PSA: a category satisfying all but the smallness condition from Giraud’s theorem…

Screenshot of text, containing the sentence "The geometrically invariant condition on T to be a theory according to this doctrine is precisely that it should be a pretopos in the sense of Grothendieck-Verdier Expose VI in Springer Lecture Notes Volume 270."

December 5, 2025December 9, 2025 ~ David Roberts ~ 6 Comments

…is called an infinitary pretopos. Once you add the existence of a small set of generators then Giraud’s theorem applies and you know you have a Grothendieck topos. A pretopos can even have a subobject classifier, but lacking cartesian closedness fail to be a topos (I’ll say just ‘pretopos’ from now on, see the end for an explanation).

Bernie Sanders "I am once again asking you"captioned meme, with text "I am once again asking you to sau thje category of condensed sets is a pretopos" and included is a footnote to "Johnstone (1977), Definition 7.38". Added to the still image of Sanders is a carefully placed copy of Johnstone's 1977 book "Topos Theory" positioned to look like it's being held just below frame, with the title and author just visible.

The oldest reference I can find is in the introduction by Lawvere to Springer LNM 445, dating to probably 1973, and he claims the idea is in SGA4 Exposé VI (I can’t find the word there, but I can absolutely believe Grothendieck and Verdier knew of the concept EDIT: prétopos are defined in Exercice 3.11 a), see the comments). And there is a formal definition included in the Baby Elephant as in the above image, from 1977. For a modern take, Lurie has a lecture titled Pretopoi with notes on his website.

Sorry to be a curmudgeon, but it’s been more than five years, and a new (very nice!) thesis just dropped that referred to a category with all the non-size conditions from Giraud’s theorem, and apparently didn’t know there was a word for these. And with a really nice classical canonical example being the category of compact Hausdorff spaces, and now the much fancier category of condensed sets (as a special case of a colimit along inverse image functors of a large diagram of Grothendieck toposes), having a precise name for such categories is helpful (‘precise’ here being actually imprecise in the original version of this blog post, see the next paragraph).

ADDED: Both Mike Shulman in the comments below and an email correspondent have prompted me to tweak the post: a vanilla pretopos of course is defined in such a way that only finite disjoint coproducts are needed. In my brain there is a spectrum of pretoposes, i.e. $\kappa$ -pretoposes, for $\kappa$ a regular cardinal or the size of the universe of all sets, and one asks for appropriately universal $<\kappa$ -sized colimits in the definition. And I am happy to leave the size parameter implicit, but this is not a universal practice, which is a mistake when trying to make a terminological point. A Grothendieck topos isn’t just (forgetting the generating set) a pretopos, but a cocomplete pretopos, and in fact an infinitary pretopos, so that its colimits of all sizes are well-behaved. Additionally, a Grothendieck topos has a geometric morphism to $\mathbf{Set}$ , which is not specified by just saying ‘pretopos’. I have another short blog post in mind, more serious, about the family of generalisations of Grothendieck toposes that I like that include pretoposes, so more about this later. But this is relevant for the category of condensed sets, too: there is a global points functor, and sets embed as discrete condensed sets.

An example of a presheaf without an associated sheaf – UPDATED

Photo of a field of harvested wheat with sheaves leaning up against each other

December 1, 2025December 26, 2025 ~ David Roberts ~ Leave a comment

[This was originally posted on Google+ on 19 June 2013. This has been lightly edited to fit the new format and for clarity. ADDED Dec 2025: it’s also wrong! I explain at the end]

I may have shared this blog posting before, but this is a really good example of when we have a large site which isn’t constrained by a small amount of data for each object: the category $\mathbf{Aff}$ of (affine) schemes with the pretopology of flat surjections. One can define a presheaf on $\mathbf{Aff}$ which has no sheafification for this pretopology, but its definition in the linked blog post explicitly uses von Neumann ordinals. I should like to write down a more structural version of this. I have some points I’d like to clear up, if you want to chip in, namely 2.-4. under ‘Some final comments’. The original source for this material is

William C. Waterhouse, Basically bounded functors and flat sheaves.
Pacific J. Math. Volume 57, Number 2 (1975), 597-610, http://projecteuclid.org/euclid.pjm/1102906018

The example as given by Waterhouse

Given an affine scheme $\mathrm{Spec}(A)$ , assign to it the set of locally constant functions from $\mathrm{Spec}(A)$ to the von Neumann cardinal of the set

$L_A := \sup\{ |k(p)| \mid p \in \mathrm{Spec}(A) \},$

the supremum of the cardinalities of residue fields at points of $\mathrm{Spec}(A)$ , such that the value at any point (which is a cardinal less than $L_A$ ) is smaller than the cardinality of the residue field at that point. This gives a functor $F \colon \mathbf{Aff}^{op} \to \mathbf{Set}$ , using the fact maps of fields are injective.

