Contramodules

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present review various definitions results related to (drawing mostly from our monographs preprints arXiv:0708.3398, arXiv:0905.2621, arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230, arXiv:1705.04960, arXiv:1808.00937) - including corings, topological associative Lie algebras groups, semicontramodules semialgebras, "contra version" Bernstein-Gelfand-Gelfand category O. Several underived manifestations comodule-contramodule correspondence phenomenon discussed.