A 2-Site for Continuous 2-Group Actions
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Topos Theory
Topos Theory Olivia Caramello Sheaves on a site Grothendieck topologies Grothendieck toposes Basic properties of Grothendieck toposes Subobject lattices Topos Theory Balancedness The epi-mono factorization Lectures 7-14: Sheaves on a site The closure operation on subobjects Monomorphisms and epimorphisms Exponentials Olivia Caramello The subobject classifier Local operators For further reading Topos Theory Sieves Olivia Caramello In order to ‘categorify’ the notion of sheaf of a topological space, Sheaves on a site Grothendieck the first step is to introduce an abstract notion of covering (of an topologies Grothendieck object by a family of arrows to it) in a category. toposes Basic properties Definition of Grothendieck toposes Subobject lattices • Given a category C and an object c 2 Ob(C), a presieve P in Balancedness C on c is a collection of arrows in C with codomain c. The epi-mono factorization The closure • Given a category C and an object c 2 Ob(C), a sieve S in C operation on subobjects on c is a collection of arrows in C with codomain c such that Monomorphisms and epimorphisms Exponentials The subobject f 2 S ) f ◦ g 2 S classifier Local operators whenever this composition makes sense. For further reading • We say that a sieve S is generated by a presieve P on an object c if it is the smallest sieve containing it, that is if it is the collection of arrows to c which factor through an arrow in P. If S is a sieve on c and h : d ! c is any arrow to c, then h∗(S) := fg | cod(g) = d; h ◦ g 2 Sg is a sieve on d. -
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 . -
Lecture Notes on Sheaves, Stacks, and Higher Stacks
Lecture Notes on Sheaves, Stacks, and Higher Stacks Adrian Clough September 16, 2016 2 Contents Introduction - 26.8.2016 - Adrian Cloughi 1 Grothendieck Topologies, and Sheaves: Definitions and the Closure Property of Sheaves - 2.9.2016 - NeˇzaZager1 1.1 Grothendieck topologies and sheaves.........................2 1.1.1 Presheaves...................................2 1.1.2 Coverages and sheaves.............................2 1.1.3 Reflexive subcategories.............................3 1.1.4 Coverages and Grothendieck pretopologies..................4 1.1.5 Sieves......................................6 1.1.6 Local isomorphisms..............................9 1.1.7 Local epimorphisms.............................. 10 1.1.8 Examples of Grothendieck topologies..................... 13 1.2 Some formal properties of categories of sheaves................... 14 1.3 The closure property of the category of sheaves on a site.............. 16 2 Universal Property of Sheaves, and the Plus Construction - 9.9.2016 - Kenny Schefers 19 2.1 Truncated morphisms and separated presheaves................... 19 2.2 Grothendieck's plus construction........................... 19 2.3 Sheafification in one step................................ 19 2.3.1 Canonical colimit................................ 19 2.3.2 Hypercovers of height 1............................ 19 2.4 The universal property of presheaves......................... 20 2.5 The universal property of sheaves........................... 20 2.6 Sheaves on a topological space............................ 21 3 Categories -
Notes on Categorical Logic
Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classifiers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean first order, finitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes infinitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as infinitary logics, generalized quantifiers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic flavour. -
Arxiv:2012.08669V1 [Math.CT] 15 Dec 2020 2 Preface
Sheaf Theory Through Examples (Abridged Version) Daniel Rosiak December 12, 2020 arXiv:2012.08669v1 [math.CT] 15 Dec 2020 2 Preface After circulating an earlier version of this work among colleagues back in 2018, with the initial aim of providing a gentle and example-heavy introduction to sheaves aimed at a less specialized audience than is typical, I was encouraged by the feedback of readers, many of whom found the manuscript (or portions thereof) helpful; this encouragement led me to continue to make various additions and modifications over the years. The project is now under contract with the MIT Press, which would publish it as an open access book in 2021 or early 2022. In the meantime, a number of readers have encouraged me to make available at least a portion of the book through arXiv. The present version represents a little more than two-thirds of what the professionally edited and published book would contain: the fifth chapter and a concluding chapter are missing from this version. The fifth chapter is dedicated to toposes, a number of more involved applications of sheaves (including to the \n- queens problem" in chess, Schreier graphs for self-similar groups, cellular automata, and more), and discussion of constructions and examples from cohesive toposes. Feedback or comments on the present work can be directed to the author's personal email, and would of course be appreciated. 3 4 Contents Introduction 7 0.1 An Invitation . .7 0.2 A First Pass at the Idea of a Sheaf . 11 0.3 Outline of Contents . 20 1 Categorical Fundamentals for Sheaves 23 1.1 Categorical Preliminaries . -
A Grothendieck Site Is a Small Category C Equipped with a Grothendieck Topology T
