Monads Defined by Involution - Preserving Adjunctions

Consider categories with involutions which fix objects, functors which preserve involution, and natural transformations. In this setting certain natural adjunctions become universal and, thereby, become constructible from abstract data. Although the formal theory of monads fails to apply and the Eilenberg-Moore category fails to fit, both are successfully adapted to this setting, which is a 2-category. In this 2-category, each monad (= triple = standard construction) defined by an adjunction is characterized by a pair of special equations. Special monads have universal adjunctions which realize them and have both underlying Frobenius monads and adjoint monads. Examples of monads which do (respectively, do not) satisfy the special equations arise from finite monoids (= semigroups with unit) which are (respectively, are not) groups acting on the category of linear transformations between finite dimensional Euclidean (= positive definite inner product) spaces over the real numbers. More general situations are exposed. This paper adapts the Eilenberg-Moore category [1], [4, p. 136] and a formal generalization [7] to a setting (categories with involution) where the ordinary construction is not appropriate. For example, to construct Euclidean (= positive definite inner product vector) spaces over the complex numbers C from those over the real numbers R, we select special Eilenberg-Moore algebras for a monad (= triple) on the category E(R) of R-linear transformations between Euclidean spaces over R, where the monad T is defined by the adjunction corresponding to the forgetful functor E(C) —*■ E(R). "Externally", the specialalgebras are those in the image of the comparison functor K: E(C) —► E(R)T; thus an algebra fails to be special if scalar multiplication by z E C with modulus \z\ = 1 does not preserve distance, for example, the algebra (R © R, h) with action / • (a, b) = (a + 2b, a b) fails. Hence, the adjunction for E(C) —» E(R) is not monadic (= tripleable), that is, not terminal with respect to all adjunctions defining the monad T. Fortunately, the adjunction is terminal in the appropriate Received by the editors September 23, 1974. AMS (MOS) subject classifications (1970). Primary 18C1S, 18D99, 18A40, 18D0S; Secondary 20C99, 20M30.