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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 3rd 2011
    • (edited Jun 5th 2011)
    Hi all. One thing I am thinking about is a 'synthetic' approach to anafunctors. Makkai, in his paper, takes a very elementary approach, using elements at will. In my paper (and in Toby's too), taking the internal road, I can't do that so blithely, but it essentially treats the theory of anafunctors as a special sort of sketch (not too dissimilar perhaps to FOLDS), but not from that point of view. However, I don't supply all proofs, neither does Toby and for that matter neither does Makkai (a fair bit of 'reader can supply the details'). I think the easiest way to see what is actually going on is to think of Cat(S) as a 2-category K (with cotensors with 2 = {0 --> 1}) equipped with a special class of 1-arrows viz a strict singleton subcanonical Grothendieck pretopology J (essentially just a pretopology on the underlying 1-category). This needs to satisfy an additional condition (**) which is 2-categorical: given fg = h where h,f in J, then g has an anafunctor pseudo-section (this condition is satisfied in a very natural way in K=Cat(S), and I would like to see if it follows from existing assumptions in general). Then one can think about spans where the left leg is a cover as 1-arrows in a bicategory K_ana, and the usual thing for 2-arrows (take the pullback of covers etc). There should be a canonical inclusion K --> K_ana, and we could even talk about the proarrow equipment that arises.

    This is a different approach - more restrictive, I believe - than Mike's work at exact completion of a 2-category (michaelshulman), where he talks about anafunctors in a 2-category (weak, by default, for him). There he talks about anafunctors in the 2-category of 2-congruences in a 2-site, and thinking of them more along the lines as in Cat(S). But I'm very interested in the relation between the two, especially if one could be derived from the other.

    One spin-off of this is that I would like to provide another model for the localisation of a 2-category. Here J needs to be weakly cofinal in the class W one wants to invert. One point of my anafunctors paper was to show that the localisation of a 2-category of internal categories had a better model that the default one constructed by Pronk, and this theorem should go through, namely K[W^-1] ~ K_ana. Note that this is (2,2)-category localisation, not (2,1)-category localisation. (As an aside, the approach to localisation via bibundles, which is even simpler to describe, wouldn't work here because that assumes one is in a (2,1)-category.)

    The one point which is a bit restrictive is that one needs covers to be an ff morphism in order to define the bicategory K_ana of anafunctors in K. (This reminds me somewhat of talking about S-local maps in a model theoretic setup, at least when the pretopology J is morally like a cover by open balls or affine schemes. But I haven't thought about this too much yet.)

    One direction this may go is if the whole game can be phrased in a suitably 2-categorical way, then perhaps similar techniques could be used to talk about localisation of higher categories (say simplicial categories), at least in special cases. For example, defining weak maps between strict higher categories or something. This is complete speculation, and not a short-term goal by any means.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 5th 2011

    Update: the condition marked (**) above is unecessary. Recall the class ffff of ff morphisms in a 2-category. Here is the full statement:

    Consider a strict 2-category KK which admits cotensors with 2=01\mathbf{2} = 0\to 1, with a subcanonical singleton pretopology JffJ\subset ff on its underlying 1-category.

    Claim: There exists a weak 2-category K JK_J with the same objects as KK where the 1-arrows of K JK_J are spans xjufyx \stackrel{j}{\leftarrow} u \stackrel{f}{\to} y and 2-arrows are diagrams

    a:fpr 1gpr 2:u× xvy a:f\circ pr_1 \Rightarrow g \circ pr_2:u\times_x v \to y

    Modulo this claim (which is just checking that composition is a well-defined functor, and that the coherence laws hold) we have:

    1. A bijective-on-objects, locally fully faithful strict 2-functor A:KK JA:K\to K_J, which
    2. sends 1-arrows in JJ to equivalences

    If J *J^* denotes the class of arrows sent to equivalences by AA, we have JJ *J \subset J^*, and any class WW in which JJ is 2-cofinal is also contained in J *J^*.

    Thus if KK admits a bicategory of fractions for WW, and JJ is 2-cofinal in WW, K[W 1]K JK[W^{-1}] \simeq K_J.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJun 5th 2011

    I am interested to discuss various issues related to bicategorical localization, but I have difficulty in understanding the rest of the wide framework you are describing above. I like very much your theorem of the relation between bicategories of fractions and bicategories of anafunctors, though I do not understand the statement that it is a better model; I mean calculi of fractions appear quite naturally in practice and I do not find them a “bad model”.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 6th 2011

    @Zoran - Have you ever tried working with Pronk’s construction? :) 2-arrows are equivalence classes of spans of spans, where the equivalence relation is generated by spans of spans of spans. I don’t find this very intuitive, and for ’geometric’ applications I find anafunctors (or ’anafunctors’, in the general 2-category case) easier to think about what things mean.

