Exponentiable Morphisms, Partial Products and Pullback Complements
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Arxiv:1705.02246V2 [Math.RT] 20 Nov 2019 Esyta Ulsubcategory Full a That Say [15]
WIDE SUBCATEGORIES OF d-CLUSTER TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the φ form φ∗ mod(Γ) in an essentially unique way, where Γ is a finite dimensional algebra and Φ −→ Γ is Φ an algebra epimorphism satisfying Tor (Γ, Γ) = 0. 1 Let F ⊆ mod(Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d- kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ∗(G ) in an essentially φ Φ unique way, where Φ −→ Γ is an algebra epimorphism satisfying Tord (Γ, Γ) = 0, and G ⊆ mod(Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories ℓ F ⊆ mod(Φ) over algebras of the form Φ = kAm/(rad kAm) . Dedicated to Idun Reiten on the occasion of her 75th birthday 1. Introduction Let d > 1 be an integer. This paper introduces and studies wide subcategories of d-abelian categories as defined by Jasso. The main examples of d-abelian categories are d-cluster tilting subcategories as defined by Iyama. -
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience. -
Projective Geometry: a Short Introduction
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g. -
Generalized Inverse Limits and Topological Entropy
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of Faculty of Science, University of Zagreb FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Zagreb, 2016. FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS Goran Erceg GENERALIZED INVERSE LIMITS AND TOPOLOGICAL ENTROPY DOCTORAL THESIS Supervisors: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016 PRIRODOSLOVNO - MATEMATICKIˇ FAKULTET MATEMATICKIˇ ODSJEK Goran Erceg GENERALIZIRANI INVERZNI LIMESI I TOPOLOŠKA ENTROPIJA DOKTORSKI RAD Mentori: prof. Judy Kennedy prof. dr. sc. Vlasta Matijevic´ Zagreb, 2016. Acknowledgements First of all, i want to thank my supervisor professor Judy Kennedy for accept- ing a big responsibility of guiding a transatlantic student. Her enthusiasm and love for mathematics are contagious. I thank professor Vlasta Matijevi´c, not only my supervisor but also my role model as a professor of mathematics. It was privilege to be guided by her for master's and doctoral thesis. I want to thank all my math teachers, from elementary school onwards, who helped that my love for math rises more and more with each year. Special thanks to Jurica Cudina´ who showed me a glimpse of math theory already in the high school. I thank all members of the Topology seminar in Split who always knew to ask right questions at the right moment and to guide me in the right direction. I also thank Iztok Baniˇcand the rest of the Topology seminar in Maribor who welcomed me as their member and showed me the beauty of a teamwork. -
Coreflective Subcategories
transactions of the american mathematical society Volume 157, June 1971 COREFLECTIVE SUBCATEGORIES BY HORST HERRLICH AND GEORGE E. STRECKER Abstract. General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and mono- coreflections. It is shown that for most categories with "reasonable" smallness and completeness conditions, each coreflection can be "split" into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have "enough" constant morphisms. 1. Introduction. The concept of reflections in categories (and likewise the dual notion—coreflections) serves the purpose of unifying various fundamental con- structions in mathematics, via "universal" properties that each possesses. His- torically, the concept seems to have its roots in the fundamental construction of E. Cech [4] whereby (using the fact that the class of compact spaces is productive and closed-hereditary) each completely regular F2 space is densely embedded in a compact F2 space with a universal extension property. In [3, Appendice III; Sur les applications universelles] Bourbaki has shown the essential underlying similarity that the Cech-Stone compactification has with other mathematical extensions, such as the completion of uniform spaces and the embedding of integral domains in their fields of fractions. In doing so, he essentially defined the notion of reflections in categories. It was not until 1964, when Freyd [5] published the first book dealing exclusively with the theory of categories, that sufficient categorical machinery and insight were developed to allow for a very simple formulation of the concept of reflections and for a basic investigation of reflections as entities themselvesi1). -
Regular Categories and Regular Logic Basic Research in Computer Science
