Table Of ContentARROWS,
STRUCTURES,
AND FUNCTORS
The Categorical Imperative
MICHAEL A. ARBIB
Department of Computer and
Information Science
University of Massachusetts at Amherst
ERNEST G. MANES
Department of Mathematics
University of Massachusetts at Amherst
ACADEMIC PRESS, INC. New York San Francisco London 1975
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Arbib, Michael A
Arrows, structures, and functors.
Bibliography: p.
Includes index.
1. Categories (Mathematics) 2. Functor theory.
3. Algebra. I. Manes, Ernest G., joint author.
II. Title.
QA169.A7 §12'.55 74-32412
ISBN 0-12-059060-3
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We shall thus have to investigate the possibility of
a categorical imperative entirely a priori, since here we
do not enjoy the advantage of having its reality given in
experience . . .
There is therefore only a single categorical imperative
and it is this: Act only on that maxim through which
you can at the same time will that it should become a
universal law.
Kant arguing, in the Groundwork of the
Metaphysic of Morals, the need for an
introductory treatment, and for the primacy
of universal constructions (see page 45 of
this volume) as The Categorical Imperative.
CONTENTS
Preface . Xi
PART I
Chapter 1 Learning to Think with Arrows
1.1 Epimorphisms and Monomorphisms 1
1.2 Products and Coproducts . 9
1.3 Coequalizers and Equalizers 16
Chapter 2 Basic Concepts of Category Theory
2.1 Vector Spaces and Posets . 24
2.2 The Definition of a Category . 29
23 Epimorphisms and Monomorphisms 33
2.4 Limits and Colimits . 41
Chapter 3 Monoids and Groups
3.1 Defining the Categories . 51
3.2 Constructions within the Categones 54
Chapter 4 Metric and Topological Spaces
4.1 Categories of Metric Spaces . . 60
4.2 Constructions in Categories of Metric Spaces .. . . 64
4.3 Topological Spaces 69
Chapter 5 Additive Categories
5.1 Vectors and Matrices in a Category . 74
5.2 Abm-Categories 77
Chapter 6 Structured Sets
6.1 The Admissible Maps Approach to Structure . . . . 85
6.2 Optimal Families 88
6.3 Examples from Automata Theory 93
CONTENTS
PART Il
Chapter 7 Functors and Adjoints
7.1 Functors 107
' 1.2 Free and Cofree 111
7.3 Natural Transformations and Adjunctions 120
Chapter 8 The Adjoint Functor Theorem
8.1 Necessary Conditions 131
8.2 Sufficient Conditions 134
Chapter 9 Monoidal and Closed Categories
9.1 Motivation from Set Theory . 139
9.2 Monoids in a Monoidal Category 144
9.3 Categories over a Monoidal Category 149
9.4 The Godement Calculus 152
Chapter 10 Monads and Algebras
10.1 Universal Algebra . 157
10.2 From Monoids to Monads . 163
10.3 Monads from Free Algebras 168
Suggestions for Further Reading 176
Index 178
PREFACE
Category theory is the mathematician’s attempt to lay bare some of the
inderlying principles common to diverse fields in the mathematical sciences. It
1as become, as well, an area of pure mathematics in its own right. Briefly, a
category is a domain of mathematical discourse characterized in a very general
way, and category theory is thus an array of tools for stating results which can
be used across a wide mathematical spectrum.
All previous texts on category theory demand considerable mathematical
maturity of the reader, assuming familiarity with many different domains of
mathematics. As such, the subject matter has been inaccessible to talented
undergraduate students, to graduate students in sciences other than pure mathe-
matics, and to professionals in control theory, automata theory, theoretical
linguistics, philosophy, mathematical biology and other fields in which category
theory is entering the literature. It is for these audiences that the present book
is designed.
The contribution of our book, then, is to build up sufficient perspective
without demanding more of the reader than a basic knowledge of sets (what is a
function? what is a cartesian product of sets?) and matrix theory (what is a
linear map? what is a direct product of vector spaces?). In short, this is by far
the most elementary category theory text in print and puts advanced books
such as Mac Lane’s “Categories for the Working Mathematician™ within the
rcach of a much wider audience.
What, then, is the categorical imperative — the set of core concepts of cate-
gory theory which should be shared by this diverse audience before they pursue
more specialized avenues tailored to their own area of interest? Our answer is
threefold. First is the ability to think with arrows: to express key concepts in
terms of mappings (we call them morphisms in the general setting of category
theory) rather than in terms of set elements. Second is the realization that col-
lections of mathematical structures find convenient characterization in terms of
arrows. Third is the use of functors as the appropriate tools with which to com-
pare different domains of mathematical discourse.
The book is divided into two parts. The first part (Chapters 1 to 6) is
devoted 1o arrows and structures: we present the most fundamental aspects of
xi
xii PREFACE
the point of view that a category models a system of ‘structures’ generalizing
specific domains of discourse such as (sets and functions) and {vector spaces
and linear maps). The analysis of such systems is in the language of ‘arrows’
f: A — B whose meaning is that fis a function or linear map respectively in
the two examples above. We introduce the reader to the art of ‘chasing’ commu-
tative diagrams as a way of constructing or verifying equalities between chains
of arrows. Structures such as monoids, metric spaces and topological spaces are
introduced from first principles and the categories induced by these structures
are used to demonstrate the diversity of possibilities inherent in generalizing
concepts from set theory to category theory. For example, we show that the
disjoint union of sets and the free product of groups are both examples of the
general categorical notion of a coproduct. Chapter 6 gives a categorical perspec-
tive on sets with structure. A special section applies many of the categorical
concepts of Part I to the study of reachability, observability and minimal reali-
zation of qutomata. This section should be especially useful to the reader who
wishes to go from the basic category theory of this book to the increasing appli-
cations in the study, e.g., of computation and control.
The second part of the book (Chapters 7 to 10) is an introduction to
functors, the third component of our categorical imperative. Just as Euclid’s
congruence of geometric figures finds modern expression in the concept of a
distance-preserving mapping from a geometric space to itself, functors provide
the appropriate concept to ‘transform’ one sort of structure into another - for
example, to formalize when two different domains of discourse are ‘essentially
the same’. Building upon the many examples of Part I we introduce natural
transformations and adjoint functors, starting from the construction of free and
cofree objects. We show that reachability and observability in automata theory
correspond to free and cofree constructions respectively. We study monoidal
categories as a framework for extending set-theoretic constructions to other
domains of discourse, noting very briefly that fopos theory — part of the current
research frontier of category theory — offers exciting new vantage points for
problems in the foundations of mathematics. Finally, we show that universal
algebra — another attempt at finding a common language for different branches
of mathematics — can itself be captured in category theory language, using the
concept of a monad. Intriguingly, the language we shall use in bridging univer-
sal algebra to category theory will be reminiscent of that used in our categorical
treatment of automata theory.
This book grew out of our need to make our work in “Category Theory
Applied to Computation and Control” accessible to an audience of computer
and system scientists who found existing introductions to category theory over-
ly formidable. We thank Joe Goguen, Stewart Bainbridge, Mitchell Wand,
Suad Alagi¢, Brian Anderson, Hartmut Ehrig and others who have contributed
