Advances in Mathematics 307 (2017) 253–287 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim Stone duality above dimension zero: Axiomatising the algebraic theory of C(X) Vincenzo Marra a,∗, Luca Reggio a,b a Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy b Université Paris Diderot, Sorbonne Paris Cité, IRIF, Case 7014, 75205 Paris Cedex 13, France a r t i c l e i n f o a b s t r a c t Article history: It has been known since the work of Duskin and Pelletier Received 17 October 2015 four decades ago that Kop, the opposite of the category of Received in revised form 4 October compact Hausdorff spaces and continuous maps, is monadic 2016 over the category of sets. It follows that Kop is equivalent Accepted 14 November 2016 to a possibly infinitary variety of algebras Δ in the sense of Available online xxxx Communicated by Ross Street Słomiński and Linton. Isbell showed in 1982 that the Lawvere– Linton algebraic theory of Δ can be generated using a finite MSC: number of finitary operations, together with a single operation primary 03C05 of countably infinite arity. In 1983, Banaschewski and Rosický secondary 06D35, 06F20, 46E25 independently proved a conjecture of Bankston, establishing a strong negative result on the axiomatisability of Kop. In Keywords: particular, Δ is not a finitary variety – Isbell’s result is best Algebraic theories possible. The problem of axiomatising Δ by equations has Stone duality remained open. Using the theory of Chang’s MV-algebras as Boolean algebras MV-algebras a key tool, along with Isbell’s fundamental insight on the Lattice-ordered Abelian groups semantic nature of the infinitary operation, we provide a finite C∗-algebras axiomatisation of Δ. Rings of continuous functions © 2016 Elsevier Inc. All rights reserved. Compact Hausdorff spaces Stone–Weierstrass Theorem Axiomatisability * Corresponding author. E-mail addresses: [email protected] (V. Marra), [email protected] (L. Reggio). http://dx.doi.org/10.1016/j.aim.2016.11.012 0001-8708/© 2016 Elsevier Inc. All rights reserved. 254 V. Marra, L. Reggio / Advances in Mathematics 307 (2017) 253–287 1. Introduction In the category Set of sets and functions, coordinatise the two-element set as 2 := {0, 1}. Consider the full subcategory T2 of Set whose objects are the finite Cartesian powers 2n. Equipped with distinguished projection functions 2n → 2 to play the rôle of variables, T2 is the Lawvere algebraic theory [35] of Boolean algebras. That is, the category [T2, Set]fpp of all functors from T2 to Set that preserve finite products is the variety of Boolean algebras. This variety is finitary, as witnessed by the fact that n only ranges over finite sets. The full subcategory of free finitely generated Boolean algebras is then dually equivalent to T2, by (Lawvere’s) construction; consequently, the full sub- category of finitely presented Boolean algebras is dually equivalent to the category Setf of finite sets. Quite generally, any finitary variety is equivalent to Grothendieck’s Ind- completion of its full subcategory of finitely presented algebras [30, Corollary VI.2.2]. Therefore the category of Boolean algebras is dually equivalent to the Pro-completion of Setf , which, by a folklore theorem, can be identified with the category St of com- pact Hausdorff zero-dimensional spaces and the continuous maps between them. This is Stone’s celebrated duality theorem for Boolean algebras [49]. The objects of St are called Stone spaces. Now enlarge 2 to the real unit interval I := [0, 1] ⊆ R, and regard I endowed with its Euclidean topology as an object in the category Top of topological spaces and continuous maps. Consider the full subcategory T of Top whose objects are the countable Cartesian powers Iκ equipped with the product (Tychonoff) topology. With distinguished projec- tions Iκ → I, the category T is then the Lawvere–Linton algebraic theory [36] of some variety of algebras. That is, the category [T, Set]cpp of all functors from T to Set that preserve countable products is a variety; but this time the variety, as described here, is infinitary [48], in that κ need not be finite. Thus, just like Boolean algebras are the models in Set of the algebraic theory of the finite Boolean cubes 2n, so are the alge- bras in [T, Set]cpp the models in Set of the infinitary algebraic theory of the metrisable Tychonoff cubes Iκ. In a terse and important paper [29] (see also [46]), Isbell showed that [T, Set]cpp is dually equivalent to K, the full subcategory of Top whose objects are compact and Hausdorff spaces. Previously, Duskin [16, 5.15.3] had shown that, indeed, Kop must be equivalent to some possibly infinitary variety of algebras. Isbell’s result fur- ther shows that such a variety can be defined using finitely many finitary operations, along with a single operation of countably infinite arity. The algebras in [T, Set]cpp freely generated by a set S have C(IS, I), the collection of all continuous functions IS → I, as their underlying sets. Isbell’s operation is then the uniformly convergent series ∞ f i , for f ∈ C(IS,I). (1) 2i i i=1 Download English Version: https://daneshyari.com/en/article/5778681 Download Persian Version: https://daneshyari.com/article/5778681 Daneshyari.com.