Simplifying the example

In fact, one can take the site to be merely the full subcategory $\mathbf{Field}^{op}$ of affine schemes which are spectra of fields, since one arrives at a contradiction assuming the existence of a sheafification by using a flat covering $\mathrm{Spec}(k') \to \mathrm{Spec}(k)$ for $k'$ and $k$ fields (in fact any field extension $k \to k'$ gives such a cover). Then locally constant functions $\mathrm{Spec}(k) \to L_k$ are merely elements of $L_k$ , or in other words, $F(\mathrm{Spec}(k))$ can be taken as $k$ itself. In other words, $F$ restricts to the forgetful functor $U \colon (\mathbf{Field}^{op})^{op} = \mathbf{Field} \to \mathbf{Set}$ . This is a very natural presheaf to consider (see comment 3 below regarding the pretopology on this subcategory).

Calculation

Let $X$ denote the constant sheaf on $\mathbf{Field}^{op}$ corresponding to a well-ordered set of the same name. Then there is a map of presheaves $i_X \colon U \to X$ which is just the inclusion for $|k| \leq X$ and the retract $k \to X$ sending $|k| - X$ to the bottom element of $X$ otherwise. Then by the universal property of sheafification, there must be a unique map $X \to U^+$ making the obvious triangle commute, where $U^+$ is the sheafification of $U$ , for any $X$ . In particular, $U \to U^+$ factors through $X$ . Now for any given $k$ take $X > |k|$ so that $U(k) \to X$ is injective, which implies that $U(k) \to U^+(k)$ is injective, and hence that $U \to U^+$ is a mono.

Now we use the fact that for any map $\mathrm{Spec}(k') \to \mathrm{Spec}(k)$ (necessarily a flat cover) we have that the equaliser of the two maps [here we’ve embedded into $\mathbf{Ring}$ , see comments below]

$k' \rightrightarrows k' \otimes_k k'$

injects into $U^+(k)$ (We can check this by applying the natural transformation $U \to U^+$ to the diagram

$k \to k' \Rightarrow k' \otimes_k k'$

(using $\Rightarrow$ for a pair of parallel arrows) and remembering $U^+(k) \to U^+(k')$ is an equaliser.) But $\mathrm{Spec}(k')$ is a point, so this equaliser is $k'$ itself (can we see this directly without going via the spectrum?)

The upshot of the preceding two paragraphs is this: $k'$ must have an injective set map into $U^+(k)$ , but $k'$ can be as large as we like, independent of $U^+$ (take say the function field over $k$ on the power set of $U^+(k)$ , which is certainly a field larger that $k$ ). Thus $U$ cannot have a sheafification.

Some final comments

I find this a nicer example, as one then doesn’t have to mess around with descriptions of cardinals and specially constructed bounded functions. Then one should be able to show without too much effort that given a presheaf on $\mathbf{Aff}$ , or even $\mathbf{Sch}$ , which restricts to the full subcategory $\mathbf{Field}^{op}$ as the forgetful functor $U$ has no sheafification for the flat pretopology, because it would give one for $U$ .
It would be nice to say that the presheaf $F$ on $\mathbf{Aff}$ above was a (nice) Kan extension of the presheaf $U$ , then one wouldn’t have to fiddle with existence of presheaves restricting to $U$ . This seems not unlikely, given the definition of $F$ using bounds on residue fields. Alternatively, we could perhaps extend point 1. to work for any Kan extension of $U$ (left/right as appropriate), rather than an extension up to isomorphism.
We find that all we are doing is taking the fact that the restriction to $\mathbf{Field}^{op}$ of the flat pretopology of $\mathbf{Aff}$ is just the maximal pretopology (actually see (*) below), where all maps are covers, and there is no weakly initial set in $\mathbf{Field}^{op}$ . Such conditions would probably give a non-sheafifiable presheaf on any $C^{op}$ for $C$ a concrete category whose image in $\mathbf{Set}$ is unbounded.
(*) This is being a little slack, as $k' \otimes_k k'$ isn’t a field, but can find a map from it to a field (as $k$ -algebras), and so we get a coverage on $\mathbf{Field}^{op}$ , rather than a Grothendieck pretopology (this shouldn’t change the calculation above). The inclusion functor $\mathbf{Field}^{op} \to \mathbf{Aff}$ is flat, so I think this means it is a morphism of sites as in Remark 2.3.7 of Sketches of an Elephant (certainly covers in $\mathbf{Field}^{op}$ are sent to covers in $\mathbf{Aff}$ ). Any thoughts on this?
On a more foundational/structural note, I would like to be able to define the maps $i_X \colon U \to X$ without choosing a well-ordering of every field, but I’m not sure I can do that, as one might not be able to get enough maps $i_X$ . Really one just needs, for any field $k$ , a sheaf $Z_k$ and a map $U \to Z_k$ such that $U(k) \to Z_k(k)$ is injective. I don’t know how to supply this if I don’t have AC, but I haven’t thought very hard. Ideas? Perhaps we can prove this by contradiction.

