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(New thread since, after a semi-cursory search, no LatestChanges thread for [path] was found.)
Added to [path] a definition of “inverse path”.
Also tried to make the definition of “Moore path” clearer. Quibblingly speaking, this term used to be defined by saying what it has, without relating it to the initial definition of “path”. I was tempted to change the definition of “path” to the one given by tom Dieck in “Algebraic Topology”, having $a$ and $b$ for the endpoints of an artbitrary interval, which in particular would make it possible to simply say “for Moore path take $a=0$, $b=n$”, but then refrained, suspecting that whoever wrote it this way set store by having path to be always a space-modelled-on $[0,1]$, which for several reasons seems more simple and systematic indeed.
Motivation is that I try to concentrate on writing an exposition of a theorem of J.A. Power, and for this, I have resolved to use a —mildly—topological writing style, and for this I in particular need to get serious about paths, and I need Moore paths.
[Incidentally, in the nLab there lives Moore path category which has much to do with a “Moore path” of the type that lives on path since its creation on September 16, 2011. Maybe one should harmonize the two “Moore path”s a little more, saying a few situating thins on either page. Yes, path already links to Moore path category, but, it seems, not the other way round. Nothing urgent here, though.]
[Incidentally, I had recourse to a footnote in path. I did not forget the advice given recently, it just seemed right here to, simultaneously,
give a reference
warn readers of some notational issues
not clutter the main text with this
and I found my hand forced by this. If this is inacceptable, you might even just say “make it into this or that format” and I’ll hopefully do so soon. Now back to pasting schemes.]
Might I suggest rather ‘reverse path’ as an inverse path would cancel out the original which is not the case? Reverse path say we go along it in the reverse direction.
Re 2: this is a nice suggestion, this was even what I intuitively called it in my writing, I was just taking it from tom Dieck.
However, one can of course view it as an honest-to-goodness inverse.
What preferences do others have ?
….. ‘one can of course view it as an honest-to-goodness inverse’ : only up to (thin) homotopy.
Re 4: thanks for pointing out. I was partly aware of that, but somehow I am still so tolerant to taking equivalence-classes, and equality of classes, or, let us say, quotienting, that I took the composition’s being equal to the zero-path after the quotienting to warrant “honest-to-goodness”.
I did not know the property “thin” though, thanks for telling me that.
I had made “reverse path” a redirect to this entry (with “path concatenation” and “constant path”), and are pointing to it this way from the lecture notes. But I failed to find the time to copy over the definitions.
re 6: Thanks for pointing out.
Having thought about it, Tim_Porter’s suggestion seems objectively better, and consistent with what you point out about the redirect. I will edit accordingly.
Why better? Mainly because it somehow warns/reminds readers of the subtleties (composition with or without reparametrizing? taking homotopy classes? etc), while “inverse” makes it seems as if there were only “the” correct notion of inverse.
Just to add context, I recently made a “recreational” 5min survey, and found that e.g. J. Baez, P. May, J. M. Møller, P. Hirschhorn, all use “inverse path”, the greater specificity of “reverse” nonwithstanding. On the other hand, “reverse path” is to be found in works of e.g. N. Gurski and D. M. Roberts.
re 6 again: only now it occurred to me what you might have meant; added a little more information to the footnote, in the form of a like to (what you probably meant by) “lecture notes”
Adopted, within path, the notation for path reversal which is used in Introduction to Topology. That seems the best notation; confusion with closure is unlikely, any path being closed (a forteriori, because every continuous image of a compact set is compact).
In path there currently are two adjectives for which I could off-hand not think of the “mot juste”: “operation” for path reversal, and “canonical” for the “kind” of group-that-is-not. In the latter case, I was tempted to say “strict group”, but this is seems not an option.
“Operation” seems fine, although I’d denote it as $P \mapsto \widebar{P}$ and not by $\widebar{\cdot}$. The word “canonical” seems inoffensive there (at least I understand the intent), although I’d consider rewording the sentence to something like “even though no group was given in which…”
Possibly I’ll have more to say later on the entry, but not now.
One of my reasons for ’reverse’ is that traversing a path takes time (in everyday life) and if afterwards you travel back in the reverse direction (at the same ’speed’!) you do not reverse time so have double the length of path/time spent in the journey. (Better to stay where you were and not waste all that time, someone might say!) I also like the ideas of directed homotopy and there paths are non-reversible in general and that is of the very essence of the theory.
One might say this is the whole difference between the “topological” meaning of “path” and the “homotopical” one.
Somewhat related observation: paths, and how they are formalized, can be used as yet another example of the concept-A-is-more-than-just-B-but-rather-B-with-a-specified-C.
There is sort-of-a-hierarchy,
pointset-ish path
parametrized path
Moore path
(Regarding the latter: I have seen authors formalize Moore paths as pairs ( $[0,\infty]\overset{P}{\rightarrow} X$ , $\ell$ ), i.e., with the length separately recorded in a formal pair; sometimes they get called measured paths; so here there are two sorts of specified structures, the $P$arametrization, and the $\ell$ength.)
EDIT: in #1 in
Maybe one should harmonize the two “Moore path”s a little more, saying a few situating thins on either page. Yes, path already links to Moore path category, but, it seems, not the other way round. Nothing urgent here, though.
I was vacillating and not communicating well.
One precise little thing to start from would be to harmonize
the definition of Moore paths as paths with domain [0,n] in path,
the definition of Moore paths as pairs in the above “Regarding the latter” formalization in Moore path category.
A more ambitious, innovative undertaking would be to take
All of these variations can be combined, of course.
in the current version of path seriously, agree upon one “correct” definition of “path”, and downplay the others.
re 11:
I’d denote it as $P \mapsto \widebar{P}$ and not by $\widebar{\cdot}$.
Now implemented.
Re the terminological discussion: it now seems to me that
[…] like many others, uses the synonym “inverse path”, even though there is no “canonical” group in which $\overline{P}$ would be a strict inverse,
should be replaced with
The use of “strict groupoid” appears to be (forgive the pun) strictly in harmony with the nLab article strict category, even in the groupoid-on-a-small-set sense (to my current way of thinking at least, homotopy classes are small sets) for obvious reasons, and the with the use of “groupoid” no explanation of why readers are being referred to strict category is necessary.
I was tempted to suggest writing “do not automatically form a canonical strict group”, but the “canonical” then suddenly seems wrong, in particular since “paths modulo homotopy” are some sort of canonical such strict group.
I will not edit this in unless someone stumbles upon this edited comment and gives a recommendation.
EDIT: replaced “automatically” by “in and of itself” which to me seems the best the English language offers at this point. Words can only do so much of course, but here this phrase seems to convey useful information, and it is better than the mechanical “automatically”.
I myself am fine with your suggested replacement, except that I prefer “term” over “synonym”.
re 16: thanks, edited in.
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