Constructive Aspects of Gelfand Duality Christopher Mulvey University of Sussex www.maths.sussex.ac.uk/Staff/CJM c.j.mulvey@cantab.net
Background In this talk, I will tell you something about Gelfand duality, seen through the eyes of the constructive world. The work that I'll be outlining dates from almost exactly thirty years ago, when it began to be realised that the construction of locales out of geometric theories could be used to provide constructive versions of mathematical results that classically had depended upon, indeed often been almost equivalent to, the Axiom of Choice. The beginnings of this work in fact lay in events of almost exactly ten years earlier, in the earliest days of topos theory. At the International Congress of Mathematicians in Nice in 1970, Lawvere had given an impressive talk outlining the significant ideas that were then unfolding and the contributions that they were likely to make. In talking about these developments he had wanted to emphasise the unity of the opposites of geometry and logic, by indicating an aspect of each to which the emerging theory applied. The first was to the internalisation of the construction of the spectrum of a commutative ring, which had been analysed by Monique Hakim to provide an important part of Grothendieck's contributions to algebraic geometry. The second was to the independence of the continuum hypothesis, that applied the topos theoretic context to yield a categorical approach to the proof.
Background Although the insights needed in each case were already present, the detail needed further development. In each case, this eventually came with the realisation of the centrality and importance of the theory of locales, of which we have already heard certain aspects, to the theory of toposes. In the case of the spectrum of a commutative ring A , Lawvere had correctly noted that the points intended were not the prime ideals of the ring, but their complements, which I shall refer to as the primes of the ring. In the constructive context of a topos, these had their own simple axiomatisation, as those subsets P satisfying the following conditions: true d 1 c P 0 c P d false a + b c P d a c P - b c P ab c P dw a c P . b c P a , b c A for all . That we have written the axioms in this form will have significance in a moment.
Background The assertion made at that time, that the spectrum Spec A of a commutative ring A in a topos was exactly its space of primes, was misguided. The correct approach to the concept of spectrum, although not yet its correct context, had already been introduced by Joyal prior to a talk given at Oberwolfach in 1972 in which he described the spectrum as being the universal support of the commutative ring, in the sense of a mapping D : A d L from A to a distributive lattice L satisfying universally the conditions: 1 L [ D ( 1 ) D ( 0 ) [ 0 L D ( a + b ) [ D ( a ) - D ( b ) D ( ab ) = D ( a ) . D ( b ) a , b c A for all . It may be noted that these are just the conditions satisfied classically by the basis for Zariski topology on the spectrum. It may also be noted that the axioms for a prime are just a transcription of these conditions into another notation.
Background With the realisation of the importance of the theory of locales, the concept of spectrum of a commutative ring A introduced by Joyal could be rephrased as a mapping D : A d L from A to a locale L satisfying universally with respect to homomorphisms the conditions: 1 L [ D ( 1 ) D ( 0 ) [ 0 L D ( a + b ) [ D ( a ) - D ( b ) D ( ab ) = D ( a ) . D ( b ) a , b c A for all . In the case of a commutative ring A in the topos of sets, the locale created is exactly that of the Zariski topology on the set of primes (or equivalently, prime ideals) of the ring. More importantly, it may be shown that in any topos the locale plays the role of the spectrum of A .
Background Although in this case the concept of the spectrum Spec A as the universal support of the commutative ring A was arrived at pragmatically, it may equally be obtained canonically as the Lindenbaum locale of the propositional geometric theory that introduces a proposition a c P a c A for each together with the axioms: true d 1 c P 0 c P d false a + b c P d a c P - b c P ab c P dw a c P . b c P a , b c A for all that we have already met. In other words, once one has the constructive form of the classical points of the spectrum, then forming the constructive version of the classical space is straightforward: one just takes the Lindenbaum locale of the theory of its constructive points, namely the locale obtained by taking the propositions generated by the theory, modulo provable equivalence within the theory, ordered by provable entailment within the theory.
