  In this paper we provide a flexible framework allowing for a unified study of
time consistency of risk measures and performance measures (also known as
acceptability indices). The proposed framework not only integrates existing
forms of time consistency, but also provides a comprehensive toolbox for
analysis and synthesis of the concept of time consistency in decision making.
In particular, it allows for in depth comparative analysis of (most of) the
existing types of time consistency -- a feat that has not be possible before
and which is done in the companion paper [BCP2016] to this one. In our approach
the time consistency is studied for a large class of maps that are postulated
to satisfy only two properties -- monotonicity and locality. The time
consistency is defined in terms of an update rule. The form of the update rule
introduced here is novel, and is perfectly suited for developing the unifying
framework that is worked out in this paper. As an illustration of the
applicability of our approach, we show how to recover almost all concepts of
weak time consistency by means of constructing appropriate update rules.
