Inverse Limit Spaces of Interval Maps
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Frobenius and the Derived Centers of Algebraic Theories
Frobenius and the derived centers of algebraic theories William G. Dwyer and Markus Szymik October 2014 We show that the derived center of the category of simplicial algebras over every algebraic theory is homotopically discrete, with the abelian monoid of components isomorphic to the center of the category of discrete algebras. For example, in the case of commutative algebras in characteristic p, this center is freely generated by Frobenius. Our proof involves the calculation of homotopy coherent centers of categories of simplicial presheaves as well as of Bousfield localizations. Numerous other classes of examples are dis- cussed. 1 Introduction Algebra in prime characteristic p comes with a surprise: For each commutative ring A such that p = 0 in A, the p-th power map a 7! ap is not only multiplica- tive, but also additive. This defines Frobenius FA : A ! A on commutative rings of characteristic p, and apart from the well-known fact that Frobenius freely gen- erates the Galois group of the prime field Fp, it has many other applications in 1 algebra, arithmetic and even geometry. Even beyond fields, Frobenius is natural in all rings A: Every map g: A ! B between commutative rings of characteris- tic p (i.e. commutative Fp-algebras) commutes with Frobenius in the sense that the equation g ◦ FA = FB ◦ g holds. In categorical terms, the Frobenius lies in the center of the category of commutative Fp-algebras. Furthermore, Frobenius freely generates the center: If (PA : A ! AjA) is a family of ring maps such that g ◦ PA = PB ◦ g (?) holds for all g as above, then there exists an integer n > 0 such that the equa- n tion PA = (FA) holds for all A. -
Planar Embeddings of Minc's Continuum and Generalizations
PLANAR EMBEDDINGS OF MINC’S CONTINUUM AND GENERALIZATIONS ANA ANUSIˇ C´ Abstract. We show that if f : I → I is piecewise monotone, post-critically finite, x X I,f and locally eventually onto, then for every point ∈ =←− lim( ) there exists a planar embedding of X such that x is accessible. In particular, every point x in Minc’s continuum XM from [11, Question 19 p. 335] can be embedded accessibly. All constructed embeddings are thin, i.e., can be covered by an arbitrary small chain of open sets which are connected in the plane. 1. Introduction The main motivation for this study is the following long-standing open problem: Problem (Nadler and Quinn 1972 [20, p. 229] and [21]). Let X be a chainable contin- uum, and x ∈ X. Is there a planar embedding of X such that x is accessible? The importance of this problem is illustrated by the fact that it appears at three independent places in the collection of open problems in Continuum Theory published in 2018 [10, see Question 1, Question 49, and Question 51]. We will give a positive answer to the Nadler-Quinn problem for every point in a wide class of chainable continua, which includes←− lim(I, f) for a simplicial locally eventually onto map f, and in particular continuum XM introduced by Piotr Minc in [11, Question 19 p. 335]. Continuum XM was suspected to have a point which is inaccessible in every planar embedding of XM . A continuum is a non-empty, compact, connected, metric space, and it is chainable if arXiv:2010.02969v1 [math.GN] 6 Oct 2020 it can be represented as an inverse limit with bonding maps fi : I → I, i ∈ N, which can be assumed to be onto and piecewise linear. -
Types Are Internal Infinity-Groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau
Types are internal infinity-groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau To cite this version: Antoine Allioux, Eric Finster, Matthieu Sozeau. Types are internal infinity-groupoids. 2021. hal- 03133144 HAL Id: hal-03133144 https://hal.inria.fr/hal-03133144 Preprint submitted on 5 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Types are Internal 1-groupoids Antoine Allioux∗, Eric Finstery, Matthieu Sozeauz ∗Inria & University of Paris, France [email protected] yUniversity of Birmingham, United Kingdom ericfi[email protected] zInria, France [email protected] Abstract—By extending type theory with a universe of defini- attempts to import these ideas into plain homotopy type theory tionally associative and unital polynomial monads, we show how have, so far, failed. This appears to be a result of a kind of to arrive at a definition of opetopic type which is able to encode circularity: all of the known classical techniques at some point a number of fully coherent algebraic structures. In particular, our approach leads to a definition of 1-groupoid internal to rely on set-level algebraic structures themselves (presheaves, type theory and we prove that the type of such 1-groupoids is operads, or something similar) as a means of presenting or equivalent to the universe of types. -
