CONVENIENT CATEGORIES of SMOOTH SPACES 1. Introduction
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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. -
The Freyd-Mitchell Embedding Theorem States the Existence of a Ring R and an Exact Full Embedding a Ñ R-Mod, R-Mod Being the Category of Left Modules Over R
The Freyd-Mitchell Embedding Theorem Arnold Tan Junhan Michaelmas 2018 Mini Projects: Homological Algebra arXiv:1901.08591v1 [math.CT] 23 Jan 2019 University of Oxford MFoCS Homological Algebra Contents 1 Abstract 1 2 Basics on abelian categories 1 3 Additives and representables 6 4 A special case of Freyd-Mitchell 10 5 Functor categories 12 6 Injective Envelopes 14 7 The Embedding Theorem 18 1 Abstract Given a small abelian category A, the Freyd-Mitchell embedding theorem states the existence of a ring R and an exact full embedding A Ñ R-Mod, R-Mod being the category of left modules over R. This theorem is useful as it allows one to prove general results about abelian categories within the context of R-modules. The goal of this report is to flesh out the proof of the embedding theorem. We shall follow closely the material and approach presented in Freyd (1964). This means we will encounter such concepts as projective generators, injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck categories, subcategories of mono objects and subcategories of absolutely pure objects. This approach is summarised as follows: • the functor category rA, Abs is abelian and has injective envelopes. • in fact, the same holds for the full subcategory LpAq of left-exact functors. • LpAqop has some nice properties: it is cocomplete and has a projective generator. • such a category embeds into R-Mod for some ring R. • in turn, A embeds into such a category. 2 Basics on abelian categories Fix some category C. Let us say that a monic A Ñ B is contained in another monic A1 Ñ B if there is a map A Ñ A1 making the diagram A B commute. -
The Category of Sheaves Is a Topos Part 2
The category of sheaves is a topos part 2 Patrick Elliott Recall from the last talk that for a small category C, the category PSh(C) of presheaves on C is an elementary topos. Explicitly, PSh(C) has the following structure: • Given two presheaves F and G on C, the exponential GF is the presheaf defined on objects C 2 obC by F G (C) = Hom(hC × F; G); where hC = Hom(−;C) is the representable functor associated to C, and the product × is defined object-wise. • Writing 1 for the constant presheaf of the one object set, the subobject classifier true : 1 ! Ω in PSh(C) is defined on objects by Ω(C) := fS j S is a sieve on C in Cg; and trueC : ∗ ! Ω(C) sends ∗ to the maximal sieve t(C). The goal of this talk is to refine this structure to show that the category Shτ (C) of sheaves on a site (C; τ) is also an elementary topos. To do this we must make use of the sheafification functor defined at the end of the first talk: Theorem 0.1. The inclusion functor i : Shτ (C) ! PSh(C) has a left adjoint a : PSh(C) ! Shτ (C); called sheafification, or the associated sheaf functor. Moreover, this functor commutes with finite limits. Explicitly, a(F) = (F +)+, where + F (C) := colimS2τ(C)Match(S; F); where Match(S; F) is the set of matching families for the cover S of C, and the colimit is taken over all covering sieves of C, ordered by reverse inclusion. -
Derived Smooth Manifolds
DERIVED SMOOTH MANIFOLDS DAVID I. SPIVAK Abstract. We define a simplicial category called the category of derived man- ifolds. It contains the category of smooth manifolds as a full discrete subcat- egory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C1-rings that is obtained by patching together homotopy zero-sets of smooth functions on Eu- clidean spaces. We show that derived manifolds come equipped with a stable normal bun- dle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived man- ifolds. In particular, the intersection A \ B of submanifolds A; B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ^ [B] = [A \ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory. Contents 1. Introduction 1 2. The axioms 8 3. Main results 14 4. Layout for the construction of dMan 21 5. C1-rings 23 6. Local C1-ringed spaces and derived manifolds 26 7. Cotangent Complexes 32 8. Proofs of technical results 39 9. Derived manifolds are good for doing intersection theory 50 10. Relationship to similar work 52 References 55 1. Introduction Let Ω denote the unoriented cobordism ring (though what we will say applies to other cobordism theories as well, e.g. -
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. -
Classifying Categories the Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories
