  The notion of comparison between system runs is fundamental in formal
verification. This concept is implicitly present in the verification of
qualitative systems, and is more pronounced in the verification of quantitative
systems. In this work, we identify a novel mode of comparison in quantitative
systems: the online comparison of the aggregate values of two sequences of
quantitative weights. This notion is embodied by comparator automata
(comparators, in short), a new class of automata that read two infinite
sequences of weights synchronously and relate their aggregate values.
  We show that aggregate functions that can be represented with B\"uchi
automaton result in comparators that are finite-state and accept by the B\"uchi
condition as well. Such $\omega$-regular comparators further lead to generic
algorithms for a number of well-studied problems, including the quantitative
inclusion and winning strategies in quantitative graph games with incomplete
information, as well as related non-decision problems, such as obtaining a
finite representation of all counterexamples in the quantitative inclusion
problem.
  We study comparators for two aggregate functions: discounted-sum and
limit-average. We prove that the discounted-sum comparator is $\omega$-regular
iff the discount-factor is an integer. Not every aggregate function, however,
has an $\omega$-regular comparator. Specifically, we show that the language of
sequence-pairs for which limit-average aggregates exist is neither
$\omega$-regular nor $\omega$-context-free. Given this result, we introduce the
notion of prefix-average as a relaxation of limit-average aggregation, and show
that it admits $\omega$-context-free comparators i.e. comparator automata
expressed by B\"uchi pushdown automata.
