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.

7 thoughts on “Category Theory I, the end in sight!”

  1. A tiny typo I’ve noticed: on page 75, line 18, the text reads “We saw. for example”, with a full stop instead of a comma.
    And I think on page 91 it’s supposed to be “its mirror image (Y×X, π2, π1)”, not “(X×Y, π2, π1)”?

    1. Thanks for the first — even the tiniest corrections appreciated!

      P. 91 is in fact correct as it stands. Maybe the use of “mirror image” is unhelpful. Look back at the original definition, if (O, \pi_1\colon O \to X, \pi_2\colon O \to Y) is a product of X with Y, then you’ll see that (O, \pi_2\colon O \to Y, \pi_1\colon O \to X) counts as a product of Y with X. Just apply that to the present case where we write X \times Y for O. (Maybe I should spell this out better!)

  2. Hi professor,

    1. I don’t understand your usage of the word “objects” in the phrase “structure comprises of objects”
    e.g. “A particular group is a structure which comprises some objects equipped with a binary operation defined on them” (page 1)
    It sounds like you are talking about the **elements** of that group. But why call these elements “objects”?

    The particular structure itself is an **object** in a category. So now you have **object** comprises of **objects**? This is confusing to me.

    2. I understand that there is distinction between the general group G and particular group *G*
    But in the phrase “The objects G equipped with a binary operation” (page 5)
    It’s weird to me that G’s plurality refers to the multiple objects (which are actually elements) in that group.

    I have always thought G’s plurality refers to its different instances: like Z, GL(n), permutations, … not what it comprises of like x, y,… in G.

    For a particular set like Z, Z’s plurality refers to its elements …,-1, 0, 1, …
    so you can say “the set Z of integers” and also “the integers Z”.

    For general structured set G (in the definition of a group), G’s plurality should refer to Z, R, … (its instances). And *G* refers to one particular instances (say, Z).
    So it makes sense to say “the group G of elements” but not really “the elements G”.
    Compare to the above: Z is instance of G, ‘integers’ is instance of ‘elements’.

    What’s wrong with this view?

    – The general group G is a single general object, which comprises of its general elements. These elements satisfy the group axioms.

    – particular instances of group G are particular objects (also called structures) *G* which appear in some category.

    Can I understand later chapters if I keep this view?

    1. (1) Call the objects which, suitably equipped with a binary function, constitute a group the group’s “elements” if you like. I prefer not to, as talk of elements characteristically goes with talk of sets, and the point of Chapter 2 is to show that we don’t need to invoke sets at least in elementary group theory.

      To be sure a group can be an object in a category. But there is nothing mysterious about one thing, a structure which is an object of a category, comprising other things, the objects which make up a group.

      (2) I don’t know what you mean by “the general group G”. And I’m not sure either what is supposed to be weird about talking of some objects being equipped with a binary function!

      I suspect you are just misunderstanding the use of plural variables. For example, when we talk of a particular set Z, “Z” is a singular term referring to one thing, a set, and not a plural term referring to many things. Whereas I was using upright “G”, for example, as a plural variable referring to many things (in the way that your “the elements”, for example, refers to many things).

      (3) You write “For general structured set G (in the definition of a group)”. But the point of Chapter 2 is to introduce group theory without talking about sets.

      (4) ” The general group G is a single general object”. I don’t know what a “general group” could be. There are lots of individual groups. There is the (general, if you like) property of being a group. There isn’t something else, as well as individual groups and the property of being a group, a “general group”.

      1. Hi professor, I’m sorry for the late reply, I haven’t had time to look at this. In retrospect, my original post was indeed quite unclear.

        (1) Before asking the question above, I actually knew you wanted to avoid the word “element” here because it has a specific technical meaning in set theory.
        and I had no issue with that.
        However my question was: why replace “elements” with the word “objects”?
        Because I thought “objects” has a specific technical meaning in category theory: it only refers to the “things” at two ends of each arrow in a category!
        So then “structure comprises of objects” to me translates into “object (in some category) comprises of objects” which confused me because it made me think of arrows inside objects
        I wouldn’t have asked this question if you had used “things”, I am comfortable with “thing comprises of other things”
        Maybe I should just take the word “objects” to mean “things”?

        (2) By “general group” I just mean “a group”, the thing defined by the group axioms/property of being a group.

        When there are things, there is also a further thing: a general thing of which all those things as instances of.
        e.g. When you make statements about Earth, Mars, Jupiter etc. you can also make statements about a further thing: a general planet, its instances are Earth, Mars, Jupiter,…
        so I say “a planet” or “a general planet” to mean the same idea.
        I suppose this sounds like set theory: when there are things, there is also a further thing: a set of those things! (set theory is just so natural to talk about generality)

        So I wrote “general group” simply to contrast “a group” with its specific instances: “a group” = “a general group” contrasts with “a specific group” like Z, GL(n) etc.

        >what is supposed to be weird about talking of some objects being equipped with a binary function

        In the context of the previous sentence “Take ‘G’ to stand for one or more objects…”
        I actually thought ‘G’ stands for specific instances of group like Z, Q, R,…
        (because these specific groups are the “objects” in category `Grp` so I preferred this interpretation)
        so saying these objects have binary function * acting on them is like saying
        “Z * Q”
        that was weird to me.
        I did infer that by “objects” you mean the things a group comprises of, as I wrote above

        >G’s plurality refers to the multiple objects (which are actually elements) in that group

        but I wanted to confirm that this is indeed the case.
        I preferred “objects” to mean different instances of a group (as this aligns with the usage of “objects” in category `Grp`) and not the things inside it.

        >I suspect you are just misunderstanding the use of plural variables. For example, when we talk of a particular set Z, “Z” is a singular term referring to one thing, a set, and not a plural term referring to many things. Whereas I was using upright “G”, for example, as a plural variable referring to many things (in the way that your “the elements”, for example, refers to many things).

        I actually understood that you use upright letters for plural variables.
        It is ok to use letter to denote plural things
        e.g. “planets P are those that …”
        “objects G are those that …”
        When I wrote “Z’s plurality refers to its elements” I understood that “Z” is a single thing but I wanted to make a point about the usage of
        “objects G equipped with…”
        I had thought that this template
        “The objects X equipped with operation…”
        causes less confusion in the case X = Z, the integers
        than when X = G, a group in general.

        In other words, I think it would be less confusing to say “the objects Z equipped with…” than to say “the objects G equipped with…”
        Because Z is not a general thing like G, so if you say “the objects Z” I know you are talking about the integers that Z comprises of.
        But when you say “the objects G” I ask myself, shouldn’t these “objects” the instances of a group like in the objects in category `Grp`?
        but the “objects” actually refer to the things a group comprises of, and not its instances!
        So I had to ask to clarify that the “objects” now are being used not as a technical word in Category Theory but rather as a generic word like “things”.

        1. I don’t think my relative use of “objects” is unusual (see also §2.2). So a category of categories has categories as its objects; some of those categories have groups as their objects; those groups will have their objects, equipped with binary functions. And so it goes.

          I don’t know what a “general group” is supposed to be. You seem to want to say that as well as all the group instances, there is something additional , a “general group”. For another sort of case, do you want to say that as well as all the individual cows, there is something else a “general cow” (as it were, a Platonic 𝛼𝜐𝜏𝜂 𝜂 cow)? I’m not sure I’d want to go down there …!

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