Instead of “The range is a collection of objects, one from each Xi”, TurnTrout should have written (and may well have meant) something like “An element of the range is a tuple of objects, one from each Xi”.
Also, instead of “A family is simply a function from an index set to a Cartesian product: I→∏i∈IXi”, you should have written “A family is simply an element of a Cartesian product ∏i∈IXi”, or else “A family is simply a function from an index set to a union: I→⋃i∈IXi such that for each i∈I, the image of i is an element of Xi.”
So, pure mathematicians have a standard way of describing this sort of thing which takes awhile to get used to but which is actually very elegant and generalizes very cleanly: instead of thinking in terms of the sets Xi, think in terms of the set X=⊔i∈IXi and the function f:X→I sending an element of X to the index of the Xi of which it’s a member. Then a family (of points in the Xi) is a section of f; that is, it’s a function g:I→X such that f∘g=idI.
The real elegance comes from generalizing I and X; for example, they could be topological spaces and you could ask for f and g to be continuous, in which case you’ve described a very general notion of continuous family of points in a continuous family of topological spaces. More on this theme here (not particularly written for accessibility, unfortunately).
Uhh… I am a pure mathematician (not that I disagree with anything you wrote but just thought I should clarify). Anyway, the paradigm you mentioned is interesting but I got the impression the book was taking more of an “everything must be defined explicitly in terms of set theory” point of view, and from that point of view I think you want to use ⋃i∈IXi rather than ∐i∈IXi, since the former actually contains the sets Xi whereas the latter only contains copies of them (of the form {i}×Xi in the standard construction). Otherwise you get that g(i) is not an element of Xi, but rather of a copy of Xi. Of course this is a nitpicking technical point, but then again it’s always seemed to me that the “everything must be defined explicitly in terms of set theory” point of view consists of a lot of nitpicking technical points…
Well, the problem with the union is that if the sets Xi aren’t disjoint then the function f I want above isn’t well-defined. Anyway, as a category theorist at heart I only care about what constructions look like up to natural isomorphism.
Yeah, I should have made more clear, I am aware of that, basically what I was trying to say is I think your paradigm might be inconsistent with what the book is trying to do (not that what the book is trying to do necessarily makes sense). Anyway like I said earlier it is an interesting paradigm, I’d actually never seen it before. By the way regarding the “continuous family of topological spaces” (by which I guess you mean the map sending a point to its fiber under a continuous map), is there a topology on the space of topological spaces such that the family is actually continuous?
Well taken, but now we’re approaching the book’s definition which, at the time, I found opaque. Let me see if I can’t massage this into something a bit more colloquial!
Hmm, I think the way you ended up writing it is still a little confusing—you never specified the relation between A and xi, and also I am not sure what it means for a function to “take” an index set."
I gave it another stab (I’m trying to see if I can synthesize your feedback without copying you word-for-word as a comprehension test); please feel free to PM me if it still doesn’t look right. I definitely appreciate the assistance tuning my technical writing, and I think it’s likely this will be helpful to future readers.
Instead of “The range is a collection of objects, one from each Xi”, TurnTrout should have written (and may well have meant) something like “An element of the range is a tuple of objects, one from each Xi”.
I did indeed mean that. Nice catch!
Also, instead of “A family is simply a function from an index set to a Cartesian product: I→∏i∈IXi”, you should have written “A family is simply an element of a Cartesian product ∏i∈IXi”, or else “A family is simply a function from an index set to a union: I→⋃i∈IXi such that for each i∈I, the image of i is an element of Xi.”
So, pure mathematicians have a standard way of describing this sort of thing which takes awhile to get used to but which is actually very elegant and generalizes very cleanly: instead of thinking in terms of the sets Xi, think in terms of the set X=⊔i∈IXi and the function f:X→I sending an element of X to the index of the Xi of which it’s a member. Then a family (of points in the Xi) is a section of f; that is, it’s a function g:I→X such that f∘g=idI.
The real elegance comes from generalizing I and X; for example, they could be topological spaces and you could ask for f and g to be continuous, in which case you’ve described a very general notion of continuous family of points in a continuous family of topological spaces. More on this theme here (not particularly written for accessibility, unfortunately).
Uhh… I am a pure mathematician (not that I disagree with anything you wrote but just thought I should clarify). Anyway, the paradigm you mentioned is interesting but I got the impression the book was taking more of an “everything must be defined explicitly in terms of set theory” point of view, and from that point of view I think you want to use ⋃i∈IXi rather than ∐i∈IXi, since the former actually contains the sets Xi whereas the latter only contains copies of them (of the form {i}×Xi in the standard construction). Otherwise you get that g(i) is not an element of Xi, but rather of a copy of Xi. Of course this is a nitpicking technical point, but then again it’s always seemed to me that the “everything must be defined explicitly in terms of set theory” point of view consists of a lot of nitpicking technical points…
Well, the problem with the union is that if the sets Xi aren’t disjoint then the function f I want above isn’t well-defined. Anyway, as a category theorist at heart I only care about what constructions look like up to natural isomorphism.
Yeah, I should have made more clear, I am aware of that, basically what I was trying to say is I think your paradigm might be inconsistent with what the book is trying to do (not that what the book is trying to do necessarily makes sense). Anyway like I said earlier it is an interesting paradigm, I’d actually never seen it before. By the way regarding the “continuous family of topological spaces” (by which I guess you mean the map sending a point to its fiber under a continuous map), is there a topology on the space of topological spaces such that the family is actually continuous?
Well taken, but now we’re approaching the book’s definition which, at the time, I found opaque. Let me see if I can’t massage this into something a bit more colloquial!
Hmm, I think the way you ended up writing it is still a little confusing—you never specified the relation between A and xi, and also I am not sure what it means for a function to “take” an index set."
Maybe something like: an element of ∏i∈IXi is just a way of assigning to each index i∈I an element of the corresponding space Xi?
I gave it another stab (I’m trying to see if I can synthesize your feedback without copying you word-for-word as a comprehension test); please feel free to PM me if it still doesn’t look right. I definitely appreciate the assistance tuning my technical writing, and I think it’s likely this will be helpful to future readers.
Looks good now.