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 6 1 Seven ideas introduced in the 1963 thesis . 8 2 DelaysandDevelopments........................... 9 3 Comments on the chapters of the 1963 Thesis . 10 4 Some developments related to the problem list in the 1968 Article . 17 5 ConcerningNotationandTerminology.................... 18 6 Outlook..................................... 20 References 20 B Functorial Semantics of Algebraic Theories 23 Introduction 24 I The category of categories and adjoint functors 26 1 Thecategoryofcategories........................... 26 2 Adjointfunctors................................. 38 3 Regular epimorphisms and monomorphisms . 58 II Algebraic theories 61 1 Thecategoryofalgebraictheories....................... 61 2 Presentationsofalgebraictheories....................... 69 III Algebraic categories 74 1 Semanticsasacoadjointfunctor........................ 74 2 Characterizationofalgebraiccategories.................... 81 IV Algebraic functors 90 1 Thealgebraengenderedbyaprealgebra................... 90 2 Algebraicfunctorsandtheiradjoints..................... 93 3 CONTENTS 4 V Certain 0-ary and unary extensions of algebraic theories 97 1 Presentations of algebras: polynomial algebras . 97 2 Monoidsofoperators..............................102 3 Rings of operators (Theories of categories of modules) . 105 References 106 C Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories 108 1 Basicconcepts..................................109 2 Methodologicalremarksandexamples....................112 3 Solvedproblems.................................115 4 Unsolvedproblems...............................118 5 Completionproblems..............................119 References 120 Part A Author’s comments 6 7 The 40th anniversary of my doctoral thesis was a theme at the November 2003 Flo- rence meeting on the “Ramifications of Category Theory”. Earlier in 2003 the editors of TAC had determined that the thesis and accompanying problem list should be made available through TAC Reprints. This record delay in the publication of a thesis (and with it a burden of guilt) is finally coming to an end. The saga began when in January 1960, having made some initial discoveries (based on reading Kelley and Godement) such as adjoints to inclusions (which I called “inductive improvements”) and fibered categories (which I called “galactic clusters” in an extension of Kelley’s colorful terminology), I bade farewell to Professor Truesdell in Bloomington and traveled to New York. My dream, that direct axiomatization of the category of categories would help in overcoming alleged set- theoretic difficulties, was naturally met with skepticism by Professor Eilenberg when I arrived (and also by Professor Mac Lane when he visited Columbia). However, the con- tinuing patience of those and other professors such as Dold and Mendelsohn, and instruc- tors such as Bass, Freyd, and Gray allowed me to deepen my knowledge and love for al- gebra and logic. Professor Eilenberg even agreed to an informal leave which turned out to mean that I spent more of my graduate student years in Berkeley and Los Angeles than in New York. My stay in Berkeley tempered the naive presumption that an impor- tant preparation for work in the foundations of continuum mechanics would be to join the community whose stated goal was the foundations of mathematics. But apart from a few inappropriate notational habits, my main acquisition from the Berkeley sojourn was a more profound acquaintance with the problems and accomplishments of 20th century logic, thanks again to the remarkable patience and tolerance of professors such as Craig, Feferman, Scott, Tarski, and Vaught. Patience began to run out when in February 1963, wanting very much to get out of my Los Angeles job in a Vietnam war “think” tank to take up a teaching position at Reed College, I asked Professor Eilenberg for a letter of recommendation. His very brief reply was that the request from Reed would go into his waste basket unless my series of abstracts be terminated post haste and replaced by an actual thesis. This tough love had the desired effect within a few weeks, turning the ta- bles, for it was now he who had the obligation of reading a 120-page paper of baroque notation and writing style. (Saunders Mac Lane, the outside reader, gave the initial ap- proval and the defence took place in Hamilton Hall in May 1963.) The hasty prepara- tion had made adequate proofreading difficult; indeed a couple of lines (dealing with the relation between expressible and definable constants) were omitted from the circulated version, causing consternation and disgust among universal algebraists who tried to read the work. Only in the new millennium did I discover in my mother’s attic the original handwritten draft, so that now those lines can finally be restored. Hopefully other ob- scure points will be clarified by this actual publication, for which I express my gratitude to Mike Barr, Bob Rosebrugh, and all the other editors of TAC, as well as to Springer- Verlag who kindly consented to the republication of the 1968 article. 