Internal Categories, Anafunctors and Localisations

November 29, 2010

Internal Categories, Anafunctors and Localisations

Posted by Mike Shulman

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This post is about my forthcoming paper, extracted from chapter 1 of my thesis:

Internal categories, anafunctors and localisations

and is also a bit of a call for examples from $n$ -category cafe visitors (see also these MO questions). I am most familiar with topological and Lie groupoids, but many interesting examples come from the world of schemes and algebraic stacks. I would like to know about these, but first I need to explain what I’m looking for. I would also appreciate to have any typos or inaccuracies pointed out. Please note that I haven’t written a final abstract yet, so what is there is just a placeholder!

Recall the notion of internal category. This comes with a natural definition of internal functor, and internal transformation, leading to a 2-category $Cat(S)$ of categories internal to $S$ . However there are a number of settings where there are not enough 1-arrows between a pair of objects. One of these is when $S = Grp$ , the category of groups. Then categories internal to $Grp$ are algebraic representations of pointed connected homotopy 2-types, but the natural hom-groupoid in $Cat(Grp)$ (yes, it’s a groupoid, as internal categories=internal groupoids here) does not represent the homotopy type of the mapping space.

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Another example, which is rather generic, is when you want to represent smooth stacks of groupoids (or topological stacks, or algebraic stacks) by Lie groupoids (or topological, etc. groupoids). Then the hom-groupoid between stacks is not the same as the hom-groupoid between the Lie groupoids (etc.).

A third example is given by orbifolds. These are well-known, by now, to be the same as proper etale Lie groupoids. There is a very well-argued review paper, Orbifolds as stacks?, that talks about the ‘correct’ 2-category of orbifolds as being a localisation of the 2-category of proper etale Lie groupoids at the class of ‘weak equivalences’.

What is a weak equivalence? I hear you ask, as there are many types of weak equivalence. here we call an internal functor $f:X \to Y$ an $E$ -equivalence if this diagram is a pullback $\begin{array}{ccc} X_1 & \to & Y_1 \\ \downarrow && \downarrow\\ X_0^2 & \rightarrow & Y_0^2 \end{array}$ and the composite map $X_0 \times_{Y_0} Y_1^{iso} \to Y_1^{iso} \to Y_0$ is in $E$ , a specified class of maps satisfying some properties. In practice $E$ will usually be the class of maps that admit local sections for a given Grothendieck pretopology $J$ . These were introduced by Bunge and Par'e for $E$ =regular epimorphisms and $S$ a finitely complete regular category. We denote the class of $E$ -equivalences by $W_E$ .

Now I don’t need to assume so much on $S$ , only binary products and the existence of $E$ . I rather move my assumptions to the 2-category $Cat(S)$ , or rather a full sub-2-category $Cat'(S)$ (which could be $Gpd(S)$ ). I assume that the objects of $Cat'(S)$ , which are internal categories, are such that pullbacks of the source and target maps exist. This is a familiar assumption from the theory of Lie groupoids, where source and target maps are assumed to be submersions. Let $E$ be a class of maps in $S$ which satisfies the following conditions:

$E$ is a Grothendieck pretopology
$E$ contains the split epimorphisms
‘Condition (S)’: If $A\to B$ is a split epimorphism and the composite $A \to B \to C$ is in $E$ , then $B\to C$ is in $E$

A class of maps which satisfies these conditions is called admissible. An example is the class of maps in a finitely complete category admitting local sections where ‘local’ means for a given pretopology $J$ .

The first main result of the paper is this:

Theorem 1: If $Cat'(S)$ admits weak pullbacks and admits base change along arrows in $E$ , then $Cat'(S)$ admits a calculus of fractions for $W_E$ , the class of $E$ -equivalences.

