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Sketches of an Elephant: A Topos Theory Compendium by Peter T.
Johnstone
(
Topos: A category modeled after the properties of the category of sets.
Topos theory is a subject that stands at the junction of
geometry, mathematical logic and theoretical computer science, and it derives
much of its power from the interplay of ideas drawn from these different areas.
Now available in this two volume set, it contains all the important information
both volumes provides. Considered to be a complete benefit for all researchers
and academics in theoretical computer science, logicians and philosophers who
study the foundations of mathematics, and those working in differential geometry
and continuum physics.
Citing the old Indian story about the blind men feeling different parts of an
elephant and coming up with divergent descriptions of the animal Johnstone says
that topos theory can be described in divergent ways depending on what part is
examined. The original conception of toposes arose in the 1960s as a
"generalized space" supporting cohomology theory, but has grown to be used by
category theory and other branches of mathematics. Addressing those already
familiar with topos theory, Johnstone offers these volumes (and a scheduled 3rd)
as a complete treatment of all of the pieces of elementary topos theory together
with fully worked-out results. Volume one discusses toposes as categories.
Toposes as spaces and theories are reserved for the second. The final volume
expected to discuss homotopy and cohomology, and toposes as mathematical
universes.
REWORK:
Very simply: if you want to know a very great deal about topos theory, buy this book. I mean, seriously, if you plan to make real work on topos theory a part of your life, then grit your teeth and come up with the money. If you do not want to know a very great deal about it, do not buy this book. You can use it at the library as a reference.
If you merely want a professional understanding of what topos theory is, then read Johnstone's earlier TOPOS THEORY. That far shorter book gives a better overview. My Amazon review of it discusses others on the subject. Most are more accessible than Johnstone's books and go more into particular aspects of the theory.
This book is a reference on all the methods, and the latest results, in topos theory. If you want the definition of "split opfibration", it is here, along with some 80 pages of background, examples, and motivation. Johnstone does an heroic job of unifying the terminology and organizing the theorems.
More than that, Johnstone has written down an expert, encyclopedic view of the subject today. It is rare for a top mathematical researcher to give so deep an account of their field. It is rare for anyone to even work out such an explicit, coherent, extensive account of the whole. Not everyone will agree with his view. Some would like to see much less of such logical topics as "allegories", others would like to see the logic more formalized from the start. But Johnstone builds a case for his choices: partly implicit in his success at explaining things this way, and partly by explicit reasons.
If you want to know that much about the subject then you want to immerse yourself in this book.
Topos Theory by Peter T. Johnstone (Academic Press)
You understand topos theory when you understand this book. I cannot say the same of any other.
Sheaves in Geometry and Logic: A First Introduction to Topos Theory
(Universitext: Springer Verlag)
by Saunders MacLane, I. Moerdijk, is a beautifully written book, a long and well
motivated book packed with well chosen clearly explained examples. Both those
authors have a rare gift for conveying an insider's view of the subject from the
start. This book is written in the best
Elementary Categories, Elementary Toposes by Colin Mclarty (Oxford Logic
Guides, 21: Oxford University Press) goes quickly and simply to the key ideas,
This text introduces categories and elementary toposes while requiring little
mathematical background. It defines the key concepts and gives complete
elementary proofs of theorems, including the fundamental theorem of toposes and
the sheafification theorem. It ends with topos theoretic descriptions of sets,
of basic differential geometry, and of recursive analysis. It is intended for
students of logic and category theory,
covers
elementary aspects of category theory and topos theory. It has few mathematical
prerequisites, and uses categorical methods throughout, rather than beginning
with set theoretical foundations. It works with key concepts such as Cartesian
closedness, adjunctions, regular categories, and the internal logic of a topos.
Full statements and elementary proofs are given for the central theorems,
including the fundamental theorem of toposes, the sheafification theorem, and
the construction of Grothendieck toposes over any topos as base. Three chapters
discuss applications of toposes in detail, namely to sets, to basic differential
geometry, and to recursive analysis. The intended readership consists of
graduate-level students in mathematics, computer science, logic, and category
theory.
