A Classification of Accessible Categories
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Balanced Category Theory II
BALANCED CATEGORY THEORY II CLAUDIO PISANI ABSTRACT. In the first part, we further advance the study of category theory in a strong balanced factorization category C [Pisani, 2008], a finitely complete category en- dowed with two reciprocally stable factorization systems such that M/1 = M′/1. In particular some aspects related to “internal” (co)limits and to Cauchy completeness are considered. In the second part, we maintain that also some aspects of topology can be effectively synthesized in a (weak) balanced factorization category T , whose objects should be considered as possibly “infinitesimal” and suitably “regular” topological spaces. While in C the classes M and M′ play the role of discrete fibrations and opfibrations, in T they play the role of local homeomorphisms and perfect maps, so that M/1 and M′/1 are the subcategories of discrete and compact spaces respectively. One so gets a direct abstract link between the subjects, with mutual benefits. For example, the slice projection X/x → X and the coslice projection x\X → X, obtained as the second factors of x :1 → X according to (E, M) and (E′, M′) in C, correspond in T to the “infinitesimal” neighborhood of x ∈ X and to the closure of x. Furthermore, the open-closed complementation (generalized to reciprocal stability) becomes the key tool to internally treat, in a coherent way, some categorical concepts (such as (co)limits of presheaves) which are classically related by duality. Contents 1 Introduction 2 2 Bicartesian arrows of bimodules 4 3 Factorization systems 6 4 Balanced factorization categories 10 5 Cat asastrongbalancedfactorizationcategory 12 arXiv:0904.1790v3 [math.CT] 27 Apr 2009 6 Slices and colimits in a bfc 14 7 Internalaspectsofbalancedcategorytheory 16 8 The tensor functor and the internal hom 20 9 Retracts of slices 24 2000 Mathematics Subject Classification: 18A05, 18A22, 18A30, 18A32, 18B30, 18D99, 54B30, 54C10. -
Notes and Solutions to Exercises for Mac Lane's Categories for The
Stefan Dawydiak Version 0.3 July 2, 2020 Notes and Exercises from Categories for the Working Mathematician Contents 0 Preface 2 1 Categories, Functors, and Natural Transformations 2 1.1 Functors . .2 1.2 Natural Transformations . .4 1.3 Monics, Epis, and Zeros . .5 2 Constructions on Categories 6 2.1 Products of Categories . .6 2.2 Functor categories . .6 2.2.1 The Interchange Law . .8 2.3 The Category of All Categories . .8 2.4 Comma Categories . 11 2.5 Graphs and Free Categories . 12 2.6 Quotient Categories . 13 3 Universals and Limits 13 3.1 Universal Arrows . 13 3.2 The Yoneda Lemma . 14 3.2.1 Proof of the Yoneda Lemma . 14 3.3 Coproducts and Colimits . 16 3.4 Products and Limits . 18 3.4.1 The p-adic integers . 20 3.5 Categories with Finite Products . 21 3.6 Groups in Categories . 22 4 Adjoints 23 4.1 Adjunctions . 23 4.2 Examples of Adjoints . 24 4.3 Reflective Subcategories . 28 4.4 Equivalence of Categories . 30 4.5 Adjoints for Preorders . 32 4.5.1 Examples of Galois Connections . 32 4.6 Cartesian Closed Categories . 33 5 Limits 33 5.1 Creation of Limits . 33 5.2 Limits by Products and Equalizers . 34 5.3 Preservation of Limits . 35 5.4 Adjoints on Limits . 35 5.5 Freyd's adjoint functor theorem . 36 1 6 Chapter 6 38 7 Chapter 7 38 8 Abelian Categories 38 8.1 Additive Categories . 38 8.2 Abelian Categories . 38 8.3 Diagram Lemmas . 39 9 Special Limits 41 9.1 Interchange of Limits . -
Covers, Envelopes, and Cotorsion Theories in Locally Presentable
COVERS, ENVELOPES, AND COTORSION THEORIES IN LOCALLY PRESENTABLE ABELIAN CATEGORIES AND CONTRAMODULE CATEGORIES L. POSITSELSKI AND J. ROSICKY´ Abstract. We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extend- ing the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples. 1. Introduction 1.1. An abelian category with enough projective objects is determined by (can be recovered from) its full subcategory of projective objects. An explicit construction providing the recovery procedure is described, in the nonadditive context, in the paper [13] (see also [21]), where it is called “the exact completion of a weakly left exact category”, and in the additive context, in the paper [22, Lemma 1], where it is called “the category of coherent functors”. In fact, the exact completion in the additive context is contained already in [19], as it is explained in [35]. Hence, in particular, any abelian category with arbitrary coproducts and a single small (finitely generated, finitely presented) projective generator P is equivalent to the category of right modules over the ring of endomorphisms of P . More generally, an abelian category with arbitrary coproducts and a set of small projective generators is equivalent to the category of contravariant additive functors from its full subcategory formed by these generators into the category of abelian groups (right modules over arXiv:1512.08119v2 [math.CT] 21 Apr 2017 the big ring of morphisms between the generators). Now let K be an abelian category with a single, not necessarily small projective generator P ; suppose that all coproducts of copies of P exist in K. -