The calculus of fractions here is a bicategorical localisation, as covered in Pronk’s article. ‘Base change along arrows in $E$ ’ may sound mysterious, but it is a simple construction. Consider an internal category $X_1 \rightrightarrows X_0$ and a map $M \to X_0$ . There is (when the displayed pullback below exists) another category denoted $X[M]$ with objects $M$ and arrows the pullback $\begin{array}{ccc} M^2\times_{X_0^2}X_1 & \to & X_1 \\ \downarrow && \downarrow\\ M^2 & \rightarrow & X_0^2 \end{array}$ Often we know that this pullback exists, but we don’t know a priori that $X[M]$ is an object of $Cat'(S)$ . Example for when it does is when $Cat'(S)$ is the 2-category of proper, etale Lie groupoids. Properness is a condition on $(s,t)$ , but etale-ness is a condition on the source and target maps, $s,t$ . When this category $X[M]$ is an object of $Cat'(S)$ for all maps in $E$ , we say $Cat'(S)$ admits base change along arrows in $E$ .

My first query is this: what classes of maps $P$ (for your favourite category) can we take the source and target maps of $X$ to belong to so that the source and target of $X[M]$ also belong to $P$ ? I’m sure there are some examples arising in the theory of algebraic stacks, but I don’t know enough algebraic geometry to figure them out.

Now once we know this localisation exists, how do we calculate it? Pronk gives a construction, but it is, to me, quite complicated. This is where anafunctors come in. (Anafunctors have had a long history at the n-Cafe!) There are all sorts of fancy foundational motivations for anafunctors, but we are only interested in the version for internal categories (although the formalism is exactly the same, the applications are different). I like to think of an anafunctor $X ⇸ Y$ , which is a span from $X$ to $Y$ , as a map from a resolution of $X$ to $Y$ . More accurately, what we need on our ambient category $S$ is a subcanonical singleton Grothendieck pretopology (‘singleton’ means that covering families are just single maps) $J$ . An anafunctor $X ⇸ Y$ is then a span $X \leftarrow X[U] \to Y$ , where $U \to X_0$ is a $J$ -cover.

There is a bicategory $Cat_{ana}(S)$ of internal categories, anafunctors and transformations of anafunctors, the construction of which was detailed by Toby Bartels in his thesis, generalises very slightly to give us a sub-bicategory $Cat'_{ana}(S)$ corresponding to $Cat'(S)$ and a canonical strict 2-functor $Cat'(S) \to Cat'_{ana}(S)$ . The second result of the paper is this:

Theorem 2: Let $Cat'(S)$ and $E$ be as in theorem 1, let $J$ be a subcanonical singleton pretopology on $S$ which is cofinal in $E$ and let $Cat'(S)$ admit base change along arrows in $J$ . Then there is an equivalence of bicategories $Cat'_{ana}(S) \simeq Cat'(S)[W_E^{-1}]$ which, up to equivalence, the identity on $Cat'(S)$ .

This allows us to focus on anafunctors if we want to invert weak equivalences.

As far as I know, theorem 1 covers pretty much all examples of localising a 2-category of internal categories or groupoids in the literature. Some of these use Hilsum-Skandalis maps/right principal bibundles instead of Pronk’s construction or anafunctors, but all these give equivalent bicategories in the end.

The paper ends with a string of comparison theorems between localisations arising from different pretopologies, and a section on size considerations. The main point of the latter is to say that if the site $(S,J)$ satisfies (an internal version of) the axiom WISC, then all these localisations are essentially locally small bicategories.

Posted at November 29, 2010 7:44 PM UTC

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Re: Internal categories, anafunctors and localisations

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Nice! I have one question, about something that’s always confused me. People often formulate this stuff in terms of a Grothendieck pretopology, as you did. But then they start talking about the maps that “admit local sections” relative to this pretopology, as you also did. Aren’t those just the maps which are coverings in the Grothendieck topology generated by the pretopology?

Posted by: Mike Shulman on November 29, 2010 9:18 PM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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Aren’t those just the maps which are coverings in the Grothendieck topology generated by the pretopology?

No. The sieve generated by the covers of an object are those maps which factor through covers, not those maps which covers factor through. This is related to this MO answer :)

When the site in question is finitely complete, then taking the pretopology of local-section-admitting maps is the saturation of the original pretopology.

Posted by: David Roberts on November 30, 2010 11:28 PM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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But as pointed out at that MO answer, saturation is part of the definition of a Grothendieck topology. So passing from a pretopology to the topology it generates doesn’t just make covering families into sieves, but also saturates it. (And, of course, given a Grothendieck topology, we say that a map is a “cover” if the sieve it generates is covering.)