Toposes, Triples and Theories (Grundlehren Der Mathematischen Wissenschaften, Band 278: Springer Verlag) by Michael Barr, Charles Wells uses the most efficient tools to get at the central theorems. This is a very readable introduction to the subject, now unfortunately out of print.
But no other book goes as concisely and comprehensively to all the aspects of toposes as this one. Category theory, algebra, logic, arithmetic, geometry, and cohomology all come in, in a well chosen perspective. This book is hard to read in places, especially at the start. It cannot serve alone as an introduction unless you are really gifted. But it remains the best single text on the subject.
Johnstone's eventual three-volume set on the current state of topos theory may well become the standard reference. But it will not replace Topos Theory as an essential introduction
Excerpt: Introduction and apologia: Four men, who had been
blind from birth, wanted to know what an elephant was like; so they asked an
elephant-driver for information. He led them to an elephant, and invited them to
examine it; so one man felt the elephant's leg, another its trunk, another its
tail and the fourth its ear. Then they attempted to describe the elephant to one
another. The first man said `The elephant is like a tree'. `No,' said the
second, `the elephant is like a snake'. `Nonsense!' said the third, `the
elephant is like a broom'. `You are all wrong,' said the fourth, `the elephant
is like a fan'. And so they went on arguing amongst themselves, while the
elephant stood watching them quietly.'
The notion of topos resembles an elephant in that it is
possible to come up with very different descriptions of what topos theory is
about, depending on the direction from which you approach the subject. It was
Andre Joyal, in his lecture at the 1981 Cambridge Summer Meeting on Category
Theory, who first drew attention to the parallel, and I am grateful to him for
allowing me to use it as the basis for the title of the present work. He listed
seven different descriptions of `what a topos is like':
(i) `A
topos is a category of sheaves on a site'
(ii) `A
topos is a category with finite limits and power-objects'
(iii) `A
topos is (the embodiment of) an intuitionistic higher-order theory'
(iv) `A
topos is (the extensional essence of) a first-order (infinitary) geometric
theory'
(v) `A
topos is a totally cocomplete object in the meta-2-category (NO U of cartesian
(i.e., finitely complete) categories'
(vi) `A
topos is a generalized space'
(vii) `A
topos is a semantics for intuitionistic formal systems'
In the 20
years that have passed since that lecture, the category-theory community has
added a few more descriptions to the list (and the theoretical computer
scientists have contributed yet others); for example,
(viii) `A
topos is a Morita equivalence class of continuous groupoids' (ix) `A topos is
the category of maps of a power allegory'
(x) `A
topos is a category whose canonical indexing over itself is complete and
well-powered'
(xi) `A
topos is the spatial manifestation of a Giraud frame' (xii) `A topos is a
setting for synthetic differential geometry' (xiii) `A topos is a setting for
synthetic domain theory',
end so on.
But the important thing about the elephant is that `however you approach it, it
is still the same animal'; this book is an attempt to demonstrate ;hat the same
is true of topos theory.
A brief
bibliography of topos theory
Sketches of an Elephant resembles an elephant in another respect: namely its
gestation period. To explain (and apologize for) this, I have to go back still
further into the history of the subject. I must also crave the reader's
indulgence for what may seem an excessively egocentric view of that history in
the paragraphs which follow.
The
original notion of topos, as a `generalized space' suitable for supporting the
exotic cohomology theories required in algebraic geometry, sprang from the
fertile brain of Alexandre Grothendieck in the early 1960s, and was developed in
his `Seminaire de Geometrie Algebrique du Bois-Marie' particularly during the
academic year 1963-64. The duplicated notes of that seminar circulated widely
among algebraic geometers and category-theorists over the next decade, until
Springer-Verlag did the world a service by publishing a revised and expanded
edition in three volumes of Lecture Notes in Mathematics in 1972 [36]. But by
then the subject had already been `reborn' in its second incarnation, as an
elementary theory having links with higher-order intuitionistic logic, through
the collaboration of Bill Lawvere and Myles Tierney during 1969-70 (which, in
turn, built upon Lawvere's work on providing a categorical foundation for I
mathematics, which had been developing since the early 1960s).
