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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2010
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   Author: Urs
   Format: Markdownadded to [[Quillen adjunction]] the statement how the SSet-enriched version presents [[adjoint (infinity,1)-functor]]s. Also indicated how one shows that a left Quillen functor prserves weak equivalences between cofibrant objects, and dually

   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

2. • CommentRowNumber2.
   • CommentAuthoradeelkh
   • CommentTimeJan 14th 2015
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   Author: adeelkh
   Format: MarkdownItexI added to both pages the new reference * [[Aaron Mazel-Gee]], _Quillen adjunctions induce adjunctions of quasicategories_, [arXiv:1501.03146](http://arxiv.org/abs/1501.03146).

   I added to both pages the new reference

   • Aaron Mazel-Gee, Quillen adjunctions induce adjunctions of quasicategories, arXiv:1501.03146.
3. • CommentRowNumber3.
   • CommentAuthorMatanP
   • CommentTimeJan 19th 2015
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   Author: MatanP
   Format: MarkdownItex@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.

   @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.

4. • CommentRowNumber4.
   • CommentAuthoradeelkh
   • CommentTimeJan 19th 2015
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   Author: adeelkh
   Format: MarkdownItexThat's good to know, thanks!

   That’s good to know, thanks!

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJan 19th 2015
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   Author: Urs
   Format: MarkdownItexThanks. I have [added](http://ncatlab.org/nlab/show/Quillen+adjunction#AssociatedInfinityAdjunction) also a pointer to HTT for the sSet-enriched case.

   Thanks. I have added also a pointer to HTT for the sSet-enriched case.

6. • CommentRowNumber6.
   • CommentAuthoramg
   • CommentTimeJan 19th 2015
   • (edited Jan 19th 2015)
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   Author: amg
   Format: MarkdownItexWow, 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](http://arxiv.org/pdf/1501.03146v1.pdf) (section 2), Hinich is more-or-less constructing the sset-enriched category ${cocart}(L^H(F^c))$ -- the "cocartesian fibration over $[1]$" associated to the functor obtained by applying hammock localization to the composite functor of relative categories $F^c : C^c \hookrightarrow C \xrightarrow{F} D$ -- and then checking that it induces an adjunction of quasicategories. At best, this identifies the resulting _left adjoint_ as being the "underlying functor of $\infty$-categories" associated to this composite $F^c$, but nowhere does he identify the resulting _right adjoint_ as being associated to the composite $G^f : D^f \hookrightarrow D \xrightarrow{G} C$. 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 ${cocart}(L^H(F^c))$, but also its cousin ${cart}(L^H(G^f))$, as well as the intermediate object $L^H(C^c+D^f)$ of Construction 2.3. Less seriously, Hinich also implicitly asserts that co/simplicial resolutions _functorially_ compute hom-spaces, i.e. that if $x^\bullet$ is a cosimplicial resolution of $x$, then for any map $y \to z$ of fibrant objects, the map ${hom}(x^\bullet,y) \to {hom}(x^\bullet,z)$ computes the induced map of hom-spaces in the underlying $\infty$-category. (Actually a full assertion of functoriality would refer to any diagram, not just to a single map $y \to z$, but that's all that Hinich uses.) In fact, the purpose of Zhen Lin Low's recent [paper](http://arxiv.org/abs/1409.8062) "Revisiting function complexes and simplicial localisation" is to show that co/simplicial resolutions do functorially compute hom-homotopytypes (of the underlying ${ho}({Top})$ 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?

   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 ${cocart}(L^H(F^c))$ – the “cocartesian fibration over $[1]$” associated to the functor obtained by applying hammock localization to the composite functor of relative categories $F^c : C^c \hookrightarrow C \xrightarrow{F} D$ – and then checking that it induces an adjunction of quasicategories. At best, this identifies the resulting left adjoint as being the “underlying functor of $\infty$-categories” associated to this composite $F^c$, but nowhere does he identify the resulting right adjoint as being associated to the composite $G^f : D^f \hookrightarrow D \xrightarrow{G} C$. 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 ${cocart}(L^H(F^c))$, but also its cousin ${cart}(L^H(G^f))$, as well as the intermediate object $L^H(C^c+D^f)$ of Construction 2.3.

   Less seriously, Hinich also implicitly asserts that co/simplicial resolutions functorially compute hom-spaces, i.e. that if $x^\bullet$ is a cosimplicial resolution of $x$, then for any map $y \to z$ of fibrant objects, the map ${hom}(x^\bullet,y) \to {hom}(x^\bullet,z)$ computes the induced map of hom-spaces in the underlying $\infty$-category. (Actually a full assertion of functoriality would refer to any diagram, not just to a single map $y \to z$, 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 ${ho}({Top})$ 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?

