An Introduction to Category Theory Lecture 1 Dr Marco Benini [email protected] Dottorato di Ricerca in Informatica e Matematica del Calcolo Università degli Studi dell’Insubria 2014 Syllabus ■ categories: examples of abstract and concrete categories, product categories, subcategories); ■ basic constructions: limits and colimits, exponentiation, subobject classifiers; ■ functors and natural transformations: Yoneda lemma, presheaves, representable functors, functor categories; ■ adjunctions: definition and examples (2 of 189) Texts Slides and supporting material: ■ All the course material is on the website: http://wp.me/P1rYpC-8W Textbooks: ■ [Borceux] F. Borceux, Handbook of Categorical Algebra I: Basic Category Theory, Cambridge University Press (1994). ISBN: 978-0521061193 The lectures are mainly based on this book ■ [MacLane] S. Mac Lane, Categories for the Working Mathematician, 2nd edition, Springer-Verlag (1998). ISBN: 978-0387984032 ■ [Goldblatt] R. Goldblatt, Topoi: The categorical analysis of logic, Elsevier (1984). ISBN: 978-0444867117 This book has been republished at a nicer price by Dover (2009). ISBN: 978-0486450261. (3 of 189) Examination The final examination for this course will be a seminar. The topic has to be decided together with the teacher, and it should expand a part of the course, possibly following the interests of the student. (4 of 189) Some points of history ■ Eilenberg and Mac Lane, 1942–1958: starting from algebraic topology, they abstracted a collection of concepts, and a new language to speak of them. Soon, it was clear that algebraic topology was just an application of a more general theory. ■ Grothendieck, 1957: Abelian categories. This is the starting point of Grothendieck’s work in algebraic geometry, which, as a by-product, will deliver the fundamental concept of topos. ■ Kan, 1958: adjoint functors. This was a fundamental result which lies behind the general and abstract treatment category theory allows. ■ Lawvere, 1963: category theory as a foundation for mathematics. Elementary topoi as the universe of mathematics, since they are, at the same time, a logical system, a mathematical theory, an algebraic framework, a space, . (5 of 189) Today ■ Entire branches of mathematics are described using, to various extents, category theory as the elected language: algebra, algebraic topology, algebraic geometry, and large parts of logic, just to name a few. ■ Since the development of denotational semantics by Scott, category theory has become a fundamental tool to describe, understand and research the meaning of computation. A fundamental part of theoretical computer science, semantics of programming languages, is strictly dominated by the categorical approach. ■ Functional programming uses, in real programming languages, categorical structures, e.g., monads in Haskell. And type theory can be really understood only within a categorical framework. ■ In contemporary research, Voevodosky launched Homotopy Type Theory, an interpretation of mathematics in a computational world, where algebraic topology is the founding basis. It uses in a fundamental way category theory and toposes. (6 of 189) Categorical style Category theory appears to be abstract and difficult. Although this is true, to some extent, the impression arises because it is, in the first place, a revolutionary approach. It is based on a few principles which are pushed to their extreme: ■ Looking systems from outside: objects have no meaning or nature, but what determines their behaviour is just the way they interact with other objects. ■ Theories have their own natural environment: it is wrong and deviating to study a theory in a framework which does not naturally and completely exploit the information in the theory. ■ Computation is a structure: computation has nothing to do with machines, but captures a fundamental property of the mathematical description of the world. ■ Mathematics talks about structures: and it does so by means of the way they are mutually related. (7 of 189) References and hints Every lesson is concluded with references to texts which may be useful to study and to get a better understanding of the content. Sometimes, reference to additional material is also given, so that the interested student has an hint where to find more information. These references, when present, are, of course, not exhaustive but just a possible starting point. In the present case, an excellent exposition on the history and philosophy of category theory can be found in J-P. Marquis, From a Geometrical Point of View, Logic, Epistemology, and the Unity of Science, 14, Springer-Verlag (2009). ISBN: 