As a
graduate student in the early 1970s, I was fortunate to `get in on the ground
floor' of elementary topos theory, by attending Tierney's summer school;
lectures [1167] at Varenna in 1971, and later Saunders Mac Lane's lectures
during his Cambridge sabbatical in autumn 1972. At that time the only general
accounts of the subject in print were the lecture notes [649] by Anders Kock and
Gavin Wraith, which were not all that widely available, and Peter Freyd's survey
article [371], whose view of the subject was idiosyncratic to say the least --
though they were joined over the next two years by the notes of Tierney's 3
lectures, already cited, and by Wraith's second account [1236].
More or
less immediately after completing my Ph.D. in 1974, I embarked on the writing of
what became the first account in book form of the elementary theory of toposes,
published by Academic Press in 1977 [504]. With the brash self-confidence of
youth, I attempted to provide between one pair of covers a full account of as
many aspects of the elephant as I could muster: the result was a book which,
though universally acknowledged as `far too hard to read, and not for the
faint-hearted' remained in demand for a long time - and continues to be widely
cited even today, despite being nearly 25 years out of date.
Each of
these books contributed a new view of the elephant, illuminating features not
visible from other angles, and thus each of them was valuable. But I hope I
shall not offend the authors listed above by saying that none of them, with the
possible exception of the last, really attempted to give a picture of the whole
animal: indeed, in several cases the concentration on one part of its anatomy is
self-evident from the title.
Meanwhile,
my own book had gone out of print in 1987, despite a corrected reprint (some
time after the original stock had run out) in 1985. There was some suggestion at
that time that Academic Press might be willing to publish a revised second
edition of the book, if I were willing to write it; but their procrastination
over, and lack of enthusiasm for, the 1985 reprint did not fill me with
enthusiasm for that course of action. Another reason for my reluctance was the
extent to which the subject had grown since the mid-1970s: since the principal
merit of my book, in comparison with the others by then available or
forthcoming, was its comprehensiveness, there would be no point in replacing it
by anything other than a comprehensive account of the subject as it stood by
then, and such an account would inevitably be very much longer than the 1977
book.
The three
blind men
By the
mid-1980s Ieke Moerdijk had also become concerned (despite his involvement as
co-author in two of the books mentioned above) about the lack of any single
account of topos theory which could properly claim to give a comprehensive
picture of the subject. However, like me he felt daunted by the prospect of
writing such a book single-handed. Sometime early in 1986, he mentioned to me
his idea that what was needed was a `handbook' type publication in which a small
team of experts would contribute extended essays on the different aspects of the
subject, under the overall supervision of a coordinating editor - he felt that I
was the ideal person to take on the latter role. I was not, at first, greatly
taken with this idea: as I saw it, the chief problem with the `handbook' style
was that, although the elephant has many different aspects, their
interconnectedness is such that it is impossible fully to appreciate any one of
them without making some reference to most of the others. So I foresaw that the
job of a coordinating editor, for such a volume, would be more than usually
nightmarish.
Nevertheless, discussions between Ieke and myself about the idea continued to
take place on the odd occasions when we met, and they soon came to involve my
former student Andrew Pitts as another possible author. The `handbook' idea
gradually metamorphosed into something more like a 'Bourbaki' volume or the
Compendium of continuous lattices [404], in which each author would have
individual responsibility for the first drafts of particular sections, but the
final text would have to be agreed by all the authors (and the individual
authorship of the different sections would not be publicly identified). At a
meeting between the three of us in December 1987, we formally agreed to go ahead
with a three-author work on that basis, and a rough table of contents -
including the division of the book into six parts, although they did not
correspond exactly to the six parts in the present table of contents - was
worked out, together with the responsibilities of the three of us for drafting
the more introductory sections. The working title Sketches of an elephant was
also agreed at this meeting.
By the late
summer of 1988 we had enough material in draft to approach publishers with a
definite proposal, and we signed an agreement with Oxford University Press in
January 1989. At the time (I blush to recall it now) we hoped that we would be
able to deliver a complete typescript to the publishers by the end of September
1991. We also estimated that the overall length of the book might be around 1000
pages; in the event, that estimate has proved more accurate than the temporal
one.
