Monads defined by involution-preserving adjunctions
Paul H. Palmquist
Trans. Amer. Math. Soc. 213 (1975), 79-87
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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.
Eilenberg and John
C. Moore, Adjoint functors and triples, Illinois J. Math.
9 (1965), 381–398. MR 0184984
Kleisli, Every standard construction is induced
by a pair of adjoint functors, Proc. Amer.
Math. Soc. 16
(1965), 544–546. MR 0177024
(31 #1289), http://dx.doi.org/10.1090/S0002-9939-1965-0177024-4
William Lawvere, Ordinal sums and equational doctrines, Sem.
on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67),
Springer, Berlin, 1969, pp. 141–155. MR 0240158
MacLane, Categories for the working mathematician,
Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol.
0354798 (50 #7275)
H. Palmquist, The double category of adjoint squares, Reports
of the Midwest Category Seminar, V (Zürich, 1970) Lecture Notes in
Mathematics, Vol. 195, Springer, Berlin, 1971, pp. 123–153. MR 0289600
-, Baer -categories (in preparation).
Street, The formal theory of monads, J. Pure Appl. Algebra
2 (1972), no. 2, 149–168. MR 0299653
- S. Eilenberg and J. C. Moore, Adjoint functors and triples, Illinois J. Math. 9 (1965), 381-398. MR 32 #2455. MR 0184984 (32:2455)
- H. Kleisli, Every standard construction is induced by a pair of adjoint functors, Proc. Amer. Math. Soc. 16 (1965), 544-546. MR 31 #1289. MR 0177024 (31:1289)
- F. W. Lawvere, Ordinal sums and equational doctrines, Seminar on Triples and Categorical Homology Theory, Lecture Notes in Math., vol. 80, Springer-Verlag, New York, 1969, pp. 141-155. MR 39 #1512. MR 0240158 (39:1512)
- S. MacLane, Categories for the working mathematician, Springer-Verlag, New York, 1971. MR 0354798 (50:7275)
- P. H. Palmquist, The double category of adjoint squares, Reports of the Midwest Category Seminar, V, Lecture Notes in Math., vol. 195, Springer-Verlag, New York, 1971. MR 44 #6788. MR 0289600 (44:6788)
- -, Baer -categories (in preparation).
- R. Street, The formal theory of monads, J. Pure Appl. Algebra 2 (1972), 149-168. MR 45 #8701. MR 0299653 (45:8701)
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