Posted by: Mike Shulman on November 30, 2010 11:41 PM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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We may be talking past each other on the definition of saturation. For me, a singleton pretopology $K$ is saturated if $a \to b \to c$ is in $J$ then $b\to c$ is in $K$ . The collection ( $J$ -epi) of $J$ -local-section-admitting maps forms a singleton pretopology and is saturated in this regard when pullbacks exist.

And, of course, given a Grothendieck topology, we say that a map is a “cover” if the sieve it generates is covering.

Ah, I see.

For me, coming from a geometric/topological background, thinking about maps that admit local sections, is easier than saying such and such a map is a cover if it generates a covering sieve from the Grothendieck topology generated by the original pretopology.

In essence, I’m just borrowing terminology. I’m not looking at sheaves, all I want is a class of maps closed under composition that are pullback stable and contain the isomorphisms. Most often these arise as ( $J$ -epi) for some pretopology $J$ , but they don’t have to.

Posted by: David Roberts on December 1, 2010 3:01 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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The collection ( $J$ -epi) of $J$ -local-section-admitting maps forms a singleton pretopology and is saturated in this regard when pullbacks exist.

Why do you need pullbacks?

I’m not looking at sheaves

No, but you are almost looking at stacks, aren’t you? Isn’t $Cat_{ana}(S)$ equivalent to the full sub-2-category of $Stacks(S)$ spanned by the stackifications of internal categories, relative to $E$ ?

(Which I think explains why it’s sufficient in general to look at singleton covers here—any internal category in any extensive category automatically being a stack for the extensive topology.)

I’m just borrowing terminology

Fair enough. On the other hand, I’m trying to avoid proliferation of terminology. I’m not really fond of calling things “good” or “nice” or “admissible.” (-: I’m wondering, what examples of “admissible” classes of maps $E$ do you have that are not derived from some Grothendieck topology? And do your constructions and theorems really depend on the pretopology, or only on the topology it generates?

Posted by: Mike Shulman on December 1, 2010 4:20 AM | Permalink | PGP Sig | Reply to this

Re: Internal categories, anafunctors and localisations

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We can take as the class $E$ of admissible maps those maps which admit local sections over a coverage $J$ . In this way, we know that if the bicategory of anafunctors defined using the coverage exists, it will be equivalent to the localisation $Cat'(S)[W_E^{-1}]$ . Actually it should be a sub-bicategory of the Pronk construction of the localisation.

I don’t know about admissible maps that are not generated in some sense by covers. But really I’m interested in when the localisation can be computed with anafunctors, so I want the class of admissible maps to be generated in this way.

Posted by: David Roberts on December 2, 2010 2:09 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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Well, that was a bit of a cheat, not looking at sheaves. Maps in ( $J$ -epi) ( $J$ -epimorphisms) are precisely the epimorphisms of representable $J$ -sheaves, as I’m sure you’re aware.

As far as stacks go, the idea is that anafunctors are the right arrows between the stacks the domain and codomain groupoids present. But proving it is not a formality.

any internal category in any extensive category automatically being a stack for the extensive topology

Really? That’s interesting.

I don’t need pullbacks to exist generally, and I really only look at those $J$ -epimorphisms are pullback stable. This class is no longer saturated, as in my previous post, but as long as it contains the split epimorphisms, then I am happy. More generally, I can drop the requirement for split epimorphisms (which doesn’t hold for surjective submersions) and ask that internal equivalences of categories are essentially surjective (in the appropriate sense).

Actually, looking over the proof of Theorem 1, pullback-stability of maps in $E$ is only used for showing that essentially surjective functors are closed under composition.

Posted by: David Roberts on December 1, 2010 4:53 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

No, I don’t have an example of admissible maps except for one derived from a pretopology.

And do your constructions and theorems really depend on the pretopology, or only on the topology it generates?

Quite probably it depends only on the topology it generates. At the end I show that several sorts of different pretopologies give equivalent localisations. I find that using a pretopology gives me something concrete to work with.

