FOX COMPLETIONS AND LAWVERE DISTRIBUTIONS. Introduction MARTA BUNGE

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1 FOX COMPLETIONS AND LAWVERE DISTRIBUTIONS MARTA BUNGE Contents Introduction. I. The Pure Topology and Branched Coverings II. Probability Distributions and Valuation Locales. III. Monoidal Structures via the Symmetric Monad References. Introduction Topology and Analysis meet in the theory of Fox completions and Lawvere distributions as two sides of the same coin. The bridge between Topology and Analysis is given by Algebra, through the classifier of Lawvere distributions and its extension to a Kock- Zoberlein monad. The general theory relating these three areas is systematically developed in a monograph [6]. Fox completions [10] were introduced in order to formalize a description of branched coverings as certain completions of unbranched coverings. On the other hand, Lawvere distributions on a topos [16, 17] were introduced in the spirit of the Riez paradigm, which models extensive quantities as linear functionals on intensive quantities. In principle, any question in the theory of complete spreads has a counterpart in the theory of distributions, and viceversa, while suggesting developments in the theory of KZ-monads [15, 21]. In this talk 1 I will give some instances of this principle, and mention some open questions. Let S be a topos. Denote by Top S the 2-category whose 0-cells are the toposes E bounded over S with a given structure geometric morphism e : E S, its 1-cells are geometric morphisms ϕ : F E commuting up to equivalence with the given structure morphisms, and the 2-cells ϕ ψ are the natural transformations η : ψ ϕ. A Lawvere distribution on E is an S-cocontinuous functor µ : E S, where E is regarded, not as a topos bounded over S, but as its frame, which is an S-cocomplete category. Notice that, for R the objects classifier in Top S, the frame of E is equivalent with the category of points the topos R E, and that S itself is the category of points of 2000 Mathematics Subject Classification:. Key words and phrases: toposes, locales, distributions, complete spreads, symmetric monad, comprehensive factorization, branched coverings, pure topology, purely skeletal geometric morphisms, probability distributions, valuation locales, kernels, Fourier transforms, descent theorems.. c Marta Bunge, Permission to copy for private use granted. 1 Symposium on Sets within Geometry, Nancy, July 26 29,

2 R. Thus, a Lawvere distribution is, quite appropriately, an extensive quantity modelled as a linear functional Points(R E ) Points(R) on an intensive quatity. It is an ongoing program to develop functional analysis based on this notion. A useful tool in the theory of Lawvere distributions on an S-bounded topos E is its topos classifier, also known as the symmetric topos M(E ), by a loose analogy with the symmetric algebra. Furthermore, M(E ) is part of a Kock-Zoeberlein monad on Top S, with unit the Dirac delta δ : Id M. For each E, δ E : E M(E ) is an essential geometric morphism. The symmetric monad M is a special case (closed, linear, complete) of KZmonad. These considerations have led to an abstract theory of extensive and intensive quantities regarded respectively as discrete M-fibrations and discrete M-opfibrations. To any geometric morphism ψ : F E whose domain f : F S is locally connected, there is associated the Lawvere distribution f! ψ on E, where f! f f are S-adjoints. A geometric morphism ψ : F E with a locally connected domain f : F S is said to be a Fox complete spread if it is equivalent to the left vertical arrow in the bicomma diagram given below, where p : S M(E ) is the point of M(E ) corresponding to the distribution f! ψ : E S on E. 2 F f SS ψ p E M(E ) δ E (1) By a general theorem of A. M. Pitts, It follows from the fact that δ E : E M(E ) is an essential geometric morphism that the top arrow f : F S in the bicomma diagram above is locally connected. Starting with ψ : F E (with locally connected domain f : F S), the square (1) may not be a bicomma object; however, there is always a comparison map ρ from F to the bicomma object. F ρ G g SS ϕ E ψ δ p M(E ) A geometric morphism ρ : F G between locally connected toposes is pure if the unique natural transformation f! ρ g! is an isomorphism. The complete spread ψ : G E in diagram (2) is named the Fox completion of ϕ : F E since the topos G may be thought of as the space of cogerms of the distribution µ = ϕ f! just as in topology. The completion of a spread geometric morphism (with locally connected domain) is an instance of the pure, complete spread factorization, which we refer to as the comprehensive factorization. (2)