Over the
next couple of years, a substantial quantity of draft material was written, with
P. T. J. working mainly on Parts A and B, I. M. on Part C and A. M. P. on Part
D. Ieke produced large quantities of draft material, principally on topics from
Chapters C2 and C3, although virtually none of it has survived into the version
that exists today; on the other hand, stylistic experts may just be able to
detect the origins of the present Sections D1.1-D1.3 in a draft written by
Andrew, even though it has been heavily rewritten by me. But it had
become
clear by early 1991 that the project was a much larger one than we had realized:
in particular, the problems of co-ordination between the three of us, and of
reaching agreement on a text that all three could be happy with, seemed
overwhelming. It was at least partly these problems that led Ieke and Andrew
independently to decide to withdraw from the project in late 1992.
I should
make it clear that, despite their not seeing the project through to completion,
I remain enormously indebted to both Ieke and Andrew. If it had not been for
their support in the early stages, I should never have dared to embark on the
writing of a work on this scale, and their encouragement kept me going until the
book had progressed far enough to acquire its own momentum. Also, their
constructive criticisms of my early drafts of Part A caught many infelicities
that I should probably not have noticed myself. And on a more mundane level,
Andrew taught me to use LATEX, without which the project would have been not
just foolhardy but downright impossible.
One blind
man
With my
other commitments to teaching, research and administration, progress was indeed
slow: often, several months would pass without my finding time to open the
files where the text of the book was stored. In addition, I often felt like
Heracles wrestling with the Hydra: every attempt to draft a new section seemed
to entail the rewriting of at least two existing ones, so that I appeared to
come no nearer to the goal of a complete text. (There was also, at times, the
fear that the subject was advancing faster than I could write it down.)
Nevertheless, progress did sometimes get made. One thing which helped to spur
the drafting, or revision, of certain sections was the chance to lecture about
the material in them, to audiences around the world, on occasions which included
the following:
-
In June 1994 I was invited to lecture on classifying toposes at Shaanxi Teachers' University, Xi'an, China, through the good offices of Wang Guo-Jun and my former student Zhao Dongsheng. This involved the polishing of material from Chapters Al, D1 and D3.
-
In May 1995, Francis Borceux invited me to lecture on locales at a summer school on `Topological Applications of Category Theory' in Alle-sur-Semois, Belgium: this involved material from Chapters C1 and C3.
-
During my sabbatical visit (arranged by Max Kelly) to Sydney, Australia, during February and March 1996, I gave a number of talks in the Sydney Category Seminar covering material from Chapters B1 and B3.
-
In July 1997, as an invited speaker at the `Workshop on Logic, Langage, Information and Computation' (commonly known as WoLLIC) in Fortaleza, Brazil, I had the opportunity (thanks to Ruy de Queiroz) to give a tutorial on categorical logic before the main conference: once again, this provided an occasion to polish the material in Chapters D1 and D2.
I mention
these meetings in order publicly to record my thanks to the organizers named:
though they were not aware of it, they helped to bring the project closer to
completion.
By the
beginning of the third millennium, however, it had become clear to me that what
was absolutely essential, if the writing of Parts A-D was ever to be finished,
was to find an extended period of time in which I could set all other work aside
and concentrate full-time on the book. (Although I had at various times begun
work on several sections from Part F, I had by then taken the decision that the
only sensible route to completion was to concentrate on the first four parts,
which were too strongly interconnected to be written separately, and to leave
the last two - which, dealing as they do with applications, are less strongly
tied to the others - for completion at a later date and publication as a
separate volume.) By great good fortune, such an opportunity arose with the
invitation to participate in the `Mathematical Logic Year' at the Institut
Mittag-Lefer in Djursholm, Sweden, for a two-month period in January-March 2001.
My gratitude to the organizers of that programme (Dag Normann, Viggo
Stoltenberg-Hansen and Jouko Vaananen), for providing me with this opportunity,
is impossible to exaggerate; and I am also deeply grateful to Kjell-Ove Widman
and the staff of the Institut Mittag-Leffler for providing such a near-perfect
working environment.
