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There is a small error in the current proof that the category of endofunctors on a Q-category is a Q-category. I am going to correct it as soon as I find my way through the notation (I used it different on the paper). It now reads
The $(C^R \dashv C^L)$-unit is the dual $C^\eta$ of the original counit $\eta$
$C^{\eta} : Id_{C^A} \to C^L \circ C^R = C^{L R}$and the counit is the dual of the original unit
$C^\epsilon : C^R\circ C^L = C^{R L}\to Id_{C^{\bar{A}}} \,.$
The wrong thing is that $C^L\circ C^R = C^{RL}$, not $C^{LR}$ and that is why the unit and counit got interchanged; they should not get interchanged, but $C^L$ and $C^R$ should. I am going to sort this out. Thus $C^\eta$ where $\eta$ is unit goes $C^\eta : Id_{C^A}\to C^{RL}$.
Edit: the correct version is now below.
Before finishing with above, made some typographic corrections and the definition of morphisms of $Q$-categories
A morphism from $\mathbb{A} : (u^* \dashv u_*) : \bar A \stackrel{\overset{u^*}{\leftarrow}}{\underset{u_*}{\to}} A$ to $\mathbb{B} : (v^* \dashv v_*) : \bar B \stackrel{\overset{v^*}{\leftarrow}}{\underset{v_*}{\to}} B$ is a triple $(\Phi,\bar{\Phi},\phi)$ where $\Phi : A\to B$, $\bar{\Phi}:\bar{A}\to\bar{B}$ are functors and $\phi:\Phi u_*\Rightarrow v_*\bar{\Phi}$ is a natural isomorphism of functors. The composition is given by
$(\Phi,\bar{\Phi},\phi)\circ(\Phi',\bar{\Phi}',\phi') = (\Phi'\Phi,\bar{\Phi}\bar{\Phi}, \bar{\Phi}'\phi\circ\phi'\Phi)$
2-cells as well
A transformation of morphisms of Q-categories is a pair $(\alpha,\bar{\alpha}):(\Phi,\bar{\Phi},\phi)\to (\Psi,\bar{\Psi},\psi)$ of natural transformations $\alpha:\Phi\to\Psi$ and $\bar{\alpha}:\bar{\Phi}\to\bar{\Psi}$ such that the diagram
$\array{ \Phi u_* & \stackrel{\phi}\longrightarrow & v_* \bar{\Psi}\\ \alpha u_*\downarrow && \downarrow v_*\bar{\alpha}\\ \Psi u_* &\stackrel{\psi}\longrightarrow& v_* \bar{\Psi} }$commutes.
New text. It should be the correct version.
The $(C^R \dashv C^L)$-unit is $C^\eta$ induced by the original unit $\eta: 1_A\to R L$
$C^{\eta} : Id_{C^A} \to C^L \circ C^R = C^{R L}$and the counit $C^\epsilon$ is induced by the original counit $\epsilon: L R\to 1_{\bar{A}}$.
$C^\epsilon : C^R\circ C^L = C^{L R}\to Id_{C^{\bar{A}}} \,.$The only thing is who is adjoint – now $C^R$ is the left adjoint. It is clear that $C^\eta$ and $C^\epsilon$ satisfy the triangle identities and that if $\eta$ is iso then the composition with $\eta$ is also iso. Thus we obtain a $Q$-categories.
In other words, since the left adjoint being a full and faithful functor is equivalent to the unit of the adjunction being an isomorphism, it follows from $L$ being full and faithful that $C^R$ is full and faithful.
The triangle identities can be obtained by expanding. For $R: \bar{A}\to A$, one has $C^R : C^A\to C^{\bar{A}}$ is given by $C^R : G\mapsto GR$, and for $L:A\to\bar{A}$ one has $C^L:F\mapsto F L$. Then $C^\eta : Id_{C^A}\to C^R C^L = C^{LR}$ has the components $(C^\eta)_G : (Id_{C^A})(G) \Rightarrow C^R C^L (G)$ given by $G \eta : G\to G L R$. Thus for each functor $G\in C^{\bar{A}}$, the composition
$G R\stackrel{G\eta R}\longrightarrow G R L R \stackrel{G R \eta}\longrightarrow G R$is the identity by the triangle identity for $L\dashv R$, but this is precisely the $G$-component of the transformation
$C^R \stackrel{C^R C^\eta}\longrightarrow C^R C^L C^R \stackrel{C^\epsilon C^R}\longrightarrow C^R.$Similarly the $F$-component of
$C^{L} \stackrel{C^\eta C^L}\longrightarrow C^L C^R C^L \stackrel{C^L C^\epsilon}\longrightarrow C^L,$for a functor $F\in C^A$ reads
$F L \stackrel{F L \eta}\longrightarrow F L R L \stackrel{F \epsilon L}\longrightarrow F L$what is again the identity by the triangle identity for $L\dashv R$.
The above text is now inserted into the proof.
Thanks. I have added some hyperlinks.
With the definition of morphisms of Q-categories we should eventually also list some properties that justify this definition. I might look into this later, right now I need to do something else.
Well, the morphisms are given by the data which we call “compatibility of endofunctor with localization”, the only thing is that it is sometimes useful to have noninvertible one. If we neglect invertibility, the compatibility has various uses like lifting the categories of quasicoherent sheaves to equivariant setup. So the form is right, the only thing I do not know of the usages of the invertibility part so far.