Introduction to Category Theory (Notes for Course Taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT)
Introduction to category theory (notes for course taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT) Alexander Yom Din February 10, 2021 It is never true that two substances are entirely alike, differing only in being two rather than one1. G. W. Leibniz, Discourse on metaphysics 1This can be imagined to be related to at least two of our themes: the imperative of considering a contractible groupoid of objects as an one single object, and also the ideology around Yoneda's lemma ("no two different things have all their properties being exactly the same"). 1 Contents 1 The basic language 3 1.1 Categories . .3 1.2 Functors . .7 1.3 Natural transformations . .9 2 Equivalence of categories 11 2.1 Contractible groupoids . 11 2.2 Fibers . 12 2.3 Fibers and fully faithfulness . 12 2.4 A lemma on fully faithfulness in families . 13 2.5 Definition of equivalence of categories . 14 2.6 Simple examples of equivalence of categories . 17 2.7 Theory of the fundamental groupoid and covering spaces . 18 2.8 Affine algebraic varieties . 23 2.9 The Gelfand transform . 26 2.10 Galois theory . 27 3 Yoneda's lemma, representing objects, limits 27 3.1 Yoneda's lemma . 27 3.2 Representing objects . 29 3.3 The definition of a limit . 33 3.4 Examples of limits . 34 3.5 Dualizing everything . 39 3.6 Examples of colimits . 39 3.7 General colimits in terms of special ones . 41 4 Adjoint functors 42 4.1 Bifunctors . 42 4.2 The definition of adjoint functors . 43 4.3 Some examples of adjoint functors . -
Sizes and Filtrations in Accessible Categories
SIZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES MICHAEL LIEBERMAN, JIRˇ´I ROSICKY,´ AND SEBASTIEN VASEY Abstract. Accessible categories admit a purely category-theoretic replace- ment for cardinality: the internal size. Generalizing results and methods from [LRV19b], we examine set-theoretic problems related to internal sizes and prove several L¨owenheim-Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large ac- cessible category has an object in all internal sizes of high-enough cofinality. We also prove that accessible categories with directed colimits have filtrations: any object of sufficiently high internal size is (the retract of) a colimit of a chain of strictly smaller objects. Contents 1. Introduction 1 2. Preliminaries 4 3. Directed systems and cofinal posets 9 4. Presentation theorem and axiomatizability 12 5. On successor presentability ranks 14 6. The existence spectrum of a µ-AEC 16 7. The existence spectrum of an accessible category 19 8. Filtrations 22 References 26 1. Introduction Recent years have seen a burst of research activity connecting accessible categories with abstract model theory. Abstract model theory, which has always had the aim of generalizing|in a uniform way|fragments of the rich classification theory of first order logic to encompass the broader nonelementary classes of structures that Date: June 5, 2019 AMS 2010 Subject Classification: Primary 18C35. Secondary: 03C45, 03C48, 03C52, 03C55, 03C75, 03E05. Key words and phrases. internal size, presentability rank, existence spectrum, accessibility spectrum, filtrations, singular cardinal hypothesis. The second author is supported by the Grant agency of the Czech republic under the grant 19-00902S. -
Arxiv:0708.2185V1 [Math.CT] 16 Aug 2007 Ro.Oragmn Sbsdo H Atta Oooyequivalen Homotopy That Fact the on Based Is Argument Our Proof
ON COMBINATORIAL MODEL CATEGORIES J. ROSICKY´ ∗ Abstract. Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke or D. Dugger. We are contributing to this endeavour by proving that weak equiv- alences in a combinatorial model category form an accessible cat- egory. We also present some new results about weak equivalences and cofibrations in combinatorial model categories. 1. Introduction Model categories were introduced by Quillen [24] as a foundation of homotopy theory. Their modern theory can found in [20] or [19]. Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly gener- ated. The latter means that both cofibrations and trivial cofibrations are cofibrantly generated by a set of morphisms. He has not published his results yet but some of them can be found in [7] or [14]. In partic- ular, [7] contains the proof of the theorem characterizing when a class W of weak equivalences makes a locally presentable category K to be a combinatorial model category with a given cofibrantly generated class C of cofibrations. The characterization combines closure properties of W together with a smallness condition saying that W satisfies the solution set condition at the generating set X of cofibrations. We will show that arXiv:0708.2185v1 [math.CT] 16 Aug 2007 these conditions are also necessary. This is based or another result of J. H. Smith saying that, in a combinatorial model category, W is always accessible and accessibly embedded in the category K→ of morphisms of K (he informed me about this result in 2002 without indicating a proof). -
