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I have been trying to polish weak homotopy equivalence by adding formal Definition/Proposition-environements. Also expanded the Idea-section and edited here and there.
The following remark used to be in the entry, but I can’t see right now how it makes sense. If I am mixed up, please clarify and I’ll re-insert it into the entry:
It is tempting to try to restate the definition as “$f$ induces an isomorphism $f_*: \pi_n(X,x) \to \pi_n(Y,f(x))$ for all $x \in X$ and $n \geq 0$,” but this is not literally correct; such a definition would be vacuously satisfied whenever $X$ is empty, without regard to what $Y$ might be. If you really want to go this way, therefore, you still must add a clause for $\Pi_{-1}$ (the truth value that states whether a space is inhabited), so the definition is no shorter.
Then, there used to be the following discussion box, which hereby I am moving from there to here. I have added a brief remark on how weak homotopy equivalences are homotopy equivalences after resolution. But maybe it deserves to be further expanded.
[ begin forwarded discussion ]
+–{.query} Is there any reason for calling these ’weak’ homotopy equivalences rather than merely homotopy equivalences? —Toby
Mike: By “these” I assume you mean weak homotopy equivalences of simplicial sets, categories, etc. My answer is yes. One reason is that in some cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of “homotopy equivalence” from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.
Toby: Interesting. I would have guessed that any weak homotopy equivalence could be strengthened to a homotopy equivalence in this sense, but maybe not.
Tim: I think the initial paragraph is somehow back to front from a philosophical point of view, as well as a historical one. Homotopy theory grew out of studying spaces up to homotopy equivalence or rather from studying paths in spaces (and integrating along them). This leads to some invariants such as homology and the fundamental group. Weak homotopy type (and it might be interesting to find out when this term was first used) is the result and then around the 1950s with the development of Whitehead’s approach (CW complexes etc.) the distinction became more interesting between the two concepts.
I like to think of ’weak homotopy equivalence’ as being ’observational’, i.e. $f$ is a w.h.e if when we look at it through the observations that we can make of it, it looks to be an ’equivalence’. It is ’top down’. ’Homotopy equivalence’ is more ’constructive’ and ’bottom up’. The idea of simple homotopy theory takes this to a more extreme case, (which is related to Toby’s query and to the advent of K-theory).
With the constructive logical side of the nLab becoming important is there some point in looking at this ’constructive’ homotopy theory as a counter balance to the model category approach which can tend to be very demanding on the set theory it calls on?
On a niggly point, the homotopy group of a space is only defined if the space is non-empty so one of the statements in this entry is pedantically a bit dodgy!
Toby: I would say that it has a homotopy group at every point, and this is true even if it is empty. You can only pretend that it has a homotopy group, period, if it's inhabited and path-connected.
Anyway, how do you like the introduction now? You could add a more extensive History section too, if you want.
Tim: It looks fine. I would add some more punctuation but I’m a punctuation fanatic!!!
With all these entries I suspect that in a few months time we will feel they need some tender loving care, a bit of Bonsai pruning!! For the moment lets get on to more interesting things.
Do you think some light treatment of simple homotopy theory might be useful,say at a historical level? =–
[ end forwarded discussion ]
The following remark used to be in the entry, but I can’t see right now how it makes sense.
The rectification seems more or less obvious: a map $f: X \to Y$ is a weak homotopy equivalence if $\pi_0(f): \pi_0(X) \to \pi_0(Y)$ is an isomorphism and if $\pi_n(f): \pi_n(X, x) \to \pi_n(Y, f(x)$ is an isomorphism for all $x \in X$ and $n \geq 1$. In other words, the mistake was to consider $\pi_0(X, x)$ in case $X$ is empty.
I have given the discussion of homotopy types that used to be there its own subsection: Relation to homotopy types and then polsihed and expanded that slightly
I have added a paragraph on Examples of non-reversible weak homotopy equivalences.
(never mind)
I didn’t see your original message, but here is a guess: Is maybe “reversible” too misleading, and too easily misread as “invertible”?
No, I erased my message (where I thought I had a simpler example than the pseudocircle) because it was in mathematical error! But since you ask, I think “reversible” gets the correct idea across.
The following remark used to be in the entry, but I can’t see right now how it makes sense.
If it really doesn’t make sense to you, then I could explain; but if you just want to know why it’s in there at all, it’s because somebody (OK, it was me) was tempted to phrase the definition that way.
There’s actually a hierarchy of ways to phrase the definition:
Starting too near the end begs the question when it comes to the homotopy hypothesis, while starting too near the beginning is unfamiliar. (I made the mistake of thinking that starting at the very beginning would simplify the phrasing, but it doesn’t.)
If you don’t remember how $\pi_n(X,x)$ is an $n$-tuply groupal set for any natural number $n$, this is covered at homotopy group (third Idea paragraph, and Examples in Low Dimensions).
