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This is maybe mainly for entertainment. But don’t forget that for newcomers there is a real issue here which may well be worth explaining:
In mathematics it happens at times that one and the same concept is given two different names to indicate a specific perspective, a certain attitude as to what to do whith such objects.
Here are examples:
A quiver is just a directed graph (pseudograph, to be explicit). But one says quiver instead of directed graph when one is interested in studying quiver representations: functors from the free category on that graph to the category of finite-dimensional vector spaces.
A presheaf is just a contravariant functor. But one says presheaf instead of contravariant functor when one is interested in studying its sheafification, or even if one is just intersted in regarding the category of functors with its structure of a topos: the presheaf topos.
(…)
Interesting. I’ve been thinking about a similar linguistic phenomenon which is that a thing might be given different names depending on what category it is considered to be an object in: frames, locales and complete heyting algebras for instance.
I’m not sure if your examples could be phrased in this way though.
I have a vague feeling that the notion shouldn’t apply to cases where the notion of morphism changes, but it’s an interesting question.
Isn’t a presheaf (unadorned, by default) specifically a contravariant functor to Set?
Feel free to adjust the wording. Our own page presheaf says:
Historically, the initial applications of presheaves and sheaves involved cases like $S = CRing$ (the category of commutative rings), $S =$Ab, $S = R$-$Mod$, etc. Later, especially with the development of topos theory, the primary importance of the category of set-valued (pre)sheaves as topos was recognized;
To be honest, this sort of thing often annoys me. Why would you choose to create confusion by introducing a new word that has the same meaning as an old word, just because you’re going to do something new with it? The whole point of abstraction is that the same structure can be used for many different things; why obscure that benefit? In some cases the reason is historical, e.g. I expect (pre)sheaves predated the general concept of “contravariant functor”. In other cases you’re abbreviating something long by something short, which is at least arguably reasonable if you’re going to be saying it a lot, although even in that case it’s better if the short word has some connection to the long one. But in general I don’t like it.
Who invented ’quiver’, anyway?
I think what annoys me even more is that often at the nLab we have shied away from “directed graph” – the term I had always used and that I found perfectly serviceable – just because other cultures use the term in a slightly different way. Thus, I’m not even sure we do use “quiver” for its attitude so much as we have used it to avoid ambiguity. I’m glad we don’t suffer the same angst over “groupoid”.
Here is the earliest paper I could find using quiver. “On Algebras of Finite Representation Type” which references this paper in German which talks about “köcher”, google translate tells me this is quiver. This references three papers but I couldn’t find quiver being mentioned. I am going to a geometry seminar tomorrow where I can ask a few experts in this field like Alistair King about this.
Yeah, I wish we would stick to “directed graph” too.
Re #7 yes: singleton pretopologies turn up all the time (for instance, in the definition of a clan a la Joyal, a calibration a la Bénabou etc) usually with an extra condition or two, but every time people have to say “consider a class of arrows closed under composition and pullback and containing all identities/isomorphisms”.
Re #11: Just to play the devil’s advocate, I would have to look up what “singleton pretopology” means, whereas I understand immediately what the longer sentence means :-).
Jon Beardsly indavertendly prodded me (here) to add one more item to the list of “concepts with an attitude”:
A field (in physics) is just a section of a fiber bundle.
But…
To be honest, this sort of thing often annoys me. Why would you choose to create confusion by introducing a new word that has the same meaning as an old word, just because you’re going to do something new with it?
I would say that often the situation is more complex than that. In two situations you may end up with the same (or equivalent in some sense) object, but what you want to express is different and it should perhaps be viewed as a lack of the formal definition that it does only incompletely cover what you want to say, e.g. the definition of an estimator actually should contain also some information about the model in which the estimated parameter lives.
Another point is that sometimes a systematic terminology can not be tailored with respect two two different areas of application, e.g. the term presheaf makes sense in sheaf theory, wheres contravariant functor makes sense in category theory.
To keep it fun, I suggest to make sure that each item in the list starts out with the exact same pattern, with the words: “An X is just a Y. But one says ’X’ in order to…”
Would ’sequence’ and ’series’ be an example?
A series is just a sequence. But one says series instead of sequence when one is interested in studying partial summations.
Or would that be stronger than ’with an attitude’ because the word ’converges’ means different things? The series $n\mapsto a_n$ converges if and only if the sequence $n\mapsto \sum_{i\lt n}a_i$ converges, which is different from the sequence $n\mapsto a_n$ converging.
Sure, I like that example. Please add it.
added this example:
A Young diagram is a partition that wants to become a Young tableau.
Thanks. That Young diagram was over-ambitious.
Thanks.
Somehow subset is not quite as satisfying an example. The good examples would have lay people say: What do you mean by X? Ah you just mean a Y. This is not so plausible for X = subset and Y = predicate.
Of course I see what you mean. Maybe we should bring some order into the list, having the more striking examples be on top and the more subtle ones further below.
Yeah I get what you mean. In certain foundations of set theory such as ETCS, an “element” is also a concept with an attitude, being simply a function out of the terminal set, which is not the case in other foundations of set theory (SEAR, MLTT, etc).
Added (here) one more item to the list:
An abstract re-writing system is just a relation on some set $X$.
However, calling this relation an abstract rewriting system indicates that one is interested in studying the behaviour of chains of related elements $x \to x_1 \to x_2 \to \cdots$ (thought of as successive stages of rewriting $x$), for instance to see if they are confluent.
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