Supremum with respect to an object

Question

In my Real Analysis class I've often seen the Upper Riemann Sum (in the definition of Riemann integrability) defined as: $$\sup_P\{U(f,P)\}$$ This seems to mean the supremum of all possible upper sums (corresponding to all possible partitions of the interval $[a,b]$. Of course, what we're changing to get the elements of this set is the partition. My question is are there other instances where the "supremum with respect to" notation is used? And am I thinking of this correctly as $P$ being a parameter here, the same way $i$ is a parameter in a sum?