Background The second application outlined by Lawvere was a sketch of a proof of the independence of the continuum hypothesis by topos theoretic means. The dénouement of this approach, in terms of the detail, also came at the Oberwolfach meeting in 1972, when it was pointed out that the concept of real number on which it was based was not quite that intended. That described by Tierney in his talk defined in the topos of sheaves on a space not the sheaf of continuous real functions, but instead the sheaf of upper semi-continuous real functions on the space. On this occasion, inadequate attention had been given to the importance constructively of x c ‘ defining a real number by describing both its lower cut and its upper cut on the . ( - x ) x c ‘ rationals. Explicitly, a real number needed to be defined as a pair of subsets of the x , rationals satisfying the conditions: . ≥ p p c - ≥ q q c x x x . p ∏ < p d p ∏ c - x . q < q ∏ d q ∏ c . . p c - q c x x x d ≥ p ∏ > p p ∏ c - x d ≥ q ∏ < q q ∏ c . . p c - q c x x . . x d p < q p < q d p c - p c - x . q c x - q c . x
Background With the constructive axiomatisation of the real numbers in place, the same approach as in the case of the spectrum may be applied to construct the locale of real numbers by writing down in any topos with natural numbers a propositional geometric theory of which the real numbers in the topos are the canonical models, hence the points of the locale described by that theory. The propositional geometric theory of real numbers is defined by introducing for each pair r , s of rational numbers in the topos a proposition x c ( r , s ) , together with the following axioms: x c ( r , s ) d false ( r , s ) [ 0 whenever ; true d - ( r , s ) x c ( r , s ) ; x c ( r , s ) d x c ( p , q ) - x c ( p ∏ , q ∏ ) ( r , s ) ü ( p , q ) - ( p ∏ , q ∏ ) whenever ; x c ( p , q ) . x c ( p ∏ , q ∏ ) d x c ( r , s ) ( p , q ) . ( p ∏ , q ∏ ) ü ( r , s ) whenever ; x c ( r , s ) dw - ( r ∏ , s ∏ ) ü ( r , s ) x c ( r ∏ , s ∏ ) . The locale of real numbers in the topos is then the Lindenbaum locale of this theory. ‘
Background Similarly, consider the propositional geometric theory defined by introducing a proposition z c ( r , s ) for each pair of complex rationals, together with the following axioms: r , s z c ( r , s ) d false ( r , s ) [ 0 whenever ; true d - ( r , s ) z c ( r , s ) ; z c ( r , s ) d z c ( p , q ) - z c ( p ∏ , q ∏ ) ( r , s ) ü ( p , q ) - ( p ∏ , q ∏ ) whenever ; z c ( p , q ) . z c ( p ∏ , q ∏ ) d z c ( r , s ) ( p , q ) . ( p ∏ , q ∏ ) ü ( r , s ) ; whenever z c ( r , s ) dw - ( r ∏ , s ∏ ) ü ( r , s ) z c ( r ∏ , s ∏ ) , in which the conditions are to be interpreted in the geometry of the complex rational plane, expressed algebraically. Then the Lindenbaum locale of the theory is the locale of complex numbers. In each case, Š the points of the relevant locale are respectively the real numbers and the complex numbers of the topos. Equivalently, these are exactly the canonical models of the respective theory.
Background The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many of the techniques used. The motivation and the outcome were linked talks at the International Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras:
Background The Gelfand duality theorem in its constructive form was proved in ten days of intensive work with Banaschewski over the summer of 1980, following similarly intense work with Wick-Pelletier on the constructive form of the Hahn-Banach theorem, which developed many of the techniques used. The motivation and the outcome were linked talks at the International Meeting on Categorical Topology held in Ottawa that summer. The ten days of work led to a widely circulated preprint of some 180 pages detailing the result, which was deemed too lengthy for publication. Over the years that followed, it was divided into four papers, inevitably coming to rather more than 180 pages, which were published individually, under the unifying title of The Spectral Theory of Commutative C*-Algebras: The Constructive Spectrum
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