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Minor matter: Some wording is offending my British linguistic sensibilities! In homotopy limit there is a title ’Homotopy colimits over simplicial diagrams’ and another ’Homotopy colimits over diagrams of spaces’. My feeling is that ’of’ is better so ’Homotopy colimits of diagrams of spaces’, but there may be a good reason why the existing wording is to be prefered. The use of ’over’ in somewhat similar titles elsewhere in theat entry does not worry me and pulling back something over something else is the usual wording in such cases.
I agree with Tim’s suggested terminology, the existing phraseology offends my Australian linguistic sensibilities also. Tim, why not just change it and see if people disagree. Else Us will probably feel compelled to do something, or else tell us to do it.
I don’t feel strongly one way or the other here, although I remember once voicing a similar complaint about “enriched over” (I prefer “enriched in”).
I don’t think “over” is wrong, but I agree that “of” is better. (I don’t mind “enriched over”, though.)
How about “limits on diagrams” and “enriched by”.
Prepositional usage has all sorts of unconscious rules and special cases. For example “in” is often used for vehicles that “enclose” you. However if you can walk around inside then “on” is preferred - e.g. taxi vs train.
@Rod: Ugh!
Rod: Have a look at some of the early pictures of trains and the wagons are mostly open so on was appropriate. Also look at trains in India!!!!!! On is certainly the right word there.
My reason for disliking ’over’ especially for colimits is that with the fibred category setting a (op)[lax limit gives a category over the original domain.There is a good argument for saying that using ‘over’ in both ways emphasises that colimits are linked to such settings, but I feel that ’colimit of a diagram’ is clearer and then you get the nice point that a colimit yields something ’over’…. . I would suggest also that a diagram ’over’ the domain category is fine as there over seems to have the sense ’indexed over’. Getting language and terminology optimised is difficult.
Heigh ho! I will change some and see how it works out.
As a first attempt, for X an object of a category C, a “Y of X” is the image of X under a functor from C to some other category, while a “Z over X” is a morphism in C from some object in C to X.
Following this principle, I would agree with Tim. A diagram over a category X is a quiver morphism from a quiver to X. While a colimit of a diagram X over a cocomplete category D is the image of the evaluation functor from to D.
How about “limits on diagrams”
I can not parse this. If diagram is a functor from a small category, then this would be limits on functors ??? You replace a functor with the limiting cone whose vertex is a constant functor. Replacing a functor with something else is operation which takes functor as an argument. So it is a blabla of a functor. Of is much better, I think.
I'm pretty sure that Rod was joking. At least, I hope so!
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