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So the main point is to say that the right derived functor of a right Quillen functor is identified with the left derived functor of a left Quillen functor.
It is immediate to say this a) on the level of homotopy categories and b) on the level of Kan-complex enriched simplicial localizations. Is there an intermediate, more purely model-category-theoretic way to express this?
Perhaps with the derived natural transformations of this paper?
These derived natural transformation go between (images under some derived Quillen functor of) left Quillen functors, right (p. 8)? The issue with the adjoint triples is to identify a left Quillen functor with a right Quillen functor, after deriving.
The ones on p8 do, but the more general ones on p20 live in a square composed of two left Quillen functors and two right Quillen functors (some of which could be identities).
Now I see, thanks. That looks like what I’d need, yes.
okay, so in the double category of model categories, the “Quillen adjoint triple” for homotopy limit/colimit is incarnated by the diagram
$\array{ && [\mathcal{C}^{op}, sSet_{Qu}]_{proj} &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ && {}^{\mathllap{ \underset{\longrightarrow}{\lim} }}\Big\downarrow &{}^{\mathllap{\eta}}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\overset{ \phantom{A}\underset{\longleftarrow}{\lim}\phantom{A} }{\longrightarrow}& sSet_{Qu} &\overset{\phantom{A}const\phantom{A}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{ id }}\Big\downarrow & {}^{\mathllap{\epsilon}}\swArrow & {}^{\mathllap{const}} \big\downarrow &\swArrow_{\mathrlap{id}}& \big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} }$maybe that kind of diagram could be taken as the general definition of “Quillen adjoint triple”.
I have made that the definiton, and spelled out the Example of the Quillen adjoint quadruple over a site with terminal object:
Let $\mathcal{C}_1, \mathcal{C}_2, \mathcal{D}$ be model categories, where $\mathcal{C}_1$ and $\mathcal{C}_2$ share the same underlying category $\mathcal{C}$, and such that the identity functor on $\mathcal{C}$ constitutes a Quillen equivalence
$\mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{C}_1$Then
a Quillen adjoint triple of the form
$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D}$is a diagram in the double category of model categories of the form
$\array{ && \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 \\ && {}^{\mathllap{ L }}\Big\downarrow &{}^{\mathllap{\eta}}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\overset{ \phantom{A}R\phantom{A} }{\longrightarrow}& \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{ id }}\Big\downarrow & {}^{\mathllap{\epsilon}}\swArrow & {}^{\mathllap{C}} \big\downarrow &\swArrow_{\mathrlap{id}}& \big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 }$such that $\eta$ is the unit of an adjunction and $\epsilon$ the counit of an adjunction, thus exhibiting Quillen adjunctions
$\array{ \mathcal{C}_1 \underoverset {\underset{C}{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }$a Quillen adjoint triple of the form
$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longrightarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longleftarrow} \\ } \mathcal{D}$is a diagram in the double category of model categories of the form
$\array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 }$such that $\eta$ is the unit of an adjunction and $\epsilon$ the counit of an adjunction, thus exhibiting Quillen adjunctions
$\array{ \mathcal{C}_2 \underoverset {\underset{C}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_1 \underoverset {\underset{R}{\longleftarrow}} {\overset{C}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }$If a Quillen adjoint triple of the first kind overlaps with one of the second kind
$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L_1 \phantom{= A_a}}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C_1 = L_2}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{R_1 = C_2}{\longrightarrow} \\ \overset{\phantom{A_a = } R_2}{\longleftarrow} \\ } \mathcal{D}$we speak of a Quillen adjoint quadruple, and so forth.
That doesn’t seem like enough: don’t you also need to require that the derived transformation of your square $id$ is an isomorphism?
Also there doesn’t seem to be any point to including the unit and counit in your “diagram” (which is not even a composable diagram in the double category). So if this is what you want, I would phrase it as “two Quillen adjunctions
$\array{ \mathcal{C}_1 \underoverset {\underset{C}{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }$such that the derived natural transformation of that square $id$ is an isomorphism.”
But it’s not really any harder to allow $\mathcal{D}$ to have two model structures as well, is it? And even to allow the underlying categories to be different and the Quillen equivalence to be a nonidentity adjunction?
you also need to require that the derived transformation of your square $id$ is an isomorphism
Sure, thanks, forgot the most important bit in all the editing. Fixed now. Also added check that this is the case in the two examples.
there doesn’t seem to be any point to including the unit and counit in your “diagram” (which is not even a composable diagram in the double category).