7. • CommentRowNumber7.
   • CommentAuthoramg
   • CommentTimeJan 19th 2015
   • (edited Jan 19th 2015)
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   Author: amg
   Format: Text(Sorry, I'm trying to use either "mbox" or "textup" to get non-italicized text in math mode, but for some reason it both de-italicizes the characters and displays the command. Hence the repeated edits of #6. But now I've removed any use of either of those commands, and for some reason the desired characters are still residually retaining their de-italicized status. Can anyone explain to me what's going on?)

   (Sorry, I'm trying to use either "mbox" or "textup" to get non-italicized text in math mode, but for some reason it both de-italicizes the characters and displays the command. Hence the repeated edits of #6. But now I've removed any use of either of those commands, and for some reason the desired characters are still residually retaining their de-italicized status. Can anyone explain to me what's going on?)
8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeJan 19th 2015
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   Author: Mike Shulman
   Format: MarkdownItexYou have to put spaces between them: `$a b c$` makes $a b c$ whereas `$abc$` makes $abc$. 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.

   You have to put spaces between them: $a b c$ makes $a b c$ whereas $abc$ makes $abc$. 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.

9. • CommentRowNumber9.
   • CommentAuthoramg
   • CommentTimeFeb 19th 2016
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   Author: amg
   Format: MarkdownItexMy aforementioned has been accepted and published; it is freely accessible [here](http://nyjm.albany.edu/j/2016/22-4.html). 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](https://ncatlab.org/nlab/show/Quillen+adjunction#AssociatedInfinityAdjunction) on Quillen adjunctions? If so, should I leave a word of explanation there as well?

   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?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2016
    • (edited Feb 19th 2016)
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    Author: Urs
    Format: MarkdownItex> Would it be reasonable to remove that reference from the nLab [page](https://ncatlab.org/nlab/show/Quillen+adjunction#AssociatedInfinityAdjunction) 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.)

    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.)

11. • CommentRowNumber11.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 19th 2016
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    Author: Zhen Lin
    Format: MarkdownItexStrange coincidence, Hinich's paper has also just been [published](http://dx.doi.org/10.4310/HHA.2016.v18.n1.a3).

    Strange coincidence, Hinich’s paper has also just been published.

12. • CommentRowNumber12.
    • CommentAuthoramg
    • CommentTimeFeb 19th 2016
    • (edited Feb 19th 2016)
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    Author: amg
    Format: MarkdownItexYes, 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 $F : C \to D$ and a right adjoint $C \leftarrow D : G$ do not necessarily have to be adjoint _to each other_. I gave the extremely concrete example of the two functors $\mathrm{Free} : \mathrm{Set} \to \mathrm{AbGrp}$ and $\mathrm{Set} \leftarrow \mathrm{AbGrp} : \mathrm{const}_{\mathrm{pt}}$. (This subtlety is why his argument was under 2 pages, whereas mine was ~20.) Anyways, I've updated the nLab page now.

    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 $F : C \to D$ and a right adjoint $C \leftarrow D : G$ do not necessarily have to be adjoint to each other. I gave the extremely concrete example of the two functors $\mathrm{Free} : \mathrm{Set} \to \mathrm{AbGrp}$ and $\mathrm{Set} \leftarrow \mathrm{AbGrp} : \mathrm{const}_{\mathrm{pt}}$. (This subtlety is why his argument was under 2 pages, whereas mine was ~20.) Anyways, I’ve updated the nLab page now.

13. • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 25th 2016
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    Author: Dmitri Pavlov
    Format: MarkdownItexHere 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?

    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?

14. • CommentRowNumber14.
    • CommentAuthorZhen Lin
    • CommentTimeFeb 25th 2016
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    Author: Zhen Lin
    Format: MarkdownItexHow 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...

    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…

15. • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 25th 2016
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    Author: Dmitri Pavlov
    Format: MarkdownItex@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.)

    @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.)

16. • CommentRowNumber16.
    • CommentAuthoramg
    • CommentTimeFeb 25th 2016
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    Author: amg
    Format: MarkdownItex@Dmitri #13: Yes, if I understand correctly your question is addressed in Remark 2.7.

    @Dmitri #13: Yes, if I understand correctly your question is addressed in Remark 2.7.