U.U.D.M. Project Report 2018:5 Classifying Categories The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories Daniel Ahlsén Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Juni 2018 Department of Mathematics Uppsala University Classifying Categories The Jordan-Holder¨ and Krull-Schmidt-Remak theorems for abelian categories Daniel Ahlsen´ Uppsala University June 2018 Abstract The Jordan-Holder¨ and Krull-Schmidt-Remak theorems classify finite groups, either as direct sums of indecomposables or by composition series. This thesis defines abelian categories and extends the aforementioned theorems to this context. 1 Contents 1 Introduction3 2 Preliminaries5 2.1 Basic Category Theory . .5 2.2 Subobjects and Quotients . .9 3 Abelian Categories 13 3.1 Additive Categories . 13 3.2 Abelian Categories . 20 4 Structure Theory of Abelian Categories 32 4.1 Exact Sequences . 32 4.2 The Subobject Lattice . 41 5 Classification Theorems 54 5.1 The Jordan-Holder¨ Theorem . 54 5.2 The Krull-Schmidt-Remak Theorem . 60 2 1 Introduction Category theory was developed by Eilenberg and Mac Lane in the 1942-1945, as a part of their research into algebraic topology. One of their aims was to give an axiomatic account of relationships between collections of mathematical structures. This led to the definition of categories, functors and natural transformations, the concepts that unify all category theory, Categories soon found use in module theory, group theory and many other disciplines. Nowadays, categories are used in most of mathematics, and has even been proposed as an alternative to axiomatic set theory as a foundation of mathematics.[Law66] Due to their general nature, little can be said of an arbitrary category. -
SHEAVES, TOPOSES, LANGUAGES Acceleration Based on A
Chapter 7 Logic of behavior: Sheaves, toposes, and internal languages 7.1 How can we prove our machine is safe? Imagine you are trying to design a system of interacting components. You wouldn’t be doing this if you didn’t have a goal in mind: you want the system to do something, to behave in a certain way. In other words, you want to restrict its possibilities to a smaller set: you want the car to remain on the road, you want the temperature to remain in a particular range, you want the bridge to be safe for trucks to pass. Out of all the possibilities, your system should only permit some. Since your system is made of components that interact in specified ways, the possible behavior of the whole—in any environment—is determined by the possible behaviors of each of its components in their local environments, together with the precise way in which they interact.1 In this chapter, we will discuss a logic wherein one can describe general types of behavior that occur over time, and prove properties of a larger-scale system from the properties and interaction patterns of its components. For example, suppose we want an autonomous vehicle to maintain a distance of some safe R from other objects. To do so, several components must interact: a 2 sensor that approximates the real distance by an internal variable S0, a controller that uses S0 to decide what action A to take, and a motor that moves the vehicle with an 1 The well-known concept of emergence is not about possibilities, it is about prediction. -
Diagram Chasing in Abelian Categories
Diagram Chasing in Abelian Categories Toni Annala Contents 1 Overview 2 2 Abelian Categories 3 2.1 Denition and basic properties . .3 2.2 Subobjects and quotient objects . .6 2.3 The image and inverse image functors . 11 2.4 Exact sequences and diagram chasing . 16 1 Chapter 1 Overview This is a short note, intended only for personal use, where I x diagram chasing in general abelian categories. I didn't want to take the Freyd-Mitchell embedding theorem for granted, and I didn't like the style of the Freyd's book on the topic. Therefore I had to do something else. As this was intended only for personal use, and as I decided to include this to the application quite late, I haven't touched anything in chapter 2. Some vague references to Freyd's book are made in the passing, they mean the book Abelian Categories by Peter Freyd. How diagram chasing is xed then? The main idea is to chase subobjects instead of elements. The sections 2.1 and 2.2 contain many standard statements about abelian categories, proved perhaps in a nonstandard way. In section 2.3 we dene the image and inverse image functors, which let us transfer subobjects via a morphism of objects. The most important theorem in this section is probably 2.3.11, which states that for a subobject U of X, and a morphism f : X ! Y , we have ff −1U = U \ imf. Some other results are useful as well, for example 2.3.2, which says that the image functor associated to a monic morphism is injective. -
Categorical Differential Geometry Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 35, No 4 (1994), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M. V. LOSIK Categorical differential geometry Cahiers de topologie et géométrie différentielle catégoriques, tome 35, no 4 (1994), p. 274-290 <http://www.numdam.org/item?id=CTGDC_1994__35_4_274_0> © Andrée C. Ehresmann et les auteurs, 1994, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET VolumeXXXV-4 (1994) GEOMETRIE DIFFERENTIELLE CATEGORIQUES CATEGORICAL DIFFERENTIAL GEOMETRY by M.V. LOSIK R6sum6. Cet article d6veloppe une th6orie g6n6rale des structures g6om6triques sur des vari6t6s, bas6e sur la tlieorie des categories. De nombreuses generalisations connues des vari6t6s et des vari6t6s de Rie- mann r.entrent dans le cadre de cette th6orie g6n6rale. On donne aussi une construction des classes caractéristiques des objets ainsi obtenus, tant classiques que generalises. Diverses applications sont indiqu6es, en parti- culier aux feuilletages. Introduction This paper is the result of an attempt to give a precise meaning to some ideas of the "formal differential geometry" of Gel’fand. These ideas of Gel’fand have not been formalized in general but they may be explained by the following example. Let C*(Wn; IR) be the complex of continuous cochains of the Lie algebra Wn of formal vector fields on IRn with coefficients in the trivial Wn-module IR and let C*(Wn,GL(n, R)), C*(Wn, O(n)) be its subcomplexes of relative cochains of Wn relative to the groups GL(n, R), 0(n), respectively. -
