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 ….]

6 thoughts on “Beginning Category Theory: NOT Chs 1 to 3”

  1. Re the Oliver and Smiley quote on p 25:

    Functions are relations. If F is a function and x is an object in F’s domain, F relates x to a particular object, y, in F’s range (or ‘codomain’). We can say that F, as a relation, holds between x and y. There is also some special terminology and notation that can be used when a relation is a function. Instead of xFy, we can write Fx = y or F(x) = y, and we can call x F’s ‘argument’ and y the ‘value of’ F at x. (In such notations, F is typically written in lower case, ‘f’.) … Syntactically, when we add a term expressing an argument to a function name, we get another ‘term’, such as ‘f(x)’; when we then add an equal-sign and a term representing the corresponding value, we get a sentence such as ‘f(x)=y’. (Terms and sentences are, of course, closely related, because terms are among the main building blocks used when making sentences.)

    [There may be some minor use/mention or similar issues there. And some rewording would be needed to handle functions of more than one argument. Nonetheless, I think the basic idea should be clear.]

    Anyway, it seems to me that you are now trying to do two different things in your Gentle Introduction: (1) provide an introduction to category theory, and (2) discuss a range of philosophical and foundational issues around set theory. And you’re doing it in a way that means the reader has to get through quite a lot of (2) before making much progress on (1).

    There was already quite a bit of (2) in the earlier, 2018, version of the Introduction. I think that was at about the upper limit of what makes sense. Now there’s more, and some of the parts that are new seem quite tendentious. (Why, btw, do you think there’s a “deep type-distinction between functions and objects”, rather than thinking that functions are a type of object? Why do you treat a quote from Church that doesn’t even use the word “object” as insisting on that distinction?)

    I think it would be better to do them as two separate documents, or at least to move much of (2) to a chapter or chapters that appear after (1).

    1. (1) On the structural issue. I wouldn’t really describe Ch. 3 as discussing “a range of philosophical and foundational issues around set theory” (that would take a book!). It’s just a dozen pages, with some low-level reminders about what set theory is supposed to do for us — which will be relevant later e.g. when I get round to (sceptical!) remarks about claims that category theory can be some sense provide an alternative foundational framework.

      (2) On the particular remark about a “deep type-distinction between functions and objects”, I realise I do need to be clearer that here it is a narrow type-theoretic sense of object that is in play here, rather than the type-promiscuous sense which I already allowed for in §2.1(c). So some tidying is needed here, and thanks for the prompt for that.

      (3) On functions vs relations: to say “F, as a relation, holds between x and y” looks just like a bald type-confusion to me! The relation that holds between x and y is the relation expressed by “F(...) = —”; and that is not the function which is expressed by “F(...)“.

      I might though add a “By my lights, though for our purposes you needn’t agree” qualifier hereabouts; as certainly I agree this is not something I want the reader to get bogged down over!

      1. (1) While it’s true that some of the philosophical and foundational issues that appear in chapter 3 are more just mentioned than discussed, I think it’s fair to say there’s a range and that they’re placed before the reader’s mind before much progress has been made on introducing category theory. Here’s a not quite complete list of what’s at least mentioned:

        What is a set? How should we understand sets?
        Whether set-talk can be paraphrased away and what significance that has.
        The idea of virtual classes, quoting Quine.
        Proper classes and how they’re treated by Kunen and by Finsler
        Plurals as a substitute for (eliminable) uses of sets
        That we shouldn’t be tempted to think of informal talk of plurals as disguised singular talk referring to sets. [Eliminability works only against sets, for some reason, not against plurals.]
        Paulo Aluffi’s use of (naive) set theory” as little more than a system of notation and terminology”
        A contrast between Aluffi’s view and one expressed by Wilfrid Hodges
        How much set theory is needed by a “competent theory about groups and the ways they interrelate”
        Whether set theory provides “mathematical universe capacious enough to contain copies of all the groups we want”
        The idea of ‘real work’ and when / whether set theory is doing some.
        When / whether talk of pairs can “be construed just as a useful but non-committal idiom, one which introduces no new objects.”
        Likewise for ordered pairs.
        The significance of product groups for when / whether pairs are genuine objects.
        The significance of being able to use different objects to play the role of pairs, and of arbitrary choices.
        Whether it’s internal make-up or how they relate by mappings that “determines objects to count as pair-objects”
        Complaints against the Kuratowski implementation of ordered pairs.
        Whether functions can be identified with their graphs.
        Are some functions ‘too big’?
        An argument that a function and its graph belong to different logical types and hence they can’t be identical, with reference to Church and Frege.
        An argument (with a quote from Terence Tao) that functions can’t be sets because a set of ordered pairs can’t do the work of taking arguments and yielding values. A threat of infinite regress.
        That parallel points can be made about identifying a relation with its extension.
        An argument (quoting Oliver and Smiley) that functions can’t be relations because of differences in their ‘natures’ and because terms are different from sentences.
        Type theory as an alternative to set theory.
        Other set theories — Mac Lane set theory, NFU — as alternatives to ZFC.
        Whether the axiom of choice is essentially set-theoretic at all.
        When modelling functions set-theoretically, how answers to questions about what binary operations are available to form groups, or about what isomorphisms there can be, interact with the axiom of choice, ‘forcing tricks’ and large cardinal axioms.

