Category Theory and Applications

A Textbook for Beginners

Marco Grandis

• Publisher: WSPC
• Year: 2018
• eBook ISBN: 9789813231085
• Topic: Mathematics
• Subtopic: Computer Science General
• Index: Mathematics

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Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots.

This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications.

Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields.

--> Contents:

• Introduction
• Categories, Functors and Natural Transformations
• Limits and Colimits
• Adjunctions and Monads
• Applications in Algebra
• Applications in Topology and Algebraic Topology
• Applications in Homological Algebra
• Hints at Higher Dimensional Category Theory
• References
• Indices

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--> Readership: Graduate students and researchers of mathematics, computer science, physics. -->
Keywords:Category TheoryReview: Key Features:

• The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topology
• The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints
• Three chapters and some extra sections are devoted to applications

1

Categories, functors and natural transformations

Categories were introduced by Eilenberg and Mac Lane [EiM] in 1945, together with the other basic terms of category theory.

1.1 Categories

We start by considering concrete categories, associated with mathematical structures. But categories are not restricted to these instances, and the theory must be developed in a general way.

Given a mathematical discipline, it may not be obvious which category or categories are best suited for its study. This questionable point is discussed in 1.1.5, 1.1.6.

1.1.1 Some examples

Loosely speaking, before giving a precise definition, a category C consists of objects and morphisms together with a (partial) composition law: given two ‘consecutive’ morphisms f : X → Y and g : Y → Z we have a composed morphism gf : X → Z. This partial operation is associative (whenever composition is legitimate) and every object X has an identity, written as idX: X → X or 1_X, which acts as a unit for legitimate compositions.

The prime example is the category Set of sets (and mappings), where:

- an object is a set,

- the morphisms f : X → Y between two given sets X and Y are the (set-theoretical) mappings from X to Y,

- the composition law is the usual composition of mappings, where (gf)(x) = g(f(x)).

The following categories of structured sets and structure-preserving mappings (with the usual composition) will often be used and analysed:

- the category Top of topological spaces (and continuous mappings),

- the category Hsd of Hausdorff spaces (and continuous mappings),

- the category Gp of groups (and their homomorphisms),

- the category Ab of abelian groups (and homomorphisms),

- the category Mon of monoids, i.e. unitary semigroups (and homomorphisms),

- the category Abm of abelian monoids (and homomorphisms),

- the category Rng of rings, understood to be associative and unitary (and homomorphisms),

- the category CRng of commutative rings (and homomorphisms),

- the category RMod of left modules on a fixed unitary ring R (and homomorphisms),

- the category K Vct ( = K Mod) of vector spaces on a commutative field K (and homomorphisms),

- the category RAlg of unitary algebras on a fixed commutative unitary ring R (and homomorphisms),

- the category Ord of ordered sets (and monotone mappings),

- the category pOrd of preordered sets (and monotone mappings),

- the category Set_• of pointed sets (and pointed mappings),

- the category Top_• of pointed topological spaces (and pointed continuous mappings),

- the category Ban of Banach spaces and continuous linear mappings.

- the category Ban₁ of Banach spaces and linear weak contractions (with norm

1).

A homomorphism of a ‘unitary structure’, like a monoid or a unitary ring, is always assumed to preserve units.

For Set_• we recall that a pointed set is a pair (X, x₀) consisting of a set X and a base-element x₀ ∈ X, while a pointed mapping f : (X, x₀) → (Y, y₀) is a mapping f : X → Y such that f(x₀) = y₀.

Similarly, a pointed topological space (X, x₀) is a space with a base-point, and a pointed map f : (X, x₀) → (Y, y₀) is a continuous mapping from X to Y such that f(x₀) = y₀. The reader may know that the category Top_• is important in Algebraic Topology: for instance, the fundamental group π₁(X, x₀) is defined for a pointed topological space.

For the categories Ban and Ban₁ it is understood that we have chosen either the real or the complex field; when usi...

Table of contents

• Cover page
• Title page
• Copyright
• Dedication
• Preface
• Contents
• Introduction
• 1 Categories, functors and natural transformations
• 2 Limits and colimits
• 3 Adjunctions and monads
• 4 Applications in Algebra
• 5 Applications in Topology and Algebraic Topology
• 6 Applications in Homological Algebra
• 7 Hints at higher dimensional category theory
• References
• Index