Categories in Categories, and Size Matters
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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 . -
A Subtle Introduction to Category Theory
A Subtle Introduction to Category Theory W. J. Zeng Department of Computer Science, Oxford University \Static concepts proved to be very effective intellectual tranquilizers." L. L. Whyte \Our study has revealed Mathematics as an array of forms, codify- ing ideas extracted from human activates and scientific problems and deployed in a network of formal rules, formal definitions, formal axiom systems, explicit theorems with their careful proof and the manifold in- terconnections of these forms...[This view] might be called formal func- tionalism." Saunders Mac Lane Let's dive right in then, shall we? What can we say about the array of forms MacLane speaks of? Claim (Tentative). Category theory is about the formal aspects of this array of forms. Commentary: It might be tempting to put the brakes on right away and first grab hold of what we mean by formal aspects. Instead, rather than trying to present \the formal" as the object of study and category theory as our instrument, we would nudge the reader to consider another perspective. Category theory itself defines formal aspects in much the same way that physics defines physical concepts or laws define legal (and correspondingly illegal) aspects: it embodies them. When we speak of what is formal/physical/legal we inescapably speak of category/physical/legal theory, and vice versa. Thus we pass to the somewhat grammatically awkward revision of our initial claim: Claim (Tentative). Category theory is the formal aspects of this array of forms. Commentary: Let's unpack this a little bit. While individual forms are themselves tautologically formal, arrays of forms and everything else networked in a system lose this tautological formality. -
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. -
10. Algebra Theories II 10.1 Essentially Algebraic Theory Idea: A
Page 1 of 7 10. Algebra Theories II 10.1 essentially algebraic theory Idea: A mathematical structure is essentially algebraic if its definition involves partially defined operations satisfying equational laws, where the domain of any given operation is a subset where various other operations happen to be equal. An actual algebraic theory is one where all operations are total functions. The most familiar example may be the (strict) notion of category: a small category consists of a set C0of objects, a set C1 of morphisms, source and target maps s,t:C1→C0 and so on, but composition is only defined for pairs of morphisms where the source of one happens to equal the target of the other. Essentially algebraic theories can be understood through category theory at least when they are finitary, so that all operations have only finitely many arguments. This gives a generalization of Lawvere theories, which describe finitary algebraic theories. As the domains of the operations are given by the solutions to equations, they may be understood using the notion of equalizer. So, just as a Lawvere theory is defined using a category with finite products, a finitary essentially algebraic theory is defined using a category with finite limits — or in other words, finite products and also equalizers (from which all other finite limits, including pullbacks, may be derived). Definition As alluded to above, the most concise and elegant definition is through category theory. The traditional definition is this: Definition. An essentially algebraic theory or finite limits theory is a category that is finitely complete, i.e., has all finite limits. -
Exercise Sheet
Exercises on Higher Categories Paolo Capriotti and Nicolai Kraus Midlands Graduate School, April 2017 Our course should be seen as a very first introduction to the idea of higher categories. This exercise sheet is a guide for the topics that we discuss in our four lectures, and some exercises simply consist of recalling or making precise concepts and constructions discussed in the lectures. The notion of a higher category is not one which has a single clean def- inition. There are numerous different approaches, of which we can only discuss a tiny fraction. In general, a very good reference is the nLab at https://ncatlab.org/nlab/show/HomePage. Apart from that, a very incomplete list for suggested further reading is the following (all available online). For an overview over many approaches: – Eugenia Cheng and Aaron Lauda, Higher-Dimensional Categories: an illustrated guide book available online, cheng.staff.shef.ac.uk/guidebook Introductions to simplicial sets: – Greg Friedman, An Elementary Illustrated Introduction to Simplicial Sets very gentle introduction, arXiv:0809.4221. – Emily Riehl, A Leisurely Introduction to Simplicial Sets a slightly more advanced introduction, www.math.jhu.edu/~eriehl/ssets.pdf Quasicategories: – Jacob Lurie, Higher Topos Theory available as paperback (ISBN-10: 0691140499) and online, arXiv:math/0608040 Operads (not discussed in our lectures): – Tom Leinster, Higher Operads, Higher Categories available as paperback (ISBN-10: 0521532159) and online, 1 arXiv:math/0305049 Note: Exercises are ordered by topic and can be done in any order, unless an exercise explicitly refers to a previous one. We recommend to do the exercises marked with an arrow ⇒ first. -