I enjoyed how the double category 2-cells allow me to display the structure involved in a “Quillen triple” as one single connected diagram (could we call it a “pasting diagram in the double category”), that’s what I tried to bring out. But the alternative that you suggest follows right after, after “thus exhibiting”.
it’s not really any harder to allow $\mathcal{D}$ to have two model structures as well, is it? And even to allow the underlying categories to be different and the Quillen equivalence to be a nonidentity adjunction?
That’s what I had initially. But the special case seems really neat, with the “2-cells with adjacent 1-identities” for the unit/counit being accompanied exactly by the remaining possible “2-cells with adjacent 1-identities” exhibiting the gluing of the two Quillen adjunction.
This feels like it really deserves to be called “Quillen adjoint triple”. The more general version might rather go by “derivable adjoint triple” or the like.(?)
No, I don’t think you could call it a pasting diagram in a double category, because it’s not composable. It wouldn’t even be a composable diagram in a 2-category. I think the more concise definition is much easier to understand than a big diagram, but YMMV I guess. I also think it obscures what’s going on, rather than being neat, to focus on the special case when certain functors happen to be identities.
Regarding naming, one criterion of sensibility is that there ought to be a theorem like “any adjoint triple between locally presentable $(\infty,1)$-categories can be presented by some Quillen adjoint triple”. Is that true for the more restricted version? For that matter, is it true for the more general version?
“any adjoint triple between locally presentable $(\infty,1)$-categories can be presented by some Quillen adjoint triple”. Is that true for the more restricted version? For that matter, is it true for the more general version?
Yeah, I was wondering, too, that’s what one should know.
I suppose adjoint triples of left/right Kan extension between Bousfield localizations of simplicial presheaves will also fit the strict definition, by direct generalization of the limit/colimit example. That probably exhausts the class of examples of adjoint triples of presentable $\infty$-categories that I know of.
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”
$\ast \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \phantom{\underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow}} \\ \underoverset{\bot}{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow} } CartSp \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } FormalCartSp \array{ \underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } SuperFormalCartSp$This induces a progression of Quillen adjoint quadruples of the form
$sSet_{Qu} \;\; \array{ \phantom{\underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ even }{\longleftarrow}} \\ \phantom{\underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \iota_{inf} }{\hookrightarrow}} \\ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ \Pi }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ Disc }{\hookrightarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \Gamma }{\longleftarrow} \\ \overset{ coDisc }{\hookrightarrow} } \;\; [CartSp^{op}, sSet_{Qu}]_{proj/inj} \;\; \array{ \phantom{\underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ even }{\longleftarrow}} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \iota_{inf} }{\hookrightarrow} \\ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ \Pi_{inf} }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ Disc_{inf} }{\hookrightarrow} \\ \underoverset{ \phantom{\phantom{{}_{Qu}}\bot_{Qu}} }{ \Gamma_{inf} }{\longleftarrow} \\ \phantom{\overset{ coDisc }{\longrightarrow}} } \;\; [FormalCartSp^{op}, sSet_{Qu}]_{proj} \;\; \array{ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ even }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ \iota_{sup} }{\hookrightarrow} \\ \underoverset{\phantom{{}_{Qu}}\bot_{Qu} }{ \Pi_{sup} }{\longleftarrow} \\ \underoverset{ \phantom{{}_{Qu}}\bot_{Qu} }{ Disc_{sup} }{\hookrightarrow} \\ \underoverset{ \phantom{\phantom{{}_{Qu}}\bot_{Qu}} }{ \Gamma_{sup} }{\longleftarrow} \\ \phantom{\overset{ coDisc }{\longrightarrow}} } \;\; [SuperFormalCartSp^{op}, sSet_{Qu}]_{proj/inj}$and so the way that the topmost left adjoints between sites keep shifting up in position, in this example, matches the alternation between a single model structure (proj) and change of model structure (proj/inj).
So at least I luck out, and the neat restrictive definition is sufficient for my current purpose.
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.
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.
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?
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.
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
$\array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} &{}^{\mathllap{?? }}\swArrow& \Big\downarrow{}^{\mathrlap{??}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 &\underset{ \phantom{AA}??\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 }$?
Quick answer: no? (-:
@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.
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.
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.
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?
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.
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.
(By “homotopy” I meant $\infty$, sorry if that wasn’t clear.)
I am trying to do without $\infty$-category theory here. Just model category theory.
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.
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”.
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.
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.
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
projective model structures “everywhere on top”
injective model structure “everywhere at the bottom”
a transition zone across a Quillen adjoint quadruple where projective turns into injective.
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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