Category Theory By: Ashley Reynolds

HISTORY OF CATEGORY THEORY  In 1942–45, Samuel Eilenberg and Saunders Mac Lane introduced categories, functors, and natural transformations as part of their work in topology, especially algebraic topology.  Their work was an important part of the transition from intuitive and geometric homology to axiomatic homology theory.  Eilenberg and Mac Lane later wrote that their goal was to understand natural transformations; in order to do that, functors had to be defined, which required categories.

History Continued  These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics.  More recent efforts to introduce undergraduates to categories as a foundation for mathematics include William Lawvere and Rosebrugh (2003) and Lawvere and Stephen Schanuel (1997) and Mirroslav Yotov (2012).

What is Category Theory  Category theory is used to formalize mathematics and its concepts as a collection of objects and arrows (also called morphisms). Category theory can be used to formalize concepts of other high-level abstractions such as set theory, field theory, and group theory.  Another definition: Category theory is a relatively young branch of mathematics, stemming from algebraic topology, and designed to describe various structural concepts from different mathematical fields in a uniform way.

Definitions  A category is a collection of data that satisfy some particular properties. So, saying that such and so forms a category is merely short for asserting that such and so satisfy all the axioms of a category.  A category is: the following data, subject to certain axioms.  A collection of things called objects. By default, A, B, C, … vary over objects.  A collection of things called morphisms, sometimes called arrows. By default, f, g, h, …, and later on α,β, φ, ψ, χ, …, very over morphisms.  A relation on morphisms and pairs of objects, called typing of the morphisms. By default, the relation is denoted f: A → B, for morphisms f and objects A, B. In this case we also say that A → B is the type of f, and that f is a morphism from A to B. In the view of the axioms below we may define the source and target by src f = A and tgt f = B whenever f: A → B.

Definitions (Cont.)  A category is: the following data, subject to certain axioms. (cont.)  A binary partial operation on morphisms, called composition. By default, f ; g is the notation of the composition of morphisms f and g. An alternative notation is g ◦ f, and even g f, with the convention f ; g = g ◦ f = g f.  For each object A a distinguished morphism, called an identity on A. By default, id A, or id when A is clear from the context, denotes the identity on object A. A category with objects X, Y, Z and morphisms f, g, g ∘ f, and three identity morphisms (not shown) 1 X, 1 Y and 1 Z.

Axioms that hold for a Category  There are three typing axioms, and two axioms for equality. The typing axioms are these:  f: A → B and f: A ′ → B′ A = A ′ and B = B ′ Unique Type  f: A → B and g: B → C f ; g : A → CComposition Type  id A : A → A Identity Type  Axioms for equality:  ( f ; g ) ; h = f ; ( g ; h)Composition- Assoc  id ; f = f = f ; id Identity

Uses of Categories  Categories can be seen in  Sets  Groups and pre-orders  Cartesian closed categories, and Topoi  Constructing new categories  Functors

Diagrams  An important tool in the practice of Category Theory is the use of diagrams for representing equations. In a diagram a morphism f Є C [a,b] is drawn as an arrow from a to b labeled as f.  A diagram commutes if the composition of the morphism along any path between two fixed objects is equal. For example, the associative and identity laws of the definition of “category” may be nicely visualized by the following diagrams:

Diagrams Cont.  Diagrams are a typical way, in Category Theory, to describe equational reasoning and turn out to be a particularly effective when dealing with several equations at a time.  In particular, assertions such as “if diagram 1 and … diagram n commutes” expression conditional statements about equalities.

Category Theory Research  The researches developed by the category theory group fit into five main lines: 1) localizations of presheaf categories and algebraic categories ; 2) non-abelian and higher order homological algebra ; 3) descent theory and Galois theories ; 4) semi-abelian and homological categories ; 5) accessible categories ;

Localizations of Presheaf Categories and Algebraic Categories  A new impulsion to the research in this area has been given by the theory of the exact completion, applied to obtain new results on (essential) localizations of monadic categories and algebraic categories, and to give a uniform approach to some classical results (Giraud characterization of topos, Gabriel-Popescu representation theorem for Grothendieck categories and Freyd representation theorems in abelian categories).  It has been used also in universal algebra, especially in connection with Malcev varieties and quasi-varieties. More recently, a complete classification of localizations and geometric morphisms of algebraic and presheaf categories has been obtained.

Non-Abelian and Higher Order Homological Algebra  A categorical approach to ring theory consists in replacing R-algebras with small enriched categories. In this line, our research group has developed a theory of Azumaya categories, together with the associated Brauer group. The more delicate treatment of the Brauer-Taylor group has also been achieved, and related new problems involving "categories without units" have been investigated.

Descent Theory and Galois Theories  The classical descent theory for ring homomorphisms admits a description in terms of monads, allowing to study it in much more general contexts. We investigate in particular a possible refinement of this classical theory in terms of the pure spectrum of the ring. We have also solved the descent problem in the case of algebraic fibrations and of internal functors in a lextensive category.

Semi-Abelian and Homological Categories  The discovery of an internal notion of crossed module opens a new perspective for the developments of this theory in the context of semi- abelian categories and offers a unified approach to the study of the homological properties of crossed modules, crossed rings and other structures internal to varieties of algebras. Encouraging results in the classification of central extensions of precrossed modules have also been obtained.

Accessible Categories  Accessible categories are the categories of models of sketchable theories. Our group has been involved in various research projects in this area: an enriched theory of accessible categories, a Morita theorem for sketches, a classification of accessible categories independent of cardinal arithmetic, a study of injectivity in accessible categories, a characterisation of von Neumann varieties, a study of filtered enriched colimits over finitely presentable bases.

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