    Aside from that, I’m aiming to show that KK JK \to K_J is a localisation without recourse to Pronk’s theorem that a bicategory that admits a calculus of fractions has a localisation. Actually I would go so far as to say that my construction is also a calculus of fractions, but one where the 2-arrows are diagrams, not equivalence classes of diagrams.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 6th 2011

    Actually I have an improvement: cotensors with 2\mathbf{2} are not required.

    Consider a strict 2-category KK with a strictly 2-subcanonical singleton pretopology JffJ\subset ff. Then KK JK \to K_J exists (modulo checking coherence as noted above) with properties as in #2.

    Here by strictly 2-subcanonical I mean K(,a):K opCatK(-,a): K^{op} \to Cat is a sheaf for JJ for all objects aKa \in K. I still require that strict pullbacks of covers exist and are covers.

    Personally, if this is the sort of structure I have on my 2-category, I would rather check this than Pronk’s axioms and work with the construction from her 1996 JPAA paper (which is marvellous, by the way).

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 6th 2011

    @Zoran - I guess a lot of #1 above is just the internal process. Ignore that if you like. Also apologies for the telegraphic style - was trying to get ideas down in the limited time I had. But if you have questions or comments please fire away!

    One thing which annoys me is that I don’t have a lot of examples to hand, even though my intuition is telling me there are some out there. I know about K=Cat(S)K = Cat(S) (or some interesting sub-2-category thereof). For example, what about 2Cat(K,Cat)2Cat(K,Cat)? Or some other thing where the object are categories of qcoh sheaves on a scheme or something?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2011

    I don’t have a lot of examples to hand

    The whole theory of Grothendieck 2-toposes is waiting to be an example! This should be the suitably left exact localizations of 2-catgeories of the form 2Cat(K,Cat)2Cat(K, Cat), yes.

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 6th 2011

    @Urs Yes! And I think that there is a lot of potential in 2-categories of topoi, model categories and derivators. When I say I don’t have examples to hand, it’s that I don’t know enough of the theory well enough to be confident in saying ’this and this and this will turn out to be fruitful applications’.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJun 6th 2011

    David, I am coming from noncommutative algebra, and we are used that in localization theory we always work with equivalence classes already for 1-categories, and I find this very convenient like various representations of rational numbers; why would this be different in dimension 2. I know that it is difficult to do computations with noncommutative fractions, and spent few years doing some very difficult examples (never published those in full), but it is very much checkable and algorithmic. I mean the properties come out when the procedures work.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 6th 2011

    Regarding the question of the relationship between the “synthetic” approach to anafunctors, which models localization, and the approach using congruences in a 2-category, which models exact completion, I think this should be naturally answered in terms of “exact completions of 2-sites”, although I don’t yet know exactly what the details should mean in the 2-dimensional case. (David, this is the sort of thing that I said I would be interested in working on in my email a while ago.)

    I can explain why I think the answer lies here by analogy with the case of 1-sites. There is a notion of the “exact completion” of a “unary site” (the “unary” condition is roughly akin to the “singleton” in David’s pretopologies), which I will talk about at CT2011. It can be constructed in two ways, one of which uses “representable profunctors” between congruences and the other of which uses “anafunctors”. When the site is a weakly lex category with trivial topology, this is its (free) exact completion as a weakly lex category, whereas when the site is a regular category with its regular topology, this is its exact completion as a regular category; in general the construction is sort of an “interpolation” between the two. If the site is an exact category with its regular topology, then its exact completion is itself, whereas if it is an exact category with a topology that contains its regular topology (i.e. all regular epis cover), then its exact completion can be identified with its localization at the monic covers (plus, a unary topology on an exact category which contains the regular topology is determined by its class of monic covers).

    By analogy, it seems that an exact 2-category equipped with a suitable class of fully-faithful morphisms ought to determine a unary (2-)topology, in such a way that the 2-exact completion of the resulting unary 2-site (constructible using anafunctors) would be identified with its localization at the given class of “ff covers.” I know that you (David) want to work with sites that are not 2-exact, but I expect the ideas will generalize.

    I also completely agree that this should generalize to higher dimensions. In fact, my original reason for thinking about exact completion of sites is that I wanted an analogous description of (∞,1)-sheaf toposes in terms of “anafunctors” between “hypercongruences”, as a means of computing their fundamental pro-∞-groupoids.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 7th 2011

    @Mike - yes, I hadn’t forgotten. One thing which I hinted at above, and would like to look into, is the relation between your anafunctors in a 2-category, and my JJ-spans (as I, perhaps temporarily, dub them) in the 2-category of congruences, and k-ary versions of congruences as in your 1-site case.

    I don’t want to stick strictly to non-(2-)exact (2-)sites, it’s just that I wanted the formulation to work with as minimal assumptions as possible (and also to help forestall comments that this has all been done before with representable profunctors).

