Category theory

Back, at last, to category theory

I am underway, at last, with the project of  improving and updating my notes on category theory. So, here are the first four chapters of Category Theory I: Notes towards a gentle introduction.

The ‘I’ in the new title signals that I am carving the old notes into Part I and Part II, and I am planning to work up Part I into a decent shape, while quite putting aside Part II for a good while. (Lichtenberg: “Just as certain writers, after first dealing their material a rough blow, say it naturally falls into two parts”.) And ‘Notes’ is a frank admission that the material still won’t be very smoothed out, and I’ll not be aiming for a polished book-style finish. ‘Gentle introduction’ means that it goes no doubt far too slowly for some.

I’d be really interested in comments on Chapter 3, which has given me a lot of grief.

Two pages on slice categories

I am intermittently thinking about categories again, and I think I got something wrong in the current set of notes. I there defined slice categories in a way that doesn’t work, at least given my initial preferred definition of categories (which is the same as Awodey’s or Riehl’s).

OK: What is an arrow in a slice category (C/X) from the object (A, f \colon A \to X) to the object (B, g \colon B \to X) — where, of course,  A, B are C-objects, and f, g are C-arrows?

Since we are constructing (C/X) from data in C the natural thing to do is to use a C-arrow j \colon A \to B which interacts appropriately with f and g, giving us a commuting triangle with f = g \circ j.

But that still leaves two options. The simpler option (1) identifies the needed C/X-arrow with j by itself. A more complex option (2) takes the needed C/X-arrow to be the whole commuting triangle, or if you like, the triple (f, j, g). In the earlier set of notes I went for the simpler (1). But I don’t think this can be right for a reason I explain. And I now note that Leinster initially goes for (2) (though his language then wobbles).

So here are two improved(?)pages on slice categories which I do hope now get things right. But let me know if I have gone off-piste!

Beginning Category Theory: Chs 1 to 16 (etc.)

A short post, just to announce another update of Beginning Category Theory. I have significantly improved the chapter on equalizers and co-equalizers, and I have also split into two, and expanded, the old chapter 15 which covered both limits in general and pullbacks/pushouts in particular. So here is a version of BCT including these newly revised chapters.

Just as before, to keep things simple, there is one long PDF here, with both the reworked chapters up to Chapter 16 and also the remaining unrevised chapters from the 2015/2018 Gentle Intro, with the division between old and new clearly flagged. The same remark applies as before: revised chapters get posted when I think they are an improvement on what went before, not when they are polished perfection! All comments and corrections as always most welcome.

My plan over the next few weeks is to rework/expand the last four chapters in Part I of BCT , and then to go back to the beginning in order to try to smooth out the (sometimes considerable) unevenness in the level of exposition/style of presentation which inevitably creeps in as you are concentrating on local revisions. What fun.

One True Logic

There’s a new book out by Owen Griffiths and Alex Paseau, One True Logic: A Monist Manifesto (OUP). As the title suggests, this argues against logical pluralism. Yes, of course, there are myriad logical systems which we can concoct and study as interesting mathematical objects. But what about the logic we actually use in reasoning about them and about other mathematical matters? Is there in fact one correct logic which tracks what really does follow from what? Our authors take a conservative line, in that they are anti-pluralist: there is indeed one true logic for in-earnest applications. They are unconservative in defending a highly infinitary logic in that role.

I’ve read the first few chapters with enjoyment and enlightenment. But I’m going to have to shelve the book for the moment, as it will be too distracting from other commitments to engage seriously with the rest of it for a while. One of the delights of somewhat senior years is finding it more difficult to think about more than one thing at a time. (“But what’s new?” murmurs Mrs Logic Matters from the wings.)


For a start, I must continue cracking on with the category theory project. I have now revised Chapters 1 to 15 of Beginning Category Theory. So here they are, in one long PDF which also includes the remaining unrevised chapters from the 2015/2018 Gentle Intro.

In this iteration there are quite a few minor changes to Chapters 1 to 13 (correcting typos, clarifying some phrasing, deleting an unnecessary section, adding a new theorem in §12.2, etc.), though there is nothing very significant there. I have also now revised Chapter 15, the first of the two general chapters on limits/colimits. This and the preceding chapter on equalisers/co-equalisers could surely do with more polishing and lightening-up in places. But as I’ve said previously, I’m including revised chapters when they are at least an improvement on what went before (I’m not waiting for final-draft perfection!).


If you are like me, you are looking for the more-than-occasional consoling distractions from the state of the wider world. Let me share one.

Of the great pianists I have had the chance to hear live over the years, the one I perhaps find the most emotionally engaging of all is Maria João Pires. Her unmannered directedness goes straight to the heart. So here she is, playing Schubert, Debussy and Beethoven, in a video recorded in Gstaad last August.

Beginning Category Theory: Chs 1 to 14 (etc.)

Back from Cornwall, and I have now revised Chapters 1 to 14 of Beginning Category Theory. So here they are.

As before, to keep things simple, there is one long PDF with both the reworked chapters and the remaining unrevised chapters from the 2015/2018 Gentle Intro. I have added a little to Ch. 2, and tidied up Ch. 12. But the main revision this time is a much improved version of Chapter 14 on equalizers and co-equalizers. I’m still not entirely happy with it, but it is a heck of a lot better than it was. Enjoy!

Famous last words, but looking ahead, I think the next few chapters will be an easier job to revise, so I hope to now pick up speed again.


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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!

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