Category theory

Beginning Category Theory: Chs 1 to 13 (etc.)

I have now re-revised Chapters 1 to 13 of Beginning Category Theory. So here they are again, as before together in one long PDF with the remaining unrevised chapters from the 2015/2018 Gentle Intro. [You may need to force a reload to get the latest version, dated April 27.]

There are significant changes in the rhetoric of Chapter 3, though the intended general position hasn’t really changed. Elsewhere there are scattered, mostly minor, changes to improve clarity and readability. I’m still far from happy with the overall tone/style: but I hope I’m edging slowly, slowly in the right direction!

OK, the next major task is to tidy up the chapter on equalizers and co-equalizers. Now, I motivated the categorial treatment of products at some length by talking more informally, and pre-categorially, about what we want from pairing schemes. But at the moment, like too many every elementary presentations, I just plonk the definitions of equalizers and co-equalizers on the table without motivational pre-amble, and then pull the rabbit out of the hat and say “oh look, quotients of equivalence relations are a special case of co-equalizers!”.

That’s not at all satisfying, and I’d like to do better (as I see Awodey does)!

Beginning Category Theory: Chs 1 to 11 (and more)

To avoid readers having to juggle two PDFs, and to keep at least some cross-references between new and old material functioning, I have decided to put the newly revised chapters together with the old unrevised chapters from the Gentle Intro into one long document. So here is Beginning Category Theory which starts with eleven revised chapters, followed by all the remaining old chapters [with prominent headline warnings about their unrevised status].

The two newly revised chapters are

  1. Pairs and products, pre-categorially [Motivational background]
  2. Categorial products introduced [Definitions, examples, and coproducts too]

Note: these early revised chapters are not final versions. Revised chapters get  incorporated when I think that they are at least better than what they replace, not when I think they are as good as they could be. So, needless to say, all comments and corrections will be very gratefully received. Onwards!

Beginning Category Theory: Chs 1 to 9

Slow progress again but, as I said before, any progress is better than none. So here are Chapters 1 to 9 of Beginning Category Theory. [As always you may need to force a reload to get the latest version.]

And no, there isn’t really a new chapter. I’ve split what was becoming a baggy chapter about kinds of arrows into two, and I hope to have made some of it a fair bit clearer and better organised. The chapters are

  1. Introduction [The categorial imperative!]
  2. One structured family of structures. [Revision about groups, and categories of groups introduced]
  3. Groups and sets [Why I don’t want to assume straight off the bat that structures are sets]
  4. Categories defined [General definition, and lots of standard examples]
  5. Diagrams [Reading commutative diagrams]
  6. Categories beget categories [Duals of categories, subcategories, products, slice categories, etc.]
  7. Kinds of arrows [Monos, epics, inverses]
  8. Isomorphisms [why they get defined as they do]
  9. Initial and terminal objects

Ch. 3 has been mildly revised again, and as I said Ch.7 has been significantly improved. Various minor typos have been corrected. And there have been quite a few small stylistic improvements (including, I’m embarrassed to say, deleting over 50 occurrences of the word “indeed” …).

Beginning Category Theory: Chs 1–8

Here now are Chapters 1 to 8 of Beginning Category Theory

The new chapter is on initial and terminal objects; there have only been minor changes to other chapters from Chapter 4 onwards. These new chapters 1 to 8 are I think a significant improvement to the corresponding Chapters 1 to 6 of the old Gentle Introduction. Or at least, they are a significant improvement in clarity of content. But I don’t think I have yet quite hit the mark as far as tone/reader-friendliness is concerned. So I need to let these pages marinade for a few days, and then return to them (particularly to the last couple of action-packed chapters) to make them a little more relaxed. Onwards!

Beginning Category Theory: Chs 1 to 7

Here now are Chapters 1 to 7 of Beginning Category Theory

The new chapter is on kinds of arrows. I have also revised Chapter 3 (now preferring to talk about implementing structures in set theory, rather than to talk of set-theoretic surrogates or proxies — the change of rhetoric isn’t really a change of view, but will I hope slightly mollify some readers!).

I should say that Beginning Category Theory is very much a work in progress, and I can imagine these early chapters getting significantly revised to better fit the later ones in content and tone. But, for all that, I’m putting them online as I go along, when I get to the point of thinking that a new chapter is at least better than the corresponding old one in the Gentle Intro!

Beginning Category Theory: Chs 1 to 6

Slow progress, but some progress is better than none. So here are Chapters 1 to 6 of Beginning Category Theory. The chapters are

  1. Introduction [The categorial imperative!]
  2. One structured family of structures. [Revision about groups, and categories of groups introduced]
  3. Groups and sets [Why I don’t want to assume straight off the bat that structures are sets]
  4. Categories defined [General definition, and lots of standard examples]
  5. Diagrams [Reading commutative diagrams]
  6. Categories beget categories [Duals of categories, subcategories, products, slice categories, etc.]

Both Chs 2 and 3 are mildly revised from the posting a week or so back. Chapters 4 to 6 are tidier versions of what were Chapters 3 and 4 in the old Category Theory: A Gentle Introduction. And so these six chapters taken together replace the first four of the Gentle Intro.

The Stage 1 plan over the coming weeks is to correct/smooth the existing content from the Gentle Intro. Stage 2 will then be to round out that content (same or closely related topics, same level, but improved examples, etc.). Stage 3 will be to push on to a look at a few more topics I want to cover.

Updated: The definition of a commutative diagram improved.

