nForum - Discussion Feed (Quillen adjoint triple) 2023-10-03T23:07:05+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "Quillen adjoint triple" (70226) https://nforum.ncatlab.org/discussion/8721/?Focus=70226#Comment_70226 2018-07-21T13:24:02+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ With this I can fix the remaining problem in #35: Since in the examples of solid model toposes of interest we happen to have not just a Quillen adjoint quadruple but a Quillen adjoint quintuple on ...

With this I can fix the remaining problem in #35: Since in the examples of solid model toposes of interest we happen to have not just a Quillen adjoint quadruple but a Quillen adjoint quintuple on the “far right”:

$\array{ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{even}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc} \\ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{ \Pi_{inf} }{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{\Pi_{sup}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {{\longleftarrow}} {\overset{Disc_{red}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{ Disc_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{\;\;id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{\;Disc_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {\underset{coDisc_{red}}{\longrightarrow}} {\overset{\Gamma_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {\phantom{\underset{id}{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} } \underoverset {\phantom{{\longleftarrow}}} {\overset{ \Gamma_{inf} }{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc}} \underoverset {\underset{}{\phantom{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu} }} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc}} \underoverset {\phantom{{\longleftarrow}}} {\overset{\Gamma_{sup}}{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc}} }$

This now implies all the 12 derived adjoint modalities, all by application of this prop..

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Urs comments on "Quillen adjoint triple" (70225) https://nforum.ncatlab.org/discussion/8721/?Focus=70225#Comment_70225 2018-07-21T13:11:11+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ added the example of Quillen adjoint quintuples on simplicial presheaves induced via Kan extension along adjoint triples (here). This has two realizations: either [&Cscr; op,sSet Qu] projA ...

added the example of Quillen adjoint quintuples on simplicial presheaves induced via Kan extension along adjoint triples (here). This has two realizations: either

$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$ $[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$ $[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$ $[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$

or

$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$ $[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$ $[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$ $[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$

This pattern continues for homotopy Kan extension along adjoint $(n+1)$-tuples: the resulting Quillen adjoint $(n+3)$-ples have

1. projective model structures “everywhere on top”

2. injective model structure “everywhere at the bottom”

3. a transition zone across a Quillen adjoint quadruple where projective turns into injective.

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Urs comments on "Quillen adjoint triple" (70213) https://nforum.ncatlab.org/discussion/8721/?Focus=70213#Comment_70213 2018-07-20T20:18:08+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ Supposing that I have it right now, I can now say “solid model topos” to be a system of Quillen adjoint triples of this form: sSet Qu Qu&bot; ...

Supposing that I have it right now, I can now say “solid model topos” to be a system of Quillen adjoint triples of this form:

$\array{ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{even}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc} \\ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj \atop loc} \\ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {{\longrightarrow}} {\overset{ \Pi_{inf} }{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {{\longrightarrow}} {\overset{\Pi_{sup}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj \atop loc} \\ sSet_{Qu} \underoverset {{\longleftarrow}} {\overset{Disc_{red}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {{\longleftarrow}} {\overset{ Disc_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {\phantom{\longrightarrow}} {\overset{\;\;id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc}} \underoverset {\phantom{\longleftarrow}} {\overset{\;Disc_{sup}}{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc}} \\ sSet_{Qu} \underoverset {\underset{coDisc_{red}}{\longrightarrow}} {\overset{\Gamma_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {\phantom{\underset{id}{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj \atop loc} } \underoverset {\phantom{{\longleftarrow}}} {\overset{ \Gamma_{inf} }{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {\underset{}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc} } }$

Here $Disc_{inf} \circ Disc_{red} \dashv \Gamma_{red} \circ \Gamma_{inf}$ composes to a Quillen adjunction, and we ask in addition that this extends to yet another Quillen adjoint triple one step down.

Unfortunately $\Pi_{inf}\circ \Pi_{sup} \dashv Disc_{sup} \circ Disc_{inf}$ is not manifestly a Quillen adjunction from just the above data. Maybe I made a mistake somewhare, or maybe it’s just not going to happen, and the right adjoint $\&$ to the $\Im$-modality will be the (only) one among the 12 which is not directly the derived functor of a Quillen functor.

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Urs comments on "Quillen adjoint triple" (70206) https://nforum.ncatlab.org/discussion/8721/?Focus=70206#Comment_70206 2018-07-20T17:51:29+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ I had had a silly variance error in the discussion of adjoint quadruples. Have fixed this now. In the process I ended up pretty much rewriting large bits of the entry. diff, v28, current

I had had a silly variance error in the discussion of adjoint quadruples. Have fixed this now. In the process I ended up pretty much rewriting large bits of the entry.