Part 1 1 Basic Theory of Categories
Part 1 1 Basic Theory of Categories 1.1 Concrete Categories One reason for inventing the concept of category was to formalize the notion of a set with additional structure, as it appears in several branches of mathe- matics. The structured sets are abstractly considered as objects. The structure preserving maps between objects are called morphisms. Formulating all items in terms of objects and morphisms without reference to elements, many concepts, although of disparate flavour in different math- ematical theories, can be expressed in a universal language. The statements about `universal concepts' provide theorems in each particular theory. 1.1 Definition (Concrete Category). A concrete category is a triple C = (O; U; hom), where •O is a class; • U : O −! V is a set-valued function; • hom : O × O −! V is a set-valued function; satisfying the following axioms: 1. hom(a; b) ⊆ U(b)U(a); 2. idU(a) 2 hom(a; a); 3. f 2 hom(a; b) ^ g 2 hom(b; c) ) g ◦ f 2 hom(a; c). The members of O are the objects of C. An element f 2 hom(a; b) is called a morphism with domain a and codomain b. A morphism in hom(a; a) is an endomorphism. Thus End a = hom(a; a): We call U(a) the underlying set of the object a 2 O. By Axiom 1, a morphism f f 2 hom(a; b) is a mapping U(a) −! U(b). We write f : a −! b or a −! b to indicate that f 2 hom(a; b). Axiom 2 ensures that every object a has the identity function idU(a) as an endomorphism. -
Accessible Categories and Model Theory: Hits and Misses Accessible Categories Workshop, Leeds
singular compactness calculus of λ-equivalences problems Accessible categories and model theory: hits and misses Accessible Categories Workshop, Leeds Tibor Beke University of Massachusetts tibor [email protected] July 17, 2018 Tibor Beke Accessible categories and model theory: hits and misses Wilfred Hodges in [Hodges 2008] singular compactness calculus of λ-equivalences problems epithet \ It would still be very welcome to be told that the whole scheme coincides with, say, something known to category theorists in another context. " Tibor Beke Accessible categories and model theory: hits and misses singular compactness calculus of λ-equivalences problems epithet \ It would still be very welcome to be told that the whole scheme coincides with, say, something known to category theorists in another context. " Wilfred Hodges in [Hodges 2008] Tibor Beke Accessible categories and model theory: hits and misses singular compactness cellular maps calculus of λ-equivalences cellular singular compactness problems beyond cellular prehistory Theorem (Schreier, 1926): Every subgroup of a free group is free. Also true for I abelian groups I abelian p-groups I over a field k: I Lie algebras (Witt) I commutative (non-associative) algebras (Shirshov) I magmas (Kurosh) Remark: there is still no complete characterization of these varieties known. Tibor Beke Accessible categories and model theory: hits and misses singular compactness cellular maps calculus of λ-equivalences cellular singular compactness problems beyond cellular conversely: Does this characterize free groups? Question Suppose G is an infinite group such that all subgroups of G of cardinality less than G, are free. (We will call such a group G almost free.) Is G free? Answer No. -
THE HEART of a COMBINATORIAL MODEL CATEGORY Introduction
Theory and Applications of Categories, Vol. 31, No. 2, 2016, pp. 31{62. THE HEART OF A COMBINATORIAL MODEL CATEGORY ZHEN LIN LOW Abstract. We show that every small model category that satisfies certain size con- ditions can be completed to yield a combinatorial model category, and conversely, every combinatorial model category arises in this way. We will also see that these constructions preserve right properness and compatibility with simplicial enrichment. Along the way, we establish some technical results on the index of accessibility of various constructions on accessible categories, which may be of independent interest. Introduction Category-theoretic homotopy theory has seen a boom in recent decades. One development was the introduction of the notion of `combinatorial model categories' by Smith[1998]. These correspond to what Lurie[2009] calls `presentable 1-categories' and are therefore a homotopy-theoretic generalisation of the locally presentable categories of Gabriel and Ulmer[1971]. The classification of locally κ-presentable categories says that each one is equivalent to the free κ-ind-completion of a κ-cocomplete small category, and Lurie proved the analogous proposition for presentable 1-categories, so it should at least seem plausible that every combinatorial model category is generated by a small model category in an appropriate sense. Indeed, the work of Beke[2000] suggests that more should be true. As stated in the abstract of op. cit., If a Quillen model category can be specified using a certain logical syntax (intuitively, `is algebraic/combinatorial enough'), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. -
