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I redirected it from model structure on an under category.
By the way: I keep seeing in the literature overcategory instead of over category . For instance in the article by Hirschhorn linked to at model structure on an over category.
Are we sure we want to have the entries named over category and so on?
Well, I like slice category, but I remember putting it over category in the days before redirects to help insure that your links to it would work.
I have put in redirects for overcategory and the like.
Oh, really, was it me who started writing "over category"? I forget. Sorry, then! :-)
Now I am interested in the special case of Top with Strom's model structure. Then there are theorems on the connection between Dold fibrations and Hurewicz fibrations, one of them is that every Dold fibration p:E -> B is
homotopy equivalent over B
with a Hurewicz fibration p:E' to B. Is this giving some light on the open question as if there is a model category structure on Top where fibrations are Dold fibrations ? Another important thing is that you can verbatim repeat the definition of Hurewitz fibration to get Dold fibration if instead of homotopies you use delayed homotopies (this is a theorem). Is there a way to use delayed homotopies to nontrivially modify the notion of cofibration ?
These are still usual cofibrations, and these do not form a model category with Dold fibrations, but maybe there is a good modified choice of cofibrations which woudl be "complementary" with Dold fibrations (maybe silly idea knowing something specific banning this choice, but to me it looks still reasonable).
That reminds me: we should add a discussion about if and how the model structure on an over category models the corresponding over quasi-category. I was about to make the obvious statement, but I'll need to check something first.
I have somewhat hastily added to model structure on an over category the argument that over a fibrant object this presents the correct over-$(\infty,1)$-category.
However, I have to dash off now and go offline. Will try to look into this again later.
have now found a few minutes to expand and polish the proof
I added a new section https://ncatlab.org/nlab/show/model+structure+on+an+over+category#quillen_adjunctions_between_slice_categories about Quillen adjunctions between slice categories.
Hm, the entry slice model structure states that slicing preserves cofibrant generation, properness, combinatoriality, but then what it means to use is preservation of simplicial model structure…
Re #14: I added Proposition 2.3, which shows that if C is a simplicial model category, then so is C/X.
added pointer to
I’ve cited a theorem in Cisinksi’s paper that proves the slice construction with a fibrant object is correct for any model category.
This makes the theorem proved in the section on derived hom-spaces redundant, and can be removed. Is there any content in the proof that should be retained on the page?
Thanks for your addition. But why would any of the proof offered on the page need to be removed. There is no harm in spelling out a proof of a special case of theorem that is proven more generally elsewhere. On the contrary. Unless I am missing something in your question?
I have merged the two subsections. Added a lead-over sentence: “We spell out a proof for the special case that $\mathcal{C}$ carries the extra structure of a simplicial model category:”
Aesthetically it seems weird for an encyclopedic reference to include the special case when it’s not a simplification of the more general case.
As a more practical note, I’ve actually found this a usability issue on the nLab from time to time where pages pay attention to a theorem written for a special case, leading me to completely miss that more general statements are available. Or when I do notice, to wind up spending a lot of time trying to understand what’s different about the special case that it would be needed addition to what is actually a strictly more general theorem – especially if there’s some restatement involved.
But, maybe it’s more a phrasing issue. I had taken as a given that we’d eventually want to remove the restatement to the special case – so I’m thinking of the question more as how to reorganize the interesting contents of the proof we’d like to retain as “here’s more interesting information!”
The text is crystal clear that there is a general case, thanks for adding that!
The general proof is not in the entry currently, is it?
I just checked out Denis-Charles’s proof. It’s nice, but the one in the entry is arguably simpler: It just observes the formula for homs in slices and pullback-power axiom in an enriched model category and it’s done; that’s pretty slick I’d think.
So I’d say: once somebody (probably you?!) writes out the more general proof into the entry, we can check again if the special case proof then feels like an annoying duplication. I don’t see how it would, but if it does, I’ll agree to remove it.
I just want to be sure that next time somebody (like myself) needs to remind themselves about how the argument works for simplicial model categories, it can still be found.
There is no harm done here, really, to readers not interested in this case, is there?
Perhaps also a useful time for a reminder that the nLab is not intended as an encyclopedia, but as a public lab book for everyone who contributes to it. So if something is useful for those people it should be kept in. Moreover, there’s also no reason for an encyclopedia not to include a special case if it is simpler or more comprehensible, at least not now that encyclopedias are digital and have essentially no space constraints.
added pointer to:
added pointer to:
I have reworked the statement about sliced Quillen adjunctions (from rev 16):
have disentangled the statement about sliced Quillen adjunctions (now this Prop.)
from that about sliced Quillen equivalences (now this Prop.)
have completed the proof of the former by adding pointer to the nature of the underlying sliced adjunctions (here)
(have not yet added proofs of the latter, which is more fiddly – but I added pointer, for what it’s worth, to a reference that at least claims one of the two cases)
I have inserted (here) the proposition & proof (taken from Quillen equivalence – Examples, but reworked a fair bit, as announced in another thread) that the left base change Quillen adjunction along a weak equivalence is a Quillen equivalence if that weak equivalence is stable under pullback.
We already had essentially that statement in what is now the following proposition, but without proof, just with a citation.
added the previously missing cross-link with model category of pointed objects (here)
Is there a citable reference for Quillen equivalences of $PSh(\mathcal{S}_{/S}, sSet)$ with $PSh(\mathcal{S}, sSet)_{/y(S)}$?
Otherwise I should type it out…
I have added missing cross-link of this Prop. with enriched slice category.
That same proposition (here) used to point to non-existent entries
“cartesian enriched model category”
“cartesian enriched categories”.
I have replaced these broken links by links to entries that do exist:
I have added (here) statement and proof that simplicial weak equivalence between fibrations over a base simplicial set are detected fiber-wise.
(I was expecting that I could cite this as a special case of some statement we already have on the $n$Lab somewhere, but if we do, then I didn’t find/remember it.)
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