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!

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 …

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.

In Vol. 12 of the Handbook of Philosophical Logic (Springer 2005), John L. Bell contributed a long piece on ‘The development of categorical logical’ (some 62 pages, plus a 10 page bibilography, with another 8 pages of background from category theory). Here’s an online version. That piece starts with a historical sketch, and then gets down to a rather hard-core, compressed, outline of some key ideas and results. I can’t imagine that anyone has found it an easy read: in fact, I’ve thought of it as not so much a source essay to start from but as a target summary, something to aim to get to eventually understand from other readings.

It is disappointing, then, to find that the main contribution on categorical logic in Landry’s collection is a 23 page piece ‘Categorical logic and model theory’ by Bell which is mostly just a series of excerpts from that old longer paper. It doesn’t range as widely, but I’m not sure that is much more readily accessible. Will a logic-minded philosopher who knows some category theory — has tackled, say, Emily Riehl’s Category Theory in Context — get much out of this as it stands? I doubt it. To my mind, then, this is an opportunity missed.

The next paper, the eighth, is again by Jean-Pierre Marquis, ‘Unfolding FOLDS: A foundational framework for abstract mathematical concepts’. The first dozen pages are mostly, as a section title has it, ‘some generalities about abstract mathematical concepts’, with arguments for the claim that traditional set theoretic foundations ‘do not encode abstract mathematical concepts properly’. Category theory, you won’t be surprised to hear, does better; and a theory embracing a hierarchy of 1-categories, 2-categories, 3-categories, …, does better still. And then, Marquis claims, Makkai’s system FOLDS — that’s first-order logic with dependent sorts — is what we need as a formal framework ‘to describe this hierarchy directly and properly’. So in the remaining dozen pages of this paper, Marquis outlines FOLDS.

The opening pages of generalities include a sample of the sort of claims made by some categorial enthusiasts which I hesitate to sign up to. For example, what are we to make of this? —

“Let G be an abstract group.” This is a common way of talking in contemporary mathematics, say in group theory. … Our mathematician certainly thinks that the abstract group G has an underlying abstract set. An abstract set is basically a set whose elements have no structure.

Really? Who, I wonder, goes in for this ‘common’ way of talking? You won’t find the phrase ‘abstract group’ anywhere in e.g. Lang’s Algebra. Nor in Aluffi’s Algebra: Chapter 0 (an interesting case, given the categorial flavouring that Aluffi likes to give to his exposition in that lovely book). You will find a few occurrences at the very beginning of Dummit and Foote’s Abstract Algebra, and they do talk on p. 13 of ‘the notion of an abstract group’. But that can be read equally as ‘the abstract notion of a group’ — the notion we get by abstracting from various cases of permutation groups, symmetry groups and the like. I don’t think they are committed to the idea that there is, as well as those more concrete groups there is another sort of thing, an abstract group made of items which (unlike permutations or symmetries) have no nature of their own. Here’s a quote from another book, a standard undergraduate text, Rotman’s First Course in Abstract Algebra: he has just moved on from talking about permutations to introducing the general notion of a group”

We are now at the precise point when algebra becomes abstract algebra. In contrast to the concrete group Sn consisting of all the permutations of the set X = {1, 2, . . ., n} under composition, we will be proving general results about groups without specifying either their elements or their operation. … It will be seen that this approach is quite fruitful, for theorems now apply to many different groups, and it is more efﬁcient to prove theorems once for all instead of proving them anew for each group encountered. For example, the next proposition and three lemmas give properties that hold in every group G. In addition to this obvious economy, it is often simpler to work with the “abstract” viewpoint even when dealing with a particular concrete group. For example, we will see that certain properties of Sn are simpler to treat without recognizing that the elements in question are permutations.

So yes, our mathematician can often proceed abstracting from a group’s details: but that doesn’t mean that “our mathematician certainly thinks” that they are dealing with an abstract group, something built from elements which actually lack structure.

But maybe, on a more careful reading, that isn’t quite the view that Marquis really wants to attribute to the contemporary mathematician. He has a footnote on p. 141, quoting the French mathematician Maurice Fréchet

In modern times it has been recognized that it is possible to elaborate full mathematical theories dealing with elements of which the nature is not specified, that is with abstract elements. A collection of these elements will be called an abstract set. … It is necessary to keep in mind that these notions are not of a metaphysical nature; that when we speak of an abstract element we mean that the nature of this element is indifferent …

So it isn’t after all that the elements of an abstract group have no structure, but rather that for certain generalizing purposes we are ignoring it. As Marquis comments, approvingly I think, here we have ‘a quote by a mathematician that specifies that the property of being abstract is epistemological rather than ontological’.

But if that’s the line, is it so obvious that good old-fashioned set-theoretic reductionism does ‘not encode abstract mathematical concepts properly’? If talk ‘the’ Klein four group is not to be construed ontologically, as talk about some Dedekind abstraction, an entity over and above all the concretely realized Klein four groups, why can’t we say: a claim of the form ‘the (abstract!) Klein four group is F’ is to be cashed out along the lines of ‘any (set-theoretic?) instantion of a Klein four group is F*’ some some suitable derived F*?