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJun 7th 2011
    • (edited Jun 7th 2011)

    David and Mike, I do not want to mess your beautiful discussion on synthetic approach which is beyond my present understanding, but want to ask David few hints (if he might know) on what may be or may not be known on 2-categorical localizations. For this comment, is there an analogue developed of the Serre-type localization at thick subcategories for abelian 2-categories a la Dupont-Vitale ? This would give a lot of examples, approached from a different direction.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 8th 2011

    @Zoran There is the preprint by Vitale, Bipullbacks and calculus of fractions, which should be useful. It is from the point of view of semiabelian categories, to some extent, but not only them. It is sort of a companion article to Butterflies in a semiabelian context. Unfortunately I don’t know of any approaches that deal exclusively with abelian 2-categories, but it may be extractible from a combination of my paper and Vitale’s (and possibly also the above discussion).

    Looking at Springer’s EOM, I see that the localisation of an abelian category at a thick subcategory can be constructed in terms of certain spans. I would definitely be interested in figuring out the analogous thing for abelian 2-categories.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeJun 8th 2011
    • (edited Jun 8th 2011)

    David, there is about 4-5 equivalent methods to describe the data for localization of abelian categories. The equivalences are interesting to study, but unfortunately the historial biases made many competent terminologies (often with the same words used for different notions) and the literature is not that easy to penetrate for an outsider. It looks to me that the localization in terms of thick subcategories is the most transparent among the approaches to the localization of abelian categories (it is also closest to the spectral theory). Thus I would like to have that approach in dimension 2.

    Thank you for reminding me of the Vitale’s paper, I never went beyond the first page and completely forgot of its existence.

    P.S. did anybody write down that given a reflective subbicategory (reflection in the sense of biadjunction whose counit is an equivalence) one has a canonically defined calculus of fractions (right or left depending on conventions) in the sense of Pronk ? Her original paper does not prove that.

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 9th 2011

    @Zoran - as far as I know, no. Since applications so far were to geometric contexts (differentiable stacks/orbifolds) and to a lesser extent, crossed modules, people (as far as I know) haven’t done looked at the ’abelian’ side of things in bicategorical localisation.

    • CommentRowNumber16.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 3rd 2012
    • (edited Oct 3rd 2012)

    Regarding my aim in the second paragraph of #4, I have now proved (at least to my own satisfaction) that my fraction construction (using general 2-categorical versions of anafunctors) is a localisation. Full statement:

    Consider a strict 2-category KK with a completely strict subcanonical singleton pretopology JffJ\subset ff (here ffff is the class of ff morphisms) and let W JffW_J\subset ff be the class of 1-arrows which have a weak section over a JJ-cover. Completely strict means that strict pullbacks exist and are again in JJ, and that representable 2-functors K(,x):K opCatK(-,x):K^{op} \to Cat are sheaves rather than just stacks.

    Claim: There exists a weak 2-category K JK_J with the same objects as KK where the 1-arrows of K JK_J are spans xjufyx \stackrel{j}{\leftarrow} u \stackrel{f}{\to} y with jJj\in J, and 2-arrows are diagrams

    a:fpr 1gpr 2:u× xvy a:f\circ pr_1 \Rightarrow g \circ pr_2:u\times_x v \to y

    such that there is a bijective-on-objects, locally fully faithful strict 2-functor A:KK JA:K\to K_J, which is a localisation of KK at W JW_J.

    The proof is direct and sidesteps Pronk’s paper. This provides an independent proof for the existence of bicategorical localisations all over the place from the last ten years or so (well, yes, they are all to do with internal groupoids/categories, but I have other applications up my sleeve!)

    I think it is possible to weaken the condition that pullbacks exist to the condition that there is a weak strict pullback (again in JJ) of a 1-arrow in JJ :P (in less confusing terms, existence of a strict cone without uniqueness), at the cost of having to choose all these squares a priori (so most naturally getting an anabicategory).

    The hard part now is to latex up all the diagrammatic notation, which I borrowed from Micah McCurdy’s work on monoidal functors, but at least this makes the proofs much shorter than relying on Pronk’s paper where they are largely omitted due to their complexity.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeOct 4th 2012

    Very nice! I look forward to seeing it. (But “weak strict pullback” — augh!)

    Also, a small trivial comment: I think being a sheaf is not strictly stronger than being a stack; they are incomparable.

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 4th 2012
    • (edited Oct 4th 2012)

    I should probably clarify the sheaf condition: it means that given a cover j:uxj\colon u\to x there is an isomorphism of categories

    j *:K(x,y)StrDesc(u,y) j^*\colon K(x,y) \to StrDesc(u,y)

    where StrDesc(u,y)StrDesc(u,y) (Strict descent data with values in yy) is the subcategory of K(u,y)K(u,y) with objects those 1-arrows f:uyf\colon u\to y whose precompositions with the two projections u× xuuu\times_x u \to u are equal, and arrows those 2-arrows a:fga\colon f\Rightarrow g which satisfy the same property. Which I imagine can be stated more succinctly as being a 𝒱\mathcal{V}-sheaf on the 𝒱\mathcal{V}-category KK, for 𝒱=Cat\mathcal{V} = Cat.