Posted by: David Roberts on December 1, 2010 4:58 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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I think it’s pretty easy to show that any internal category in an extensive category is a stack for the extensive topology. First notice that any internal category is a prestack (i.e. its hom-presheaves are sheaves) for any subcanonical topology, since its hom-presheaves are representable. Thus the only thing to check for stackiness is that gluings exist for descent data. For instance, for the covering family $\{A \to A\sqcup B, B\to A\sqcup B\}$ we need to prove that given maps $A\to C$ and $B\to C$ which are isomorphic over their intersection, we can construct a map $A\sqcup B\to C$ . But since coproducts are disjoint in an extensive category, their intersection is the initial object, so isomorphism over that is a non-condition. So we can just use the universal property of the coproduct to induce the desired gluing.

the idea is that anafunctors are the right arrows between the stacks the domain and codomain groupoids present. But proving it is not a formality.

Hmm, I always thought it should be straightforward. On the one hand, stackification inverts weak equivalences, so every anafunctor determines a morphism of stackifications. On the other hand, a functor $\hat{C} \to \hat{D}$ of stackifications should be equivalent (since stackification is a left 2-adjoint) to a functor $C\to \hat{D}$ , which is (by the Yoneda lemma) an object of $\hat{D}(C_0)$ equipped with “action data” over $C_1$ . But by the construction of stackifications, an object of $\hat{D}(C_0)$ is determined by a cover $p:M\to C_0$ and a map $M\to D_0$ with descent data over the kernel of $p$ . It then seems that the “action data” over $C_1$ should be equivalent to making $M\to D_0$ into a functor $C[M] \to D$ , so that we get an anafunctor $C \leftarrow C[M] \to D$ . Is it tricky to make that precise? Or am I missing a subtlety somewhere?

Posted by: Mike Shulman on December 1, 2010 5:15 AM | Permalink | PGP Sig | Reply to this

Re: Internal categories, anafunctors and localisations

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Or am I missing a subtlety somewhere?

I haven’t given it any thought, but I can go on Pronk’s article, where she shows that certain sober topological groupoids correspond to certain topological stacks. The hard part I think was showing the condition EF3 in what is proposition 6.5 in my paper, that the map from the 2-category of internal groupoids to that of stacks is locally fully faithful.

And when you said extensive (pre)topology I thought you mean a superextensive (pre)topology. That is too good to be true.

Posted by: David Roberts on December 1, 2010 5:44 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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I still feel like it ought to be possible to get a theory of anafunctors without assuming that any pullbacks exist, only the sort of “there exists a covering family that factors through” sort of condition that appears in the definition of a coverage or a Grothendieck topology (in terms of pullbacks of sieves). But I seem to recall there were some technical difficulties somewhere.

Posted by: Mike Shulman on December 1, 2010 5:18 AM | Permalink | PGP Sig | Reply to this

Re: Internal categories, anafunctors and localisations

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I think that it is possible to form anafunctors in a category when you have a subcanonical singleton coverage, but you need to (at the very least) choose a filler for each diagram $A \to B \leftarrow U$ for $U \to B$ a cover so as to compose anafunctors. When proving coherence you’ll need to be a bit more clever, but I think it goes through.

Posted by: David Roberts on December 1, 2010 5:40 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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Mike wrote:

I still feel like it ought to be possible to get a theory of anafunctors without assuming that any pullbacks exist, only the sort of “there exists a covering family that factors through” sort of condition that appears in the definition of a coverage or a Grothendieck topology (in terms of pullbacks of sieves).

David replied:

when you have a subcanonical singleton coverage you need to (at the very least) choose a filler for each diagram $A \to B \leftarrow U$ for $U \to B$ a cover so as to compose anafunctors.

David, this is preciely what Mike is referring to. The “filler” that you mean is that depicted here in the definition of coverage. There it is stated generally, you can specify to singleton covering families to get your situation.

