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303 pages , Originally published: Lecture Notes in Mathematics, Volume 80, Springer-Verlag, 1969.

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Seminar on Triples and Categorical Homology Theory Lecture Notes in Mathematics, Volume 80 Edited by Beno Eckmann and Myles Tierney Originally published: Lecture Notes in Mathematics, Volume 80, Springer--Verlag, 1969. MSC 2000: 18C05, 18C15, 18E25, 18G10 Keywords: Triples, Homology, Equational Categories Republished in: Reprints in Theory and Applications of Categories, No. 18 (2008) pp. 1--303. Authors' Contents: "Introduction p.6 F. E. J. Linton: An Outline of Functorial Semantics p.11 1. Introduction to algebras in general categories. . . . . . . . . . . . . . 11 2. General plan of the paper. . . . . . . . . . . . . . . . . . . . 15 3. Preliminary structure-semantics adjointness relation. . . . . . . . . . 16 4. Full images and the operational structure-semantics adjointness theorem. . . 19 5. Remarks on Section 4. . . . . . . . . . . . . . . . . . . . . 20 6. Two constructions in algebras over a clone. .. . . . . . . 22 7. Constructions involving triples. . . .. . . . . . . . . . . . . . . 24 8. Codensity triples. . . . . . . . . . . . . . . . . 27 9. The isomorphism theorem . . . . . . . . . . . . . . . . . .. . . . 32 10. Structure and semantics in the presence of a triple. . . . . . . . 36 11. Another proof of the isomorphism theorem . . . . . . . . . 39 F. E. J. Linton: Applied Functorial Semantics, II 44 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 44 1. The precise triple--ableness theorem. . . . . . . . . . . . . . . . . 44 2. When A has enough kernel pairs. . .. . . . . . . . . . . . . . . . 47 3. When A is very like f sets g. . .. . . . . . . . . . 50 4. Proof of Theorem 4. . . . . . . . . . . . . . . . . . . . . . . . . . 53 5. Applications. . . . . . . . . . . . . . . . . . . . 56 F. E. J. Linton: Coequalizers in Categories of Algebras 61 1. Constructions using coequalizers of reflexive pairs. . . . . . . . . . 62 2. Criteria for the existence of such coequalizers. . . . . . . . . . . . . 67 F. E. J. Linton: Coequalizers in Categories of Algebras p.61 Ernest Manes: A Triple Theoretic Construction of Compact Algebras 73 1. Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . 73 2. Algebras over a triple. . . . . . . . . . . . . . . . . . 78 3. Birkhoff subcategories . . . . . . . . . . . . . . . . . . 81 4. The category K(T;eT) .. . . . . . . . . . . . . . . . . . . . . . . 83 5. Compact spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6. Operations . . . . . . . . . . . . . . . . . . . 88 7. Compact algebras . . . .. . . . . . . . . . . . . . . . . . . . . 92 Jon Beck: Distributive Laws 95 1. Distributive laws, composite and lifted triples . 2. Algebras over the composite triple . . . . . . . . . . . . . . . 101 3. Distributive laws and adjoint functors . . . . .. . . . . 104 4. Examples . . . . . . . . . . . . . . . 107 5. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 110 F. William Lawvere: Ordinal Sums and Equational Doctrines 113" http://www.tac.mta.ca/tac/reprints/articles/18/tr18.dvi http://www.tac.mta.ca/tac/reprints/articles/18/tr18.ps http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf

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http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html Originally published in : Lecture Notes in Mathematics, Volume 80, Springer-Verlag, 1969. Republished in: Reprints in Theory and Applications of Categories, No. 18 (2008) pp. 1--303.

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