  We prove axiomatic characterizations of several important multiwinner rules
within the class of approval-based committee choice rules. These are voting
rules that return a set of (fixed-size) committees. In particular, we provide
axiomatic characterizations of Proportional Approval Voting, the
Chamberlin--Courant rule, and other Thiele methods. These rules share the
important property that they satisfy an axiom called consistency, which is
crucial in our characterizations.
