Category theory

Category Theory II, update

Well, this is really going at such a cracking pace!

I’ve managed to revise just one more chapter of Category Theory II in the last fortnight.

That’s mostly because I went back and tinkered again with the first four chapters of Part II. And I’m still not super-happy with the fifth chapter, introducing hom-functors. But it is better than it was, and eliminates a foolish mistake or two, so I guess that’s progress.

Drat: a ‘small’ mistake in Category Theory I

Just spotted an elementary thinko in Category Theory I.

I carelessly define a finite limit as a limit over a diagram with a finite number of objects. But standardly, it should of course be “with a finite number of objects and finite number of arrows”. And I define a small limit as a limit over a diagram with no more than set-many objects: it should of course be “with no more than set-many objects and set-many arrows”.

No great harm done as the definitions do little work in Category Theory I but were really for future use in Part II. Very annoying all the same. And the twin mistakes have been there in however many earlier iterations, and no-one has pointed them out.

I’ll have to start a corrections page …

More revised category theory chapters

Short version: I’ve revised the first four chapters of Category Theory II.

Longer version for the three people who might care …! I’ve reorganized the early chapters of Part II into what is I think a more logical sequence, though that means there will be, for a moment, some broken cross-references and other oddities. And the first four chapters in the new order are significantly rewritten.

In keeping with what is, after all, supposed to be a gentle introduction, I’ve deleted some sections which were distractingly convoluted. In fact, as I get to work (slowly!) on Part II, my ambitions for it are becoming a bit more modest …

I used to tell grad students that their theses needn’t be great, they just needed to be finished. Likewise, I need to sternly tell myself, for these notes!

Category Theory I, the paperback!

So, at last, the paperback version of Category Theory I is available. Sound the trumpets!

Just put the title or the ISBN 1916906370 in the search field of your local Amazon store. And yes, at least for the moment, this is only available as a print-on-demand book from Amazon. But that keeps the price as low as possible — about three coffees. So treat yourself!

The Amazon KDP system makes later editorial revisions particularly easy and cost-free for me, another plus. As I’ve said before, I’m still thinking of this first paperback version as a revisable beta version: it’s there for those — like me — who prefer to work with printed copy once a text gets past a certain length. All comments, corrections, and suggestions for improvement are still most welcome. The PDF of course continues to be freely downloadable: the short Preface and Introduction to that PDF will tell you about the book project, in the most unlikely event that you are reading this but don’t know about it!

Category Theory I — one last revised draft

I didn’t intend to put another version of Category Theory I online before the positively final version for paperback printing was available. But I’m going to be caught up with other things including a short family holiday for a few days, so I am after all going to post this almost-final version now. What’s left to do? Finish the index of notation, design a cover, get things into the Amazon print-on-demand system. The paperback should be out by about August 10.

I’ve corrected some typos. But the main changes are that I have deleted  §16.7 on ‘naming’ arrows, and the whole of what was the short but dense Chapter 24 on power objects. Both seemed in retrospect a bit out of place in what is billed as ‘Notes towards a gentle introduction’. The material in §16.7 wasn’t mentioned again (I hope!); and the details in Chapter 24 also go far beyond what is really needed at this stage.

The latest version of Category Theory I, with Category Theory II updated to keep theorem numbering etc. in sync, can be downloaded here.

Category Theory I, a complete draft

So, at long last, a full draft of Category Theory I is online.

There are now thirteen added pages of content, plus an index of definitions. The main substantive change is the added last chapter on ‘the elementary theory of the category of sets’.

I’ll draw breath, do a quick-ish end-to-end read for consistency, and aim to get a print-on-demand paperback set up by the end of the month. That won’t be to fix the text once and for all: I’m thinking of this as a beta version, and an easily revisable paperback will just be there for those (including me!) who find a 228 page printed book easier to work from and comment on than an onscreen PDF.

Of course (and you know what’s coming next, because I’ve said it before), if you have been meaning to drop me a note with comments/suggestions/corrections, then now — yes, really now — is the time to do so!

Current versions of Category Theory I and II can be downloaded here.

Updated Jul 17: Revised version with a few very minor corrections to pp.1-100.

Category Theory I, the end in sight!

A further revised version of Category Theory I is now online. The main substantive changes are in the last few chapters. In particular, the short Chapter 24 on power objects is much improved.

There are also quite a few corrections of typos and thinkos — I should particularly thank Ruiting Jiang of the Queen’s College Oxford for comments.

What’s left to do before I paperback these notes? Add a final chapter on ‘the elementary theory of the category of sets’ as all the pieces are in place to cover that and an add an index. So to repeat what I said a couple of weeks ago, but with a tad more urgency, if you have been meaning to drop me a note with comments/suggestions/corrections, then now — yes, really now — is the time to do so!

Current versions of Category Theory I and II can be downloaded here.

Category theory Notes updated again (and again)

A revised version of Category Theory I is now online. The main substantive change is a rather more consistent handling of plural notation; but there are a couple more theorems and some scattered improvements in explanations and a few typos corrected.

There is a corresponding new version of Category Theory II but the only changes are ones to automatically update cross references to Category Theory I.

I plan over June to turn Category Theory I into a very cheap paperbacked Big Red Logic book (though it will of course still be free to download as a PDF). The text still won’t be set in stone — in fact, I’ll call it a β-version. But it will be a lot nicer to work from.