3 The aim of this lecture is to discuss three areas in which I envisage as useful to investigate certain open questions arising from any of the three interconnected areas mentioned in this Introduction. The basic references are [6, 10, 13, 15, 16, 17, 21]. Additional references will be indicated in each section. In addition to the questions posed here, which are in fact programs of research, several specific open problems are given as exercises in [6] The Pure Topology and Branched Coverings One can argue that historically singular coverings precede locally constant coverings in the theory of Riemann surfaces, and that only on account of a desired connection with the fundamental group had additional assumptions been made on the maps, assumptions that in practice have the effect of reducing singular coverings to locally constant coverings. In fact, the familiar concept of locally constant covering is a topological concept formed from the analytical concept of a Riemann surface, or rather, that part of the Riemann surface remaining after the branch points have been deleted. In this section we briefly review what is known about branched coverings in the context of singular coverings (or complete spreads). The theory itself is in need of further developments. We point here to some specific issues in this area that could benefit from further investigation. The pure monomorphisms in an S-bounded topos E are the dense monomorphisms for a topology in E. We refer to the topology of pure monomorphisms as the pure topology in E, and to its sheaves as pure-sheaves. We denote the subtopos of pure-sheaves by E p E. If E is locally connected, so is E p is locally connected and it is the smallest pure subtopos of E. Let R denote the real numbers. Then (Sh(R)) p = Sh(R) because a pure inclusion of open intervals must be an equality. However, (Sh(R 2 )) p is a proper subtopos of Sh(R 2 ). For instance, the complement in R 2 of a curve is a pure-closed open subset of R 2. On the other hand, a punctured plane is a pure subset of R 2. However, (Sh(R n )) p (for n > 1) has no points. Indeed, a point x : Set Sh(R n ) p would give a point of R n, which we may delete producing a pure open subset V = R n {x} R n. But Sh(R n ) p is a subtopos of Sh(R n )/V. This contradicts the assumption. The category B(E ) of branched coverings over E is canonically equivalent to the category C (E p ) of locally constant coverings of its smallest pure subtopos E p. For instance, if a V is the complement of a single knot in S 3, then the knot group C (Sh(S 3 )/V )can be regarded as a full subcategory of C (Sh(S 3 ) p ). Hence, any two knot groups have an intersection, by which we mean their overlap in C (Sh(S 3 ) p ). The proof of the main theorem B(E ) = C (E p ) resorts to an intrinsic characterization of branched coverings which involves the pure analogue of the so-called skeletal geometric

4 morphisms. Explicilty, a geometric morphism is purely skeletal if its inverse image part preserves pure monomorphisms. Equivalently, a geometric morphism ψ : F E is purely skeletal if and only if it restricts to smallest pure subtoposes 4 F p E p F E ψ (3) The intrinsic characterization states that a geometric morphism is a branched covering if and only if it is a purely skeletal complete spread which in addition is a purely locally constant covering. The category of purely skeletal complete spreads over a locally connected topos E is equivalent to the category of complete spreads over E p. The following examples illustrate the notion of a purely skeletal geometric morphism. The balloon s shadow map S 2 S 2 is a purely skeletal complete spread which is not a branched covering. On the other hand, the image part of this map S 2 D, where D is a closed disk, is a purely locally constant covering and a complete spread, but it is not purely skeletal (eg., consider the interior of D). Both these maps exhibit foldings but in different ways. Intuitively, the additional two conditions on a complete spread together rule out foldings in a complete spread. It is of interest to identify the distributions corresponding to branched coverings. We only have some partial results in this direction. Let C, J denote a locally connected site for E, so that E = Sh(C, J). Consider the assignment Φ, which to a sheaf X in E, assigns the distribution corresponding to the spread completion of the local homeomorphism E /X E. This is the distribution Φ(X) = X e! : E S whose value at an object Y of E is Φ(X)(Y ) = e! (X Y ). Define d(µ)(c) = {nat. trans. h c.e! µ}. Then d(µ) is a sheaf, said to be the density (or interior) of µ. d : Dist S (S, E ) E is a functor right adjoint to Φ : E Dist S (S, E ). An object of the form d(µ) for some distribution µ is said to be a density object. A distribution µ on a locally connected topos E with e! e is called absolutely continuous if there exists an object X of E such that µ = X e!, meaning that, for every object Y of E, µ(y ) = e! (X Y ), natural in Y. Density objects in a locally connected topos E are pure-sheaves. E p is the smallest subtopos of E containing the density objects. A traditional branched covering of a locally connected topos E is a complete spread