        They’re interesting. Do they need to be placed before the reader at this stage of the journey, though, in an introduction to category theory? (In an introduction to philosophical and foundational issues about set theory and category theory, OTOH, it would more clearly make sense.)

        (2) I find it hard to work out what you mean by ‘object’.

        To me, ‘object’ is a general term. Alternatives of similar generality are ‘thing’ and ‘entity’. (‘Gadget’ OTOH has a connotation of being a device or tool and so works less well as a general term.) To me, it seems natural to think that functions are a kind of object. I don’t think there is an intuitive type-distinction between objects and functions (p 26). Since you say there is an intuitive type-distinction, I have to try to work out what ‘object’ has to mean for that to be true, and I find that difficult.

        What §2.1(c) seems to be allowing is a ‘relative’ sense of ‘object’, rather than a type-promiscuous sense.

        In the 2018 version of the Gentle Introduction, you say in (b) on p 18

        The labels ‘objects’ and ‘arrows’ for the two kinds of data are quite standard. But note that the ‘objects’ in categories needn’t be objects at all in the logician’s familiar strict sense, i.e. in the sense which contrasts objects with entities like relations or functions. There are perfectly good categories whose ‘objects’ – in the sense of their first type of data – are actually relations, and other categories where they are functions.

        So it looks like people doing category theory are happy to regard functions as objects. I don’t know how many people would find “the logician’s familiar strict sense” familiar. It’s at least not familiar to me.

        (3) I’m not doing metaphysics, trying to say what functions and relations Really Are. I just think it’s reasonable to think of a function as a kind of relation, and that Oliver and Smiley (who do seem to be doing metaphysics) don’t have a very good argument.

        Here’s why I don’t think there’s a type-confusion. ‘F’ is a name I’m using to refer to an entity. (I’ll avoid the word “object”.) I’m saying that the entity is a function, and that functions are relations. F (the entity) is therefore also a relation. (Like Felix, a cat, is an animal as well as a cat.) Since F is a function, we can use the language and notations we used with functions: argument, value of, F(x), F(x)=y. Since F is a relation, we can also use the language and notations we use with relations: holds, xFy. Consequently, we can say that F, as a relation, holds between x and y.

        What’s the type confusion? We can use a different expression / notation when talking about F as a function then when talking about F as a relation. That doesn’t mean the relation and the function must be two different things (so that they can have disjoint types) rather than one thing talked about in two different ways. Humans are animals even though they have some capabilities that other animals don’t and even though we say some things when talking about humans that we don’t when talking about other animals.

        In practice, people often do treat functions and relations as distinct, for a variety of reasons. It lets them use a simpler notion of type, for example: one that doesn’t include subtypes. Or they don’t have any need to treat functions as relations and find it convenient to treat them as distinct to align with the syntactic differences in the expressions they’ll use with them. I don’t mind things like that.

        1. There’s a lot here. On (1) I agree that Chapter 3 was over-stuffed, and more importantly it wasn’t clear why I was fussing about some of the issues. I hope the chapter’s shorter replacement will at least meet these worries.

          On (2) and (3) I think there are two different issues – (i) the correctness of a certain Fregean orthodoxy about objects vs functions vs relations, and (ii) whether it is a good policy to tangle with this in the present context. I’m fully signed up to (i) but not to (ii). The new version of the chapter will still touch on (i) but I hope in a way that makes it clear that the main thrust of the chapter doesn’t depend on it.

          As to “‘F’ is a name I’m using to refer to an entity” I’d for a start object that it’s not a name. But this isn’t the place to pursue that!

          1. If functions aren’t objects, then I don’t know what you mean by “object”. When you say, for example, “in many elementary contexts informal talk of a set doesn’t really carry any serious commitment to there being any additional object over and above those many things”, what does that mean?

            Or “Merely virtual classes are not objects in their own right” — not objects in the way that functions aren’t objects?

            Re objecting to “‘F’ is a name” — I said in my original comment that there may be some minor use/mention or similar issues. I don’t think such things should make it difficult to understand what I’m saying about the entity I called ‘F’.

  2. > I wanted to be reminded of a different Russia

    I’m sorry, but then Russia were occupying Poland, Grand Duchy of Lithuania (mainly contemporary Belarus) and bunch of other countries. And in some years after Home of the Gentry publication it drown Poland and Belarus in blood (during January Uprising).

    So it was no different. (And just to make it clear, Belarus is under Russian occupation today, but their propaganda tries to make Belarusians guilty together with them).

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