Contents 5 Grothendieck topologies 1 6 Exactness properties 10 7 Geometric morphisms 17 8 Points and Boolean localization 22 5 Grothendieck topologies A Grothendieck site is a small category C equipped with a Grothendieck topology T . A Grothendieck topology T consists of a collec- tion of subfunctors R ⊂ hom( ;U); U 2 C ; called covering sieves, such that the following hold: 1) (base change) If R ⊂ hom( ;U) is covering and f : V ! U is a morphism of C , then f −1(R) = fg : W ! V j f · g 2 Rg is covering for V. 2) (local character) Suppose R;R0 ⊂ hom( ;U) and R is covering. If f −1(R0) is covering for all f : V ! U in R, then R0 is covering. 3) hom( ;U) is covering for all U 2 C . 1 Typically, Grothendieck topologies arise from cov- ering families in sites C having pullbacks. Cover- ing families are sets of maps which generate cov- ering sieves. Suppose that C has pullbacks. A topology T on C consists of families of sets of morphisms ffa : Ua ! Ug; U 2 C ; called covering families, such that 1) Suppose fa : Ua ! U is a covering family and y : V ! U is a morphism of C . Then the set of all V ×U Ua ! V is a covering family for V. 2) Suppose ffa : Ua ! Vg is covering, and fga;b : Wa;b ! Uag is covering for all a. Then the set of composites ga;b fa Wa;b −−! Ua −! U is covering. 3) The singleton set f1 : U ! Ug is covering for each U 2 C . -
Notes on Topos Theory
Notes on topos theory Jon Sterling Carnegie Mellon University In the subjectivization of mathematical objects, the activity of a scientist centers on the articial delineation of their characteristics into denitions and theorems. De- pending on the ends, dierent delineations will be preferred; in these notes, we prefer to work with concise and memorable denitions constituted from a common body of semantically rich building blocks, and treat alternative characterizations as theorems. Other texts To learn toposes and sheaves thoroughly, the reader is directed to study Mac Lane and Moerdijk’s excellent and readable Sheaves in Geometry and Logic [8]; also recommended as a reference is the Stacks Project [13]. ese notes serve only as a supplement to the existing material. Acknowledgments I am grateful to Jonas Frey, Pieter Hofstra, Ulrik Buchholtz, Bas Spiers and many others for explaining aspects of category theory and topos theory to me, and for puing up with my ignorance. All the errors in these notes are mine alone. 1 Toposes for concepts in motion Do mathematical concepts vary over time and space? is question is the fulcrum on which the contradictions between the competing ideologies of mathematics rest. Let us review their answers: Platonism No. Constructivism Maybe. Intuitionism Necessarily, but space is just an abstraction of time. Vulgar constructivism No.1 Brouwer’s radical intuitionism was the rst conceptualization of mathematical activity which took a positive position on this question; the incompatibility of intuition- ism with classical mathematics amounts essentially to the fact that they take opposite positions as to the existence of mathematical objects varying over time. -
Notes/Exercises on the Functor of Points. Due Monday, Febrary 8 in Class. for a More Careful Treatment of This Material, Consult
Notes/exercises on the functor of points. Due Monday, Febrary 8 in class. For a more careful treatment of this material, consult the last chapter of Eisenbud and Harris' book, The Geometry of Schemes, or Section 9.1.6 in Vakil's notes, although I would encourage you to try to work all this out for yourself. Let C be a category. For each object X 2 Ob(C), define a contravariant functor FX : C!Sets by setting FX (S) = Hom(S; X) and for each morphism in C, f : S ! T define FX (f) : Hom(S; X) ! Hom(T;X) by composition with f. Given a morphism g : X ! Y , we get a natural transformation of functors, FX ! FY defined by mapping Hom(S; X) ! Hom(S; Y ) using composition with g. Lemma 1 (Yoneda's Lemma). Every natural transformation φ : FX ! FY is induced by a unique morphism f : X ! Y . Proof: Given a natural transformation φ, set fφ = φX (id). Exercise 1. Check that φ is the natural transformation induced by fφ. This implies that the functor F that takes an object X of C to the functor FX and takes a morphism in C to the corresponding natural transformation of functors defines a fully faithful functor from C to the category whose objects are functors from C to Sets and whose morphisms are natural transformations of functors. In particular, X is uniquely determined by FX . However, not all functors are of the form FX . Definition 1. A contravariant functor F : C!Sets is set to be representable if ∼ there exists an object X in C such that F = FX . -
A Bivariant Yoneda Lemma and (∞,2)-Categories of Correspondences
A bivariant Yoneda lemma and (∞,2)-categories of correspondences* Andrew W. Macpherson August 12, 2021 Abstract A well-known folklore states that if you have a bivariant homology theory satisfying a base change formula, you get a representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula . Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for bivariant functors. Moreover, conditioned on the existence of a 2-dimensional Grothendieck construction, I provide a proof of the aforementioned universal property. The methods, morally speaking, employ the ‘internal logic’ of higher category theory: they make no explicit use of any particular model. Contents 1 Introduction 2 2 Higher category theory 7 2.1 Homotopy invariance in higher category theory . .............. 7 2.2 Models of n-categorytheory .................................. 9 2.3 Mappingobjectsinhighercategories . ............. 11 2.4 Enrichmentfromtensoring. .......... 12 2.5 Bootstrapping n-naturalisomorphisms . 14 2.6 Fibrations ...................................... ....... 15 2.7 Grothendieckconstruction . ........... 17 2.8 Evaluationmap................................... ....... 18 2.9 Adjoint functors and adjunctions -