Okay, so that first clause in your item (1) was missing!
If you don’t remember how π n(X,x) is an n-tuply groupal set for any natural number n,
If I didn’t remember that, I should go home and do something else than I am doing, such as Macrame maybe. ;-)
Well, sometimes people forget that for $n = 0$.
Saying “isomorphism of n-tuply groupal sets” is not really necessary, though, since $\pi_n(f,x)$ is automatically a morphism of n-tuply groupal sets, so its being an isomorphism of such is equivalent to its being merely a bijection.
It seems to me that the funny thing is that you can’t start any lower:
is not a correct definition. Maybe this hierarchy should be discussed in the entry somewhere.
Maybe this is a good opportunity to advertise my favorite definition of a weak homotopy equivalence. A map of spaces $f : X \to Y$ is a weak homotopy equivalence if for every natural number $m$ and a diagram
$\begin{matrix} \partial I^m & \overset{u}{\longrightarrow} & X \\ \downarrow & & \, \downarrow f \\ I^m & \underset{v}{\longrightarrow} & Y \end{matrix}$there exist maps $w : I^m \to X$ and $H : I^m \times I \to Y$ such that $w | \partial I^m = u$ and $H$ is a homotopy from $f w$ to $v$ over $\partial I^m$. We obtain a definition of $k$-equivalence simply by restricting this condition to $m \le k$.
This definition is completely basepoint-free and it doesn’t refer to homotopy groups. In my experience it has the advantage that many basic properties of weak homotopy equivalences (and $k$-equivalences) can be verified using this definition without ever mentioning homotopy groups and the proofs tend to be neater than the ones that use the classical definition directly.
Ah, that’s nice. And it is a vague sense dual to a trivial Dold fibration, where the homotopy is in the upper triangle, not the lower triangle.
Hi Karol,
that characterization is used quite widely at least in model category theory circles. I guess it was introduced in
I have added a note under Equivalent characterizations.
its being an isomorphism of such is equivalent to its being merely a bijection
Sure, but why complicate matters by bringing in the category of sets? If one is having trouble with the proposition that a certain group homomorphism is an isomorphism, then a handy theorem to use may be that it is so iff the underlying function is a bijection, but that fact is not the point.
It seems to me that the funny thing is that you can’t start any lower
That’s because the concept of $(-1)$-tuply groupal set doesn’t make sense, as far as I know. (At least, it’s not in k-tuply monoidal n-category.) Thus, there is such a thing as $\Pi_{-1}(X)$ (which is a $(-1)$-groupoid), but no such thing as $\pi_{-1}(X,x)$ (which would be a $(-1)$-tuply groupal set).
but why complicate matters by bringing in the category of sets?
Only at the nForum would you hear a complaint that bringing in the category of sets complicates matters vis a vis the category of “$n$-tuply groupal sets”. (-:
@13 in the case of topological spaces is the “technical lemma” in section 9.6 of A concise course in algebraic topology. It, or something closely related to it, is also called the HELP lemma and apparently dates back at least to 1973, Boardman & Vogt “Homotopy invariant structures on topological spaces”. I agree that it is quite nice; it’s also quite similar to the solution-set condition in Smith’s theorem for generating combinatorial model category structures.
Thanks for these pointers. I have added them here to the entry.
@14 They’re not quite dual. The notion of lifting property is self-dual, but here it splits into two distinct notions which are dual to each other: “lifting property up to under-homotopy” and “lifting property up to over-homotopy”. Morphisms characterized by such properties behave differently depending on whether they’re on the side of a strictly commuting triangle or on the side of a homotopy commuting one. The former are somewhat rigid and behave like (co)fibrations (for example Dold fibrations), the latter tend to be homotopy invariant and behave more like weak equivalences. I would be interested in learning about general theory of such lifting properties and classes of maps characterized by them, but I’m not aware of any such results.
@18 You’re right. The funny thing is that when I was first reading A concise course in algebraic topology a few years ago I spent a lot of time trying to understand what this lemma is about and finally gave up. Only now I realize that this is exactly the thing I learned to appreciate in the meantime. The problem is that this “technical lemma” is awfully overloaded, in my mind this is five or six separate lemmas.
The problem is that this “technical lemma” is awfully overloaded, in my mind this is five or six separate lemmas.
To me it is clearly a single lemma, but stated in a very confusing way because of the insistence on using one diagram that strictly commutes rather than talking about homotopies that live in squares and triangles.
I would definitely have an easier time understanding this lemma if it were stated using homotopy commutative diagrams, but I do think that its proof mixes up a few different lines of reasoning and would benefit from being split up into a few lemmas.
would benefit from being split up into a few lemmas.
You could have a go at it in the $n$Lab entry. Would do the world a service, I am sure.
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