# Category theory

## Notes on Basic Category Theory, v.8a/8b

It’s a balancing act. On the one hand, I don’t want to annoy readers with over-frequent announcements of minor revisions. On the other hand, I don’t want to keep propagating flawed versions when I have an improved offering in hand!

Anyway, I’ve been reading through the first 11 chapters of the Notes making some minor corrections and other changes. I’ve also had some much appreciated corrections of a few mistakes in later chapters from Alessandro Stecchina. Since I think there will be something of a pause before I can press on to re-read the rest of the Notes, here’s an interim update, to version 8a version 8b of the Notes.

## Notes on Category Theory, v.8

At long last — more or less exactly nine months since I started intermittently writing them — there is a first complete version of my Notes on Category theory (as they are now called). Or at least, the Notes are complete in the sense that I don’t intend to press on to add further chapters on significant new topics like monads or abelian categories. Rather the current plan is to leave these notes in more or less their present form, for all their shortcomings (and I hope in due course to start writing a differently organized, more discursive, bigger and better version).  Still, I would very much like to hear about errors of one kind or another. And I’ll no doubt issue occasional “maintenance upgrades” when I hear about mistakes or spot passages which really won’t do —  and perhaps I might add more illustrative examples or even new sections here or there to round out the treatment of existing topics where the coverage in retrospect seems too skimpy.

Since the previous version, I have expanded the chapter on some general results about adjunction, and  added a chapter on adjunctions and limits. This has entailed quite a bit of going back to earlier chapters, adding material to smooth the route to later theorems. I finish up by waving my hands at, though not elucidating the content of, the Adjoint Functor Theorems, General and Special.  But it is a non-trivial expositional task to explain these (the technical proofs aren’t hard; what isn’t so easy is to see is the motivation for the various new concepts — like the ‘solution set condition’ — which they involve). I’m not sure I yet have a sufficiently good grip on the place of these theorems in the scheme of things to give an illuminating account of the motivations. So I’m at the moment shirking the task of trying to explain more.

But in any case, the Adjoint Functor Theorems arguably sit on one of the boundaries  between basic category theory and the beginnings of more serious stuff. So given the intended limited remit of the Notes (now highlighted by calling them notes on Basic Category Theory), the Theorems mark a reasonable point at which to stop for now.

So, with that by way of preamble, here is the new version of the Notes (190 pages). [Link updated to version 8a] Enjoy!

## Notes on Category Theory, v.7

Progress seems to have been a bit slow for various reasons, but I have now added two short-ish chapters to the Notes on Category Theory. One is a chapter on Exponentials added between the chapters on limits and the start of the chapters on Galois connections and adjunctions. The other is second chapter on adjunctions, showing how to generalize certain results we saw in the special case of Galois connections to apply to adjunctions more generally.  I have also added some material earlier to make some of the new later proofs work more smoothly.  Here then is the latest version of the Notes, some 170+ pages.

As I have stressed before, I’m myself learning as I go along from the project of writing these Notes. So as more light dawns, I of course can see how I could have arranged/explained earlier material rather better (OK, a LOT better). But for the moment, the plan remains to add a few more chapters, before eventually starting over and trying to get everything into a more ideal shape.

## Notes on Category Theory v.6b

Illness (not so great) followed by fortnight’s holiday (really excellent) stopped work on category theory for a while. Very slowly getting back to it. But in the meantime, a number of people have very kindly been sending corrections to the last version of the Notes. I have also tinkered in minor ways, improving the last chapter in particular. There are just about enough changes to warrant another “maintenance upgrade”, making some of the needed repairs and improvements.

I hope to have a couple more chapters on adjoints ready for prime time later in the month — but finding a neat expository path through the material is a challenge, so don’t hold your breath!

The plan at the moment is for another five or six more chapters in total to round off Part I of these Notes, on basic category theory (I’m not sure yet whether I also need to say anything about monads for what is going to follow). And then — having got the bit between my teeth — I’d like to continue, by discussing some logic and set theory in a categorial way in Part II of the Notes. Promises, promises.

## Notes on Category Theory v.6a

A number of people have very kindly sent corrections to the last version of Notes on Category Theory. There were some possibly confusing typos and also a downright wrong proof. Embarrassing. So here’s a “maintenance upgrade”, making some needed repairs.

[Added: a few more needed corrections are noted in the comments below.]

## Notes on Category Theory v.6

Here is an updated version of my on-going Notes on Category Theory, now over 150 pp. long. I have added three new chapters, at last getting round to the high point of any introduction to category theory, i.e. the discussion of adjunctions. Most people seem to just dive in, things can get a bit hairy rather quickly, and only later do they mention, more or less in passing, the simpler special case of Galois connections between posets (which transmute into adjunctions between poset categories). If there’s some novelty in the Notes at this point, it’s in doing things the other way around. We first have a couple of chapters on Galois connections — one defining and illustrating this simple idea, the other discussing a special case of interest to the logic-minded. Only then do we get round to generalizing in a rather natural way. We then find e.g. that two equivalent standard definitions of Galois connections generalize to two standard definitions of adjunctions (presented without that background, it isn’t at all so predicatable that the definitions of adjunctions should come to the same).  I do think this way in to the material is pretty helpful: I’ll be interested, eventually, in knowing how readers find it.