Ok, I can’t pursue this further here. But I’m flagging up that Marquis’s opening discussion about ‘abstract mathematical concepts’, while hinting at interesting issues, to my mind goes too fast to be very satisfactory.

What about the more technical part of the paper, on FOLDS? I can only say that I didn’t find the exposition inviting or leaving me wanting to find out more …

Marquis’s paper may or may not be a success, but at least I can see why the editor thought it belonged in this volume. Not so for the next two papers. A sixty page heavy technical paper by Kohei Koshida on ‘Categories and modalities’ belongs in one of the specialist journals. Similarly for forty pages on ‘Proof theory and the cut rule’ by J. Cockett and R. Seely. Unless I am much mistaken, I imagine the number of readers who will be interested and comprehending will be the same as for most such technical articles. Tiny.

The next paper in Landry’s collection is a reprint of a short 2014 Phil. Math. paper by Steve Awodey, ‘Structuralism, invariance, and univalence’. You can download it here.

Awodey’s question concerns the content we can give to what he calls the Principle of Structuralism, that isomorphic objects are identical, when a naive reading makes that Principle simply false. And as the title suggests, Awodey wants to argue that Voevodsky’s Univalence Axiom is the way to make sense of the Principle. This does involve skating at great speed over some ideas from type theory, such as Propositions-as-Types. My sense is that the typical logic-and-phil-maths-minded philosopher is still pretty hazy when it comes to such type-theoretic ideas, at least once we step beyond Simple Type Theory. However, Awodey’s vigorous advocacy here and elsewhere should perhaps prompt more of us to put in the effort to get up to speed (though sadly, as I’ve complained before, the literature isn’t exactly over-supplied with good introductions to type theory!). So: short but thought-provoking.

The fifth paper is ‘Category theory and foundations’ by Michael Ernst. This might rather naturally have been the first paper in the collection, as it is a lucid state-of-play assessment of debates – some long-running — about the foundational role that category theory may or may not have.

In a previous paper, Ernst himself proved a technical result which in fact changed the state-of-play: he showed that some plausible conditions on what could count as an unlimited all-encompassing category (which all categories belong to) can’t be jointly satisfied. This means that e.g. the complaint that ZFC (plus, for example, some large cardinal axioms) fails to cope with an all-encompassing category is no strike against standard set-theoretic foundations: no consistent theory meets that impossible goal. There is a nice short presentation of Ernst’s result here.

But equally, Adrian Matthias’s well-known arguments that categorial foundations in the shape of ETCS are inadequate for mathematical purposes fail to strike a fatal blow, given the fact that we can extend ETCS in categorially motivated ways to recover a theory which can do everything that can be done by ZFC (or indeed by ZFC plus some large cardinal axioms). Also see again McLarty’s piece in this collection and references there.

So, arguably, both a conventional set theoretic framework and a deviant categorial framework starting from ETCS can be developed into adequate technical foundations for mathematical practice (in some useful sense of ‘foundations’).

But what about the complaint that category theory is not really autonomous but makes use of notions of operation/morphism and collection/objects-combined-into-a-structure which need ultimately to be elucidated in set-theoretic terms? The potentially most interesting part of Ernst’s paper, his §5, discusses this autonomy issue: but to be frank I found the discussion a bit thin. E.g. he touches on Lawvere’s supposedly Cantorian “bag-of-dots” conception of sets which I have always found opaque: and I can’t say I was really helped here. However, that said, this piece is overall definitely worth reading, with useful pointers to relevant debates.

The sixth paper is on ‘Canonical maps’ by Jean-Pierre Marquis. The idea is that there is, out there in mathematical practice, a notion of canonical map which has a reasonably definite use and which can be illuminatingly analysed in categorial terms. There’s a great deal of initial arm-waving but nowhere near enough examples for me (or you?) to latch onto the supposed notion under examination. So I found this entirely unsatisfying. But you can download the paper here, and see if you can get more out of it.

Categories for the Working Philosopher, edited by Elaine Landry (OUP), was first published in 2017, and I briefly noted this collection when it came out as seeming to be a very mixed bag. Since my mind is back on matters categorial, and the book has been recently paperbacked, I thought I’d take another look.

Landry describes the first eleven essays as concerning the “use of category theory for mathematical, foundational, and logical purposes”. The remaining seven pieces are on scattered applications of category theory (though there seems to be nothing on why it has come to be of such central concern to theoretical computer science). I’ll probably only comment here on the earlier essays, and I’ll take them in the published order, starting with the first three.

‘The roles of set theories in mathematics’ by Colin McLarty takes up what is by now a familiar theme — that ZFC radically overshoots as an account of what ‘ordinary mathematics’ requires by way of background assumptions about sets. McLarty in particular takes a look at the set-theoretic preliminaries expounded in two classic texts, by Munkres on topology and Lang on algebra, and comments on their relatively modest character. Now, neither Munkres or Lang gives a regimented summary of his set-theoretic assumptions in axiomatic form: but if we did so, what would it look like? Arguably like ETCS, the theory so neatly presented in Tom Leinster’s well-known ‘Rethinking set theory’ whose aim is, as he puts it, to show that “simply by writing down a few mundane, uncontroversial statements about sets and functions, we arrive at an axiomatization that reflects how sets are used in everyday mathematics.”