BRICS Basic Research in Computer Science BRICS LS-98-2 C. Butz: Regular Categories and Regular Logic Regular Categories and Regular Logic Carsten Butz BRICS Lecture Series LS-98-2 ISSN 1395-2048 October 1998 Copyright c 1998, BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Lecture Series publica- tions. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory LS/98/2/ Regular Categories and Regular Logic Carsten Butz Carsten Butz [email protected] BRICS1 Department of Computer Science University of Aarhus Ny Munkegade DK-8000 Aarhus C, Denmark October 1998 1Basic Research In Computer Science, Centre of the Danish National Research Foundation. Preface Notes handed out to students attending the course on Category Theory at the Department of Computer Science in Aarhus, Spring 1998. These notes were supposed to give more detailed information about the relationship between regular categories and regular logic than is contained in Jaap van Oosten’s script on category theory (BRICS Lectures Series LS-95-1). Regular logic is there called coherent logic. -
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. -
Basic Category Theory and Topos Theory
Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object . -
The Hecke Bicategory
Axioms 2012, 1, 291-323; doi:10.3390/axioms1030291 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication The Hecke Bicategory Alexander E. Hoffnung Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA; E-Mail: [email protected]; Tel.: +215-204-7841; Fax: +215-204-6433 Received: 19 July 2012; in revised form: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012 Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Keywords: Hecke algebras; categorification; groupoidification; Yang–Baxter equations; Zamalodchikov tetrahedron equations; spans; enriched bicategories; buildings; incidence geometries 1. Introduction Categorification is, in part, the attempt to shed new light on familiar mathematical notions by replacing a set-theoretic interpretation with a category-theoretic analogue. Loosely speaking, categorification replaces sets, or more generally n-categories, with categories, or more generally (n + 1)-categories, and functions with functors. By replacing interesting equations by isomorphisms, or more generally equivalences, this process often brings to light a new layer of structure previously hidden from view. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
• Rotations • Camera Calibration • Homography • Ransac
Agenda • Rotations • Camera calibration • Homography • Ransac Geometric Transformations y 164 Computer Vision: Algorithms andx Applications (September 3, 2010 draft) Transformation Matrix # DoF Preserves Icon translation I t 2 orientation 2 3 h i ⇥ ⇢⇢SS rigid (Euclidean) R t 3 lengths S ⇢ 2 3 S⇢ ⇥ h i ⇢ similarity sR t 4 angles S 2 3 S⇢ h i ⇥ ⇥ ⇥ affine A 6 parallelism ⇥ ⇥ 2 3 h i ⇥ projective H˜ 8 straight lines ` 3 3 ` h i ⇥ Table 3.5 Hierarchy of 2D coordinate transformations. Each transformation also preserves Let’s definethe properties families listed of in thetransformations rows below it, i.e., similarity by the preserves properties not only anglesthat butthey also preserve parallelism and straight lines. The 2 3 matrices are extended with a third [0T 1] row to form ⇥ a full 3 3 matrix for homogeneous coordinate transformations. ⇥ amples of such transformations, which are based on the 2D geometric transformations shown in Figure 2.4. The formulas for these transformations were originally given in Table 2.1 and are reproduced here in Table 3.5 for ease of reference. In general, given a transformation specified by a formula x0 = h(x) and a source image f(x), how do we compute the values of the pixels in the new image g(x), as given in (3.88)? Think about this for a minute before proceeding and see if you can figure it out. If you are like most people, you will come up with an algorithm that looks something like Algorithm 3.1. This process is called forward warping or forward mapping and is shown in Figure 3.46a. -
Math 535 - General Topology Additional Notes
Math 535 - General Topology Additional notes Martin Frankland September 5, 2012 1 Subspaces Definition 1.1. Let X be a topological space and A ⊆ X any subset. The subspace topology sub on A is the smallest topology TA making the inclusion map i: A,! X continuous. sub In other words, TA is generated by subsets V ⊆ A of the form V = i−1(U) = U \ A for any open U ⊆ X. Proposition 1.2. The subspace topology on A is sub TA = fV ⊆ A j V = U \ A for some open U ⊆ Xg: In other words, the collection of subsets of the form U \ A already forms a topology on A. 2 Products Before discussing the product of spaces, let us review the notion of product of sets. 2.1 Product of sets Let X and Y be sets. The Cartesian product of X and Y is the set of pairs X × Y = f(x; y) j x 2 X; y 2 Y g: It comes equipped with the two projection maps pX : X × Y ! X and pY : X × Y ! Y onto each factor, defined by pX (x; y) = x pY (x; y) = y: This explicit description of X × Y is made more meaningful by the following proposition. 1 Proposition 2.1. The Cartesian product of sets satisfies the following universal property. For any set Z along with maps fX : Z ! X and fY : Z ! Y , there is a unique map f : Z ! X × Y satisfying pX ◦ f = fX and pY ◦ f = fY , in other words making the diagram Z fX 9!f fY X × Y pX pY { " XY commute.