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.)

]]>added mentioning of the example of the Borel model structure (here)

]]>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 missing cross-link of this Prop. with *enriched slice category*.

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…

]]>added the previously missing cross-link with *model category of pointed objects* (here)

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.

]]>spelled out the example (here) of induced Quillen adjunctions on pointed objects

]]>added statement and proof of the left base change Quillen adjunction (here) and cross-linked with the respective discussion at *proper model category* (there)

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)

added pointer to:

- Peter May, Kate Ponto, Thm. 15.3.6 in:
*More concise algebraic topology*, University of Chicago Press (2012) (ISBN:9780226511795, pdf)

added pointer to:

- Zhi-Wei Li,
*A note on the model (co-)slice categories*, Chinese Annals of Mathematics, Series B volume 37, pages 95–102 (2016) (arXiv:1402.2033, doi:10.1007/s11401-015-0955-z)

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.

]]>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?

]]>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!”

]]>But on re-reading I still found it a little weird, so I took the liberty of adding this line:

]]>this proof was written in 2011 when no comparable statement seemed to be available in the literature

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:”

]]>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’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?

]]>added pointer to

- Philip Hirschhorn, Theorem 7.6.5 of:
*Model Categories and Their Localizations*, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

Re #14: I added Proposition 2.3, which shows that if C is a simplicial model category, then so is C/X.

]]>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…

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.

]]>have now found a few minutes to expand and polish the proof

]]>