{"results":[{"id":"arxiv_2105.11736","title":"Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology","authors":[{"name":"Mamta Balodi"},{"name":"Abhishek Banerjee"}],"abstract":"We replace a ring with a small $\\mathbb C$-linear category $\\mathcal{C}$, seen as a ring with several objects in the sense of Mitchell. We introduce Fredholm modules over this category and construct a Chern character taking values in the cyclic cohomology of $\\mathcal C$. We show that this categorified Chern character is homotopy invariant and is well-behaved with respect to the periodicity operator in cyclic cohomology. For this, we also obtain a description of cocycles and coboundaries in the cyclic cohomology of $\\mathcal C$ (and more generally, in the Hopf-cyclic cohomology of a Hopf module category) by means of DG-semicategories equipped with a trace on endomorphism spaces.","source":"arXiv","year":2021,"language":"en","subjects":["math.CT"],"url":"https://arxiv.org/abs/2105.11736","pdf_url":"https://arxiv.org/pdf/2105.11736","is_open_access":true,"published_at":"2021-05-25T07:59:44Z","score":65},{"id":"arxiv_1310.1329","title":"A note on permutation twist defects in topological bilayer phases","authors":[{"name":"Jürgen Fuchs"},{"name":"Christoph Schweigert"}],"abstract":"We present a mathematical derivation of some of the most important physical quantities arising in topological bilayer systems with permutation twist defects as introduced by Barkeshli et al. in cond-mat/1208.4834. A crucial tool is the theory of permutation equivariant modular functors developed by Barmeier et al. in math.CT/0812.0986 and math.QA/1004.1825.","source":"arXiv","year":2013,"language":"en","subjects":["hep-th","cond-mat.str-el","math.QA"],"doi":"10.1007/s11005-014-0719-9","url":"https://arxiv.org/abs/1310.1329","pdf_url":"https://arxiv.org/pdf/1310.1329","is_open_access":true,"published_at":"2013-10-04T16:13:22Z","score":57},{"id":"arxiv_1112.5213","title":"The formal theory of Tannaka duality","authors":[{"name":"Daniel Schäppi"}],"abstract":"A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a field extension. If we are working over an arbitrary commutative ring rather than a field, the categories of representations cease to be abelian. We provide a list of sufficient conditions which ensure that an additive tensor category is equivalent to the category of representations of an affine groupoid scheme acting on an affine scheme, or, more generally, to the category of representations of a Hopf algebroid in a symmetric monoidal category. In order to do this we develop a \"formal theory of Tannaka duality\" inspired by Ross Street's \"formal theory of monads.\" We apply our results to certain categories of filtered modules which are used to study p-adic Galois representations.","source":"arXiv","year":2011,"language":"en","subjects":["math.CT","math.AG"],"url":"https://arxiv.org/abs/1112.5213","pdf_url":"https://arxiv.org/pdf/1112.5213","is_open_access":true,"published_at":"2011-12-22T00:48:54Z","score":55},{"id":"arxiv_1105.5186","title":"On monoidal functors between (braided) Gr-categories","authors":[{"name":"Nguyen Tien Quang"},{"name":"Nguyen Thu Thuy"},{"name":"Pham Thi Cuc"}],"abstract":"In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors between categorical groups.","source":"arXiv","year":2011,"language":"en","subjects":["math.CT"],"url":"https://arxiv.org/abs/1105.5186","pdf_url":"https://arxiv.org/pdf/1105.5186","is_open_access":true,"published_at":"2011-05-26T01:33:58Z","score":55},{"id":"arxiv_1105.5187","title":"Cohomological Classification of Ann-categories","authors":[{"name":"Nguyen Tien Quang"}],"abstract":"The notion of Ann-categories is a categorification of the ring structure. Regular Ann-categories were classified by Shukla algebraic cohomology. In this article, we state and prove the precise theorem on classification for the general case due to Mac Lane cohomology for rings. And an application for classification problem of ring extensions is also introduced.","source":"arXiv","year":2011,"language":"en","subjects":["math.CT"],"url":"https://arxiv.org/abs/1105.5187","pdf_url":"https://arxiv.org/pdf/1105.5187","is_open_access":true,"published_at":"2011-05-26T01:43:45Z","score":55},{"id":"doaj_10.2168/LMCS-7(1:15)2011","title":"Semantics