5 5 ϕ : D E with interior dϕ whose support is S E /dϕ D D E /S E ϕ (4) such that (a) dϕ S is locally constant, (b) S is a pure subobject of 1, and (c) E /dϕ D is pure (so that ϕ is the spread completion of dϕ). Fox remarks that condition (c) rules out foldings, whereas (b) rules out cuts. Notice that (a) says that the distribution of a traditional branched covering of E is absolutely continuous. On the other hand, not every absolutely continuous distribution corresponds to a branched covering. The topos C (E p ), equivalent to the branched group B(E ), is like a coproduct of the groups C (E /V ) for the pure subobjects V 1. On the other hand, C (E p )/H, obtained as the quotient of C (E p ) by the so-called Hurewitz action by homeomorphisms, is like a colimit of the C (E /V ). Questions 1. What sort of distributions on a locally connected topos E correspond to general branched coverings over E? For which toposes E are the absolutely continuous distributions precisely those corresponding to the (traditional) branched coverings of E? 2. The pure topology is contained in the dense (or double negation) topology. Every topos has a smallest dense subtopos, which is a Boolean topos. Similarly, every locally connected topos has a smallest pure subtopos. E p is the smallest subtopos of E containing the density objects. Characterize intrinsically those toposes of the form E p for some locally connected topos E. Develop a theory of purely skeletal geometric morphisms. What sorts of foldings does it rule out from complete spreads? 3. G. Janelidze asked (private communication) whether there is a Galois theory (in his sense) corresponding to branched coverings. Related to this is the following question. Assume that E is locally simply connected (that is, a single U 1 splits all locally constant objects). Is E p is locally simply connected? If not, what can one say about C (E p )/H? Additional references [2, 7, 11, 12].

6 2. Probability Distributions and Valuation Locales 6 The symmetric topos M(E ) is a sort of generalization of the lower power locale P L (X). For any locale X in S, the topos Sh(P L (X)) of sheaves on the lower power locale of X is equivalent to the localic reflection of the symmetric topos M(Sh(X)). However, the same is true of the bagdomain topos B L (E ). This is basically because first, M(S) = B L (S) = S[U], where S[U] is the object classifier, and second, because the Sierpinski locale (subobjects of 1) is the localic reflection of the object classifier. The lower bagdomain topos B L (E ) is the classifier of Riemann sums meaning bags of points. In particular, toposes E for which all distributions are Riemman sums are precisely those toposes E for which the domain topos of every complete spread over E is locally connected and furthermore the connected components functor preserves pullbacks. Perhaps less understood are what were called probability distributions. A probability distribution on a topos E is a distribution that preserves the terminal object. For any topos E over S, there is a subtopos T(E ) of M(E ) that classifies probability distributions. There is a factorization E δ M(E ) δ T(E ) where δ is essential and satisfies δ! (1) = 1. An X -point p : X M(E ), where X is an S-topos, factors through T(E ) iff in the bicomma object i F γ XX ψ E M(E ) δ the locally connected γ is connected, in which case the resulting inside square with T(E ) is a bicomma object. In particular, the complete spreads over E corresponding to probability distributions on E are precisely those with connected, locally connected domain. Let E denote an arbitrary topos over S. The connection between the three toposes M(E ), B L (E ), and T (E ) is reflected, firstly, in the following diagram depicting the two canonical factorizations of the unit δ : E M(E ) E B L (E ) T(E ) M(E ) p

7 7 and secondly in the equivalence M(E ) B L (T(E )), natural in E. The valuation locale on a locale X has as as points the continuous valuations on the frame of X, that is, functions m : O(X) R from the frame of X to the lower reals from 0 to infinity with reverse order, such that m is Scott continuous, m(0) = 0 and m has the modular law m(u) + m(v ) = m(u V ) + m(u V ). It is related to the probabilistic power domain of computer science. A KZ-monad M on Top S is said to be additive if for any two toposes E and F, the coproduct injections ι E : E E + S F and ι F : F E + S F have coreflection M-adjoints ρ E, ρ F, and for any coproduct diagram in K, below left, X M(Z) r X M(X) Y ι Y Z ι X r Y M(Y ) the right-hand diagram is a product diagram. Let M be the symmetric monad. A geometric morphism over S admits an M-adjoint if and only if it is S-essential. The symmetric monad M is additive. We refer to M(E + S F ) = M(E ) S M(F ) as the Waelbroeck theorem for toposes. Note that the topos frame of E + S F is the underlying category product E F, and that this is not the topos product E S F. Questions. 1. The lower power domain P L (X) of a locale X in a topos S is the localic reflection of the symmetric monad M(E ) and classifies weakly closed sublocales on X with open domain. What is the localic reflection of the topos T(E )? What does it classify? S. Vickers asked (private communication) roughly what is the connection between the classifier of probability (Lawvere) distributions on locales and the valuation locale of computer science? More relevant than a direct comparison, since the settings are different, would be to examine their respective explanatory powers. 2. The probability KZ-doctrine T is not additive since given any two probability distributions µ : E S and λ : F S, the pairing µ, λ : E + S F S given by µ, λ (E, F ) = µ(e) + λ(f ) no longer preserves 1. Is there a suitable correction to the Waelbroeck theorem formula for probability distributions? 3. Lawvere has formalized and generalized completions of the cohesive type, like the Fox completion. Is there a comprehensive factorization whose second factors are the cohesive geometric morphisms? What can one say about cohesive toposes of the probabilistic sort? Are they well represented by simply imposing the condition p! (1) = 1 on the given data?