978-1-4020-9383-8 CC BY: $\ C Dr Marco Benini 2014 (8 of 189) An Introduction to Category Theory Lecture 2 Dr Marco Benini [email protected] Dottorato di Ricerca in Informatica e Matematica del Calcolo Università degli Studi dell’Insubria 2014 Categories I Definition 2.1 (Category) A category C is a structure C O,Hom, ,1 such that Æ h ± i ■ O is a class of objects, denoted as ObjC. ■ Hom is an operation assigning to each pair A, B of objects a set Hom(A,B), sometimes denoted as C(A,B), of arrows or morphisms. Given an arrow f Hom(A,B), we say and write that A is the domain of f (dom(f ) dom 2f A), B is the codomain of f (cod(f ) cod f B), Æ Æ Æ Æ and we usually write f : A B instead of f Hom(A,B). ! 2 ■ is a partial operation, called composition, assigning to each pair of arrows± f and g such that cod f dom g, an arrow g f such that dom(g f ) dom f and cod(g Æf ) cod g. ± ± Æ ± Æ , ! (10 of 189) Categories II , (Category) ! ■ Moreover, satisfies the associative law: for any arrows f , g, h such that the following± compositions are defined, h (g f ) (h g) f . ± ± Æ ± ± ■ 1 is an operation, called identity, assigning to each object P an arrow 1P such that dom(1P ) P cod(1P ). Æ Æ Moreover, 1P satisfies the identity law: for any arrow f : P Q, ■ ! 1Q f f f 1P . ± Æ Æ ± Definition 2.2 Given a category C O,Hom, ,1 , we say that C is small if O is a set. Æ h ± i Sometimes, the definition of category is relaxed, requiring that Hom(A,B) is just a class for every A,B ObjC. In this case, the categories satisfying our previous definition are called2 locally small, while the relaxed categories are called large. (11 of 189) Categories III A substantial point in the definition of categories is that objects have no nature by themselves: what really defines a category is the way objects are linked to each other through arrows. In fact, it is possible to define categories without requiring objects at all. Moreover, arrows have no nature either. They are just links between objects. As far as we are able to render a mathematical structure S through these basic entities, and to reduce the fundamental properties to equality of arrows, S can be effectively described in categorical terms. It is a fact that most mathematical entities can be reduced in this way, and in many cases this description is a deep and powerful one. (12 of 189) Concrete categories I Concrete categories are, generally speaking, based on sets. Example 2.3 (Set) The category Set has sets as objects and (total) functions between them as arrows. Specifically, Set O,Hom, ,1 and Æ h ± i ■ O is the proper class of all sets; ■ Hom(D,C) is the set of total functions from D to C; ■ is the usual composition of functions: given f : D E and g : E C, g± f : D C is (g f )(x) g(f (x)), for all x D; ! ! ± ! ± Æ 2 1P : P P is the identity on P, i.e., for all x P, 1P (x) x. ■ ! 2 Æ It is immediate to see that the associative and the identity laws both hold. [Exercise] Prove it. (13 of 189) Concrete categories II There is a subtlety in the definition of Set: each function corresponds to many arrows in Set, in fact, given f : D C and g : D E such that for all x D, f (x) g(x), it does not follow that! f g, unless!C E. 2 Æ Æ Æ Arrows are defined along with their domain and codomain, as we did. Also, most of the times, composition and identities are obvious from the context; in these cases, it is customary to define the category specifying only the objects and the arrows. Sometimes, when also the arrows (objects) are clear from the context, just the objects (arrows) are specified. (14 of 189) Concrete categories III Definition 2.4 (Poset) A preorder is a pair P, P such that P is a binary relation on P which is reflexive and transitive.h · i · A partially ordered set or poset is a preorder P, P where the relation is also anti-symmetric. h · i An order preserving function or monotone function from the poset (preorder) P, P to the poset (preorder) Q, Q is a function f : P Q such that, if h · i h · i ! p P p0, then f (p) Q f (p0). · · Posets and preorders are an example of algebraic structure, that is, a set, the universe, plus a number of operations acting on the universe. (15 of 189) Concrete categories IV Example 2.5 (Poset) The category Poset has all the posets as objects and all the monotone functions between them as arrows. Example 2.6 (Preorder) The category Preorder has all the preorders as objects and all the monotone functions between them as arrows. [Exercise] Check that Poset and Preorder are categories. (16 of 189) Concrete categories V Example 2.7 (Mon) The category Mon has all the monoids as objects and their homomorphisms as arrows.