I mean, we should eventually state how given a morphism of Q-categories we get corresponding morphisms of Q-sheaf-categories, etc.
Oh, yes…
Zoran,
in the context of Q-categories, is there any discussion of stability or not of the formally étale morphisms with respect to a given Q-category under pullback and retracts?
Given a Q-category $\mathbb{A}$ I can see pullback stability of $\mathbb{A}$-formally étale morphisms under the condition that the functor $u^* : A \to \bar A$ preserves pullbacks. That’s at least sufficient for the applications that I am currently looking at. But what about retracts?
Oh, never mind, I can see stability under retracts, too.
I’ll write it out on the $n$Lab later.
Given a Q-category $\mathbb{A}$ I can see pullback stability of $\mathbb{A}$-formally étale morphisms under the condition that the functor $u^* : A \to \bar A$ preserves pullbacks.
Sorry, under that condition formally étale morphisms are reflected under pullback, maybe not necessarily preserved.
This and a handful of other statements and proofs I have now typed up at formally etale morphism.
Could you have a careful look at my argument that they are stable under retract? This uses a lemma I have added to retract. It looks easy enough, but somehow I am worried that I am mixed up about something. (Should have taken more sleep last night.)
As you can see, I am trying to get hold of the list of properties required of a collection of “admissible morphisms” in the sense of geometry (for structured (infinity,1)-toposes). So the only condition still missing now in my list to show that the general abstract Rosenberg-Kontsevich formally étale morphisms always form an admissibility structure is their pullback stability. And the generalization of the arguments to the $\infty$-category case.
I will look into your proof later today, or tomorrow. It looks you are digging out something very interesting. I am first to prove something else today (also related to (co)reflective categories but in different direction) what I isolated last night.
Sorry, under that condition formally étale morphisms are reflected under pullback, maybe not necessarily preserved.
Sorry again, the original statement was true after all. I can’t distinguish left and right anymore.
Hey Zoran,
thanks for your mail while the Forum was down. I’ll reply here now.
Right, in section 5 of Rosenberg-Kontsevich they discuss pullback stability.
By the way, their discussion does make the same assumption, somewhat implicitly at that point, as I was making above an by now in the entry (formally etale morphism): that what I wrote $u^*$ (which is what they write $\hat u^*$) preserves products:
because they consider the case (first line of 5) that we are in a Q-category of copresheaves on another Q-category. That automatically gives not just an adjoint pair, but an adjoint quadruple.
However, in the entry formally etale morphism I was using a more minimalistic definition, where I am just demanding an adjoint triple necessary to write what in their article is diagram (1) on p. 21. So my assumption that the leftmost adjoint preserves products is satisfied in their setup, where it is indeed a right adjoint.
The notation is a bit of a problem here, with all the decorated $u$s floating around, with hats and cohats and subscripts and superscripts (I’d dare say it is not even fully consistent in their article always), so I’ll not write out more details on the comparison unless you want me to. There is also a shift of two adjoint triples against each other that comes from the fact that their Q-category of infinitesimal thickening really has a thrid adjoint, too, and they choose the lower adjoint pair where I choose the upper adjoint pair to characterize the infinitesimal thickening. It’s a bit tedious to sort this all out.
But I think I am happy. I think I have enough data now to write out a fully abstract discussion of locally ringed cohesion with open maps determined by KR-type formal étale morphisms. I’ll discuss that in another thread.
I have added a remark about the subtlety that I just mentioned to Q-category: here.
I added the paragraph:
The Q-category factoring a fully faithful factor
Any fully faithful functor among small categories $F: A\to B$ factors canonically into the composition $A\stackrel{u^*}\to \bar{A}\hookrightarrow B$ where $\bar{A}\subset B$ is the full subcategory of $B$ whose objects are all $b$ in $Ob B$ such that $a\mapsto B(F(a),b)$ is a representable functor $A^{op}\to Set$, and $u^*$ is the corestriction of $F$ to $\bar{A}$. This corestriction makes sense: $F$ is fully faithful, hence $B(F(a),F(a)) = B(a,a)$, i.e. $F(a)\in \bar{A}$ for all $a$ in $Ob A$. For each $b\in \bar{A}$, define now $u_*(b)$ as the functor representing $B(F(-),b)$, i.e. by $\bar{A}(u^*(a),b) = B(F(a),b) \cong B(a,u_*(b))$ (KR NcSpaces A1.1.1). This relation on objects extends to an adjunction $u^*\dashv u_*$ with $u^*$ fully faithful.
The entry is a bit of a mess. It starts off with a $Q^\circ$-category $\mathbb{C}$, but then shifts to talk of presheaves on $A$ and $\bar{A}$, and some previously unmentioned forms of $u$. Then in section 3, it reverts to $Q$ rather than $Q^\circ$-categories, and these same $u$ maps are now acting between $A$ and $\bar{A}$ rather than presheaves as earlier.
I think this is intentional. If $A$ in adjunction with $\overline{A}$ is a Q-category, then the presheaves over $A$ and over $\overline{A}$ have an induced adjunction which again makes them into a Q-category. If you look for a sheaves in a Q-category you understand that this Q-category is already a category of presheaves then the sheaves are special presheaves, hence certain objects in $A$. On the other hand if we talk about objects on a Q-category, then we mean the sheaves in the associated Q-category of presheaves. In a majority of situations one has the former (in) case.
There are some additions in the idea section where the inconsistent notation is used. I’ll try to correct.
I improved the clarity of motivation section a bit.
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