The following preprint is available Cubical monads and their symmetries Marco Grandis Abstract. This work is concerned with a setting for homotopical algebra based on a cylinder endofunctor I, and more precisely on the notion of diad (I, d-, d+, e, g-, g+), or cubical monad, or I-category with connections, possibly enriched with symmetries, reversion r: I -> I and interchange s: I^2 -> I^2. Generalised symmetries, relative to involutive endofunctors R, S and applying for instance to cubical objects (with connections) or differential graded algebras, are also considered. A few basic developments of this setting are given here. In particular: The associated monads (C-, d, g), (C+, d, g) obtained by collapsing one base of the cylinder and called lower- and upper-cone; their isomophism, in the presence of a reversion; their homotopical invariance, in the presence of an interchange; and the extension of these results to generalised symmetries. A second paper will show how the Puppe sequence of a map works in this setting, as a consequence of connections and interchange. Marco Grandis Dipartimento di Matematica Universita' di Genova Via L.B. Alberti 4, Italia grandis@cartesio.dima.unige.it ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++