Coinverters and categories of fractions for categories with structure

A category of fractions is a special case of acoinverter in the 2-categoryCat. We observe that, in a cartesian closed 2-category, the product of tworeflexive coinverter diagrams is another such diagram. It follows that an equational structure on a categoryA, if given by operationsA ⁿ A forn N along with natural transformations and equations, passes canonically to the categoryA [^–1] of fractions, provided that is closed under the operations. We exhibit categories with such structures as algebras for a class of 2-monads onCat, to be calledstrongly finitary monads.The first and third authors gratefully acknowledge the support of the Australian Research Council.     

Co-rectangular bands and cosheaves in categories of algebras

Conditions on a categoryC are studied which imply that every structure of rectangular band on an objectS ofC arises from a unique product decompositionS=S ₁×S ₂, especially in the case whereC is the opposite of a category of algebras.Sheaves on Stone spaces with values in opposites of categories of algebras are examined.The analog of the bounded Boolean power constructionR[B]^* forR an object of a general category is described.This work was done while the author was partly supported by NSF contract DMS 85-02330.Presented by R. S. Pierce.     

On the equationx t=x t+q in categories

We will show that a categoryC divides a monoid that satisfies the equationx ^t=x ^t+q iff each monoid inC satisfies the same equation.     

Relation categories and coproduct congruence categories in universal algebra

It is known that a categoryV-Rel ofadmissible relations can be formed for any variety of algebrasV, such that morphismsAB correspond to subalgebras ofA x B. We adapt the relation category construction of Hilton and Wu to categoriesC with finite limits and colimits and an image factorization system. The existence ofC-Rel and a dualcograph constructionC-Cogr are proved equivalent to certain stability properties of pullbacks or pushouts forC. For algebraic varietiesV,V -Cogr exists iffV satisfies the amalgamation property (AP) and the congruence extension property (CEP). MorphismsAB inV-Cogr correspond to congruences on the coproductA + B. It is showed that congruence permutability (CP), the intersection property for amalgamations (IPA), the Hamiltonian property, and the property that congruences 6 are determined by the equivalence class [0] can be given characterizations in terms of interlocked pullbacks and pushouts in such a categoryC. A new property IDA (intersections determine amalgamations) is defined, which is dual to CP in this context. Familiar results, such as CP implies congruence modularity, can be proved in such categories. Dually, ifV satisfies AP, CEP, IPA and IDA, it has modular lattices of subalgebras. These results are related to order duality for Su and Con. (For certain varietiesV, the subalgebras ofA are in one-one correspondence with the morphisms below 1_A inV -Rel orV-Cogr, and the congruences correspond to the morphisms above 1_A.) IfV is pointed (eachA in V has a smallest trivial subalgebra), then a category formulation is obtained for: CP implies the Jónsson-Tarski decomposition properties. The dual shows that pointed varieties satisfying IDA have a restricted form, with pointed unary varieties and varieties ofR-modules as special cases.Dedicated to Bjarni Jónsson on his 70th birthday ntprbPresented by G. McNulty.     

The near centre of Doi-Hopf module categoryA M (H) C

It is known that any strict tensor category (C⊗I) determines a braided tensor categoryZ(C), the centre ofC. WhenA is a finite dimension Hopf algebra, Drinfel’d has proved thatZ(_A M) is equivalent to _D(A) M as a braided tensor category, whereA ^M is the left A-module category andD(A) is the Drinfel’d double ofA. For a braided tensor category, the braidC _U,v is a natural isomorphism for any pair of object (U,V) in. If weakening the natural isomorphism of the braidC _U,V to a natural transformation, thenC _U,V is a prebraid and the category with a prebraid is called a prebraided tensor category. Similarly it can be proved that any strict tensor category determines a prebraided tensor category Z∼ (C), the near centre of. An interesting prebraided tensor structure of the Yetter-Drinfel’d category _C*A YD ^C*A given, whereC # A is the smash product bialgebra ofC andA. And it is proved that the near centre of Doi-Hopf module _A M(H) ^C is equivalent to the Yetter-Drinfel’ d _C*A YD ^C*A as prebraided tensor categories. As corollaries, the prebraided tensor structures of the Yetter-Drinfel’d category _A YD ^A , the centres of module category and comodule category are given.     

