Monads defined by involution-preserving adjunctions

Author:
Paul H. Palmquist

Journal:
Trans. Amer. Math. Soc. **213** (1975), 79-87

MSC:
Primary 18C15

DOI:
https://doi.org/10.1090/S0002-9947-1975-0376811-8

MathSciNet review:
0376811

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Abstract: 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0376811-8

Keywords:
Category with involution,
monad,
adjunction,
2-category,
-monad,
Eilenberg-Moore category,
Kleisli category,
2-category,
adjoint monad,
Frobenius monad,
-category

Article copyright:
© Copyright 1975
American Mathematical Society