Elementary Introduction to Representable Functors and Hilbert Schemes
PARAMETER SPACES BANACH CENTER PUBLICATIONS, VOLUME 36 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1996 ELEMENTARY INTRODUCTION TO REPRESENTABLE FUNCTORS AND HILBERT SCHEMES STEINARILDSTRØMME Mathematical Institute, University of Bergen All´egt.55, N-5007 Bergen, Norway E-mail: [email protected] 1. Introduction. The purpose of this paper is to define and prove the existence of the Hilbert scheme. This was originally done by Grothendieck in [4]. A simplified proof was given by Mumford [11], and we will basically follow that proof, with small modifications. The present note started life as a handout supporting my lectures at a summer school on Hilbert schemes in Bayreuth 1991. I want to emphasize that there is nothing new in this paper, no new ideas, no new points of view. To get an idea of a model for the Hilbert scheme, recall how the lines in the projec- tive plane are in 1-1 correspondence with the points of another variety: the dual plane. Similarly, plane conic sections are parameterized by the 5-dimensional projective space of ternary quadratic forms. More generally, hypersurfaces of degree d in P n are naturally 0 n parameterized by the projective space associated to H (P ; OP n (d)). A generalization of the dual projective plane in another way: linear r-dimensional subspaces of P n are parameterized by the Grassmann variety G(r; n). The Hilbert scheme provides a generalization of these examples to parameter spaces for arbitrary closed subschemes of a given variety. Consider the classification problem for algebraic space curves, for example. It consists of determining the set of all space curves, but also to deal with questions like which types of curves can specialize to which other types, or more generally, which algebraic families of space curves are there? (An algebraic 3 family is a subscheme Z ⊆ P × T where T is a variety and each fiber Zt for t 2 T is a space curve. -
Math 161: Category Theory in Context Problem Set 31 Due: March 3, 2015
Math 161: Category theory in context Problem Set 31 due: March 3, 2015 Emily Riehl Exercise 1. In the presence of a full and faithful functor F : C!D prove that objects x; y 2 C are isomorphic if and only if Fx and Fy are isomorphic in D. Exercise 2. For each of the three functors 0 1 ! / 2 o / 1 between the categories 1 and 2, describe the corresponding natural transformations be- tween the covariant functors Cat ! Set represented by these categories. Exercise 3. A functor F defines a subfunctor of G if there is a natural transformation α: F ) G whose components are monomorphisms. In the case of G : Cop ! Set a subfunctor is given by a collection of subsets Fc ⊂ Gc so that each G f : Gc ! Gc0 restricts to define a function F f : Fc ! Fc0. Provide an elementary characterization of the elements of a subfunctor of the representable functor C(−; c). Exercise 4. As discussed in Section 2.2, diagrams of shape ! are determined by a count- ably infinite family of objects and a countable infinite sequence of morphisms. Describe op the Yoneda embedding y: !,! Set! in this manner (as a family of !op-indexed functors and natural transformations). Prove, without appealing to the Yoneda lemma, that y is full and faithful. Exercise 5. Use the defining universal property of the tensor product to prove that (i) k ⊗k V V for any k-vector space V (ii) U ⊗k (V ⊗k W) (U ⊗k V) ⊗k W for any k-vector spaces U; V; W. -
Category Theory
University of Cambridge Mathematics Tripos Part III Category Theory Michaelmas, 2018 Lectures by P. T. Johnstone Notes by Qiangru Kuang Contents Contents 1 Definitions and examples 2 2 The Yoneda lemma 10 3 Adjunctions 16 4 Limits 23 5 Monad 35 6 Cartesian closed categories 46 7 Toposes 54 7.1 Sheaves and local operators* .................... 61 Index 66 1 1 Definitions and examples 1 Definitions and examples Definition (category). A category C consists of 1. a collection ob C of objects A; B; C; : : :, 2. a collection mor C of morphisms f; g; h; : : :, 3. two operations dom and cod assigning to each f 2 mor C a pair of f objects, its domain and codomain. We write A −! B to mean f is a morphism and dom f = A; cod f = B, 1 4. an operation assigning to each A 2 ob C a morhpism A −−!A A, 5. a partial binary operation (f; g) 7! fg on morphisms, such that fg is defined if and only if dom f = cod g and let dom fg = dom g; cod fg = cod f if fg is defined satisfying f 1. f1A = f = 1Bf for any A −! B, 2. (fg)h = f(gh) whenever fg and gh are defined. Remark. 1. This definition is independent of any model of set theory. If we’re givena particuar model of set theory, we call C small if ob C and mor C are sets. 2. Some texts say fg means f followed by g (we are not). 3. Note that a morphism f is an identity if and only if fg = g and hf = h whenever the compositions are defined so we could formulate the defini- tions entirely in terms of morphisms.