Now, ‘ETCS’ is of course short for the ‘Elementary Theory of the Category of Sets’ developed by Lawvere and its original idiom was categorial. But Leinster is at pains to point out that the axioms of this set theory, in his presentation, do not involve any essentially categorial notions — they just talk about sets and functions (without reducing the functions to sets, by the way). McLarty explores a little further in a helpful and interesting way, e.g. telling us more about how variant stronger set theories can be seen as expansions of ETCS.

But I can imagine some thinking that the interest here is primarily set-theoretic rather than in the (apparently dispensable?) categorial idiom. I doubt that McLarty sees it like that! — but then those readers would have probably liked to hear more from him here about quite how he does see ETCS to be more intertwined with category theory.

The next paper is, by my lights, pretentious arm-waving by David Corfield on ‘Reviving the philosophy of geometry’: you can give this a miss.

Next up, Michael Shulman writing about ‘Homotopy type theory: a synthetic approach to higher equalities’: you can download the paper here. Now, Shulman has elsewhere shown a considerable gift for what you might call the higher exposition — explaining, unifying, organising difficult material. His much-referenced paper ‘Set theory for category theory‘ is indispensable. But this present paper is pretty hard going — surely rather too compressed and allusive (would someone who had never before encountered Einstein’s hole argument get what is going on? more to the point, is the supposedly crucial distinction between the rules of a type theory and axioms of a set theory made transparently clear?). I’ll certainly not try to summarize, then. But despite its density, the piece is helpful enough to be worth struggling with, if you want to get some glimmer of what the programme of homotopy type theory/univalent foundations is.

However, as Shulman tells us at the outset, HoTT is a “surprising synthesis of constructive intensional type theory and abstract homotopy theory” and neither of those is essentially categorial from the off. Indeed, in the foundational text, Homotopy Type Theory: Univalent Foundations of Mathematics, category theory doesn’t get mentioned until p. 377! So we might wonder: interesting though HoTT’s programme is as “a new foundation for mathematics and logic” (in Landry’s words), in what sense exactly is this programme essentially categorial in nature?

The two Parts of my notes on category theory are now two separate PDFs, and all the chapters in Part I have now been revised (with the last chapters significantly rewritten since the last update). So here they are!

Part I is still very much work in progress, but there are more than enough major improvements since the last version to make it worthwhile to post this update. I think the structure of Part I, the overall order in which topics are tackled, should remain stable from now on: but I’ll certainly want to return to improve the treatment of some of those topics. So all comments on the current version will be immensely useful.

I have added a short new chapter to Category Theory: Notes towards a gentle introduction. Just four-and-a-bit pages on “power objects” (i.e. objects which behave in a certain way as a powerset does in the category Set). This is really just an exercise in making the definition look tolerably well-motivated. I don’t want/need to run very far with the idea: I just want to have it in play for when I get round to first introducing the notion of a topos.

So the end of Part I of the Notes is in sight. I need to have a chapter expanding the current half-chapter on natural number objects, and another chapter expanding the current half-chapter on group objects. Then the Part will finish with a new chapter introducing the notion of a topos (all the ingredients are in place) and looking ahead, motivating the transition to Part II. As all my predictions about how long things will take turn out to be useless, I won’t venture one. But whenever I do get to the end of Part I, the plan is to pause and work through it again making it as clear and unified in level and approach as I can, rather than rush on to Part II.

Post-covid chez Logic Matters continuous to be exceedingly tedious. Tests may come back negative, but the fatigue continues with only very slow improvement. Par for the course we are told. If we had the energy to be really annoyed, we’d be really annoyed …

I have not been exactly full of logical energy! But I’ve been up to a little light re-editing of the already-revised chapters of Category Theory: Notes towards a gentle introduction. I’ve repaired some typos and a few thinkos, slightly speeded up a few sections which added unnecessary complications, and even added a couple of minor theorems. The main changes are at the very outset in the pre-categorial chapter on groups, and in the first two chapters on products.

Some of the typos/thinkos in the previous version are annoying enough for me to want to put the current corrected notes online right away. So I have …

Otherwise covid chez Logic Matters wends its tedious way along what seems to be the now most common trajectory, where the main continuing effect is considerable fatigue outlasting mild-ish cold-like symptoms. Could be a lot worse. Back to music and other postings, I hope, next week.

Another update to Category Theory: Notes towards a gentle introduction. This time, I have rewritten and indeed doubled the length of the chapter on subobjects and subobject classifiers. Much improved, I hope! I’ve also added two or three needed results at the appropriate places earlier in the notes, so that the proofs in this newly revised chapter can be snappy, and corrected some random typos I happened to notice.

(I’ll be pausing on categories now for a few weeks, with a holiday break coming up.)