Quantum doubles in monoidal categories

Let H be a Hopf algebra in a rigid symmetric monoidal category C then the evaluation map τis a convolution-invertible skew pairing. In the previous paper, we constructed a Hopf algebra D(H)=H ? _r H ^?cop in C. In this paper, we first show that D(H) is a quasitriangular Hopf algebra in C. Next, let H be an ordinary triangular finite-dimensional Hopf algebra. Then one can form quasitriangular Hopf algebras B(H,H) and B(H,D(H)) (in a rigid braided monoidal category) by Majid’s method associated to the ordinary Hopf algebra maps H →H and i_H H → D(H), where D(H) is the Drin-fePd quantum double. We show that D (B(H,H)) and B(H,D(H)) are isomorphic Hopf algebras in the braided monoidal category.     

A homological characterization of hyperbolic groups

A finitely presented group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0=⁽¹⁾ ₂(G, ℝ), where H ⁽¹⁾ _* (resp. ⁽¹⁾ _*) denotes the ℓ₁-homology (resp. reduced ℓ₁-homology). If Γ is a graph, then every ℓ₁ 1-cycle in Γ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group G is hyperbolic iff H ⁽¹⁾ ₁(G,ℝ)=0. Oblatum: 30-IV-1997 & 14-V-1998 / Published online: 14 January 1999     

Isoperimetric inequalities in potential theory

Given a non-empty bounded domainG in ⁿ ,n2, letr ₀(G) denote the radius of the ballG ₀ having center 0 and the same volume asG. The exterior deficiencyd _e (G) is defined byd _e (G)=r _e (G)/r ₀(G)–1 wherer _e (G) denotes the circumradius ofG. Similarlyd _i (G)=1–r _i (G)/r ₀(G) wherer _i (G) is the inradius ofG. Various isoperimetric inequalities for the capacity and the first eigenvalue ofG are shown. The main results are of the form CapG(1+cf(d _e (G)))CapG ₀ and ₁(G)(1+cf(d _i (G)))₁(G ₀),f(t)=t ³ ifn=2,f(t)=t ³/(ln 1/t) ifn=3,f(t)=t ^(n+3)/2 ifn4 (for convex G and small deficiencies ifn3).     

A conglomerate of exponential supercategories of the category of finitely generated topological spaces

Summary For every ordinaln > 1 we define a categoryT _n of topological spaces in ech's sense which is isomorphic to a category ofn-ary monorelational systems. We show that every categoryT _n is an exponential supercategory of the categoryB of finitely generated topological spaces, which means that well-behaved function spacesG ^H can be defined inT _n wheneverG B.     

Simplicial cells in arrangements of hyperplanes

Let be an arrangement of n hyperplanes in P ^d , C() its cell complex, and H any hyperplane of . It is proved: (1) If is not a near pencil then there are at least n–d–1 simplicial d-cells of C(), each having no facet in H. (2) There are at least d+1 simplicial d-cells of C(), each having a facet in H.Material for this paper was taken from the author's doctoral dissertation.     

A generalization of Ostrowski's theorem on ultrametric inequalities

Summary We first characterize all the ultrametric functionsf on (assuming both thatf(–x)=f(x) and thatf(x)=0 if and only ifx=0) and then, among these functions, we describe those satisfying the functional equationf(x)·f(x ^–1)=1, for all nonzerox in .Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

The cohomology of homotopy categories and the general linear group

A homotopy categoryC (of co-H-groups resp.H-groups) represents an element C in the third cohomology ofC. This element determines all Toda brackets and secondary homotopy operations inC. Moreover, in caseC =VS ⁿ consists of all one-point unions ofn-spheres, the bracket is actually a /2-generator which restricts to Igusa's class(1) in casen3; an explicit new cocycle for(1) is obtained by automorphisms of free nil(2)-groups.     

