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we had created limit, reflected limit, preserved limit, but not lifted limit. I have now created a stub for the last one, for completeness.
Would be good to harmonize and cross-relate these four entries more…
There’s at least a typo in lifted limit (two occurrences of $\mathcal{D}$), but more to the point, I don’t quite follow what this entry wants to say. Do you have a source?
There is the concept of “initial lift” or “final lift” which is used in topological concrete category.
Finally, can I ask you again why page titles are bold-faced among related concepts? Is it the result of rapid copy-and-paste?
a typo
Fixed, thanks.
a source?
I won’t dig through Borceux or the like now to check if the terminology appears there, but that it’s being commonly used is easily confirmed with a little help from Google. For what it’s worth, Wikipedia has is here.
Once on the CatTheory mailing list Aleks Kissinger suggested a concise way to state all the lifting/creating/reflecting/preserving business in a form that the $n$Lab should eventually provide, here.
why page titles are bold-faced among related concepts?
When you switch through the following example pages, does it become clear?
The notion is clear to me now; thanks. I inserted a link to the comment made by Aleks Kissinger.
does it become clear?
I’m not sure, but if I’m reading you correctly, then it is a case of rapid copy-and-paste. It seems to me we discussed this once before, and if I recall correctly you had said those bold-faced entries don’t belong and one should feel free to erase them. Is that true? If it isn’t true, I’d like to know why they should be there.
Curiously, the bold-face comes from having the time to edit after the copy-and-paste, namely to turn the link into an unlinked bold-face. But if the point doesn’t become clear, please feel invitedto erase them.
(I’m not trying to be difficult, honest. I just don’t understand what the point could be, saying that a concept is related to itself.)
Returning to the mathematical point, Aleks’ definitions don’t seem correct to me. His version of “creates”, for instance, doesn’t seem to match the standard one which also involves a quantification over diagrams in the domain. Same with “lifts”, so that in particular I think the version now at lifted limit is wrong. One published reference, if we want one, is arXiv:1104.2111 (Def. 3.7).
Maybe it’s too late here, but I don’t get #7. Both Aleks and the little sentence that I put into the entry speak about all diagrams at once… while it’s the reference you point to that considers the case where one kind of diagram is fixed! In either case, this seems a trivial matter of choice of wording to me.
No, the point is that to say that $F:C\to D$ lifts limits of shape $J$, say, is to say that for every functor $G:J\to C$ and every limiting cone over $F\circ G$, there is a limiting cone over $G$ that maps onto it. The definition as it is phrased now only says that for every functor $G':J\to D$ and limiting cone over $G'$, there is a functor $G:J\to C$ with $G' = F\circ G$ and a limiting cone over $G$ that maps onto it. It’s not enough for some such $G$ to exist; it must be the case for all such $G$.
The current definition of a lifted limit is not invariant under isomorphisms.
Specifically, a limiting cone for F∘J in D can be the image of a limiting cone for J in C, but a cone isomorphic to F∘J need not be, since its apex need not be in the image of F.
Should the definition be adjusted to say that any limiting cone for F∘J is isomorphic to the image of a limiting cone for J in C?
I think it would be okay to include both options as variants. The strict version is often reasonable in practice, when the functor is generally an isofibration.
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