  Norm-compatible families of cohomology classes for Shimura varieties, and
other arithmetic symmetric spaces, play an important role in Iwasawa theory of
automorphic forms. The aim of this note is to give a systematic approach to
proving "vertical" norm-compatibility relations for such classes (where the
level varies at a fixed prime p), treating the case of Betti and \'etale
cohomology at once, and revealing an unexpected relation to the theory of
spherical varieties. This machinery can be used to construct many new examples
of norm-compatible families, potentially giving rise to new constructions of
both Euler systems and p-adic L-functions: examples include families of
algebraic cycles on Shimura varieties for U(n) x U(n+1) and U(2n) over the
p-adic anticyclotomic tower.