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Mike Shulman comments on "Quillen adjoint triple" (70182) https://nforum.ncatlab.org/discussion/8721/?Focus=70182#Comment_70182 2018-07-19T21:00:06+00:00 2023-10-03T23:07:04+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Expand Idea section to mention other possibilities in the definition, and remark that we could just as well use (&infin;,1)(\infty,1)-categories. diff, v22, current

Expand Idea section to mention other possibilities in the definition, and remark that we could just as well use $(\infty,1)$-categories.

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Mike Shulman comments on "Quillen adjoint triple" (70181) https://nforum.ncatlab.org/discussion/8721/?Focus=70181#Comment_70181 2018-07-19T20:51:48+00:00 2023-10-03T23:07:04+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Well, but we should at least tell the reader on this page that these model-categorical shenanigans are not actually required, and in particular that any seeming-arbitrary restrictions like “having ...

Well, but we should at least tell the reader on this page that these model-categorical shenanigans are not actually required, and in particular that any seeming-arbitrary restrictions like “having the same weak equivalences” or “only working in 3/4 cases” come only from the choice of machinery rather than being fundamental to the result.

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Urs comments on "Quillen adjoint triple" (70178) https://nforum.ncatlab.org/discussion/8721/?Focus=70178#Comment_70178 2018-07-19T16:58:41+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ Why? To give readers a comprehensive account without sending them through the &infin;\infty-literature. To have a clean 1-category theoretic construction. To know how to build models of modal ...

Why?

To give readers a comprehensive account without sending them through the $\infty$-literature. To have a clean 1-category theoretic construction. To know how to build models of modal HoTT in model categories.

it seems like it would be more nPOV to first do it with $\infty$-category theory, and then relegate the model-categorical version to a subsection or a sub-page.

That would be for an entry with a different title. This here is titled “Quillen adjoint triple” and not “$\infty$-adjoint triple”.

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Mike Shulman comments on "Quillen adjoint triple" (70174) https://nforum.ncatlab.org/discussion/8721/?Focus=70174#Comment_70174 2018-07-19T13:49:00+00:00 2023-10-03T23:07:04+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Why? In any case, it seems like it would be more nPOV to first do it with &infin;\infty-category theory, and then relegate the model-categorical version to a subsection or a sub-page.

Why?

In any case, it seems like it would be more nPOV to first do it with $\infty$-category theory, and then relegate the model-categorical version to a subsection or a sub-page.

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Urs comments on "Quillen adjoint triple" (70171) https://nforum.ncatlab.org/discussion/8721/?Focus=70171#Comment_70171 2018-07-19T08:24:52+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ I am trying to do without &infin;\infty-category theory here. Just model category theory.

I am trying to do without $\infty$-category theory here. Just model category theory.

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Mike Shulman comments on "Quillen adjoint triple" (70168) https://nforum.ncatlab.org/discussion/8721/?Focus=70168#Comment_70168 2018-07-18T22:22:41+00:00 2023-10-03T23:07:04+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ (By “homotopy” I meant &infin;\infty, sorry if that wasn’t clear.)

(By “homotopy” I meant $\infty$, sorry if that wasn’t clear.)

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Mike Shulman comments on "Quillen adjoint triple" (70167) https://nforum.ncatlab.org/discussion/8721/?Focus=70167#Comment_70167 2018-07-18T22:22:12+00:00 2023-10-03T23:07:04+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ But the properties that you’re talking about look like they should follow by abstract nonsense of (&infin;,1)(\infty,1)-category theory once we have an &infin;\infty-adjoint triple, without ...

But the properties that you’re talking about look like they should follow by abstract nonsense of $(\infty,1)$-category theory once we have an $\infty$-adjoint triple, without needing to futz about with model categories any more.

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Urs comments on "Quillen adjoint triple" (70164) https://nforum.ncatlab.org/discussion/8721/?Focus=70164#Comment_70164 2018-07-18T16:49:20+00:00 2023-10-03T23:07:04+00:00 Urs https://nforum.ncatlab.org/account/4/ Here I am after control of the &infin;\infty-adjoint triple, whose properties are seen not by the set-valued hom in the homotopy category, but the simplicial-set valued derived hom.

Here I am after control of the $\infty$-adjoint triple, whose properties are seen not by the set-valued hom in the homotopy category, but the simplicial-set valued derived hom.

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Mike Shulman comments on "Quillen adjoint triple" (70163) https://nforum.ncatlab.org/discussion/8721/?Focus=70163#Comment_70163 2018-07-18T13:37:50+00:00 2023-10-03T23:07:05+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ I don’t understand what you’re trying to do here. Once we know that we have an adjoint triple at the homotopy level, doesn’t everything we want follow from abstract nonsense there?

I don’t understand what you’re trying to do here. Once we know that we have an adjoint triple at the homotopy level, doesn’t everything we want follow from abstract nonsense there?

