Coherent extensions and relational algebras

Bunge, Marta C.

doi:10.1090/S0002-9947-1974-0344305-0

Coherent extensions and relational algebras
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by Marta C. Bunge PDF

Trans. Amer. Math. Soc. 197 (1974), 355-390 Request permission

Abstract:

The notion of a lax adjoint to a 2-functor is introduced and some aspects of it are investigated, such as an equivalent definition and a corresponding theory of monads. This notion is weaker than the notion of a 2-adjoint (Gray) and may be obtained from the latter by weakening that of 2functor and replacing the adjointness equations by adding 2-cells satisfying coherence conditions. Lax monads are induced by and resolve into lax adjoint pairs, the latter via 2-categories of lax algebras. Lax algebras generalize the relational algebras of Barr in the sense that a relational algebra for a monad in Sets is precisely a lax algebra for the lax monad induced in Rel. Similar considerations allow us to recover the T-categories of Burroni as well. These are all examples of lax adjoints of the “normalized” sort and the universal property they satisfy can be expressed by the requirement that certain generalized Kan extensions exist and are coherent. The most important example of relational algebras, i.e., topological spaces, is analysed in this new light also with the purpose of providing a simple illustration of our somewhat involved constructions.

References

Michael Barr, Relational algebras, Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, Vol. 137, Springer, Berlin, 1970, pp. 39–55. MR 0262140
Jean Bénabou, Introduction to bicategories, Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77. MR 0220789
Marta C. Bunge, Relative functor categories and categories of algebras, J. Algebra 11 (1969), 64–101. MR 236238, DOI 10.1016/0021-8693(69)90102-1
Marta C. Bunge, Bifibration induced adjoint pairs, Reports of the Midwest Category Seminar, V (Zurich, 1970) Lecture Notes in Math., Vol. 195, Springer, Berlin, 1971, pp. 70–122. MR 0292904
Albert Burroni, $T$-catégories (catégories dans un triple), Cahiers Topologie Géom. Différentielle 12 (1971), 215–321 (French). MR 308236
Samuel Eilenberg and John C. Moore, Adjoint functors and triples, Illinois J. Math. 9 (1965), 381–398. MR 184984
John W. Gray, The $2$-adjointness of the fibred category construction, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 457–492. MR 0269717
Peter J. Huber, Homotopy theory in general categories, Math. Ann. 144 (1961), 361–385. MR 150184, DOI 10.1007/BF01396534
Daniel M. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958), 294–329. MR 131451, DOI 10.1090/S0002-9947-1958-0131451-0
Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
Ernest Manes, A triple theoretic construction of compact algebras, Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, pp. 91–118. MR 0244342
Ross Street, Two constructions on lax functors, Cahiers Topologie Géom. Différentielle 13 (1972), 217–264. MR 347936