of Higher-Order Recursion Schemes","authors":[{"name":"Jiri Adamek"},{"name":"Stefan Milius"},{"name":"Jiri Velebil"}],"abstract":"Higher-order recursion schemes are recursive equations defining new\noperations from given ones called \"terminals\". Every such recursion scheme is\nproved to have a least interpreted semantics in every Scott's model of\n\\lambda-calculus in which the terminals are interpreted as continuous\noperations. For the uninterpreted semantics based on infinite \\lambda-terms we\nfollow the idea of Fiore, Plotkin and Turi and work in the category of sets in\ncontext, which are presheaves on the category of finite sets. Fiore et al\nshowed how to capture the type of variable binding in \\lambda-calculus by an\nendofunctor H\\lambda and they explained simultaneous substitution of\n\\lambda-terms by proving that the presheaf of \\lambda-terms is an initial\nH\\lambda-monoid. Here we work with the presheaf of rational infinite\n\\lambda-terms and prove that this is an initial iterative H\\lambda-monoid. We\nconclude that every guarded higher-order recursion scheme has a unique\nuninterpreted solution in this monoid.","source":"DOAJ","year":2011,"language":"","subjects":["Logic","Electronic computers. Computer science"],"doi":"10.2168/LMCS-7(1:15)2011","url":"https://lmcs.episciences.org/1177/pdf","pdf_url":"https://lmcs.episciences.org/1177/pdf","is_open_access":true,"published_at":"","score":55},{"id":"arxiv_1004.1825","title":"A Geometric Construction for Permutation Equivariant Categories from Modular Functors","authors":[{"name":"Till Barmeier"},{"name":"Christoph Schweigert"}],"abstract":"Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for Z/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper arXiv:0812.0986 [math.CT].","source":"arXiv","year":2010,"language":"en","subjects":["math.QA","math.CT"],"url":"https://arxiv.org/abs/1004.1825","pdf_url":"https://arxiv.org/pdf/1004.1825","is_open_access":true,"published_at":"2010-04-11T17:48:59Z","score":54},{"id":"doaj_10.2168/LMCS-6(3:24)2010","title":"Weak omega-categories from intensional type theory","authors":[{"name":"Peter LeFanu Lumsdaine"}],"abstract":"We show that for any type in Martin-L\\\"of Intensional Type Theory, the terms\nof that type and its higher identity types form a weak omega-category in the\nsense of Leinster. Precisely, we construct a contractible globular operad of\ndefinable composition laws, and give an action of this operad on the terms of\nany type and its identity types.","source":"DOAJ","year":2010,"language":"","subjects":["Logic","Electronic computers. Computer science"],"doi":"10.2168/LMCS-6(3:24)2010","url":"https://lmcs.episciences.org/1062/pdf","pdf_url":"https://lmcs.episciences.org/1062/pdf","is_open_access":true,"published_at":"","score":54},{"id":"arxiv_0707.0835","title":"The Euler characteristic of a category as the sum of a divergent series","authors":[{"name":"Tom Leinster"}],"abstract":"The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one (math.CT/0610260).","source":"arXiv","year":2007,"language":"en","subjects":["math.CT","math.AT"],"url":"https://arxiv.org/abs/0707.0835","pdf_url":"https://arxiv.org/pdf/0707.0835","is_open_access":true,"published_at":"2007-07-05T17:12:39Z","score":51},{"id":"arxiv_math/0403214","title":"Vertically Iterated Classical Enrichment","authors":[{"name":"Stefan Forcey"}],"abstract":"We propose a recursive definition of V-n-categories and their morphisms. We show that for V k-fold monoidal the structure of a (k-n)-fold monoidal strict (n+1)-category is possessed by V-n-Cat. This article is a completion of the work begun in math.CT/0306086, and the initial sections duplicate the beginning of that paper.","source":"arXiv","year":2004,"language":"en","subjects":["math.CT","math.AT","math.QA"],"url":"https://arxiv.org/abs/math/0403214","pdf_url":"https://arxiv.org/pdf/math/0403214","is_open_access":true,"published_at":"2004-03-12T18:31:54Z","score":50},{"id":"arxiv_math/0002216","title":"About the globular homology of higher dimensional automata","authors":[{"name":"Philippe