….and why this is wrong.

Sadly, the above reasoning falls over right at the part where I claimed “(this shouldn’t change the calculation above)”. It’s true that one can get a singleton coverage on $\mathbf{Field}^{op}$ as described, where every inclusion of fields gives a cover in the opposite category (‘singleton’ in the sense that every covering family consists of a single arrow). The definition of (singleton) coverage doesn’t require pullbacks, merely ‘filler’ squares with the same ‘parallel arrow also is a cover’ condition. For the replacement of the “consider the pullback of $(\iota\colon k\to k')^{op}$ along itself”, we can actually fill the square with the identity map of $k'$ (in the opposite cat) on the remaining two sides. More generally, any maps $(k' \to K)^{op}$ can be used to fill in the square, using the fact every arrow is a cover. From now on I’ll drop all the ‘op’ business for simplicity.

So when we come to consider the sheaf condition for $U$ , the correct definition to use for the general coverage we have here is that for any filling square as suitable, and descent data for the presheaf, it glues uniquely. I have a small quibble in that the Elephant only defines a compatible family (i.e. descent data) for a general coverage as a subordinate clause in the definition of sheaf. But it amounts to this in the current context: descent data for $\iota\colon k\to k'$ is an element $x\in U(k')=k'$ , such that when a span $k' \stackrel{i}{\leftarrow} K \stackrel{j}{\to} k'$ fills the square (i.e. $i\circ \iota=j\circ \iota$ ) then $i(x)=j(x)$ . The sheaf condition is then that given such descent data, we have that there is a unique $x_0\in k$ such that $x=\iota(x_0)$ . Since this condition quantifies over fillers of the square, we can consider the subclass where $K=k'$ , $j=\mathrm{id}_{k'}$ and thus that $i=\alpha\in\mathrm{Gal}(k'|k)$ . The only way that we can have $\alpha(x)=x$ for all elements of the Galois group is that $x$ comes from the (underlying set of the) base field $k$ .

Thus while Waterhouse’s example is not a sheaf, and it doesn’t have a sheafification, its restriction $U$ to the category of fields is already a sheaf.

A game strategy that depends on the Continuum Hypothesis?

A visualisation of the countable ordinal omega to the power of omega

November 19, 2025November 19, 2025 ~ David Roberts ~ 2 Comments

Thanks to @jeanas on Mathstodon, here’s a fun pair of problems, the first easy, the second not so much.

Suppose Sam and Joe are playing a game where Sam is given two finite ordinals, and has to give one to Joe. Joe then is allowed to make finitely-many guesses at the other one, and they both win if a correct guess is made. One can find an always-winning strategy for this game.

Now suppose Sam and Joe are given three ordinals, not necessarily finite. Sam has to give Joe two of them, and Joe has to make finitely-many guesses at the third. Otherwise the game is the same The challenge (as I understand it) is now to show that there is a winning strategy if and only if all three ordinals are countable.

