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added to Quillen adjunction the statement how the SSet-enriched version presents adjoint (infinity,1)-functors.
Also indicated how one shows that a left Quillen functor prserves weak equivalences between cofibrant objects, and dually
I added to both pages the new reference
@adeelkh: There is a previous reference for the result you gave a pointer to: it is already in Hinich, Proposition 1.5.1 http://arxiv.org/pdf/1311.4128v3.pdf.
That’s good to know, thanks!
Thanks. I have added also a pointer to HTT for the sSet-enriched case.
Wow, I remember taking a look at that paper when it came out – I can’t believe I forgot its contents!
However, I don’t think that the argument given there actually proves the stated result. In the notation of my own paper (section 2), Hinich is more-or-less constructing the sset-enriched category – the “cocartesian fibration over ” associated to the functor obtained by applying hammock localization to the composite functor of relative categories – and then checking that it induces an adjunction of quasicategories. At best, this identifies the resulting left adjoint as being the “underlying functor of -categories” associated to this composite , but nowhere does he identify the resulting right adjoint as being associated to the composite . As I mention in the middle of page 4 (“Now, it is actually not so hard to show…”), the bulk of the difficulty of my own paper is in proving that this adjunction actually does have the expected right adjoint. This is achieved by constructing not just , but also its cousin , as well as the intermediate object of Construction 2.3.
Less seriously, Hinich also implicitly asserts that co/simplicial resolutions functorially compute hom-spaces, i.e. that if is a cosimplicial resolution of , then for any map of fibrant objects, the map computes the induced map of hom-spaces in the underlying -category. (Actually a full assertion of functoriality would refer to any diagram, not just to a single map , but that’s all that Hinich uses.) In fact, the purpose of Zhen Lin Low’s recent paper “Revisiting function complexes and simplicial localisation” is to show that co/simplicial resolutions do functorially compute hom-homotopytypes (of the underlying enriched category of the hammock localization of a model category), but there he assumes his model categories come equipped with functorial factorizations, which is a stronger requirement than Hinich or I use.
As I mentioned, I was certainly unaware of this paper when I wrote my own. My inclination here is to update my own paper by turning the prose “Now, it is not so hard to show…” into a numbered Remark and referring to Hinich’s result there. Given what I’ve said above, does this seem reasonable to you all?
You have to put spaces between them: $a b c$
makes whereas $abc$
makes . iTex “saves you work” by automatically treating juxtaposed letters as part of the same identifier. You can always put a \mathrm
around them for emphasis, but iTeX doesn’t understand \mbox
or \textup
, and it also treats commands it doesn’t know as identifiers to be displayed.
My aforementioned has been accepted and published; it is freely accessible here. In Remark 2.3, I explain concisely why Hinich’s argument falls short. Would it be reasonable to remove that reference from the nLab page on Quillen adjunctions? If so, should I leave a word of explanation there as well?
Would it be reasonable to remove that reference from the nLab page on Quillen adjunctions? If so, should I leave a word of explanation there as well?
Yes!
(And don’t hesitate to add pointer to your article, too.)
Strange coincidence, Hinich’s paper has also just been published.
Yes, that is strange and it is a coincidence! I corresponded at length with Hinich about this, but he remained unconvinced. The key issue is just that a left adjoint and a right adjoint do not necessarily have to be adjoint to each other. I gave the extremely concrete example of the two functors and . (This subtlety is why his argument was under 2 pages, whereas mine was ~20.) Anyways, I’ve updated the nLab page now.
Here is what is probably a stupid question, but let me ask it anyway: suppose in a Quillen adjunction one of the functors (left or right) preserves all weak equivalences. Then one can induce an adjunction of quasicategories in a similar fashion, but without peforming the (co)fibrant replacements for one of the categories (namely, the source of the functor that preserves all weak equivalences). Is the resulting adjunction of quasicategories the “same” as the adjunction obtained from the general construction?
How do you get actual structure of the adjunction? The easiest way is to do transport of structure from the Quillen adjunction, in which case it is the same by construction…
@ZhenLin: I have in mind the same construction as in Mazel-Gee’s paper, but with the functor F^c replaced by F, provided that F preserves all weak equivalences. (Here F^c = F∘Q, where Q is a cofibrant replacement functor.)
@Dmitri #13: Yes, if I understand correctly your question is addressed in Remark 2.7.
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