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expanded the entry cofinal functor: formal definition, list of equivalent characterizations and textbook reference.
How do you feel about my suggestion to rename that page?
I am allright with renaming the page. I tried to implement your suggestion already by slightly changing the wording of the entry.
But I do find it a bit unfortunate that “final functor” and “initial functor” then really have two diferent meanings: one in terms of preservation of limits/colimits, the other as initial/final object in the functor category.
The term “cofinal”, while not well motivated, at least indicates that it is something different than a terminal functor.
But it’s all a mess, anyway. Who is responsible for that terminology?!
I’ve never felt any desire to refer to a terminal object in a functor category as a “terminal functor.” Maybe because when I talk about functor categories in such a way that I care about limits and colimits in them, I’m always calling them either diagrams or presheaves. (-:
added more properties and a trivial example to cofinal functor
There is also cofinal diagrams, which I don’t know what to do with.
There is also cofinal diagrams, which I don’t know what to do with.
I hadn’t been aware of this. I have now edited this as follows.
After the sentence
In most cases where the word “cofinal” is used, it seems to be the case D′ is a subdiagram of D in whatever sense of subdiagram appears suitable.
I have added the following sentence
In that case the cofinality is equivalent to the inclusion functor being a cofinal functor.
I have also moved the whole paragraph containing this from a “Notes”-section up to the definition, where I think it belongs.
Then I have changed the entry title from plural to singular.
Whoa, I’ve never heard that usage of “cofinal diagrams”. Anyone know a reference? I wonder whether we could use instead something like “jointly final diagrams” to better match final functor?
I wonder whether it is correct to make the title of that page singular, since as the term is used on that page it doesn’t actually make sense to talk about “one cofinal diagram”.
I wonder whether it is correct to make the title of that page singular, since as the term is used on that page it doesn’t actually make sense to talk about “one cofinal diagram”.
True. I have changed it back.
I agree with Mike on the title (which is actually what I moved it to from cofinal, which is now an adjective page).
added to cofinal functor at Examples the statement that every right adjoint functor is cofinal.
added to final functor the Example that the inclusion of the cospan diagram into its cocone is initial, and the remark that this implies that fiber products in a slice are preserved by dependent sum.
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