermeta implementation for creating an active transformation.