Category Theory and Diagrammatic Reasoning 3 Universal Properties, Limits and Colimits
Category theory and diagrammatic reasoning 13th February 2019 Last updated: 7th February 2019 3 Universal properties, limits and colimits A division problem is a question of the following form: Given a and b, does there exist x such that a composed with x is equal to b? If it exists, is it unique? Such questions are ubiquitious in mathematics, from the solvability of systems of linear equations, to the existence of sections of fibre bundles. To make them precise, one needs additional information: • What types of objects are a and b? • Where can I look for x? • How do I compose a and x? Since category theory is, largely, a theory of composition, it also offers a unifying frame- work for the statement and classification of division problems. A fundamental notion in category theory is that of a universal property: roughly, a universal property of a states that for all b of a suitable form, certain division problems with a and b as parameters have a (possibly unique) solution. Let us start from universal properties of morphisms in a category. Consider the following division problem. Problem 1. Let F : Y ! X be a functor, x an object of X. Given a pair of morphisms F (y0) f 0 F (y) x , f does there exist a morphism g : y ! y0 in Y such that F (y0) F (g) f 0 F (y) x ? f If it exists, is it unique? 1 This has the form of a division problem where a and b are arbitrary morphisms in X (which need to have the same target), x is constrained to be in the image of a functor F , and composition is composition of morphisms. -
Change of Base for Locally Internal Categories
Journal of Pure and Applied Algebra 61 (1989) 233-242 233 North-Holland CHANGE OF BASE FOR LOCALLY INTERNAL CATEGORIES Renato BETTI Universitb di Torino, Dipartimento di Matematica, via Principe Amedeo 8, IO123 Torino, Italy Communicated by G.M. Kelly Received 4 August 1988 Locally internal categories over an elementary topos 9 are regarded as categories enriched in the bicategory Span g. The change of base is considered with respect to a geometric morphism S-r 8. Cocompleteness is preserved, and the topos S can be regarded as a cocomplete, locally internal category over 6. This allows one in particular to prove an analogue of Diaconescu’s theorem in terms of general properties of categories. Introduction Locally internal categories over a topos & can be regarded as categories enriched in the bicategory Span& and in many cases their properties can be usefully dealt with in terms of standard notions of enriched category theory. From this point of view Betti and Walters [2,3] study completeness, ends, functor categories, and prove an adjoint functor theorem. Here we consider a change of the base topos, i.e. a geometric morphismp : 9-t CF. In particular 9 itself can be regarded as locally internal over 8. Again, properties of p can be expressed by the enrichment (both in Spangand in Span 8) and the relevant module calculus of enriched category theory applies directly to most calculations. Our notion of locally internal category is equivalent to that given by Lawvere [6] under the name of large category with an &-atlas, and to Benabou’s locally small fibrations [l]. -
Category Theory Course
Category Theory Course John Baez September 3, 2019 1 Contents 1 Category Theory: 4 1.1 Definition of a Category....................... 5 1.1.1 Categories of mathematical objects............. 5 1.1.2 Categories as mathematical objects............ 6 1.2 Doing Mathematics inside a Category............... 10 1.3 Limits and Colimits.......................... 11 1.3.1 Products............................ 11 1.3.2 Coproducts.......................... 14 1.4 General Limits and Colimits..................... 15 2 Equalizers, Coequalizers, Pullbacks, and Pushouts (Week 3) 16 2.1 Equalizers............................... 16 2.2 Coequalizers.............................. 18 2.3 Pullbacks................................ 19 2.4 Pullbacks and Pushouts....................... 20 2.5 Limits for all finite diagrams.................... 21 3 Week 4 22 3.1 Mathematics Between Categories.................. 22 3.2 Natural Transformations....................... 25 4 Maps Between Categories 28 4.1 Natural Transformations....................... 28 4.1.1 Examples of natural transformations........... 28 4.2 Equivalence of Categories...................... 28 4.3 Adjunctions.............................. 29 4.3.1 What are adjunctions?.................... 29 4.3.2 Examples of Adjunctions.................. 30 4.3.3 Diagonal Functor....................... 31 5 Diagrams in a Category as Functors 33 5.1 Units and Counits of Adjunctions................. 39 6 Cartesian Closed Categories 40 6.1 Evaluation and Coevaluation in Cartesian Closed Categories. 41 6.1.1 Internalizing Composition................. 42 6.2 Elements................................ 43 7 Week 9 43 7.1 Subobjects............................... 46 8 Symmetric Monoidal Categories 50 8.1 Guest lecture by Christina Osborne................ 50 8.1.1 What is a Monoidal Category?............... 50 8.1.2 Going back to the definition of a symmetric monoidal category.............................. 53 2 9 Week 10 54 9.1 The subobject classifier in Graph................. -
A Concrete Introduction to Category Theory
