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FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES AND SOME ALGEBRAIC PROBLEMS IN THE CONTEXT OF FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES F. WILLIAM LAWVERE c F. William Lawvere, 1963, 1968. Permission to copy for private use granted. Contents A Author’s comments 1 2 3 4 5 6 6 Seven ideas introduced in the 1963 thesis . . . Delays and Developments . . . . . . . . . . . Comments on the chapters of the 1963 Thesis Some developments related to the problem list Concerning Notation and Terminology . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in the 1968 Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References B I 8 9 10 17 18 20 20 Functorial Semantics of Algebraic Theories 23 Introduction 24 The 1 2 3 26 26 38 58 category of categories and adjoint functors The category of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Regular epimorphisms and monomorphisms . . . . . . . . . . . . . . . . . II Algebraic theories 61 1 The category of algebraic theories . . . . . . . . . . . . . . . . . . . . . . . 61 2 Presentations of algebraic theories . . . . . . . . . . . . . . . . . . . . . . . 69 III Algebraic categories 74 1 Semantics as a coadjoint functor . . . . . . . . . . . . . . . . . . . . . . . . 74 2 Characterization of algebraic categories . . . . . . . . . . . . . . . . . . . . 81 IV Algebraic functors 90 1 The algebra engendered by a prealgebra . . . . . . . . . . . . . . . . . . . 90 2 Algebraic functors and their adjoints . . . . . . . . . . . . . . . . . . .