Etude de la categorie des algebres de Hopf commutatives connexes sur un corps

Let k be a perfect field of characteristic p0; the categoryH of connected abelian Hopf algebras over k is abelian and locally noetherian. Technics of locally noetherian categories are used here to obtain Krull and homological dimensions ofH (which are respectively 1 and 2), and a decomposition ofH in a product of categories. First we have, whereH ^– is the category of Grassman algebras, andH ⁺ consists of Hopf algebras which are zero in odd degrees; then we prove thatH ⁺ itself is a product of isomorphic categoriesH _n, n^*, and we give an equivalence betweenH _n and a category of modules. This is compared to some results of algebraic geometry about Greenberg modules.     

Critical points for spread-out self-avoiding walk,percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in ^d, oriented percolation in ^d×₊, and the contact process in ^d, with p D(·) being the coupling function whose range is proportional to L. For percolation, for example, each bond is independently occupied with probability p D(y–x). The above models are known to exhibit a phase transition when the parameter p varies around a model-dependent critical point p_c. We investigate the value of p_c when d>6 for percolation and d>4 for the other models, and L1. We prove in a unified way that p_c=1+C(D)+O(L^–2d), where the universal term 1 is the mean-field critical value, and the model-dependent term C(D)=O(L^–d) is written explicitly in terms of the random walk transition probability D. We also use this result to prove that p_c=1+cL^–d+O(L^–d–1), where c is a model-dependent constant. Our proof is based on the lace expansion for each of these models.     

Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow in Z ^d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z ^d we let our (random) graph cal C cal (Z ^d,p) be the graph obtained from Z ^d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p _c(Z ^d), simple random walk on cal C cal (Z ^d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x ² energy _e f(e)² is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p _c(Z ^d), then cal C cal (Z ^d,p) supports a nonzero flow f such that the x ^q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z ^d. In this paper we close the gap by showing that if d3 and Z ^d supports a flow of finite H energy then the infinite percolation cluster on Z ^d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.     

Deforming Curves in Jacobians to Non-Jacobians II: Curves in <Emphasis Type="Italic">C</Emphasis><Superscript>(<Emphasis Type="Italic">e</Emphasis>)</Superscript>, 3⩽ <Emphasis Type="Italic">e</Emphasis> ⩽ <Emphasis Type="Italic">g</Emphasis>−3*

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic Jacobian JC will deform with JC to a non-Jacobian. We apply these methods to a particular class of curves in symmetric powers C^(e) of C where 3⩽ e ⩽ g−3. More precisely, given a pencil g¹_d of degree d on C, let X be the curve parametrizing divisors of degree e in divisors of g¹_d (see the paper for the precise scheme-theoretical definition). Under certain genericity assumptions on the pair (C, g¹_d), we prove that if X deforms infinitesimally out of the Jacobian locus with JC then either d=2e, dim H⁰ (g¹_d)=e or d=2e+1, dim H⁰ (g¹_d)=e+1. The analogous result in the case e=2 without genericity assumptions was proved earlier. *This material is based upon work partially supported by the National Science Foundation under Grant No. DMS-0071795. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

Quasi-coproducts and accessible categories with wide pullbacks

We establish a 2-categorical duality involving the 2-category A of all -accessible categories with wide pullbacks, also known as locally -polypresentable categories, and of functors preserving -filtered colimits and wide pullbacks. Commutation of wide pullbacks with so-called quasi-coproducts in Set is the basic ingredient to this duality, which leads to a full characterization of categories of type Wdpb Filt (A, Set)=A The first author acknowledges financial assistance from a special research grant of the Faculty of Arts at York University. The second author is partially supported by an NSERC operating grant.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Intersection properties of boxes. Part I: An upper-bound theorem

LetP be a family ofn boxes inR ^d (with edges parallel to the coordinate axes). Fork=0, 1, 2, …, denote byf _k (P) the number of subfamilies ofP of sizek+1 with non-empty intersection. We show that iff _r (P)=0 for somer≦n, then where thef _k (n, d, r) are ceg196rtain definite numbers defined by (3.4) below. The result is best possible for eachk. Fork=1 it was conjectured by G. Kalai (Israel J. Math.48 (1984), 161–174). As an application, we prove a ‘fractional’ Helly theorem for families of boxes inR ^d .