  Photonic crystals (PhCs) have emerged as a popular platform for realizing
various topological phases due to their flexibility and potential for device
applications. In this article, we present a comprehensive classification of
topological bands in one- and two dimensional photonic crystals, with and
without time-reversal symmetry. Our approach exploits the symmetry
representations of field eigenmodes at high-symmetry points in momentum space,
allowing for the efficient design of a wide range of topological PhCs. In
particular, we show that the complete classification provided here is useful
for diagnosing photonic crystal analogs of obstructed atomic limits, fragile
phases, and stable topological phases that include bands with Dirac points and
Chern numbers.