On Colimits in Various Categories of Manifolds
ON COLIMITS IN VARIOUS CATEGORIES OF MANIFOLDS DAVID WHITE 1. Introduction Homotopy theorists have traditionally studied topological spaces and used methods which rely heavily on computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what is required. For this reason, a homotopy theorist need the freedom to make any construction he wishes, e.g. products, subobjects, quotients, gluing towers of spaces together, dividing out by group actions, moving to fixed points of group actions, etc. Over the past 50 years tools have been developed which allow the study of many different categories. The key tool for this is the notion of a model category, which is the most general place one can do homotopy theory. Model categories were introduced by the Fields Medalist Dan Quillen, and the best known reference for model categories is the book Model Categories by Mark Hovey. Model categories are defined by a list of axioms which are all analogous to properties that the category Top of topological spaces satisfies. The most important property of a model category is that it has three distinguished classes of morphisms–called weak equivalences, cofibrations, and fibrations–which follow rules similar to the classes of maps with those names in the category Top (so the homotopy theory comes into play by inverting the weak equivalences to make them isomorphisms). The fact that homotopy theorists need to be able to do lots of constructions is contained in the axiom that M must be complete and cocomplete. -
VII Sheaf Theory
VII Sheaf Theory Claudia Centazzo and Enrico M. Vitale 1. Introduction To write a few lines of introduction to a few pages of work on a real corner stone of mathematics like sheaf theory is not an easy task. So, let us try with . two introductions. 1.1. First introduction: for students (and everybody else). In the study of ordinary differential equations, when you face a Cauchy problem of the form n (n) 0 (n¡1) (i) (i) y = f(x; y; y ; : : : ; y ) ; y (x0) = y0 you know that the continuity of f is enough to get a local solution, i.e. a solution defined on an open neighborhood Ux0 of x0: But, to guarantee the existence of a global solution, the stronger Lipschitz condition on f is required. P n In complex analysis, we know that a power series an(z ¡ z0) uniformly converges on any compact space strictly contained in the interior of the convergence disc. This is equivalent to the local uniform convergence: for any z in the open disc, there is an open neighborhood Uz of z on which the series converges uniformly. But local uniform convergence does not imply uniform convergence on the whole disc. This gap between local uniform convergence and global uniform convergence is the reason why the theory of Weierstrass analytic functions exists. These are only two simple examples, which are part of everybody’s basic knowl- edge in mathematics, of the passage from local to global. Sheaf theory is precisely meant to encode and study such a passage. Sheaf theory has its origin in complex analysis (see, for example, [18]) and in the study of cohomology of spaces [8] (see also [26] for a historical survey of sheaf theory). -
Toposes Toposes and Their Logic
Building up to toposes Toposes and their logic Toposes James Leslie University of Edinurgh [email protected] April 2018 James Leslie Toposes Building up to toposes Toposes and their logic Overview 1 Building up to toposes Limits and colimits Exponenetiation Subobject classifiers 2 Toposes and their logic Toposes Heyting algebras James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Products Z f f1 f2 X × Y π1 π2 X Y James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coproducts i i X 1 X + Y 2 Y f f1 f2 Z James Leslie Toposes e : E ! X is an equaliser if for any commuting diagram f Z h X Y g there is a unique h such that triangle commutes: f E e X Y g h h Z Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers Suppose we have the following commutative diagram: f E e X Y g James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers Suppose we have the following commutative diagram: f E e X Y g e : E ! X is an equaliser if for any commuting diagram f Z h X Y g there is a unique h such that triangle commutes: f E e X Y g h h Z James Leslie Toposes The following is an equaliser diagram: f ker G i G H 0 Given a k such that following commutes: f K k G H ker G i G 0 k k K k : K ! ker(G), k(g) = k(g).