I’ll try now to make a general comment, of a sort that I have made several times before in similar discussions, in the hope that it is taken in the right spirit:

As long as we are not regarding non-invertible higher morphisms, the theory of localization by generalized anafunctors, 2-anafunctors, … $\infty$ -anafunctors is old. It is Dwyer-Kan’s theory of simplicial localization. This says says that a generalizd $n$ -anafunctor is a zig-zag of $n$ -functors of the form

$\array{ &&&& \hat A_3 &\to& A \\ && && \downarrow^{\mathrlap{\simeq}} \\ && \hat A_2 &\to& A_3 \\ && \downarrow^{\mathrlap{\simeq}} \\ \hat X &\to& A_1 \\ \downarrow^{\mathrlap{\simeq}} \\ X }$

You can directly see that the condition that such a zig-zag may be reduced to a single span is precisely that you can fill all the squares here to get

$\array{ \hat {\hat{\hat X}}&\to&&\to& \hat A_3 &\to& A \\ \downarrow^{\mathrlap{\simeq}}&& && \downarrow^{\mathrlap{\simeq}} \\ \hat{\hat X} &\to& \hat A_2 &\to& A_3 \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \hat X &\to& A_1 \\ \downarrow^{\mathrlap{\simeq}} \\ X }$

There are analogous localization results now for the situation where you start with an $(k,1)$ -category and localize.

For that reason I would think that for your article you should stress that your theory also does handle the genuine 2-category case, where we do allow non-invertible 2-morphisms. In other words, while it is good to mention that your theory constructs for instance the $(2,1)$ -category of Lie groupoids, the case that is not treated by standard localization theory is that of the 2-category of Lie categories. I know you mention this in your text, but I would think this might be emphasized more to make clear that you are after a $(\infty,2)$ -categorical generalization of standard $(\infty,1)$ -categorical localization theory.

Posted by: Urs Schreiber on December 1, 2010 10:33 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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For that reason I would think that for your article you should stress that your theory also does handle the genuine 2-category case, where we do allow non-invertible 2-morphisms.

Good point, Urs. I’ll have to put that in.

But I’ve heard no-one complain when people write down special cases of the above result, say in manifolds or groups, by going, ‘oh, that is achieved by taking a partial decategorification of a simplicial localisation of an (2,1)-category’. :-) (tongue removed from cheek)

Posted by: David Roberts on December 2, 2010 2:17 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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If you hang out here, you’ll hear that all the time.

And for good reason: a whole lot of 1-categorical or (2,1)-categorical mathematics is being subsumed by $(\infty,1)$ -categorical mathematics, so in the long run it doesn’t make sense to keep working on the low-dimensional stuff. This is one reason I quit working on $n$ -category theory: Urs convinced me that now it’s time to move up to $(\infty,1)$ -categories, or $(\infty,n)$ -categories — but I didn’t have the patience, or energy, or whatever it takes, to master the machinery involved. But if I were a decade or two younger, and felt like my career were just starting instead of beginning to slowly wind down, I’d definitely dive into the $(\infty,n)$ stuff. That’s where it’s at, when it comes to ‘higher-dimensional mathematics’.

Posted by: John Baez on December 2, 2010 3:36 AM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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a whole lot of 1-categorical or (2,1)-categorical mathematics is being subsumed by (∞,1)-categorical mathematics, so in the long run it doesn’t make sense to keep working on the low-dimensional stuff.

I don’t exactly disagree, but I don’t want anyone who reads this to feel that there’s nothing valuable to be contributed to higher category theory nowadays unless and until you understand ∞-stuff. I do definitely agree that it’s worth the patience and energy to get into the ∞-stuff (I’m still working on it myself), but I think there is still valuable work to be done in low dimensions.

I do suspect that some non-∞-categorical subjects, such as strictification theory for pseudoalgebraic structures, may be a casualty of ∞-categorical mathematics (despite the fact that I just wrote a paper about it). For instance, while the possibilities of developing Gray-monad theory analogously to 2-monad theory sound fun to me (the only paper on the subject, as far as I know, is John Power’s “three-dimensional monad theory”), my enthusiasm is tempered by the realization that such methods are unlikely to generalize beyond dimension 3. One might consider particularly the theorem that the 2-category of strict algebras and pseudomorphisms for a strict 2-monad admits strict PIE-limits, and its Gray-categorical analogue. From a “fully non-evil” perspective what we care about are non-strict 2-limits (aka “bilimits”), and while this theorem does in fact give us such limits in the 2- and 3-dimensional cases, I wouldn’t really expect its methods to generalize.