1 Seven ideas introduced in the 1963 thesis 8 1. Seven ideas introduced in the 1963 thesis (1) The category of categories is an accurate and useful framework for algebra, geometry, analysis, and logic, therefore its key features need to be made explicit. (2) The construction of the category whose objects are maps from a value of one given functor to a value of another given functor makes possible an elementary treatment of adjointness free of smallness concerns and also helps to make explicit both the existence theorem for adjoints and the calculation of the specific class of adjoints known as Kan extensions. (3) Algebras (and other structures, models, etc.) are actually functors to a background category from a category which abstractly concentrates the essence of a certain general concept of algebra, and indeed homomorphisms are nothing but natural transformations between such functors. Categories of algebras are very special, and explicit axiomatic characterizations of them can be found, thus providing a general guide to the special fea- tures of construction in algebra. (4) The Kan extensions themselves are the key ingredient in the unification of a large class of universal constructions in algebra (as in [Chevalley, 1956]). (5) The dialectical contrast between presentations of abstract concepts and the abstract concepts themselves, as also the contrast between word problems and groups, polynomial calculations and rings, etc. can be expressed as an explicit construction of a new adjoint functor out of any given adjoint functor. Since in practice many abstract concepts (and algebras) arise by means other than presentations, it is more accurate to apply the term “theory”, not to the presentations as had become traditional in formalist logic, but rather to the more invariant abstract concepts themselves which serve a pivotal role, both in their connection with the syntax of presentations, as well as with the semantics of rep- resentations. (6) The leap from particular phenomenon to general concept, as in the leap from coho- mology functors on spaces to the concept of cohomology operations, can be analyzed as a procedure meaningful in a great variety of contexts and involving functorality and natu- rality, a procedure actually determined as the adjoint to semantics and called extraction of “structure” (in the general rather than the particular sense of the word). (7) The tools implicit in (1)–(6) constitute a “universal algebra” which should not only be polished for its own sake but more importantly should be applied both to constructing more pedagogically effective unifications of ongoing developments of classical algebra, and to guiding of future mathematical research. In 1968 the idea summarized in (7) was elaborated in a list of solved and unsolved problems, which is also being reproduced here. 2 Delays and Developments 9 2. Delays and Developments The 1963 acceptance of my Columbia University doctoral dissertation included the con- dition that it not be published until certain revisions were made. I never learned what exactly those revisions were supposed to be. Four years later, at the 1967 AMS Summer meeting in Toronto, Sammy had thoroughly assimilated the concepts and results of Func- torial Semantics of Algebraic Theories and had carried them much further; one of his four colloquium lectures at that meeting was devoted to new results in that area found in col- laboration with [Eilenberg & Wright, 1967]. In that period of intense advance, not only Eilenberg and Wright, but also [Beck, 1967], [B´enabou, 1968], [Freyd, 1966], [Isbell, 1964], [Linton, 1965], and others, had made significant contributions. Thus by 1968 it seemed that any publication (beyond my announcements of results [Lawvere, 1963, 1965]) should not only correct my complicated proofs, but should also reflect the state of the art, as well as indicate more systematically the intended applications to classical algebra, alge- braic topology, and analysis. A book adequate to that description still has not appeared, but Categories and Functors [Pareigis, 1970] included an elegant first exposition. Ernie Manes’ book called Algebraic Theories, treats mainly the striking advances initiated by Jon Beck, concerning the Godement-Huber-Kleisli