LIMITS and COLIMITS I I
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Category of G-Groups and Its Spectral Category
Communications in Algebra ISSN: 0092-7872 (Print) 1532-4125 (Online) Journal homepage: http://www.tandfonline.com/loi/lagb20 Category of G-Groups and its Spectral Category María José Arroyo Paniagua & Alberto Facchini To cite this article: María José Arroyo Paniagua & Alberto Facchini (2017) Category of G-Groups and its Spectral Category, Communications in Algebra, 45:4, 1696-1710, DOI: 10.1080/00927872.2016.1222409 To link to this article: http://dx.doi.org/10.1080/00927872.2016.1222409 Accepted author version posted online: 07 Oct 2016. Published online: 07 Oct 2016. Submit your article to this journal Article views: 12 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=lagb20 Download by: [UNAM Ciudad Universitaria] Date: 29 November 2016, At: 17:29 COMMUNICATIONS IN ALGEBRA® 2017, VOL. 45, NO. 4, 1696–1710 http://dx.doi.org/10.1080/00927872.2016.1222409 Category of G-Groups and its Spectral Category María José Arroyo Paniaguaa and Alberto Facchinib aDepartamento de Matemáticas, División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Mexico, D. F., México; bDipartimento di Matematica, Università di Padova, Padova, Italy ABSTRACT ARTICLE HISTORY Let G be a group. We analyse some aspects of the category G-Grp of G-groups. Received 15 April 2016 In particular, we show that a construction similar to the construction of the Revised 22 July 2016 spectral category, due to Gabriel and Oberst, and its dual, due to the second Communicated by T. Albu. author, is possible for the category G-Grp. -
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. -
Diagram Chasing in Abelian Categories
Diagram Chasing in Abelian Categories Daniel Murfet October 5, 2006 In applications of the theory of homological algebra, results such as the Five Lemma are crucial. For abelian groups this result is proved by diagram chasing, a procedure not immediately available in a general abelian category. However, we can still prove the desired results by embedding our abelian category in the category of abelian groups. All of this material is taken from Mitchell’s book on category theory [Mit65]. Contents 1 Introduction 1 1.1 Desired results ...................................... 1 2 Walks in Abelian Categories 3 2.1 Diagram chasing ..................................... 6 1 Introduction For our conventions regarding categories the reader is directed to our Abelian Categories (AC) notes. In particular recall that an embedding is a faithful functor which takes distinct objects to distinct objects. Theorem 1. Any small abelian category A has an exact embedding into the category of abelian groups. Proof. See [Mit65] Chapter 4, Theorem 2.6. Lemma 2. Let A be an abelian category and S ⊆ A a nonempty set of objects. There is a full small abelian subcategory B of A containing S. Proof. See [Mit65] Chapter 4, Lemma 2.7. Combining results II 6.7 and II 7.1 of [Mit65] we have Lemma 3. Let A be an abelian category, T : A −→ Ab an exact embedding. Then T preserves and reflects monomorphisms, epimorphisms, commutative diagrams, limits and colimits of finite diagrams, and exact sequences. 1.1 Desired results In the category of abelian groups, diagram chasing arguments are usually used either to establish a property (such as surjectivity) of a certain morphism, or to construct a new morphism between known objects. -
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. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
Category Theory
CMPT 481/731 Functional Programming Fall 2008 Category Theory Category theory is a generalization of set theory. By having only a limited number of carefully chosen requirements in the definition of a category, we are able to produce a general algebraic structure with a large number of useful properties, yet permit the theory to apply to many different specific mathematical systems. Using concepts from category theory in the design of a programming language leads to some powerful and general programming constructs. While it is possible to use the resulting constructs without knowing anything about category theory, understanding the underlying theory helps. Category Theory F. Warren Burton 1 CMPT 481/731 Functional Programming Fall 2008 Definition of a Category A category is: 1. a collection of ; 2. a collection of ; 3. operations assigning to each arrow (a) an object called the domain of , and (b) an object called the codomain of often expressed by ; 4. an associative composition operator assigning to each pair of arrows, and , such that ,a composite arrow ; and 5. for each object , an identity arrow, satisfying the law that for any arrow , . Category Theory F. Warren Burton 2 CMPT 481/731 Functional Programming Fall 2008 In diagrams, we will usually express and (that is ) by The associative requirement for the composition operators means that when and are both defined, then . This allow us to think of arrows defined by paths throught diagrams. Category Theory F. Warren Burton 3 CMPT 481/731 Functional Programming Fall 2008 I will sometimes write for flip , since is less confusing than Category Theory F. -
Families of Sets and Extended Operations Families of Sets