However, the way I was originally told about this game was in terms of picking real numbers, not ordinals, and that the answer was dependent on the Continuum Hypothesis being true or not—which is famously not provable nor refutable from the ZFC axioms of set theory. The hint was that one should use the well-ordering principle, which immediately clued me in that the phrasing in terms of real numbers was a red herring. Using the Axiom of Choice, then if (and only if) the CH is true, then there is a bijection $\mathbb{R} \stackrel{\simeq}{\to} \omega_1$ , where $\omega_1$ is the set of all countable ordinals. Since the solution to the first problem really uses the well-orderedness of the set of all finite ordinals (namely, there are no infinitely-long, strictly descending sequences of ordinals, finite or otherwise), this is a big hint for the second. Otherwise, just stating the problem as being about real numbers hides the solution behind both AC and CH, and there are no useful order properties of the reals with their usual ordering here. It feels like a gotcha to state this in terms of real numbers, when thinking up the actual winning strategy is more fun once you have the non-obvious secret step revealed, since all the information is now available. It’s not the strategy that’s independent of ZFC, but rather the translation trick that requires a strong axiom that’s also independent from the other axioms that make up the ZF fragment.

The ATLAS of Finite Groups is sound

Cover of the book "ATLAS of Finite Groups"

September 27, 2025September 29, 2025 ~ David Roberts ~ Leave a comment

I missed this at the end of last year, but I had been periodically looking to see if it had been done: the last gap in verifying that the ATLAS of Finite Groups is bug-free has been filled! (Added: not formally verified, I might add, just in the old-fashioned sense of the tables being replicated independently, and this now being reproducible) This book was a landmark in computational group theory first published in 1985, but the software that ran the computations on finite group representations (especially the larger finite simple groups that were listed) was not exactly something that could be re-run on modern systems. My recollection was that a lot of the computations in the 1970s were essentially only done once, and then relied on ever after. The data from the ATLAS was included in more modern systems, but sometimes not recomputed. Here’s what that generally looks like in the book:

Image of the character table of the Mathieu group M12 from the ATLAS of Finite Groups — The character table of the Mathieu group $M_{12}$

The character table of the Mathieu group $M_{12}$

As a result, Jean-Pierre Serre raised the issue at the end of an April 2015 talk of when proofs depended on the ATLAS. He thought that depending on the Classification of Finite Simple Groups was alright, because everyone knows that that was a theorem whose status was understood to be conditional on people having assembled the thousands of papers correctly without holes. And the second generation proof team is working through and collating an improved and unified proof.

Just over ten years ago I asked for the status of the independent verification of the ATLAS, prompted by seeing Serre’s talk. And there was already at that time some progress, but by 2016 we had the survey

T. Breuer, G. Malle, and E. A. O’Brien, Reliability and reproducibility of Atlas information, Contemporary Mathematics 694 (2017) pp 21–31, doi:10.1090/conm/694/13960, arXiv:1603.08650

Thomas Breuer has been a major player in the efforts to independently verify the ATLAS data, and he appears as (co)author on all the rest of the papers I give. You can consult his page on the matter for gritty details such as run-times on how long it takes to verify/recompute character tables, and Magma files used.

Thomas Breuer, Constructing the ordinary character tables of some Atlas groups using character theoretic methods, arXiv:1604.00754,
Thomas Breuer, Kay Magaard, Robert Wilson, Verification of the ordinary character table of the Baby Monster, J. Algebra 561 (2020) pp 111–130, doi:10.1016/j.jalgebra.2019.06.047, arXiv:1902.07758
Thomas Breuer, Kay Magaard, Robert A. Wilson, Some steps in the verification of the ordinary character table of the Baby Monster group, arXiv:1902.06823.

These three papers got us to the point of having verified everything except the character table for the Monster group, the largest of the sporadic finite simple groups (but by no mean the ‘largest’ finite simple group: exceptional groups of Lie type over small-ish finite fields can be enormous, not to mention the sizes of finite simple groups are obviously unbounded). The third paper was technical with code, supporting the second paper.

Now there had been corrections to the ATLAS over the years: typos and whatnot happen. Finite group theorist Derek Holt wrote, when saying how for a book he coauthored they tried to independently check all the ATLAS data they needed to use,

…we found remarkably few errors in the ATLAS. I think there might have been one or two very small and minor inaccuracies in some of the structure descriptions, which we reported to the authors, and I think they might have known about them already

From memory a list of corrections to the original edition of the ATLAS was included in An Atlas of Brauer Characters, published a decade after the original ATLAS. This has a subsequent list of corrections to typos, but nothing drastic. In any case, Serre spoke in April 2017 and brought up the point again, and it seem that the Breuer–Malle–O’Brien paper had addressed his concerns, modulo, at that time, the Monster, the Baby Monster (now done by Breuer–Magaard–Wilson above) , and the double cover of the Baby Monster (now done in the solo Breuer paper above).