Gaucher"}],"abstract":"We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of math.CT/9902151 disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets.","source":"arXiv","year":2000,"language":"en","subjects":["math.CT","cs.OH","math.AT"],"url":"https://arxiv.org/abs/math/0002216","pdf_url":"https://arxiv.org/pdf/math/0002216","is_open_access":true,"published_at":"2000-02-25T15:38:22Z","score":50},{"id":"arxiv_math/0109021","title":"Structures in higher-dimensional category theory","authors":[{"name":"Tom Leinster"}],"abstract":"This paper, written in 1998, aims to clarify various higher categorical structures, mostly through the theory of generalized operads and multicategories. Chapters I and II, which cover this theory and its application to give a definition of weak n-category, are largely superseded by my thesis (math.CT/0011106), but Chapters III and IV have not appeared elsewhere. The main result of Chapter III is that small Gray-categories can be characterized as the sub-tricategories of the tricategory of 2-categories, homomorphisms, strong transformations and modifications; there is also a conjecture on coherence in higher dimensions. Chapter IV defines opetopes and a category of n-pasting diagrams for each n, which in the case n=2 is a definition of the category of trees.","source":"arXiv","year":2001,"language":"en","subjects":["math.CT"],"url":"https://arxiv.org/abs/math/0109021","pdf_url":"https://arxiv.org/pdf/math/0109021","is_open_access":true,"published_at":"2001-09-04T17:04:06Z","score":50},{"id":"arxiv_math/0602463","title":"Flatness, preorders and general metric spaces (revised)","authors":[{"name":"Vincent Schmitt"}],"abstract":"We use a generic notion of flatness in the enriched context to define various completions of metric spaces -- enrichments over [0,\\infty] -- and preorders -- enrichments over 2. We characterize the weights of colimits commuting in [0,\\infty] with the terminal object and cotensors. These weights can be intrepreted in metric terms as peculiar filters, the so-called filters of type 1. This generalizes Lawvere's correspondence between minimal Cauchy filters and adjoint modules. We obtain a metric completion based on the filters of type 1 as an instance of the free cocompletion under a class of weights defined by G.M. Kelly. Another class of flat presheaves is considered both in the metric and the preorder context. The corresponding completion for preorders is the so-called dcpo-completion.","source":"arXiv","year":2006,"language":"en","subjects":["math.CT","math.MG"],"url":"https://arxiv.org/abs/math/0602463","pdf_url":"https://arxiv.org/pdf/math/0602463","is_open_access":true,"published_at":"2006-02-21T14:05:57Z","score":50},{"id":"arxiv_math/0610552","title":"Tensor envelopes of regular categories","authors":[{"name":"Friedrich Knop"}],"abstract":"We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows:   1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories.   2. Using lattice theory, we give a simple numerical criterion for the vanishing of N.   3. We determine all degree functions for which T(A,d) is Tannakian. As a result, we are able to interpolate the representation categories of many series of profinite groups such as the symmetric groups S_n, the hyperoctahedral groups S_n\\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed finite field.   This paper generalizes work of Deligne, who first constructed the interpolating category for the symmetric groups S_n. It also extends (and provides proofs for) a previous paper math.CT/0605126 on the special case of abelian categories.","source":"arXiv","year":2006,"language":"en","subjects":["math.CT","math.RT"],"doi":"10.1016/j.aim.2007.03.001","url":"https://arxiv.org/abs/math/0610552","pdf_url":"https://arxiv.org/pdf/math/0610552","is_open_access":true,"published_at":"2006-10-18T13:17:04Z","score":50},{"id":"arxiv_math/0208222","title":"On the representation theory of Galois and Atomic Topoi","authors":[{"name":"Eduardo J. Dubuc"}],"abstract":"We elaborate on the representation theorems of topoi as topoi of discrete actions of various kinds of localic groups and groupoids. We introduce the concept of \"proessential point\" and use it to give a new characterization of pointed Galois topoi. We establish a hierarchy of connected topoi:   [1. essentially pointed Atomic = locally simply connected],   [2. proessentially pointed Atomic = pointed Galois],   [3. pointed Atomic].   