# 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. 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 …!

2. 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 is a product of with , then you’ll see that counts as a product of with . Just apply that to the present case where we write for . (Maybe I should spell this out better!)

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