Neural Subgraph Matching
Neural Subgraph Matching NEURAL SUBGRAPH MATCHING Rex Ying, Andrew Wang, Jiaxuan You, Chengtao Wen, Arquimedes Canedo, Jure Leskovec Stanford University and Siemens Corporate Technology ABSTRACT Subgraph matching is the problem of determining the presence of a given query graph in a large target graph. Despite being an NP-complete problem, the subgraph matching problem is crucial in domains ranging from network science and database systems to biochemistry and cognitive science. However, existing techniques based on combinatorial matching and integer programming cannot handle matching problems with both large target and query graphs. Here we propose NeuroMatch, an accurate, efficient, and robust neural approach to subgraph matching. NeuroMatch decomposes query and target graphs into small subgraphs and embeds them using graph neural networks. Trained to capture geometric constraints corresponding to subgraph relations, NeuroMatch then efficiently performs subgraph matching directly in the embedding space. Experiments demonstrate that NeuroMatch is 100x faster than existing combinatorial approaches and 18% more accurate than existing approximate subgraph matching methods. 1.I NTRODUCTION Given a query graph, the problem of subgraph isomorphism matching is to determine if a query graph is isomorphic to a subgraph of a large target graph. If the graphs include node and edge features, both the topology as well as the features should be matched. Subgraph matching is a crucial problem in many biology, social network and knowledge graph applications (Gentner, 1983; Raymond et al., 2002; Yang & Sze, 2007; Dai et al., 2019). For example, in social networks and biomedical network science, researchers investigate important subgraphs by counting them in a given network (Alon et al., 2008). -
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0. -
A Few Points in Topos Theory
A few points in topos theory Sam Zoghaib∗ Abstract This paper deals with two problems in topos theory; the construction of finite pseudo-limits and pseudo-colimits in appropriate sub-2-categories of the 2-category of toposes, and the definition and construction of the fundamental groupoid of a topos, in the context of the Galois theory of coverings; we will take results on the fundamental group of étale coverings in [1] as a starting example for the latter. We work in the more general context of bounded toposes over Set (instead of starting with an effec- tive descent morphism of schemes). Questions regarding the existence of limits and colimits of diagram of toposes arise while studying this prob- lem, but their general relevance makes it worth to study them separately. We expose mainly known constructions, but give some new insight on the assumptions and work out an explicit description of a functor in a coequalizer diagram which was as far as the author is aware unknown, which we believe can be generalised. This is essentially an overview of study and research conducted at dpmms, University of Cambridge, Great Britain, between March and Au- gust 2006, under the supervision of Martin Hyland. Contents 1 Introduction 2 2 General knowledge 3 3 On (co)limits of toposes 6 3.1 The construction of finite limits in BTop/S ............ 7 3.2 The construction of finite colimits in BTop/S ........... 9 4 The fundamental groupoid of a topos 12 4.1 The fundamental group of an atomic topos with a point . 13 4.2 The fundamental groupoid of an unpointed locally connected topos 15 5 Conclusion and future work 17 References 17 ∗e-mail: [email protected] 1 1 Introduction Toposes were first conceived ([2]) as kinds of “generalised spaces” which could serve as frameworks for cohomology theories; that is, mapping topological or geometrical invariants with an algebraic structure to topological spaces. -
Some Planar Embeddings of Chainable Continua Can Be
Some planar embeddings of chainable continua can be expressed as inverse limit spaces by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Montana State University © Copyright by Susan Pamela Schwartz (1992) Abstract: It is well known that chainable continua can be expressed as inverse limit spaces and that chainable continua are embeddable in the plane. We give necessary and sufficient conditions for the planar embeddings of chainable continua to be realized as inverse limit spaces. As an example, we consider the Knaster continuum. It has been shown that this continuum can be embedded in the plane in such a manner that any given composant is accessible. We give inverse limit expressions for embeddings of the Knaster continuum in which the accessible composant is specified. We then show that there are uncountably many non-equivalent inverse limit embeddings of this continuum. SOME PLANAR EMBEDDINGS OF CHAIN ABLE OONTINUA CAN BE EXPRESSED AS INVERSE LIMIT SPACES by Susan Pamela Schwartz A thesis submitted in partial fulfillment of the requirements for the degree of . Doctor of Philosophy in Mathematics MONTANA STATE UNIVERSITY Bozeman, Montana February 1992 D 3 l% ii APPROVAL of a thesis submitted by Susan Pamela Schwartz This thesis has been read by each member of the thesis committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. g / / f / f z Date Chairperson, Graduate committee Approved for the Major Department ___ 2 -J2 0 / 9 Date Head, Major Department Approved for the College of Graduate Studies Date Graduate Dean iii STATEMENT OF PERMISSION TO USE . -