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Urs comments on "Quillen adjoint triple" (70158) https://nforum.ncatlab.org/discussion/8721/?Focus=70158#Comment_70158 2018-07-18T09:23:28+00:00 2023-10-03T23:07:05+00:00 Urs https://nforum.ncatlab.org/account/4/ I have written out discussion (here) of how simplicial Quillen adjoint triples induce “derived adjoint modalities”, such that on suitably fibrant/cofibrant objects the derived modal operators are ...

I have written out discussion (here) of how simplicial Quillen adjoint triples induce “derived adjoint modalities”, such that on suitably fibrant/cofibrant objects the derived modal operators are represented by the ordinary modal operators.

Strangely, I can make this work only in 3 of 4 cases: For Quillen adjoint triples of the form

$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longrightarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longleftarrow} \\ } \mathcal{D}$

it works for a) $C$ being fully faithful and b) $L$ and $R$ being fully faithful, but for Quillen adjoint triples of the form

$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D}$

it seems to work only for the case that $C$ is fully faithful.

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Urs comments on "Quillen adjoint triple" (70139) https://nforum.ncatlab.org/discussion/8721/?Focus=70139#Comment_70139 2018-07-17T14:06:06+00:00 2023-10-03T23:07:05+00:00 Urs https://nforum.ncatlab.org/account/4/ Another thought: Given a Quillen adjoint triple L&dashv;C&dashv;RL \dashv C \dashv R, what we should really ask is if the derived unit of &Ropf;C&SmallCircle;&Lopf;L\mathbb{R}C ...

Another thought:

Given a Quillen adjoint triple $L \dashv C \dashv R$, what we should really ask is if the derived unit of $\mathbb{R}C \circ \mathbb{L}L$ satisfies its universal property for pre-composition on derived hom-spaces.

This is easy to see if in additon to $\mathcal{C}_1 \underoverset{\underset{id}{\longrightarrow}}{\overset{id}{\longleftarrow}}{\bot} \mathcal{C}_2$ being a Quillen equivalence, we also assume that $\mathcal{C}_1$ and $\mathcal{C}_2$ have the same class of weak equivalences. Of course this further assumption is again met for the case of left/right homotopy Kan extension. (Hm, or is this automatic, in general?)

I made a simple note on this here.

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DavidRoberts comments on "Quillen adjoint triple" (70138) https://nforum.ncatlab.org/discussion/8721/?Focus=70138#Comment_70138 2018-07-17T07:30:37+00:00 2023-10-03T23:07:05+00:00 DavidRoberts https://nforum.ncatlab.org/account/42/ @Mike - ah, ok :-) I had a stupidly wrong guess at first, then realised the didn’t work, but then couldn’t quickly see what would prevent some other filler for that gap.

@Mike - ah, ok :-) I had a stupidly wrong guess at first, then realised the didn’t work, but then couldn’t quickly see what would prevent some other filler for that gap.

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Mike Shulman comments on "Quillen adjoint triple" (70136) https://nforum.ncatlab.org/discussion/8721/?Focus=70136#Comment_70136 2018-07-17T05:49:58+00:00 2023-10-03T23:07:05+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Quick answer: no? (-:

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DavidRoberts comments on "Quillen adjoint triple" (70131) https://nforum.ncatlab.org/discussion/8721/?Focus=70131#Comment_70131 2018-07-17T01:21:43+00:00 2023-10-03T23:07:05+00:00 DavidRoberts https://nforum.ncatlab.org/account/42/ Naive quick question: could the diagram &Cscr; 2 &longrightarrow;AAidAA &Cscr; 1 &longrightarrow;AAidAA &Cscr; 1 id&downarrow; id&swArr; &downarrow; C ...

Naive quick question: could the diagram

$\array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 }$

be extended to some

$\array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} &{}^{\mathllap{?? }}\swArrow& \Big\downarrow{}^{\mathrlap{??}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 &\underset{ \phantom{AA}??\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 }$

?

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Mike Shulman comments on "Quillen adjoint triple" (70125) https://nforum.ncatlab.org/discussion/8721/?Focus=70125#Comment_70125 2018-07-16T18:28:43+00:00 2023-10-03T23:07:05+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ That’s… a good question. It’s not obvious to me! Maybe his theorem needs to be extended to a localization of the double category of model categories.

That’s… a good question. It’s not obvious to me! Maybe his theorem needs to be extended to a localization of the double category of model categories.

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Urs comments on "Quillen adjoint triple" (70124) https://nforum.ncatlab.org/discussion/8721/?Focus=70124#Comment_70124 2018-07-16T18:22:29+00:00 2023-10-03T23:07:05+00:00 Urs https://nforum.ncatlab.org/account/4/ Thanks, Mike, for the input! I’ll try to further think about this later. For the moment, I have a different kind of question: Can we see that a Quillen adjoint triple between combinatorial model ...

