In large scale distributed systems, shared objects provide a valuable abstraction of communication. However, these objects can only be used reliably if they are specified precisely. Until now, a lot of work has been done on shared memory, to the detriment of other objects. This paper aims at extending this work to any update-query abstract data type, the types in which the operations are updates or queries. A shared object should be fully specified by two complementary aspects: a sequential specification that defines how the updates influence the queries, and a consistency criterion that discriminates which distributed histories are eligible according to the sequential specification. This paper formalizes the notions of sequential specification and consistency criterion. It then extends the definition of many consistency criteria, including causal consistency, to all update-query abstract data types. It also explores the notion of composability for consistency criteria and proves that no consistency criterion between pipelined consistency and sequential consistency is composable, which includes causal consistency.