I plan to add one more chapter, on ‘the elementary theory of the category of sets’ as all the pieces are in place to cover that. I have a fairly short current list of previous episodes I want to improve, though I will no doubt find more; I want to add more signalling about which more techie bits can be skipped by those wanting a less proof-heavy (initial) read; I’ll add an index. So if you have been meaning to drop me a note with comments/suggestions/corrections now is the time to do so!

Current versions of Category Theory I and II can be downloaded here.

Added June 6: Minor revisions and typo-corrections for  Chapters 1 to 6 uploaded.

Categories for the Working Philosopher: 15–18

I’ll be brief. I’m going to skip the fifteenth piece, ‘Application of Categories to Biology and Cognition’ by Andrée Ehresmann: this reader made absolutely nothing of it. The next piece by David Spivak on ‘Categories as mathematical models’ (downloadable here) is pretty empty of serious content, the notion of ‘model’ in play being hopelessly vague. This is followed by Hans Halvorson and Dimitris Tsementzis on ‘Categories of scientific theories’ (downloadable here) which proceeds at such a stratospheric level of abstraction as to cast no light at all on the sort of issues in the philosophy of science that back in the day used to interest me. The final paper is by our editor, Elaine Landry, ‘Structural realism and category mistakes’ is disappointing in a different way. Landry has written thought-provoking pieces about category theory elsewhere (e.g. here and here): but this present piece has the flavour of a narrower-interest journal article replying to particular target papers rather than the sort of more general-interest essay appropriate for this sort of collection.

Heavens! Haven’t I been curmudgeonly? But I confess I started pretty sceptical about claims about the wider significance of category theory (once we go beyond the world of pure mathematics/logic — and perhaps functional programming): and on the evidence of this book, I remain as sceptical. And happy enough to be so: there is some lovely maths in e.g. the Elephant as far as I understand it, and lovely maths is good enough for me!

If you want to read more judicious(?) responses to Categories for the Working Philosopher, you could try Neil Barton’s review or the review by Chris Kapulkin and Nicholas Teh. Obviously, those reviewers are nicer and more generous than I am! But life is short …

Categories for the Working Philosopher: 11–14

In fact I am only going to really comment (and that only briefly) on one of these four papers. For two of them relate to quantum mechanics; and to my great regret I quite lost my grip on such matters many years back. Here, Samson Abransky writes on ‘Contextually: On the borders of paradox’; and Bob Coecke and Aleks Kissinger contribute ‘Categorical quantum mechanics I: causal quantum processes’. Both papers can be downloaded from the arXiv and you can chase them up. And if you want to know more about Bob Coecke and Aleks Kissinger’s take on quantum mechanics, they have a very large book Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning (CUP, 2017) whose opening chapters are pretty accessible.

Back in the Landry collection, another paper is an eight-page note by the late Joachim Lambek on a ‘Six-dimensional Lorentz category’ (again the piece is downloadable). This, however, seems quite out of place in this volume. And indeed the author himself concludes ‘The two extra dimensions of time had been introduced for the sake of mathematical elegance and I have not settled on their physical meaning. For a while I had hoped that they might help to incorporate the direction of the spin axis, but did not succeed to make this idea work’ — hardly a ringing endorsement of the project. Best forgotten as far as I can see.

Which leaves James Owen Weatherall’s ‘Category theory and the foundations of classical space-time theories’: again, a version of this paper is downloadable.

I don’t know quite what Elaine Landry asked of her contributors. In her preface, however, she writes that ‘this book aims to bring the concepts of category theory to philosophers working in [a variety of] areas … Moreover, it aims to do this in a way that is accessible to a general audience.’ And Weatherall’s piece is indeed clear and engaging. But does he actually show categorial ideas doing essential work?

His topic is various classical field theories which have, in an intuitive sense, “excess content” (they are, as it is said, gauge theories), and the aim is to use categorial ideas to analyse this notion of excess content. Without going into details here, the discussion is interesting and persuasive about the differences between various gauge theories. He sums up:

I have reviewed several cases in which representing a scientific theory as a category of models is useful for understanding the structure associated with a theory. In the context of classical space–time structure, the category theoretic machinery merely recovers relationships that have long been appreciated by philosophers of physics; these cases are perhaps best understood as litmus tests for the notion of “structure” described here. In the other cases, the new machinery appears to do useful work. It helps crystalize the sense in which [versions of classical Newtonian gravity and of electromagnetic theory] have excess structure, in a way that clarifies an important distinction between these theories and other kinds of gauge theories, such as Yang–Mills theory and general relativity. It also clarifies the relationship between various formulations of physical theories that have been of interest to philosophers because of their alleged parsimony. These results seem to reflect real progress in our understanding of these theories — progress that apparently required the basic category theory used here.

But the last claim does seem to overshoot. The basic category theory in question is just the invocation of the notion of a functor as a map between different models and their automorphisms, plus the idea that different functors can preserve different amounts of information, a general idea which is entirely available to someone who has met no category theory at all. In fact, Weatherall himself admits as much at the beginning of his paper:

Although some of the results I describe in the body of the chapter are non-trivial, the category theory I use is elementary and, arguably, appears only superficially.

He does give a promissory note that there are cases in the same neck of the woods to the ones which he discusses ‘where category theory plays a much deeper role’. But as things stand in this paper itself, the category theory indeed seems inessential.

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