The speed
with which everything fell into place, once I had the opportunity for
uninterrupted work on the book, greatly surprised me. Although the proportion
of the total text which was written in Sweden is not so great, the number of
sections which reached (more or less) their final form during my time there is
very large: about half of those in Part B, almost all in Part C apart from
Chapter C1, and most of those in Chapters D4 and D5. (In addition, a good deal
of material was added to the supposedly complete draft of Part A.) Within a
month of my return to Cambridge, I was able to produce a complete text of the
first four parts, except for three sections in Part D. Copies of this draft were
circulated to a number of colleagues (including Ieke Moerdijk and Andrew Pitts,
as well as Martin Hyland, Anders Kock and Gavin Wraith) for their comments and
suggestions; the latter (which were, in many cases, of considerable value) were
taken into account in the final revision of the text which took place in
August-September 2001, after the writing of the last three sections had been
completed.
At the time
of writing these words, Parts E and F of the original plan remain to be written.
In view of my experience so far, I am making no promises about how long it will
take to write them; but I hope that interested readers will not have to wait as
long for the third volume as they have done for the first two.
How to read
Sketches of an Elephant.
`Begin at
the beginning,' the King said gravely, `and go on till you come to the end; then
stop.’
Whilst the
King of Hearts's advice to the White Rabbit is generally sound, it is not
particularly helpful in relation to this book. At the December 1987 meeting to
which I have referred, the three of us took a number of decisions about the
overall structure and content of the book: despite the many changes of detail
which have taken place since then, the general principles which we agreed have
by and large survived unchanged. Since some understanding of these principles is
important for anyone who seeks to obtain information from this book, I shall
describe them here.
The first
and most important is that the book is not a textbook. It is not addressed to
those who are trying to learn about topos theory for the first time, but rather
to those who already have some acquaintance with the subject and who wish to
deepen their understanding, or to learn about aspects of it which they have not
previously encountered. In keeping with this, there are no `exercises for the
reader' (which are often the lazy author's way of leaving out the proofs of
results that he can't be bothered to write out in full): instead, all results
needed in the book (with the exception of a few results from other areas of
mathematics, for which references to standard works are given) are proved in
full, and all examples are fully worked out. This is not to say that the book
cannot be used as a course text: it would, for example, be possible to base an
introductory course on the purely category-theoretic side of topos theory on
Chapters Al, A2 and A4, although the instructor would have to be fairly
selective about the specialized material which has been inserted in those
chapters for future reference, and would also probably want to leaven the
presentation with some material from Parts C and/or D.
Secondly,
the book is not intended to be read sequentially. The whole point of the
six-part structure is that the book is seeking to present several different
approaches to topos theory, so where one starts reading will depend on the line
of approach one wishes to follow. Most readers, however experienced, will
probably wish to start by at least skimming through Chapter Al, in order to
familiarize themselves with the book's conventions on categorical notation and
terminology (about which I shall have more to say below); but thereafter a
reader who already knows the basic category theory of toposes may well decide to
jump to Chapter B1, C1 or D1, depending on the approach he wishes to take.
However, almost any reader will have to be prepared to refer backwards and/or
forwards a good deal. Although it would have been theoretically possible to
write entirely independent accounts of the geometric and logical aspects of
topos theory in Parts C and D (and that was more or less our original
intention), in fact the two are so thoroughly interconnected that such a course
would have involved an immense amount of duplication (and of course increased
the overall length of the book).
There is
some duplication, which is quite deliberate: for example, the treatment of
natural number objects in Section A2.5 overlaps with that in Section D5.1, the
construction of classifying toposes is treated in two different ways in
Sections B4.2 and D3.1, and exponentiability of toposes is treated in Sections
B4.3 and C4.4. And there are numerous smaller instances, where an individual
result is proved in two different ways, using material from different areas of
the book. But in general, if there is an optimal proof of a particular theorem,
and it clearly belongs (say) in Part D, there does not seem to be any point in
writing it out a second time in Part C if it happens to be needed there. So the
reader who wishes to see the full picture must be prepared to skip around from
one part to another, when necessary. (On the other hand, a reader who is willing
to `take on trust' the results quoted from other parts of the book should be
able to follow any given chapter as a (more or less) connected narrative.)