So we this is what we now cover:

1. Categories defined
2. Duality, kinds of arrows (epics, monics, isomorphisms …)
3. Functors
4. More about functors and categories
5. Natural transformations (with rather more than usual on the motivation)
6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
7. Categories of categories: issues of size
8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
9. An aside on Cayley’s Theorem
10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
11. Representables (definitions, examples, universal elements, the category of elements).
12. First examples of limits (terminal objects, products, equalizers and their duals)
13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
15. Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)
16. [NEW] Galois connections (warming up for the general discussion of adjoint functors by looking at a special case, functions that form a Galois connection)
17. [NEW] An aside for logicians, concerning a well-known example, the Galois connection between syntax and semantics.
18. [NEW] Adjoints introduced. [Two different definitions of adjoint functors, generalizing two different definitions of Galois connections; some examples of adjunctions; a proof that the two definitions are equivalent.]

There will certainly be a few more chapters on adjoints. But don’t hold your breath, with a family holiday coming up and some other commitments. I haven’t decided yet whether eventually to add a chapter or two on monads (for monads seem a standard next topic to cover — e.g. in the last main segment of the Part III Tripos category theory course this year, the last chapter of Awodey’s book). Watch this space.

## Notes on Category Theory v.5

Here is an updated version of my on-going Notes on Category Theory, now 130 pp. long. I have done an amount of revision/clarification of earlier chapters, and added two new chapters — inserting a new Ch. 7 on categories of categories and issues of size (which much expands and improves some briefer remarks in earlier versions), and adding at the end Ch. 15 saying something about how functors can interact with limits. There’s quite a bit more that could be said in this last chapter, and I’ll have to decide in due course whether to expand the chapter, or return to the additional topics later, or indeed to only mention some of those topics in the end (I’m trying to keep things at a modestly introductory level). But for the moment I’ll leave things like this and move on to a block of chapters on adjoints and adjunctions. So here’s where we’ve got to:

1. Categories defined
2. Duality, kinds of arrows (epics, monics, isomorphisms …)
3. Functors
4. More about functors and categories
5. Natural transformations (with rather more than usual on the motivation)
6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
7. [New] Categories of categories: issues of size
8. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
9. An aside on Cayley’s Theorem
10. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
11. Representables (definitions, examples, universal elements, the category of elements).
12. First examples of limits (terminal objects, products, equalizers and their duals)
13. Limits and colimits defined (cones, limit cones: pullbacks etc.)
14. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).
15. [New] Functors and limits (preserving, reflecting, creating limits: hom-functors preserve limits, etc.)

Don’t hold your breath for the chapters on adjoints, though. After a very busy time for various reasons, I’ve a couple of family holidays coming up!

## Category theory again

The category theory page here has been much expanded with links to (i) some online lecture notes, and (ii) some books which are freely (and legitimately!) available online in one form or another.

I am not at all aiming to include everything that is available out there: but on the other hand, if I have missed something good, do please let me know!

## Notes on Category Theory v.4

I wasn’t able to do much work in January for family reasons, but levels of concentration and energy are returning. So, much later than I’d hoped, here at last is an updated version of the Notes on Category Theory (still very partial though now 109 pp.). There are three newly added chapters, and some minor tinkering earlier. We now cover:

1. Categories defined
2. Duality, kinds of arrows (epics, monics, isomorphisms …)
3. Functors
4. More about functors and categories (and the category of categories!)
5. Natural transformations (with rather more than usual on the motivation)
6. Equivalence of categories (again with a section on motivation, why we want ‘equivalence’ rather than full isomorphism)
7. The Yoneda embedding (shown to indeed be an embedding by using an easy restricted version of the Yoneda Lemma)
8. An aside on Cayley’s Theorem
9. The Yoneda Lemma (how to get to the full-dress version by two conceptually easy steps from the restricted version).
10. Representables (definitions, examples, universal elements, the category of elements).
11. First examples of limits (terminal objects, products, equalizers and their duals)
12. Limits and colimits defined (cones, limit cones: pullbacks and upshots)
13. The existence of limits (in particular, having finite products and equalizers implies having all finite limits).

Coming soon: a further chapter on limits (on functors and the preservation of limits) and then a chapter on Galois connections as a nice gentle lead-in to the chapters on adjoints.

## What is the category Set?

We all know that we quickly encounter issues of size when we start category theory. Categories, we are told, are collections of objects and morphisms satisfying certain axioms. But we very soon meet, among our very first examples of categories, categories like Set whose objects are too many for the collection of them to itself be a set. So what kind of ‘collection’ do we have here?

But here’s another issue about that initial example of the category Set which is ignored by many writers of introductory texts.