As of 2021, only the Monster remained, but there was a report on progress from December 2020 on this case, which said it had been done modulo believing the character tables of three subgroups of the Monster (with complex names I won’t write here). One of these tables had been re-done in 2010 by Barraclough and Wilson. But now, or rather, as of December 2024, all of this has now been finished, and there are two papers on the arXiv with the details:

Thomas Breuer, Kay Magaard, Robert A. Wilson, Verification of the conjugacy classes and ordinary character table of the Monster, arXiv:2412.12182
Thomas Breuer, Kay Magaard, Robert A. Wilson, Some steps in the verification of the ordinary character table of the Monster group, arXiv:2412.09313

The first paper is more theoretical, and the second paper contains details of code and so forth, and appendices that give details on how the character tables of the three Monster subgroups mentioned earlier are checked. There are no surprises, the ATLAS turned out to be ok in the end, but it’s nice to know that nothing got broken in the meantime that needed fixing. Now, ten years after Serre complained in the first talk above, we can be much much more confident that citing the ATLAS in a proof is not going to cause grief down the road.

Glue work instead of pasting diagrams

A little girl applying glue to paper, to make a crown

August 20, 2025August 20, 2025 ~ David Roberts ~ Leave a comment

[this is an old draft that I just had sitting on my laptop, I’m not sure it’s even finished, and ~~I can’t remember the talk I mention – if someone can find it that would be great!~~ I just want to post this rather than not]

One particular thing that I (now not-so) recently learned a name for is “glue work”, crucial infrastructure-style contributions to the research community in one’s area,but which aren’t measured and aren’t really counted towards anything like actually getting the next job/promotion—at least not compared to the usual beans that get counted. One can do something like build a really good resource that’s widely used and makes research better for many, many people in a diffuse way, but it’s not a research paper in a high-reputation journal. But as a contribution to research it has just as wide a reach, if not wider. The talk I watched on that (held by the Topos Institute) mentioned how such work is expected of more senior people, and if junior people do it, it is detrimental to their career because it “doesn’t count”. I highly recommend watching the talk, “Sociotechnical infrastructure for mathematics research“, by Steven Clontz [thanks to Paul Fabel on mathstodon for figuring out this was the talk].

I did a whole bunch of this as a junior academic, and I thought it would show me to be a well-rounded researcher, with lots of interesting and diverse contributions to my field. Perhaps it had the effect of harming my (relatively few, for reasons) goes at being a more successful academic researcher—at least as measured by university jobs/grants. It’s a hard call, but letting people like past me know “if you make these positive contributions to your research community, you are likely reducing your chances” is important. I find it a bit sad.

[image source: https://www.pexels.com/video/a-little-girl-applying-glue-on-paper-6978130/ I think that more often than not, glue work is undertaken by women, with the effects on their careers as you can guess]

BB(5)=47,176,870: BB(6) is … astronomically larger

A visualisation of the evolution of the state of the tape of the current longest-known running 6-state 2-symbol Turing Machine, 1RB1RA 1RC1RZ 1LD0RF 1RA0LE 0LD1RC 1RA0RE, together with a tabulation of the rules it runs by

July 9, 2025 ~ David Roberts ~ Leave a comment

Just over a year ago, the Busy Beaver Challenge group managed to prove and formally verify that the fifth Busy Beaver number is indeed 47,176,870. You can read the Quanta Magazine article on that for background on what that is, and why it’s a fun human story as well as a little bit of a big deal. Or else have a look at the Wikipedia page for a less journalistic take. In 2022 it was found that the next BB number, BB(6), was no less than $10 \uparrow \uparrow 15$ , namely an exponential stack of fifteen 10s. For comparison, $10\uparrow\uparrow 2$ is already more than 200 times larger than BB(5), and increasing that index at the end makes this function grow superexponentially. In fact, about as fast as the Ackermann function, which grows so fast it isn’t primitive recursive (though still computable, unlike BB).

