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Update: the condition marked (**) above is unecessary. Recall the class $ff$ of ff morphisms in a 2-category. Here is the full statement:
Consider a strict 2-category $K$ which admits cotensors with $\mathbf{2} = 0\to 1$, with a subcanonical singleton pretopology $J\subset ff$ on its underlying 1-category.
Claim: There exists a weak 2-category $K_J$ with the same objects as $K$ where the 1-arrows of $K_J$ are spans $x \stackrel{j}{\leftarrow} u \stackrel{f}{\to} y$ and 2-arrows are diagrams
$a:f\circ pr_1 \Rightarrow g \circ pr_2:u\times_x v \to y$Modulo this claim (which is just checking that composition is a well-defined functor, and that the coherence laws hold) we have:
If $J^*$ denotes the class of arrows sent to equivalences by $A$, we have $J \subset J^*$, and any class $W$ in which $J$ is 2-cofinal is also contained in $J^*$.
Thus if $K$ admits a bicategory of fractions for $W$, and $J$ is 2-cofinal in $W$, $K[W^{-1}] \simeq K_J$.
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”.
@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 $K \to K_J$ 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.
Actually I have an improvement: cotensors with $\mathbf{2}$ are not required.
Consider a strict 2-category $K$ with a strictly 2-subcanonical singleton pretopology $J\subset ff$. Then $K \to K_J$ exists (modulo checking coherence as noted above) with properties as in #2.
Here by strictly 2-subcanonical I mean $K(-,a): K^{op} \to Cat$ is a sheaf for $J$ for all objects $a \in K$. 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).
@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 $K = Cat(S)$ (or some interesting sub-2-category thereof). For example, what about $2Cat(K,Cat)$? Or some other thing where the object are categories of qcoh sheaves on a scheme or something?
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 $2Cat(K, Cat)$, yes.
@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’.
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.
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.
@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 $J$-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).
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.
@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.
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.
@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.
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 $K$ with a completely strict subcanonical singleton pretopology $J\subset ff$ (here $ff$ is the class of ff morphisms) and let $W_J\subset ff$ be the class of 1-arrows which have a weak section over a $J$-cover. Completely strict means that strict pullbacks exist and are again in $J$, and that representable 2-functors $K(-,x):K^{op} \to Cat$ are sheaves rather than just stacks.
Claim: There exists a weak 2-category $K_J$ with the same objects as $K$ where the 1-arrows of $K_J$ are spans $x \stackrel{j}{\leftarrow} u \stackrel{f}{\to} y$ with $j\in J$, and 2-arrows are diagrams
$a:f\circ pr_1 \Rightarrow g \circ pr_2:u\times_x v \to y$such that there is a bijective-on-objects, locally fully faithful strict 2-functor $A:K\to K_J$, which is a localisation of $K$ at $W_J$.
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 $J$) of a 1-arrow in $J$ :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.
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.
I should probably clarify the sheaf condition: it means that given a cover $j\colon u\to x$ there is an isomorphism of categories
$j^*\colon K(x,y) \to StrDesc(u,y)$where $StrDesc(u,y)$ (Strict descent data with values in $y$) is the subcategory of $K(u,y)$ with objects those 1-arrows $f\colon u\to y$ whose precompositions with the two projections $u\times_x u \to u$ are equal, and arrows those 2-arrows $a\colon f\Rightarrow g$ which satisfy the same property. Which I imagine can be stated more succinctly as being a $\mathcal{V}$-sheaf on the $\mathcal{V}$-category $K$, for $\mathcal{V} = Cat$.
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