I have at
least done my best to make the cross-references easy to use. All numbered
references (definitions, lemmas, theorems, examples, etc.) within a given
section are numbered in a single sequence: thus `B3.4.5' denotes the fifth
numbered reference (in fact a corollary) in the fourth section of the third
chapter of Part B. For references within a given Part, the letter is omitted:
thus if the corollary just mentioned is cited elsewhere in Part B, it appears as
`3.4.5'. If a lemma, proposition, theorem, corollary or scholium is subdivided
into parts, they are normally labelled (i), (ii), (iii) and so on, whereas
definitions, examples and remarks are subdivided (a), (b), (c),.... The symbol D
is used to denote the end of a proof; if it appears at the end of the statement
of a result, it means either that no proof is given, or that the proof is
contained in the discussion which precedes the statement.
At the
December 1987 meeting we adopted the principle that, for every major result to
be included in the book, we should seek to find the optimal proof's although
that should not preclude us from giving alternative proofs if they helped to
shed additional light on the result. (I hope it is unnecessary to add that I am
conscious of having not always succeded in this aim.) Partly for this reason, we
decided that we would not attempt to give detailed attributions for all the
results in the book, since the optimal proof is often not the original one, but
has been arrived at through successive polishing of the latter by several hands.
Nevertheless, I have attempted to compile a comprehensive bibliography of topos
theory, and to give attributions where credit is clearly due to a single source.
(I considered attempting to classify the bibliography entries by subject area,
as was done in [404], for example; but the task seemed too large and ultimately
of doubtful value.) And in most sections I have provided a list of references at
the end of the section for those who wish to pursue the topic(s) covered in the
section in greater detail.
We also
recognized that the choice of terminology and notation, in a book of this
nature, was of immense importance. It is impossible to be entirely consistent
about notation, in a subject of this size: there simply aren't enough different
letters and typefaces for one to avoid re-using some of them in mutually
contradictory ways. But I have done my best to ensure local consistency of
notation, within each chapter. As regards terminology, I have consciously
attempted to use this book to disparage some of the sloppier pieces of
inappropriate usage which are common among category-theorists; whether it will
have any influence in this respect of course remains to be seen, but readers who
are familiar with the existing terminology may well find some of my choices
rather surprising at first. I have, however, listed alternatives to my chosen
terminology in the index, where they appear in italics, and in most cases my
reasons for discarding them will be found in the text at the points to which the
index entries refer.
A
comprehensive index is also of great importance in a book of this nature. I have
done my best to provide one: in addition to listing alternative terminology in
italics, as already mentioned, it also contains entries for many of the
principal theorems, particularly those which occur in several different versions
in different places, and I hope that this will assist the specialist in finding
his way quickly to the information which he wants. One point which should be
mentioned about the index is that, where an index entry begins with a
mathematical symbol, it is listed in the alphabetical order as if the name of
the symbol had been spelled out: thus, for example, `alpha-pretopos' will be
found between `sifted coverage' and `signature'. In addition to the general
index, there is also an index of standard notation, ordered alphabetically on
the same principle.
Sketches of an Elephant: A Topos Theory Compendium (2 Volume Set) by Peter
T. Johnstone (Oxford University Press)
CONTENTS: VOLUME ONE: A. Toposes as Categories A1. Regular
and cartesian closed categories A2. Toposes--basic theory A3. Allegories A4.
Geometric morphisms--basic theory B. 2-Categorical Aspects of Topos Theory B1.
Indeexed categories and fibrations B2. Internal and locally internal categories
B3. Toposes over a base B4. BTop/S as a 2-Category; Bibliography, Index of
notation, General index
VOLUME TWO: C. Toposes as Spaces C1. Sheaves on a locale C2. Sheaves on a site
C3. Classes of geometric morphisms C4. Local compactness and exponentiability
C5. Toposes as groupoids D. TOPOSES AS THEORIES D1. First-order categorical
logic D2. Sketches D3. Classifying toposes D4. Higher-order logic
D5. Aspects of finiteness; Bibliography, Index of notation, General index
Headline 3
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