Beginning Category Theory: NOT Chs 1 to 3

I wanted to be reminded of a different Russia. And so picked up our old Penguin copy of Turgenev’s Home of the Gentry to start re-reading. And it has fallen quite to pieces. Which somehow seems rather symbolic.

We must all distract ourselves from the dire state of the world for some of the time as best we can. Mathematics still works for me: as Russell remarks, “it has nothing to do with life and death and human sordidness”. So I have been starting working again on my notes on category theory which, as I’ve said before, are downloaded rather embarrassingly often given their current half-baked state. It will help keep my mind off other things, trying to get them into better shape.

Things are going slowly, as I need to do a lot of (re)reading. But for those who might like the distraction, here are the first three chapters (under 30 pages). Chapter 3 is mostly new, and the previous chapters have been significantly revised.

[Update: the Preface has now been revised too.]

[Further update: Hmmmmmmmm. I think a more radical rethink of the opening chapters is needed …. so I’ve dropped the link, and am banging my head on the desk ….]

Telling your monos from your epis

Reposting from many many moons ago ….

Ok, so how do you remember which are the epimorphisms, which are the monomorphisms, and which way around the funny arrows get used?

Since the textbooks don’t seem eager to offer helpful mnemonics, I offer a forgetful world the following: go by alphabetical proximity!

L-for-left goes with M-for-mono, and P-almost-for-epi goes almost next to R-for-right. OK?

But what does that mean? Simple.

A mono is of course a left-cancellable morphism, and you signal one using an arrow with an extra decoration (a tail) on the left.

Dually, an epi is a right-cancellable morphism, and you signal one of those using an arrow with an extra decoration (another head) on the right.

Easy, huh? Well, it works for me — and these days, I’m grateful for all the props I can get. You can thank me later.

Category Theory, without too many tears?

So, I’ve pressed the “publish this” button on the Amazon KDP system for Beginning Mathematical Logic, and will let the world know when it gets through the review process and goes live.

And for my next Big Red Logic Book? A few years back I put together some notes on category theory (running to almost 300 pages). And despite their very rackety half-baked form, they are downloaded startlingly often — almost a thousand times in January.  Ye gods! So it’s decision time. Do I let them continue to stand as they are, despite their unsatisfactory, unfinished, form? Or do I try to make them more respectable, and round them out into a more polished book form?

On the one hand, I’m really pretty embarrassed to leave the notes online in their current state. On the other hand, having putting the work in earlier, I’m reluctant to trash them. Which leaves the remaining option, of getting down to more work and and making a better fist of it. So here goes …

I think I’ll take it in three phases. First, go through the current version, correcting all the typos and thinkos I’ve been told about, improving the presentation wherever I can, to make a better version of the existing  material. Second, get back to doing quite a lot more reading and rereading in category theory, so I can round out the existing material e.g. with more examples, more applications, and additional closely related ideas at the same level. Then third, decide which further topics (if any) I should add at the end to make a more satisfying book (though given the current length, I wouldn’t want the book to extend its reach very far into new territory).

That should all take quite a while, but (I hope) not too many tears. Watch this space for occasional updates.

Book note: Ralf Krömer’s Tool and Object

It is inevitably going to be difficult to write illuminatingly about the history of category theory. For this is entangled with the distinctly complicated history of mid-twentieth-century topology. Colin McLarty sees the difficulty like this:

For even a rough understanding of [just some of] the problems [topologists] faced we would have to go into the array of homology theories at the time and the forefront of 1940s abstract algebra, and we would do this without using category theory, and we would waste a lot of time on things category theory has now made much easier. We could give a few trivial examples just before reversing the order of discovery to define categories, functors, and natural transformations but precisely the examples serious enough to have motivated the definitions are too hard to be worth giving now without benefit of categorical hindsight.

Is that too pessimistic? Well, this much is surely true. It would take rather exceptional expositional skills, combined with an exceptional depth of mathematical understanding, to be able to helpfully isolate and explore critical moments in the development of category theory, while doing this in a way that is both interestingly detailed and yet also still quite widely accessible.

For different reasons, it is a challenge too to write illuminatingly about the philosophy of category theory. Working out what is really novel about categorical concepts and approaches isn’t easy. Working out in what senses category theory does or does not provide a new kind of foundation for mathematics isn’t easy. And we are not exactly  helped by the fact that some category theorists are wont to make distractingly sweeping claims about the philosophical significance of what they are up to, claims which are hard to deconstruct. So it would take a different set of skills, beginning with a serious feel for the philosophy of mathematics more generally, to tackle the philosophy of category theory.

It is highly ambitious of anyone, then, to take on writing a book which is intended to be both ‘A History and Philosophy of Category Theory’. But that’s the subtitle of Ralf Krömer’s 2007 book Tool and Object. This has been on my ‘must read one day’ list for quite a while. I’ve at last had time to take a serious look at it. How well does Krömer succeed at the daunting dual task?

I found the book a very considerable disappointment, even allowing for the difficulties we’ve just mentioned. Life being short and all that, I’ve decided against a chapter-by-chapter commentary here, as it would take a lot more time than it would be worth, either for me as writer or for you as reader. But in headline terms, the philosophical bits are just far too arm-waving for someone of my analytic tastes; and I found the historical mathematical exposition just too unhelpful, even for someone coming to the party with a decent amount of mathematical background. The exception, perhaps, is Chapter 6, ‘Categories as sets: problems and solutions’ which is more closely focused on one familar issue, and is quite a useful guide to some of the discussions on “The possibilities and problems attendant on the construction of a set-theoretical foundation for CT and the relevance of such foundations”. And forgive me if I leave it at that. Your mileage may vary of course; but I can’t recommend the rest of this book.

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