Cauchy Graph Embedding
Cauchy Graph Embedding Dijun Luo [email protected] Chris Ding [email protected] Feiping Nie [email protected] Heng Huang [email protected] The University of Texas at Arlington, 701 S. Nedderman Drive, Arlington, TX 76019 Abstract classify unsupervised embedding approaches into two cat- Laplacian embedding provides a low- egories. Approaches in the first category are to embed data dimensional representation for the nodes of into a linear space via linear transformations, such as prin- a graph where the edge weights denote pair- ciple component analysis (PCA) (Jolliffe, 2002) and mul- wise similarity among the node objects. It is tidimensional scaling (MDS) (Cox & Cox, 2001). Both commonly assumed that the Laplacian embed- PCA and MDS are eigenvector methods and can model lin- ding results preserve the local topology of the ear variabilities in high-dimensional data. They have been original data on the low-dimensional projected long known and widely used in many machine learning ap- subspaces, i.e., for any pair of graph nodes plications. with large similarity, they should be embedded However, the underlying structure of real data is often closely in the embedded space. However, in highly nonlinear and hence cannot be accurately approx- this paper, we will show that the Laplacian imated by linear manifolds. The second category ap- embedding often cannot preserve local topology proaches embed data in a nonlinear manner based on differ- well as we expected. To enhance the local topol- ent purposes. Recently several promising nonlinear meth- ogy preserving property in graph embedding, ods have been proposed, including IsoMAP (Tenenbaum we propose a novel Cauchy graph embedding et al., 2000), Local Linear Embedding (LLE) (Roweis & which preserves the similarity relationships of Saul, 2000), Local Tangent Space Alignment (Zhang & the original data in the embedded space via a Zha, 2004), Laplacian Embedding/Eigenmap (Hall, 1971; new objective. -
The Elliptic Drinfeld Center of a Premodular Category Arxiv
The Elliptic Drinfeld Center of a Premodular Category Ying Hong Tham Abstract Given a tensor category C, one constructs its Drinfeld center Z(C) which is a braided tensor category, having as objects pairs (X; λ), where X ∈ Obj(C) and λ is a el half-braiding. For a premodular category C, we construct a new category Z (C) which 1 2 i we call the Elliptic Drinfeld Center, which has objects (X; λ ; λ ), where the λ 's are half-braidings that satisfy some compatibility conditions. We discuss an SL2(Z)-action el on Z (C) that is related to the anomaly appearing in Reshetikhin-Turaev theory. This construction is motivated from the study of the extended Crane-Yetter TQFT, in particular the category associated to the once punctured torus. 1 Introduction and Preliminaries In [CY1993], Crane and Yetter define a 4d TQFT using a state-sum involving 15j symbols, based on a sketch by Ooguri [Oog1992]. The state-sum begins with a color- ing of the 2- and 3-simplices of a triangulation of the four manifold by integers from 0; 1; : : : ; r. These 15j symbols then arise as the evaluation of a ribbon graph living on the boundary of a 4-simplex. The labels 0; 1; : : : ; r correspond to simple objects of the Verlinde modular category, the semi-simple subquotient of the category of finite dimen- πi r 2 sional representations of the quantum group Uqsl2 at q = e as defined in [AP1995]. Later Crane, Kauffman, and Yetter [CKY1997] extend this~ definition+ to colorings with objects from a premodular category (i.e. -
Strong Inverse Limit Reflection
