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I added to walking structure a 2-categorical theorem that implies that usually “the underlying X of the walking X is the initial X”. This fact seems like it should be well-known, but I don’t offhand know a reference for it, can anyone give a pointer?
Unsure whether this is any help, but similar in spirit seem some characterizations of indiscrete categories. Internal pointer: indiscrete category, Section 2. External pointer: e.g. page 13 in pdf of this talk: http://benedikt-ahrens.de/talks/Fields_HoTT_2016.pdf.
I don’t see any relationship.
I realized that the above claim that “the underlying X of the walking X is the initial X” was a little overbroad; the situation is a bit subtler than that. The problem is that the “underlying set” functor C(I,−) of a monoidal category is in general only lax monoidal, whereas the universal property of a “walking X qua monoidal category” is relative to strong monoidal functors. The argument works much better for things like categories with products, where the relevant categorical structure is preserved by the “underlying set” functor C(1,−). I think we can conclude things about some walking Xs qua monoidal category, by applying the argument to multicategories instead and using the fact that a multicategory embeds fully-faithfully in the monoidal category that it freely generates (since if X is something that makes sense in a multicategory, then the walking X qua monoidal category is the free monoidal category generate by the walking X qua multicategory). But for other walking Xs qua monoidal category the claim doesn’t even make sense. I’ve updated the page walking structure with this.
Adding the interval groupoid as the walking isomorphism (page to be created).
Adding walking equivalence as an example.
Are we rules lawyering to this extent, now?
While I personally prefer “free-standing” to “walking”, there are instances of “walking” in the literature too, so I do not think the literature alone is enough to determine what this page should be called.
We should first add to the article some references in the literature for each of the names listed in the article. Only after should we decide which of the names to choose as the title of the article, since this will decide the fate of a whole series of articles on the nLab.
A whole bunch of walking structures are defined in
{#HLOR24} Amar Hadzihasanovic, Félix Loubaton, Viktoriya Ozornova, Martina Rovelli A model for the coherent walking ω-equivalence (arXiv:2404.14509)
{#OR24} Viktoriya Ozornova, Martina Rovelli, What is an equivalence in a higher category, Bulletin of the London Mathematical Society, Volume 56, Issue 1, January 2024, Pages 1-58 (doi:10.1112/blms.12947)
and renamed page back to “walking structure” as it is clearly the more used term in category theory literature compared to “free-living structure”, and to parallel walking morphism, walking isomorphism, walking equivalence, walking adjoint equivalence, etc already defined in the nLab.
Also removed the paragraph about “walking” being colloquial; it isn’t, it’s a technical term used in category theory.
Ramsay MacDonald
Sorry. I have never liked the use of ’walking’ used as here, as it does not mean anything helpful (to me) and requires the ’backstory’ to try to make sense of it. I would not call it a ’technical term’ either. It is meant as a sort of in-joke (which I may not get, but that is not important.) I would say it is used colloquially although it is then usually defined, as it should be. I recall being completely mystified by ’walking’ when I first met it. Saying it means ’archetypal’ might help. It does NOT relate to walking in the usual meaning of the term however. It is possibly also too Americal-English language specific.
Jokish terms can be helpful so please do not eliminate all attempts at fun in the use of categorical language. (Some jokes have worthy ancestry such at Kittygory as a term for a small category in Freyd’s early book on Abelian Categories.) On the other hand joke terms are best if they do suggest the definition, and are not too language specific, i.e. they might get in the way of non-native English speakers understanding the concept.
That literature you found, from last year, surely picked the term up from the nLab.
Okay, fine, if that is the case now we’ll have to live with it.
But your deletions to the entry (rev. 26) are too sweeping:
The alternative (more sane) terminology must not be suppressed and the comment on pronounciation has no reason to go away.
I have expanded the note on terminology as follows:
The term walking is believed to have been introduced by James Dolan, was used colloquially for several years and eventually got enshrined, for better or worse, in various nLab articles (such as this one here, originating in 2010), from which the recent literature cited above probably picked it up.
The idea of “walking” here is as in “John is a walking almanac” or “Eugene Levy is a walking pair of eyebrows” (which may be the original motivation given by Dolan).
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