These topoi are the classifying topos of, respectively: 1. discrete groups, 2. prodiscrete localic groups, and 3. general localic groups.   We analyze also the unpoited version, and show that for a Galois topos, may be pointless, the corresponding groupoid can also be considered, in a sense, the groupoid of \"points\". In the unpointed theories, these topoi classify, respectively: 1. connected discrete groupoids, 2. connected (may be pointless) prodiscrete localic groupoids, and 3. connected groupoids with discrete space of objects and general localic spaces of hom-sets, when the topos has points (we do not know the class of localic groupoids that correspond to pointless connected atomic topoi).   We comment and develop on Grothendieck's galois theory and its generalization by Joyal-Tierney, and work by other authors on these theories.","source":"arXiv","year":2002,"language":"en","subjects":["math.CT"],"url":"https://arxiv.org/abs/math/0208222","pdf_url":"https://arxiv.org/pdf/math/0208222","is_open_access":true,"published_at":"2002-08-28T15:41:31Z","score":50},{"id":"arxiv_math/0501331","title":"On automorphisms of categories of free algebras of some varieties","authors":[{"name":"Boris Plotkin"},{"name":"Grigori Zhitomirski"}],"abstract":"Given a variety of universal algebras. A method is suggested for describing automorphisms of a category of free algebras of this variety. Applying this general method all automorphisms of such categories are found in two cases: 1) for the variety of all free associative K-algebras over an infinite field K and 2) for the variety of all representations of groups in unital R-modules over a commutative associative ring R with unit. It turns out that they are almost inner in a sense.","source":"arXiv","year":2005,"language":"en","subjects":["math.RA","math.CT"],"url":"https://arxiv.org/abs/math/0501331","pdf_url":"https://arxiv.org/pdf/math/0501331","is_open_access":true,"published_at":"2005-01-20T17:38:32Z","score":50},{"id":"arxiv_math/0111204","title":"From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories","authors":[{"name":"Michael Mueger"}],"abstract":"We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A=F-Vect, where F is a field. An object X in A with two-sided dual X^ gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with Obj(E)={X,Y} such that End_E(X,X) is equivalent to A and such that there are J: Y-\u003eX, J^: X-\u003eY producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to E. We define weak monoidal Morita equivalence (wMe) of tensor categories and establish a correspondence between Frobenius algebras in A and tensor categories B wMe A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence of A and B implies (for A,B semisimple and spherical or *-categories) that A and B have the same dimension, braided equivalent `center' (quantum double) and define the same state sum invariants of closed oriented 3-manifolds as defined by Barrett and Westbury. If H is a finite dimensional semisimple and cosemisimple Hopf algebra then H-mod and H^-mod are wMe. The present formalism permits a fairly complete analysis of the quantum double of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205.","source":"arXiv","year":2001,"language":"en","subjects":["math.CT","math.OA"],"url":"https://arxiv.org/abs/math/0111204","pdf_url":"https://arxiv.org/pdf/math/0111204","is_open_access":true,"published_at":"2001-11-19T17:38:54Z","score":50},{"id":"arxiv_math/0111205","title":"From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors","authors":[{"name":"Michael Mueger"}],"abstract":"We are concerned with the center (=quantum double) of tensor categories and prove generalizations of several results proven previously for quantum doubles of Hopf algebras. We consider F-linear tensor categories C with simple unit and finitely many isomorphism classes of simple objects. We assume that C is either a *-category (i.e. there is a positive *-operation on the morphisms) or semisimple and spherical over an algebraically closed field F. In the latter case we assume that dim C=sum_i d(X_i)^2 is non-zero, where the summation runs over the isomorphism classes of simple objects. We prove: (i) Z(C) is a semisimple spherical (or *-) category. (ii) Z(C) is weakly monoidally Morita equivalent (in the sense of math.CT/0111204) to C X C^op. This implies dim Z(C)=(dim C)^2. (iii) We analyze the simple objects of Z(C) in terms of certain finite dimensional algebras, of which Ocneanu's tube algebra is the smallest. We prove the conjecture of Gelfand and Kazhdan according to which the number of simple objects of Z(C) coincides with the dimension of the state space H_{S^1\\times S^1} of the torus in the triangulation TQFT built from C. (iv) We prove that Z(C) is modular and we compute the Gauss sums Delta_+/-(Z(C))=sum_i theta(X_i)^{+/- 1}d(X_i)^2=dim C. (v) Finally, if C is already modular then Z(C)\\simeq C X C~, where C~ is the tensor category C with the braiding c~_{X,Y}=c_{Y,X}^{-1}.","source":"arXiv","year":2001,"language":"en","subjects":["math.CT","math.OA"],"url":"https://arxiv.org/abs/math/0111205","pdf_url":"https://arxiv.org/pdf/math/0111205","is_open_access":true,"published_at":"2001-11-19T16:54:20Z","score":50},{"id":"arxiv_math/0403152","title":"Enrichment over iterated monoidal categories","authors":[{"name":"Stefan Forcey"}],"abstract":"Joyal and Street note in their paper on braided monoidal categories [Braided tensor categories, Advances in Math. 102(1993) 20-78] that the 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception that they mention is the case in which V is symmetric, which leads to V-Cat being symmetric as well. The symmetry in V-Cat is based upon the symmetry of V. The motivation behind this paper is in part to describe how these facts relating V and V-Cat are in turn related to a categorical analogue of topological delooping. To do so I need to pass to a more general setting than braided and symmetric categories -- in fact the k-fold monoidal categories of Balteanu et al in [Iterated Monoidal Categories, Adv. Math. 176(2003) 277-349]. It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts. The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way. In the next paper I indicate how this process may be iterated by enriching over V-Cat, along the way defining the 3-category of categories enriched over V-Cat. In future work I plan to make precise the n-dimensional case and to show how the group completion of the nerve of V is related to the loop space of the group completion of the nerve of V-Cat.   This paper is an abridged version of `Enrichment as categorical delooping I' math.CT/0304026.","source":"arXiv","year":2004,"language":"en","subjects":["math.CT","math.AT","math.QA"],"doi":"10.2140/agt.2004.4.95","url":"https://arxiv.org/abs/math/0403152","pdf_url":"https://arxiv.org/pdf/math/0403152","is_open_access":true,"published_at":"2004-03-09T11:32:35Z","score":50},{"id":"arxiv_math/0403164","title":"Flatness, accessibility and metric spaces","authors":[{"name":"Vincent Schmitt"}],"abstract":"This paper studies a notion of parameterized flatness in the enriched context: p-flatness where the parameter p stands for a class of presheaves. One obtains a completion of a category A by considering the category F_p(A) of p-flat presheaves over A. The completion is related to the free cocompletion under a class of colimits defined by Kelly. We define a notion of Q-accessible categories where Q is the class of p-flat indexes. For a category A, for p = P0 the class of all presheaves, F_P0(A) is the Cauchy-completion of A. Two classes P1 and P2 of interest for general metric spaces and prorders are considered. The F_P1- and F_P2- flatess are characterized yielding non-symmetric completions of metric spaces a la Cauchy involving non-symmetric filters.","source":"arXiv","year":2004,"language":"en","subjects":["math.CT"],"url":"https://arxiv.org/abs/math/0403164","pdf_url":"https://arxiv.org/pdf/math/0403164","is_open_access":true,"published_at":"2004-03-09T20:32:30Z","score":50}],"total":2,"page":1,"page_size":20,"sources":["arXiv","DOAJ"],"query":"math.CT"}