A CONCRETE INTRODUCTION TO CATEGORIES WILLIAM R. SCHMITT DEPARTMENT OF MATHEMATICS THE GEORGE WASHINGTON UNIVERSITY WASHINGTON, D.C. 20052 Contents 1. Categories 2 1.1. First Definition and Examples 2 1.2. An Alternative Definition: The Arrows-Only Perspective 7 1.3. Some Constructions 8 1.4. The Category of Relations 9 1.5. Special Objects and Arrows 10 1.6. Exercises 14 2. Functors and Natural Transformations 16 2.1. Functors 16 2.2. Full and Faithful Functors 20 2.3. Contravariant Functors 21 2.4. Products of Categories 23 3. Natural Transformations 26 3.1. Definition and Some Examples 26 3.2. Some Natural Transformations Involving the Cartesian Product Functor 31 3.3. Equivalence of Categories 32 3.4. Categories of Functors 32 3.5. The 2-Category of all Categories 33 3.6. The Yoneda Embeddings 37 3.7. Representable Functors 41 3.8. Exercises 44 4. Adjoint Functors and Limits 45 4.1. Adjoint Functors 45 4.2. The Unit and Counit of an Adjunction 50 4.3. Examples of adjunctions 57 1 1. Categories 1.1. First Definition and Examples. Definition 1.1. A category C consists of the following data: (i) A set Ob(C) of objects. (ii) For every pair of objects a, b ∈ Ob(C), a set C(a, b) of arrows, or mor- phisms, from a to b. (iii) For all triples a, b, c ∈ Ob(C), a composition map C(a, b) ×C(b, c) → C(a, c) (f, g) 7→ gf = g · f. (iv) For each object a ∈ Ob(C), an arrow 1a ∈ C(a, a), called the identity of a. -
Basic Category Theory
Basic Category Theory TOMLEINSTER University of Edinburgh arXiv:1612.09375v1 [math.CT] 30 Dec 2016 First published as Basic Category Theory, Cambridge Studies in Advanced Mathematics, Vol. 143, Cambridge University Press, Cambridge, 2014. ISBN 978-1-107-04424-1 (hardback). Information on this title: http://www.cambridge.org/9781107044241 c Tom Leinster 2014 This arXiv version is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). Licence information: https://creativecommons.org/licenses/by-nc-sa/4.0 c Tom Leinster 2014, 2016 Preface to the arXiv version This book was first published by Cambridge University Press in 2014, and is now being published on the arXiv by mutual agreement. CUP has consistently supported the mathematical community by allowing authors to make free ver- sions of their books available online. Readers may, in turn, wish to support CUP by buying the printed version, available at http://www.cambridge.org/ 9781107044241. This electronic version is not only free; it is also freely editable. For in- stance, if you would like to teach a course using this book but some of the examples are unsuitable for your class, you can remove them or add your own. Similarly, if there is notation that you dislike, you can easily change it; or if you want to reformat the text for reading on a particular device, that is easy too. In legal terms, this text is released under the Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). The licence terms are available at the Creative Commons website, https:// creativecommons.org/licenses/by-nc-sa/4.0. -
Categorical Notions of Fibration
CATEGORICAL NOTIONS OF FIBRATION FOSCO LOREGIAN AND EMILY RIEHL Abstract. Fibrations over a category B, introduced to category the- ory by Grothendieck, encode pseudo-functors Bop Cat, while the special case of discrete fibrations encode presheaves Bop → Set. A two- sided discrete variation encodes functors Bop × A → Set, which are also known as profunctors from A to B. By work of Street, all of these fi- bration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to Cat, unexpectedly, the dual two-sided codiscrete cofibra- tions are necessary to model V-profunctors internally to V-Cat. These notes were initially written by the second-named author to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in the fall of 2010 at the University of Chicago. A few years later, the first-named author joined to expand and improve the internal exposition and external references. Contents 1. Introduction 1 2. Fibrations in 1-category theory 3 3. Fibrations in 2-categories 8 4. Fibrations in bicategories 12 References 17 1. Introduction Fibrations were introduced to category theory in [Gro61, Gro95] and de- veloped in [Gra66]. Ross Street gave definitions of fibrations internal to an arbitrary 2-category [Str74] and later bicategory [Str80]. Interpreted in the 2-category of categories, the 2-categorical definitions agree with the classical arXiv:1806.06129v2 [math.CT] 16 Feb 2019 ones, while the bicategorical definitions are more general. In this expository article, we tour the various categorical notions of fibra- tion in order of increasing complexity. -
HIGHER-DIMENSIONAL ALGEBRA V: 2-GROUPS 1. Introduction
Theory and Applications of Categories, Vol. 12, No. 14, 2004, pp. 423–491. HIGHER-DIMENSIONAL ALGEBRA V: 2-GROUPS JOHN C. BAEZ AND AARON D. LAUDA Abstract. A 2-group is a ‘categorified’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G × G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak’ and ‘coherent’ 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an ∼ ∼ inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y = 1 = y ⊗ x. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inversex ¯ and isomorphisms ix:1→ x⊗x¯, ex:¯x⊗x → 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an ‘improvement’ 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the ‘fundamental 2-group’ of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G ( ∈ Z)havingG as its group of objects and U(1) as the group of automorphisms of its identity object.