Families of Sets and Extended Operations Families of Sets When dealing with sets whose elements are themselves sets it is fairly common practice to refer to them as families of sets, however this is not a definition. In fact, technically, a family of sets need not be a set, because we allow repeated elements, so a family is a multiset. However, we do require that when repeated elements appear they are distinguishable. F = {A , A , A , A } with A = {a,b,c}, A ={a}, A = {a,d} and 1 2 3 4 1 2 3 A = {a} is a family of sets. 4 Extended Union and Intersection Let F be a family of sets. Then we define: The union over F by: ∪ A={x :∃ A∈F x∈A}= {x :∃ A A∈F∧x∈A} A∈F and the intersection over F by: ∩ A = {x :∀ A∈F x∈A}= {x :∀ A A∈F ⇒ x∈A}. A∈F For example, with F = {A , A , A , A } where A = {a,b,c}, 1 2 3 4 1 A ={a}, A = {a,d} and A = {a} we have: 2 3 4 ∪ A = {a ,b , c , d } and ∩ A = {a}. A∈F A∈F Theorem 2.8 For each set B in a family F of sets, a) ∩ A ⊆ B A∈F b) B ⊆ ∪ A. A∈F Pf: a) Suppose x ∈ ∩ A, then ∀A ∈ F, x ∈ A. Since B ∈ F, we have x ∈ B. Thus, ∩ A ⊆ B. b) Now suppose y ∈ B. Since B ∈ F, y ∈ ∪ A. Thus, B ⊆ ∪ A. Caveat Care must be taken with the empty family F, i.e., the family containing no sets. -
The Left and Right Homotopy Relations We Recall That a Coproduct of Two
The left and right homotopy relations We recall that a coproduct of two objects A and B in a category C is an object A q B together with two maps in1 : A → A q B and in2 : B → A q B such that, for every pair of maps f : A → C and g : B → C, there exists a unique map f + g : A q B → C 0 such that f = (f + g) ◦ in1 and g = (f + g) ◦ in2. If both A q B and A q B 0 0 0 are coproducts of A and B, then the maps in1 + in2 : A q B → A q B and 0 in1 + in2 : A q B → A q B are isomorphisms and each others inverses. The map ∇ = id + id: A q A → A is called the fold map. Dually, a product of two objects A and B in a category C is an object A × B together with two maps pr1 : A × B → A and pr2 : A × B → B such that, for every pair of maps f : C → A and g : C → B, there exists a unique map (f, g): C → A × B 0 such that f = pr1 ◦(f, g) and g = pr2 ◦(f, g). If both A × B and A × B 0 are products of A and B, then the maps (pr1, pr2): A × B → A × B and 0 0 0 (pr1, pr2): A × B → A × B are isomorphisms and each others inverses. The map ∆ = (id, id): A → A × A is called the diagonal map. Definition Let C be a model category, and let f : A → B and g : A → B be two maps. -
Learning from Streams
Learning from Streams Sanjay Jain?1, Frank Stephan??2 and Nan Ye3 1 Department of Computer Science, National University of Singapore, Singapore 117417, Republic of Singapore. [email protected] 2 Department of Computer Science and Department of Mathematics, National University of Singapore, Singapore 117417, Republic of Singapore. [email protected] 3 Department of Computer Science, National University of Singapore, Singapore 117417, Republic of Singapore. [email protected] Abstract. Learning from streams is a process in which a group of learn- ers separately obtain information about the target to be learned, but they can communicate with each other in order to learn the target. We are interested in machine models for learning from streams and study its learning power (as measured by the collection of learnable classes). We study how the power of learning from streams depends on the two parameters m and n, where n is the number of learners which track a single stream of input each and m is the number of learners (among the n learners) which have to find, in the limit, the right description of the target. We study for which combinations m, n and m0, n0 the following inclusion holds: Every class learnable from streams with parameters m, n is also learnable from streams with parameters m0, n0. For the learning of uniformly recursive classes, we get a full characterization which depends m only on the ratio n ; but for general classes the picture is more compli- cated. Most of the noninclusions in team learning carry over to nonin- clusions with the same parameters in the case of learning from streams; but only few inclusions are preserved and some additional noninclusions hold. -
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). -
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. -
Comparison of Waldhausen Constructions
COMPARISON OF WALDHAUSEN CONSTRUCTIONS JULIA E. BERGNER, ANGELICA´ M. OSORNO, VIKTORIYA OZORNOVA, MARTINA ROVELLI, AND CLAUDIA I. SCHEIMBAUER Dedicated to the memory of Thomas Poguntke Abstract. In previous work, we develop a generalized Waldhausen S•- construction whose input is an augmented stable double Segal space and whose output is a 2-Segal space. Here, we prove that this construc- tion recovers the previously known S•-constructions for exact categories and for stable and exact (∞, 1)-categories, as well as the relative S•- construction for exact functors. Contents Introduction 2 1. Augmented stable double Segal objects 3 2. The S•-construction for exact categories 7 3. The S•-construction for stable (∞, 1)-categories 17 4. The S•-construction for proto-exact (∞, 1)-categories 23 5. The relative S•-construction 26 Appendix: Some results on quasi-categories 33 References 38 Date: May 29, 2020. 2010 Mathematics Subject Classification. 55U10, 55U35, 55U40, 18D05, 18G55, 18G30, arXiv:1901.03606v2 [math.AT] 28 May 2020 19D10. Key words and phrases. 2-Segal spaces, Waldhausen S•-construction, double Segal spaces, model categories. The first-named author was partially supported by NSF CAREER award DMS- 1659931. The second-named author was partially supported by a grant from the Simons Foundation (#359449, Ang´elica Osorno), the Woodrow Wilson Career Enhancement Fel- lowship and NSF grant DMS-1709302. The fourth-named and fifth-named authors were partially funded by the Swiss National Science Foundation, grants P2ELP2 172086 and P300P2 164652. The fifth-named author was partially funded by the Bergen Research Foundation and would like to thank the University of Oxford for its hospitality.