Thanks, Mike, for the input!

I’ll try to further think about this later. For the moment, I have a different kind of question:

Can we see that a Quillen adjoint triple between combinatorial model categories maps to an adjunction under Renaudin’s 2-localization?

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Mike Shulman comments on "Quillen adjoint triple" (70094) https://nforum.ncatlab.org/discussion/8721/?Focus=70094#Comment_70094 2018-07-13T20:47:56+00:00 2023-10-03T23:07:05+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ Let’s see, if CC and DD are locally presentable and U:C&rightarrow;DU:C\to D is a right adjoint, then if D f&subseteq;DD_f\subseteq D is a small generating full subcategory, the restricted ...

Let’s see, if $C$ and $D$ are locally presentable and $U:C\to D$ is a right adjoint, then if $D_f\subseteq D$ is a small generating full subcategory, the restricted Yoneda embedding $D \to [D_f^{op},Set]$ is fully faithful and has a left adjoint. So the composite $C \to D \to [D_f^{op},Set]$ is a right adjoint and lands inside $D$, while its left adjoint restricts on $D$ to the left adjoint $F$ of $U$. Now let $C_f\subseteq C$ be a small generating full subcategory that contains $F(D_f)$; then since every presheaf is the colimit of representables and left adjoints preserve colimits, the left adjoint $[D_f^{op},Set] \to C$ factors as left Kan extension $[D_f^{op},Set] \to [C_f^{op},Set]$ along the restriction $F_f: D_f \to C_f$ followed by reflection into $C$ (i.e. weighted colimits). Thus, if we present $C$ and $D$ with model structures on these presheaf categories, the given adjunction should be presented by the left Kan extension Quillen adjunction.

Not sure yet how to extend this to adjoint triples though.

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Mike Shulman comments on "Quillen adjoint triple" (70089) https://nforum.ncatlab.org/discussion/8721/?Focus=70089#Comment_70089 2018-07-13T15:53:36+00:00 2023-10-03T23:07:05+00:00 Mike Shulman https://nforum.ncatlab.org/account/3/ I know that there are various theorems about how every geometric morphism between Grothendieck toposes can be induced by a morphism of sites as long as we choose appropriate sites for the two ...

I know that there are various theorems about how every geometric morphism between Grothendieck toposes can be induced by a morphism of sites as long as we choose appropriate sites for the two toposes. What sort of theorems are there like this for adjunctions between locally presentable categories? I feel like I should know the answer to that.

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Urs comments on "Quillen adjoint triple" (70088) https://nforum.ncatlab.org/discussion/8721/?Focus=70088#Comment_70088 2018-07-13T15:40:47+00:00 2023-10-03T23:07:05+00:00 Urs https://nforum.ncatlab.org/account/4/ For what it’s worth, the restriction to changing the model structure only on one of the two sides also goes along well with the examples of “&infin;\infty-solid ...

For what it’s worth, the restriction to changing the model structure only on one of the two sides also goes along well with the examples of “$\infty$-solid sites”

$\ast \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \phantom{\underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow}} \\ \underoverset{\bot}{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow} } CartSp \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } FormalCartSp \array{ \underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } SuperFormalCartSp$

$sSet_{Qu} \;\; \array{ \phantom{\underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ even }{\longleftarrow}} \\ \phantom{\underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \iota_{inf} }{\hookrightarrow}} \\ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ \Pi }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ Disc }{\hookrightarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \Gamma }{\longleftarrow} \\ \overset{ coDisc }{\hookrightarrow} } \;\; [CartSp^{op}, sSet_{Qu}]_{proj/inj} \;\; \array{ \phantom{\underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ even }{\longleftarrow}} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \iota_{inf} }{\hookrightarrow} \\ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ \Pi_{inf} }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ Disc_{inf} }{\hookrightarrow} \\ \underoverset{ \phantom{\phantom{{}_{Qu}}\bot_{Qu}} }{ \Gamma_{inf} }{\longleftarrow} \\ \phantom{\overset{ coDisc }{\longrightarrow}} } \;\; [FormalCartSp^{op}, sSet_{Qu}]_{proj} \;\; \array{ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ even }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \iota_{sup} }{\hookrightarrow} \\ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ \Pi_{sup} }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ Disc_{sup} }{\hookrightarrow} \\ \underoverset{ \phantom{\phantom{{}_{Qu}}\bot_{Qu}} }{ \Gamma_{sup} }{\longleftarrow} \\ \phantom{\overset{ coDisc }{\longrightarrow}} } \;\; [SuperFormalCartSp^{op}, sSet_{Qu}]_{proj/inj}$