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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)

    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 (CRCL)-unit is the dual Cη of the original counit η

    Cη:IdCACLCR=CLR

    and the counit is the dual of the original unit

    Cε:CRCL=CRLIdCˉA.

    The wrong thing is that CLCR=CRL, not CLR and that is why the unit and counit got interchanged; they should not get interchanged, but CL and CR should. I am going to sort this out. Thus Cη where η is unit goes Cη:IdCACRL.

    Edit: the correct version is now below.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011

    Before finishing with above, made some typographic corrections and the definition of morphisms of Q-categories

    A morphism from 𝔸:(u*u*):ˉAu*u*A to 𝔹:(v*v*):ˉBv*v*B is a triple (Φ,ˉΦ,ϕ) where Φ:AB, ˉΦ:ˉAˉB are functors and ϕ:Φu*v*ˉΦ is a natural isomorphism of functors. The composition is given by

    (Φ,ˉΦ,ϕ)(Φ,ˉΦ,ϕ)=(ΦΦ,ˉΦˉΦ,ˉΦϕϕΦ)
    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)

    2-cells as well

    A transformation of morphisms of Q-categories is a pair (α,ˉα):(Φ,ˉΦ,ϕ)(Ψ,ˉΨ,ψ) of natural transformations α:ΦΨ and ˉα:ˉΦˉΨ such that the diagram

    Φu*ϕv*ˉΨαu*v*ˉαΨu*ψv*ˉΨ

    commutes.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)

    New text. It should be the correct version.

    The (CRCL)-unit is Cη induced by the original unit η:1ARL

    Cη:IdCACLCR=CRL

    and the counit Cε is induced by the original counit ε:LR1ˉA.

    Cε:CRCL=CLRIdCˉA.

    The only thing is who is adjoint – now CR is the left adjoint. It is clear that Cη and Cε satisfy the triangle identities and that if η is iso then the composition with η 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 CR is full and faithful.

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)

    The triangle identities can be obtained by expanding. For R:ˉAA, one has CR:CACˉA is given by CR:GGR, and for L:AˉA one has CL:FFL. Then Cη:IdCACRCL=CLR has the components (Cη)G:(IdCA)(G)CRCL(G) given by Gη:GGLR. Thus for each functor GCˉA, the composition

    GRGηRGRLRGRηGR

    is the identity by the triangle identity for LR, but this is precisely the G-component of the transformation

    CRCRCηCRCLCRCεCRCR.

    Similarly the F-component of

    CLCηCLCLCRCLCLCεCL,

    for a functor FCA reads

    FLFLηFLRLFεLFL

    what is again the identity by the triangle identity for LR.

    The above text is now inserted into the proof.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2011

    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.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2011
    • (edited Apr 25th 2011)

    I mean, we should eventually state how given a morphism of Q-categories we get corresponding morphisms of Q-sheaf-categories, etc.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeApr 25th 2011

    Oh, yes…

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011

    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?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011
    • (edited Apr 29th 2011)

    Given a Q-category 𝔸 I can see pullback stability of 𝔸-formally étale morphisms under the condition that the functor u*:AˉA preserves pullbacks. That’s at least sufficient for the applications that I am currently looking at. But what about retracts?

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011

    Oh, never mind, I can see stability under retracts, too.

    I’ll write it out on the nLab later.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011
    • (edited Apr 29th 2011)

    Given a Q-category 𝔸 I can see pullback stability of 𝔸-formally étale morphisms under the condition that the functor u*:AˉA 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 -category case.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeApr 29th 2011

    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.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011

    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.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011
    • (edited Apr 29th 2011)

    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 ˆu*) 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 us 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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2011

    I have added a remark about the subtlety that I just mentioned to Q-category: here.

    • CommentRowNumber18.
    • CommentAuthorzskoda
    • CommentTimeJun 2nd 2011

    I added the paragraph:

    The Q-category factoring a fully faithful factor

    Any fully faithful functor among small categories F:AB factors canonically into the composition Au*ˉAB where ˉAB is the full subcategory of B whose objects are all b in ObB such that aB(F(a),b) is a representable functor AopSet, and u* is the corestriction of F to ˉA. This corestriction makes sense: F is fully faithful, hence B(F(a),F(a))=B(a,a), i.e. F(a)ˉA for all a in ObA. For each bˉA, define now u*(b) as the functor representing B(F(),b), i.e. by ˉA(u*(a),b)=B(F(a),b)B(a,u*(b)) (KR NcSpaces A1.1.1). This relation on objects extends to an adjunction u*u* with u* fully faithful.

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 14th 2017

    The entry is a bit of a mess. It starts off with a Q-category , but then shifts to talk of presheaves on A and ˉA, and some previously unmentioned forms of u. Then in section 3, it reverts to Q rather than Q-categories, and these same u maps are now acting between A and ˉA rather than presheaves as earlier.

    • CommentRowNumber20.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2017
    • (edited Sep 14th 2017)

    I think this is intentional. If A in adjunction with ˉA is a Q-category, then the presheaves over A and over ˉA 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.

    • CommentRowNumber21.
    • CommentAuthorzskoda
    • CommentTimeSep 14th 2017

    I improved the clarity of motivation section a bit.

    • CommentRowNumber22.
    • CommentAuthorzskoda
    • CommentTimeAug 22nd 2024

    Indicated that this is a concept with an attitude.

    diff, v34, current