However, if you can prove some new result about 2- or 3-categories, which it is reasonable to expect to generalize to (∞,2)- or (∞,3)-categories, then I think that is a worthwhile contribution. Even if the interest of mathematicians as a whole will eventually turn to the (∞,-) version, it’s always easier to think of, and prove, new things first in easier cases. For instance, by way of comparison to the case of PIE-limits of pseudo-morphisms, characterizing the class of limits admitted by the 2-category of algebras and lax or colax morphisms for a 2-monad (a theorem which Steve Lack and I are one day going to get around to publishing) is, I think, a worthwhile endeavor. This is because the analogous question for (∞,2)-monads is just as interesting and nontrivial, and an answer to the 2-categorical question is likely to suggest the correct answer (and maybe even the correct methods) for the (∞,2)-case.

A different use of low-categorical technology is that sometimes, higher categories can be usefully described in terms of lower categories. I am thinking particularly of derivators and equipments.

Posted by: Mike Shulman on December 2, 2010 10:04 PM | Permalink | PGP Sig | Reply to this

Re: Internal categories, anafunctors and localisations

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but I think there is still valuable work to be done in low dimensions.

I think it’s one more example of the general dichotomy:

(nice big category of very general objects)

$\leftrightarrow$

(small non-nice category of concrete objects)

You want to develop your concepts in $(\infty,n)$ -category theory. But in concrete cases it is worthwhile and of interest to explicitly cut down to $(r,n)$ -categories for low $r$ as much as possible.

Doing low-dimensional category theory is extra work, not something made superfluous by the higher category theory. But the higher category theory context can prevent you from going down blind alleys with the low dimensional constructions.

Posted by: Urs Schreiber on December 2, 2010 10:32 PM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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But I’ve heard no-one complain when people write down special cases of the above result,

For what it’s worth, I have complained about that.

But the good thing is, your article is a good response to such complaints: because your construction should be a first step into new territory that has not been studied much (as far as I am aware):

the entire theory of presentations of $(\infty,1)$ -categories and their homotopy categories by 1-categories with weak equivalences should have an analog theory of presentations of $(\infty,2)$ -categories and their homotopy 2-categories by 2-categories with weak equivalences.

You describe such a homotopy 2-category of a 2-category with weak equivalences, for special cases, by a 2-categorical variant of a calculus of fractions method. That’s very good. I think if you highlight this in your introduction in addition to pointing out the applications to the $(2,1)$ -case, the number of people impressed by your result will roughly double.

Posted by: Urs Schreiber on December 2, 2010 7:14 PM | Permalink | Reply to this

Re: Internal categories, anafunctors and localisations

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You describe such a homotopy 2-category of a 2-category with weak equivalences, for special cases, by a 2-categorical variant of a calculus of fractions method. That’s very good. I think if you highlight this in your introduction in addition to pointing out the applications to the (2,1)-case, the number of people impressed by your result will roughly double.

I should point out that it is Pronk’s bicategory of fractions, but what I show is perhaps a case of particular interest.

I am very interested in the more general situation, but time has so far not permitted me to work on it.

Posted by: David Roberts on December 4, 2010 5:14 AM | Permalink | Reply to this

Re: Internal Categories, Anafunctors and Localisations

Sorry for the off-topic comment, but did the domain for ncatlab.org expire? It now links to one of the domain-squatter-type pages.

Posted by: walt on November 30, 2010 9:09 PM | Permalink | Reply to this

Re: Internal Categories, Anafunctors and Localisations

Something went wrong, anyway. Andrew links to a temporary alternative site here.

Posted by: Tom Leinster on November 30, 2010 9:32 PM | Permalink | Reply to this

Re: Internal Categories, Anafunctors and Localisations

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See here. It’s being worked on.

Posted by: Mike Shulman on November 30, 2010 9:37 PM | Permalink | Reply to this

Re: Internal Categories, Anafunctors and Localisations

Just to let you know, the paper is now on the arXiv. Comments are still welcome and encouraged. In particular, Urs (if you read this), did I do justice regarding your earlier comments on higher categorical localisation?

Posted by: David Roberts on January 13, 2011 1:24 AM | Permalink | Reply to this

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