8 8 Additional references: [4, 5, 8, 18, 19, 20] 3. Monoidal Structures via the Symmetric Monad It is an ongoing program (suggested by Lawvere) to develop Functional Analysis without the use of Hilbert space. This, of course is a huge enterprise, of which in [6] we have barely touched the surface. In this connection the Symmetric Monad on Top S is a valuable tool. We begin by reviewing briefly what we will need about it here. Our basic reference is [6] and references therein. A KZ-adjointness F U between 2-categories G : A B ; F : B A has units η B : B GF (B) and counits ɛ A : F G(A) A such that F (η B ) ɛ F (B) ; G(ɛ A ) η G(A). (5) The unit is a 2-cell in B,and the counit is a 2-cell in A. As part of the definition, we assume that these 2-cells are isomorphisms. A KZ-adjointness induces a special sort of 2-monad on B that we shall call a KZ-monad (M, δ, µ) = (GF, η, GɛF ). We have a canonical 2-cell and adjoints M(δ B ) δ M(B), M(δ B ) µ B δ M(B). (6) We are interested in the symmetric KZ-monad M on Top S. We begin by considering its algebraic side. We have an adjoint pair of 2-functors U : Frm S Coc S ; Σ : Coc S Frm S where U is the forgetful 2-functor from the 2-category Frm S of frames of Grothendieck toposes (over S or if you wish over Set), and Σ is its left 2-adjoint [1, 4]. Σ(E ) classifies distributions on E via the unit δ E, an essential geometric morphism thought of as the Dirac delta. Intuitively, Σ freely adds left exact structure to a distribtution, so that the result is )the inverse part of) a geometric morphism. The lex completion of a category is naturally involved in its construction. We now take the geometric point of view and regard Σ as a 2-functor Σ : Coc S op = Dist S Top S. There is then a KZ-adjointness U Σ.

9 9 It induces a KZ-monad in Top S : (M, δ, µ) = (ΣU, η, ΣɛU) which we refer to as the symmetric monad on Top S. By convention, we denote a KZmonad by its functor part, in this case M = ΣU. We shall say that a 1-cell ϕ in K admits an M-adjoint if M(ϕ) has a right adjoint. If ϕ admits an M-adjoint, we denote the right adjoint of M(ϕ) by r ϕ. In the case of the symmetric monad M on Top S, a geometric morphism admits an M-adjoint if and only if it is S-essential. An M-bifibration is a span ff ψ that arises in a bicomma object F f XX ψ E M(E ) δ E for some p. If an M-bifibration has an S-essential domain f, then we have a functor : K (X, M(E )) Span M (X, E ) such that (p) = δ E p. On the other hand, we may associate with any span ff ψ with f S-essential, the 1-cell Σ f (δ E ψ) = M(ψ) r f δ F. This 1-cell is the left extension as the notation indicates. This defines a functor Λ : Span M (X, E ) K (X, M(E )). Λ, and is full and faithful. Thus, we may equivalently regard M-bifibrations as 1-cells X M(E ).These are precisely 1-cells X E in the so-called Kleisli category for M, denoted K M. We are mostly interested in the following two special kinds of M-bifibrations. A discrete M-opfibration (intensive quantity) is a geometric morphism X f E that appears in bicomma object X f EE p x S q M(S) δ S for some geometric morphism q.

10 A discrete M-fibration (extensive quantity) is a geometric morphism Y ψ E that appears in bicomma object Y y SS ψ E M(E ) δ E for some geometric morphism p. There is an action of discrete opfibrations in discrete fibrations : we may multiply a discrete opfibration f : X E on a discrete fibration ψy E to form a discrete fibration f.ψ as follows. In the diagram Mf M(X ) M(E ) p 10 Mx M(S) M(δ S ) Mq M 2 (S) we have M(f) r f. Now define f ψ to be the discrete fibration corresponding to the 1-cell f ψ : S p r f M(E ) M(X ) M(f) M(E ), (7) where p corresponds to ψ. Consider the case of the topos S. In this case, the unit is δ S : S M(S) R, where R is the object classifier in Top S. The object of S that corresponds to the geometric morphism δ S is the terminal object 1, i.e., the unit is the Dirac delta supported on 1. The line R is an M-algebra, in fact it is a free one, and δ S is a homomorphism. The evaluation of a distribution can be understood in terms of the structure map : M(R) R of this M-algebra. Indeed, if S p M(E ) is the geometric morphism corresponding to a distribution µ : E S, and if f : E R is the geometric morphism corresponding to an object X of E, then the evaluation µ(x) is the object of S whose corresponding geometric morphism is the composite S p M(E ) M(f) M(R) R.