Strong Inverse Limit Reflection Scott Cramer March 4, 2016 Abstract We show that the axiom Strong Inverse Limit Reflection holds in L(Vλ+1) assuming the large cardinal axiom I0. This reflection theorem both extends results of [4], [5], and [3], and has structural implications for L(Vλ+1), as described in [3]. Furthermore, these results together highlight an analogy between Strong Inverse Limit Reflection and the Axiom of Determinacy insofar as both act as fundamental regularity properties. The study of L(Vλ+1) was initiated by H. Woodin in order to prove properties of L(R) under large cardinal assumptions. In particular he showed that L(R) satisfies the Axiom of Determinacy (AD) if there exists a non-trivial elementary embedding j : L(Vλ+1) ! L(Vλ+1) with crit (j) < λ (an axiom called I0). We investigate an axiom called Strong Inverse Limit Reflection for L(Vλ+1) which is in some sense analogous to AD for L(R). Our main result is to show that if I0 holds at λ then Strong Inverse Limit Reflection holds in L(Vλ+1). Strong Inverse Limit Reflection is a strong form of a reflection property for inverse limits. Axioms of this form generally assert the existence of a collection of embeddings reflecting a certain amount of L(Vλ+1), together with a largeness assumption on the collection. There are potentially many different types of axioms of this form which could be considered, but we concentrate on a particular form which, by results in [3], has certain structural consequences for L(Vλ+1), such as a version of the perfect set property. -
Arxiv:1204.3607V6 [Math.KT] 6 Nov 2015 At1 Ar N Waldhausen and Pairs 1
ON THE ALGEBRAIC K-THEORY OF HIGHER CATEGORIES CLARK BARWICK In memoriam Daniel Quillen, 1940–2011, with profound admiration. Abstract. We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a simple universal property. Using this, we give new, higher categorical proofs of the Approximation, Additivity, and Fibration Theorems of Waldhausen in this context. As applications of this technology, we study the algebraic K-theory of associative rings in a wide range of homotopical contexts and of spectral Deligne–Mumford stacks. Contents 0. Introduction 3 Relation to other work 6 A word on higher categories 7 Acknowledgments 7 Part 1. Pairs and Waldhausen ∞-categories 8 1. Pairs of ∞-categories 8 Set theoretic considerations 9 Simplicial nerves and relative nerves 10 The ∞-category of ∞-categories 11 Subcategories of ∞-categories 12 Pairs of ∞-categories 12 The ∞-category of pairs 13 Pair structures 14 arXiv:1204.3607v6 [math.KT] 6 Nov 2015 The ∞-categories of pairs as a relative nerve 15 The dual picture 16 2. Waldhausen ∞-categories 17 Limits and colimits in ∞-categories 17 Waldhausen ∞-categories 18 Some examples 20 The ∞-category of Waldhausen ∞-categories 21 Equivalences between maximal Waldhausen ∞-categories 21 The dual picture 22 3. Waldhausen fibrations 22 Cocartesian fibrations 23 Pair cartesian and cocartesian fibrations 27 The ∞-categoriesofpair(co)cartesianfibrations 28 A pair version of 3.7 31 1 2 CLARK BARWICK Waldhausencartesianandcocartesianfibrations 32 4. The derived ∞-category of Waldhausen ∞-categories 34 Limits and colimits of pairs of ∞-categories 35 Limits and filtered colimits of Waldhausen ∞-categories 36 Direct sums of Waldhausen ∞-categories 37 Accessibility of Wald∞ 38 Virtual Waldhausen ∞-categories 40 Realizations of Waldhausen cocartesian fibrations 42 Part 2. -
RESEARCH STATEMENT 1. Introduction My Interests Lie
RESEARCH STATEMENT BEN ELIAS 1. Introduction My interests lie primarily in geometric representation theory, and more specifically in diagrammatic categorification. Geometric representation theory attempts to answer questions about representation theory by studying certain algebraic varieties and their categories of sheaves. Diagrammatics provide an efficient way of encoding the morphisms between sheaves and doing calculations with them, and have also been fruitful in their own right. I have been applying diagrammatic methods to the study of the Iwahori-Hecke algebra and its categorification in terms of Soergel bimodules, as well as the categorifications of quantum groups; I plan on continuing to research in these two areas. In addition, I would like to learn more about other areas in geometric representation theory, and see how understanding the morphisms between sheaves or the natural transformations between functors can help shed light on the theory. After giving a general overview of the field, I will discuss several specific projects. In the most naive sense, a categorification of an algebra (or category) A is an additive monoidal category (or 2-category) A whose Grothendieck ring is A. Relations in A such as xy = z + w will be replaced by isomorphisms X⊗Y =∼ Z⊕W . What makes A a richer structure than A is that these isomorphisms themselves will have relations; that between objects we now have morphism spaces equipped with composition maps. 2-category). In their groundbreaking paper [CR], Chuang and Rouquier made the key observation that some categorifications are better than others, and those with the \correct" morphisms will have more interesting properties. Independently, Rouquier [Ro2] and Khovanov and Lauda (see [KL1, La, KL2]) proceeded to categorify quantum groups themselves, and effectively demonstrated that categorifying an algebra will give a precise notion of just what morphisms should exist within interesting categorifications of its modules.