The issue can be posed, bluntly, like this: when the category Set is mentioned early on in an introductory text what category is that?

Typically no explanation is given at this stage. But of course which category we are dealing with depends on our set theory. For an NF-iste, the category of NFsets has very different properties from what is intended (for a start NFsets is not cartesian closed and later we learn that  Set is). But fair enough, in an intro book you aren’t going to mention that in the opening pages! Rather the authors are, surely, intending to point to stuff that their beginning readers can be assumed to know about, and are saying, “Hey, you in fact already know about some categories, for example …” The charitable reading, then, is that authors are relying on their readers to think of  Set as comprising the sets they already know and love.

But what sets are those?

Well, for many readers (if they have done any more than had a bit of set-notation waved at them) the sets they know and love are the  pure sets of the cumulative hierarchy — pure in that there are no urlements, no memberless entities in the universe of sets other than the empty set. If set theories with urelements are mentioned in passing in an intro set-theory course, it is usually only to be dismissed and forgotten about. So: in the absence of special explicit signals to the contrary, we might well reasonably take the category Set mentioned in the very early pages to be a category of pure sets of the usual hierarchy (or at least the hierarchy up to some inaccessible, or whatever). Just sometimes this is made explicit. Thus, Horst Schubert in his terse but very good and clear Categories (§3.1) writes “One has to be aware that the set theory used here has no “primitive (ur-) elements”; elements of sets, or classes …, are always themselves sets.”

But then what are we to make, a bit later in our introductory books, of e.g. the usual presentation of the Yoneda embedding as $latex \mathcal{Y}\colon \mathscr{C} \to [\mathscr{C}^{op}, \mathbf{Set}]$. Putting it this way, if you look at the details, assumes that hom-collections $latex \mathscr{C}(A, B)$ for $latex A, B \in \mathscr{C}$ actually live in $latex \mathbf{Set}$. And since such a hom-collection is a set of $latex \mathscr{C}$-morphisms, that assumes that the $latex \mathscr{C}$-morphisms — irrespective of what the [small] category $\mathscr{C}$ is — must live in the world of pure sets too. [Sure, we may want the relevant hom-collections to be set-sized in the Yoneda embedding case — but being no bigger than set-sized is one thing, living in the universe of pure sets is something else!]

But do we really want to assume that morphisms are always pure sets?

Might we not be looking to category theory for a story about how different bits of the mathematical universe hang together which need not presuppose some over-arching, all-in, set-theoretic reductionism, and so in particular doesn’t presuppose from day one that all morphisms are pure sets?

Now, as I noted, the foundational sections you often meet early in category theory books worry away about questions of size (sets vs classes etc.). But the present worry is orthogonal to all that, and is in a way more basic. If we want to make no assumption that the denizens of different bits of the mathematical universe are all cut from the same cloth, we won’t want to slip into assuming that sets of these denizens are all pure sets. So in particular, do we really want to assume that a collection of morphisms (hom-set) must actually live in Set, if that’s the category mentioned back almost on p.1 of the book which is naturally read (and sometimes explicitly said to be) a category of pure sets? It may be that Set eventually gets to play a special role in the mathematical universe, with other most other categories being representable inside it: but surely this should be something to be shown later, not an assumption to be built in from the start.

In Chapter 3 of his fine Basic Category Theory, Tom Leinster makes the right moves here. Back track. Ask again: what are the sets that, before we start into category theory, we know and love and which are pointed to as constituting an exemplar of a category? Not the sets of pure ZFC (which mathematicians other than set theorists could even spell out the axioms which are supposed to tell you what these are?) but rather what we might call the sets of the working mathematician — sets of naturals, perhaps, or sets of reals, or sets of points in a space, or a set of continuous functions, or … And these are sets with ur-elements, elements that are not themselves presumed to be sets. (Or if we ascend to talk of sets of sets of naturals, for example, it remains the case that when we descend again through members of members … everything typically bottoms out with primitive non-sets.)

We assume various operations are permissible on these sets, and we can codify the permissible operations (for the purposes of the non-set-theorist, falling short of ZFC). Now Leinster associates this with Lawvere’s codification of the Elementary Theory of the Category of Sets (ETCS). But of course, Zermelo was in the same game, aiming to codify the principles of set-building that the working mathematician actually needs, aiming to provide enough to do the usual constructions but not enough to threaten paradox, and Zermelo’s original set theory too allowed for ur-elements. But then, if Set is to be a category of sets-with-urelements, built up using the principles working mathematicians actually need (whatever they are) — which indeed makes some of appeals to Set in category theory much more natural — then this really needs to be said (as Leinster does, and others don’t, and some actually deny).

Of course, we could go Schubert’s way and take hom-sets to be pure sets, so that morphisms and the other apparatus of category theory are all pure sets. But seeing category theory in this way as simply in the service of an across-the-board set-theoretic reductionism would surely be an unhappy way of looking at things: category theory seemed to promise more than that!

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