11 11 A suggestive notation for the value of the distribution f µ on some geometric morphism g : E R corresponding to an object Y of E is gd(f µ) = (g.f)d(µ) Questions. 1. The terms kernel and Fourier transform [9] could also be used in our setting. Any geometric morphism k : F M(E ) (a morphism in the Kleisli category of the KZ-monad M) is said to be the kernel of its extension k = Σ δf (k) : M(F ) M(E ), and the latter is said to be the Fourier transform of ψ. k F M(F ) δ F M(E ) Develop this point of view having the intended application in mind. 2. In [14], frames (of locales) A in a topos S are characterized in the monoidal category sl of suplattices (in S), as those commutative monoids satisfying the formula (a A ((a 1) (a 2 = a)). This characterization is not internal to the monoidal category sl,, 1 as it employs the diagonal map A A A which is not part of the monoidal structure. By contrast, the characterization of frames as algebras for the symmetric monad on sl [4] is internal to it. Morever, it is extensible to topos frames regarded in a suitable full subcategory of Coc S. The question of establishing the descent theorem of Joyal and Tierney stating that open surjections are of effective descent without the use of tensor products and exclusively in terms of the symmetric monad was sketched in [3], but the details have not yet completely been written up. If done in sufficient generality of a 2-category K and a suitable KZ-monad on it, one should be able to prove a general descent theorem of which the effective descent property of both open surjections of locales and S-essential surjections of toposes are derived as instances. Additional references : [1, 4, 9, 14, 16]. k References [1] M. Bunge, Cosheaves and distributions on toposes, Algebra Universalis 34: , 1995.

12 [2] M. Bunge, Galois groupoids and covering morphisms in topos theory, Proceedings of the Fields Institute: Workshop on Descent, Galois Theory and Hopf algebras Fields Institute Communications, American Mathematical Society (2008) [3] M. Bunge, A characterization of frames as suplattices without the use of the tensor product, Invited lecture, CMS Summer Meeting 2010, Fredericton, N.B., June [4] M. Bunge and A. Carboni, The symmetric topos, J. Pure Appl. Alg , [5] M. Bunge and J. Funk, Constructive theory of the lower power locale, Math. Struct. in Comp. Science 6 (1996), [6] M. Bunge and J. Funk, Singular Coverings of Toposes, LNM 1890, Springer, [7] M. Bunge and J. Funk, An intrinsic characterization of branched coverings, Contemporary Mathematics 431 (2007) [8] T. Coquand and B. Spitters, Integrals and valuations, Journal of Logic and Analysis1 (2008) [9] B. Day, Monoidal Functor Categories and Graphic Fourier Transforms, Theory and Applications of Categories 25 (5) (2011) [10] R. H. Fox, Covering spaces with singularities, R. H. Fox et al. (editors), Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, Princeton University Press, Princeton 1957, [11] J. Funk, The Hurwitz action and braid group orderings, Theory and Applications of Categories 9-1 (2001) [12] G. Janelidze, Pure Galois theory in categories, J. Algebra 132 (1990) [13] P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Clarendon Press, Oxford, [14] A. Joyal and M. Tierney, An extension of the Galois theory of Grothendieck, Memoirs of the Amer.Math.Soc [15] A. Kock, Monads for which structure are adjoints to units, J. Pure Appl. Alg.104 (1975), Princeton University Press, Princeton 1957, [16] F. W. Lawvere, Measures on Toposes, Workshop on Categorical Methods in Geometry Aarhus, [17] F. W. Lawvere, Categories of space and quantity, The Space of Mathematics, W.de Gruyter, Berlin-New York, 1992, pp

13 [18] F. W. Lawvere, Axiomatic Cohesion, Theory and Applications of Categories 19-3 (2007) [19] S. Vickers, Topology via Logic, Cambridge University Press, [20] S. Vickers A localic theory of lower and upper integrals, Mathematical Logic Quarterly54 (1) (2008) [21] V. Zoeberlein, Doctrines on 2-categories, Math. Zeit(1976) Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada H3A 2K6 marta.bunge@mcgill.ca