The recent news is that first, a lower bound for BB(6) was found, namely $10\uparrow\uparrow 11,010,000$ , by finding a 6-state Turing Machine that will halt sometime after running for that many steps (and, even better, formally verifying this behaviour). This is a mind-boggling number. Recall that even the measly $10\uparrow\uparrow 3 = 10^{10,000,000,000}$ , compared to which the number of particles in the observable universe is a rounding error away from zero. But then even more recently, a Turing Machine with 6 states was found that ran for much, much …. much longer than that before halting. Namely

$2\uparrow\uparrow2\uparrow\uparrow2\uparrow\uparrow10$

many steps. o.O Using the next iteration of Knuth’s up-arrow notation, this number blows $2\uparrow\uparrow\uparrow 5$ out of the water. It’s hard to conceive of this number as a concrete actual natural number, one is left resorting to imagining exponential towers whose height is also measured in exponential towers.

For people who know famous big numbers, no, this last lower bound is not bigger than Graham’s number $g_{64}$ . But we just don’t know (yet?) if BB(6) is bigger than $g_{64}$ . At some point, BB(k) will be (vastly) bigger than $g_k$ in the recursive definition that extends the definition of Graham’s number, for all k past some threshold, and that threshold may well be 6 or 7, but it is no bigger than 14, since BB(14) is bigger than Graham’s number, which is bigger than $g_{14}$ . If anyone knows tighter bounds on when $BB(k) > g_k$ , I’d love to hear about it! However, it should be noted that BB(14) beating Graham’s number is certainly more impressive than learning that BB(64) beats it, merely because the BB function has blown past the $g$ -function before that point. To point to the current lower bound of BB(7), we need much more iteration of the up-arrow notation to say how long the current champion machine runs before halting, namely

$2\uparrow^{11}2\uparrow^{11}3$

steps. One of the former world-record holders for longest-running Busy Beavers has made a $1000 bet that BB(7) will be proved bigger than Graham’s number in the next ten years, and the region of doubt is that it will be proved in time, not so much that the inequality is wrong. In fact, there’s a lot of interesting discussion in the comments to Aaronson’s blog post, where I was sent by Bill Gasarch’s blog post on the news for the details.

Infinite descent in ancient times

January 21, 2025 ~ David Roberts ~ Leave a comment

Check out this quote from Euclid

But if it is composite, some number measures it. Thus, if the investigation is continued in this way, then some prime number will be found which measures the number before it, which also measures A. If it is not found, then an infinite sequence of numbers measures the number A, each of which is less than the other, which is impossible in numbers.

This brings Euclid’s Proposition 31 of Book 7 of “Elements” to the logical conclusion: every (positive whole) number (bigger than 1 [*]) has a prime number that divides it, aka a prime divsor.

This may well be the oldest proof relying essentially on the technique known as ‘induction’, or rather, a related technique known as ‘infinite descent’ (the ‘each of which is less than the other’ part). I only just learned this from KP Hart on MathOverflow, and I will use it if I ever have to talk about axiomatic arithmetic to students. Or maybe just mention it anyway, because I like it.

What is really cool is that this uses the well-founded property of the natural numbers (or rather, those greater than 1), and but doesn’t mention the usual dreaded ‘induction step’ that is so mysterious to students. There’s no ‘now prove it for n+1’, because the definition of composite number means you actually know nothing about the size of any hypothetical divisor.

[*] The idea that 1 is to be considered a ‘number’ was slower to develop in mathematics than you might think!

Morphisms between abelian differentiable gerbes

November 10, 2024 ~ David Roberts ~ Leave a comment

Following on from last post I want to talk about the appropriate notion of morphism between the objects I defined. Recall that these are Lie groupoids $X$ with a map to the manifold $M$ satisfying some properties ( $X_0 \to M$ and $X_1 \to X_0\times_MX_0$ are surjective submersions), and then equipped with a little bit of extra structure. We will fix a bundle of abelian Lie groups $\mathcal{A} \to M$ (where this projection map is a surjective submersion). Then the extra structure is that we have a specified isomorphism between the bundle of Lie groups $\Lambda X$ (see the previous post for notation) and the pullback of $\mathcal{A}$ along $X_0\to M$ , such that that the conjugation action of $X$ on $\Lambda X$ factors through the resulting descent data for $\Lambda X$ in the guise of its encoding of an action of the Čech groupoid $\check{C}(X_0)$ on $\mathcal{A}$ . This is the definition of an abelian $\mathcal{A}$ -bundle gerbe.

