November was a busy month. A wonderful week out in Vienna (not to mention the time out before and after, getting ready and recovering from the excitements!). Reading for the 2016 version of the Teach Yourself Logic guide to be done.
The Gentle Introduction to matters categorial has, therefore, not had anywhere near as much attention as I would have liked. Pardon, gentles all.
I have, however, uploaded a new version — now 171 pages, with two short new chapters (one either side of the chapter on exponentials). Prompted by Paolo Giarrusso, to whom many thanks, I’ve completely recast what was a short and potentially misleading section on subobjects into a standalone chapter. And I’ve added another chapter on some mathematical gadgets to be found inside categories, namely group objects and natural number objects (though as you’ll see the current version stops short of giving a clean proof of the final stated theorem on recursion). I have also tinkered elsewhere with the first dozen chapters — the ones preceding the introduction of functors — at quite a few scattered points, not least in sorting some typos. The later chapters from functors onwards, by contrast, remain untouched. As I say, not the state that I’d like the Gentle Introduction to have got to. But I won’t have much time to look at it again over the next few weeks (for various nice reasons), so I thought I should at least make available such improvements/corrections/ additions as I have been able to make. Here, then, is the latest version.
So please do continue to “piece out our imperfections with your thoughts” and let me know of corrections, obscurities, possible improvements etc. And I “your humble patience pray/Gently to hear, kindly to judge” these efforts so far!
5 thoughts on “‘But pardon, and gentles all …’”
First, thanks for the new discussion of subobjects (and for mentioning me), I like it better. The prose there feels somehow different and leaves me uneasy though, but I can’t really explain it — if it helps, reading it feels more like watching an action movie, while the rest feels more gently paced. I’m not sure that’s bad, so feel free to ignore.
About “Functors and limits” (currently chap. 15): starting at least from theorem 85, it seems that by “all limits” you implicitly mean “all small limits” — otherwise I doubt the proof goes through. I’d be fine with this implicit usage, it might be standard (dunno), but I don’t see where it is introduced, and I don’t know if it applies to earlier theorems (say, Theorem 84). Actually, this probably starts at Theorem 54: it talks about “all limits”, but the previous Definition 62 only introduces “all small limits”. Categories Work 2nd ed. (page 116, Sec. V.4, Thm. 1) says that hom-sets preserve all limits, in particular all small limits.
Another thing we spent a while checking during our meeting: if F preserves a limit, it must maps a limit not to an arbitrary limit, but to a limit of _the same shape_ — so to preserve a product A × B, F(A × B) must be the product FA × FB (up to isomorphism).
While this should be obvious, I like your writing because unlike mathematicians you know this is not a good argument ;-). In particular, Definition 85 could introduce “the shape of a limit” as being J (this can’t come any earlier), and Def. 86 could highlight that FL is a limit of the same shape.
BTW, somehow, I’m not finding the equation F(A × B) ~= FA × FB (for F preserving products) among examples.
Relatedly, your examples tend to focus on concrete exercises, which is maybe fine (do you get those as exercises from Johnstone?) — but maybe at the expense of more typical examples like F(A × B) = FA × FB? Most textbooks focus too much on the second kind of examples, but both are needed I guess.
Many thanks for this. It will be a while before I can do the necessary rewriting (holiday in Florence, then work on the TYL Guide, Christmas …!). But this is very helpful and I really appreciate your comments.
I’d like to thank you for putting the Gentle Introduction online
for free. The reading group at Tübingen has been appreciative of
the many examples and of the helpful warnings, an important one
being that the images of functors need not be categories.
Page 106, proof of 11.3(4), end of first paragraph.
“X^Y × Z = Z” should be “X^Y × Y = Y”,
“ev : X^Y × Z → X” should be “ev : X^Y × Y → X”.
TWO DEFINITIONS OF DIAGRAMS
§8.1(a) defines a diagram as a bunch of labeled objects with
arrows between them. Definition 84 defines a diagram as a
functor. The 2nd paragraph on page 147 argues that diagrams in
the sense of §8.1(a) are always functors, because each
non-transitive path “A → B → C” can be split into two arrows
“A → B₁” and “B₂ → C” in some category.
I agree that diagrams in the sense of §8.1(a) are functors, but
the reason given above appears faulty. Let D be the functor
obtained by splitting “A → B → C”, with the preorder
“0 → 1, 2 → 3” as domain. Suppose D 0 = A, D 1 = B, D 2 = B
and D 3 = C. Then a cone of D can be an object Y together with
c₀ : Y → A,
c₁, c₂ : Y → B,
c₃ : Y → C.
The cone looks like this:
A → B B → C
Now this is a cone of D even if c₀ ≠ c₁. However, if c₀ ≠ c₁,
then this is not a cone of the original diagram “A → B → C”.
To convert a diagram in the sense of §8.1(a) to a functor,
consider taking the functor’s domain to be the reflexive
transitive closure of the diagram. Due to the laws of
composition, the extra arrows should not impose any new
constraint on cones.
The Hom-functors defined in chapter 16 are, in general, neither
full nor faithful. Given a category C, is it possible to isolate
a subcategory H of SET such that every Hom-functor of C is a full
and faithful (co-/contravariant) functor into H?
H seems to be related to, and different from, the arrow category
of C. Is it a standard notion?
Many, many thanks for this!!
I was still confused on diagrams vs functors, but apparently at least Wikipedia agrees (in different words): to build the index category J, just construct the free category over the source graph (that is, its reflexive transitive closure).