Filtered Derived Categories and Their Applications
Filtered derived categories and their applications D. Kaledin∗ Contents 1 f-categories. 1 1.1 Definitions and examples. 1 1.2 Reflecting the filtration. 3 1.3 The comparison functor. 5 1.4 Further adjunctions. 7 1.5 Morphisms. 9 2 End(N)-action. 12 2.1 Prototype: End(N)-action on functor categories. 12 2.2 N-categories. 15 2.3 Uniqueness. 16 2.4 Full embeddings. 19 2.5 Bifibrations. 22 3 Λ∞-categories. 28 3.1 The category Λ∞. ........................ 28 3.2 Unbounded maps. 30 3.3 The bifibration ΛD: preliminaries. 32 3.4 The bifibration ΛD: the construction. 35 3.5 Functoriality. 37 Introduction. 1 f-categories. 1.1 Definitions and examples. Out starting point is a version of a definition given by Beilinson in [B]. ∗Partially supported by grant NSh-1987.2008.1 1 Preliminary version – please do not distribute, use at your own risk 2 Definition 1.1. An f-category is a triple hD, F, si of a triangulated cate- gory D, a fully faithful left-admissible triangulated embedding F : D → D, called the twist functor, and a morphism s : F → Id from F to the identity functor Id : D → D, such that 2 (i) for any M ∈ D, we have sF (M) = F (sM ): F (M) → F (M), and ⊥ (ii) for any M ∈ F (D), the cone of the map sM : F (M) → M lies in F (D)⊥. The quotient category D/F (D) is called the core of the f-category hD, F, si and denoted by C(D,F ), or simply C(D) when there is no danger of confu- sion. -
Dirichlet Polynomials Form a Topos
Dirichlet Polynomials form a Topos David I. Spivak David Jaz Myers November 5, 2020 Abstract One can think of power series or polynomials in one variable, such as %(y) = 2y3 + y + 5, as functors from the category Set of sets to itself; these are known as polynomial functors. Denote by PolySet the category of polynomial functors on Set and natural transformations between them. The constants 0, 1 and operations +, × that occur in %(y) are actually the initial and terminal objects and the coproduct and product in PolySet. Just as the polynomial functors on Set are the copresheaves that can be written ∞ y as sums of representables, one can express any Dirichlet series, e.g. ==0 = ,asa coproduct of representable presheaves. A Dirichlet polynomial is a finite Dirichlet series, that is, a finite sum of representables =y. We discuss how bothÍ polynomial functors and their Dirichlet analogues can be understood in terms of bundles, and go on to prove that the category of Dirichlet polynomials is an elementary topos. 1 Introduction Polynomials %(y) and finite Dirichlet series (y) in one variable y, with natural number coefficients 08 ∈ N, are respectively functions of the form = 2 1 0 %(y) = 0=y +···+ 02y + 01y + 00y , arXiv:2003.04827v2 [math.CT] 4 Nov 2020 y y y y (1) (y) = 0= = +···+ 022 + 011 + 000 . The first thing we should emphasize is that the algebraic expressions in (1) can in fact be regarded as objects in a category, in fact two categories: Poly and Dir. We will explain the morphisms later, but for now we note that in Poly, y2 = y × y is a product and 2y = y+y is a coproduct, and similarly for Dir. -
The Limit-Colimit Coincidence for Categories
The Limit-Colimit Coincidence for Categories Paul Taylor 1994 Abstract Scott noticed in 1969 that in the category of complete lattices and maps preserving directed joins, limit of a sequence of projections (maps with preinverse left adjoints) is isomorphic to the colimit of those left adjoints (called embeddings). This result holds in any category of domains and is the basis of the solution of recursive domain equations. Limits of this kind (and continuity with respect to them) also occur in domain semantics of polymorphism, where we find that we need to generalise from sequences to directed or filtered diagrams and drop the \preinverse" condition. The result is also applicable to the proof of cartesian closure for stable domains. The purpose of this paper is to express and prove the most general form of the result, which is for filtered diagrams of adjoint pairs between categories with filtered colimits. The ideas involved in the applications are to be found elsewhere in the literature: here we are concerned solely with 2-categorical details. The result we obtain is what is expected, the only remarkable point being that it seems definitely to be about pseudo- and not lax limits and colimits. On the way to proving the main result, we find ourselves also performing the constructions needed for domain interpretations of dependent type polymorphism. 1 Filtered diagrams The result concerns limits and colimits of filtered diagrams of categories, so we shall be interested in functors of the form FCCatcp I! where is a filtered category and FCCatcp is the 2-category of small categories with filtered colimits,I functors with right adjoints which preserve filtered colimits and natural transformations.