Now, given two Lie groupoids $X\to \mathrm{disc}(M)$ and $Z\to \mathrm{disc}(M)$ over $M$ (with the surjective submersions as above) we can consider internal functors $X\to Z$ between them, commuting with the projections to $M$ . Such a functor is a map of (generalised bundle) gerbes over $M$ , but we want to consider more specifically maps of abelian $\mathcal{A}$ -bundle gerbes.

The cartoon analogy to keep in mind as to what could happen without further conditions is if you had a pair of principal $G$ -bundles $P \to M$ and $Q\to M$ , and a map $f\colon P\to Q$ over $M$ , but it was only equivariant in a weak way, namely that $f(pg) = f(p)\phi(g)$ where $\phi$ is a group automorphism of $G$ . Abelian $\mathcal{A}$ -bundle gerbes are principal 2-bundles for the 2-group (or bundle of 2-groups) $\mathcal{A} \rightrightarrows 1$ , but the way in which they inherit this action because of the isomorphism $\Lambda X\simeq \mathcal{A}$ , and functors automatically commute with the action of $\Lambda X$ on $X$ . So instead of having to impose a condition analogous to $f(pg) = f(p)\phi(g)$ , this comes for free, in that from a functor (over $M$ ) $f\colon X\to Y$ between abelian $\mathcal{A}$ -bundle gerbes you get an endomorphism of bundles of groups of the pullback of $\mathcal{A}$ along $\pi_X\colon X_0\to M$ . This arises as the composite $\pi_X^* \mathcal{A} \simeq \Lambda X \to f_0^*\Lambda Y \simeq f_0^*\pi_Y^*\mathcal{A} \simeq \pi_X^* \mathcal{A}$ .

So the requirement that we are going to impose is going to firstly ensure this endomorphism of $\pi_X^*\mathcal{A}$ is an automorphism, which we can then ask is actually descendable, so that it arises from an automorphism of $\mathcal{A}$ down on $M$ (in principle, one could ask the endomorphism descends without asking it’s an automorphism, but not today). But, secondly we are going to ask that the automorphism of $\mathcal{A}$ is in fact the identity map. This is the analog of asking the group automorphism $\phi$ above is the identity map on $G$ . However, we need to specify these conditions explicitly in a way that makes sense in the picture of bundle gerbes as internal groupoids.

So, it’s been a long way around to give the intuition (and this blog post has been long delayed, due to work commitments and other things), but the final definition is in fact rather easy. All that we ask is that $\Lambda X \to f_0^*\Lambda Y$ is a pullback, so that the square witnessing the fact $\Lambda X$ is a pullback of $\mathcal{A}$ along $\pi_X$ is the pasting of the similar square for $\Lambda Y$ and the square witnessing the fact $\Lambda X$ is a pullback of $\Lambda Y$ .

One then has to check that from this definition, and the condition that the descent data for $\pi_X^*\mathcal{A}$ “comes from” the adjoint action of $X_1$ on $\Lambda X$ , means that the resulting automorphism of $\pi_X^*\mathcal{A}$ descends to the identity map.

Given the above, one can then go back and examine what happens if you drop the requirement the bundle gerbe is abelian. Certainly the definition given (that $\Lambda X \simeq f_0^*\Lambda Y$ ) still makes sense even when we only ask that we have a “ $\mathcal{G}$ -bundle gerbe”, for any bundle of Lie groups $\mathcal{G}$ on $M$ .

The final point, I think, is that a morphism of abelian $\mathcal{A}$ -bundle gerbes is a functor of Lie groupoids satisfying two conditions: commuting with the functors down to the base, and the pullback condition discussed above. In particular there is no extra data that we need to supply to define a morphism.

What does this tell us in the case of a vanilla (abelian) $U(1)$ -bundle gerbe, as originally defined by Michael Murray? (Though note that in the paper, everything’s done with $\mathbb{C}^\times$ -bundles instead) He defined a notion of morphism of bundle gerbes $X\to Y$ , which amounts to a functor over $M$ such that $X_1 \to Y_1$ is a map of $U(1)$ -bundles covering the map $X_0\times_M X_0 \to Y_0\times_M Y_0$ . If I take $\mathcal{A}= M\times U(1)$ , then as last time, a bundle gerbe on $M$ is a Lie groupoid $X\to M$ over $M$ equipped with a trivialisation $\Lambda X\simeq X_0\times U(1)$ as a bundle of groups, satisfying conditions. Given two such, say $X$ and $Y$ , a morphism between them in my sense is a functor over $M$ such that the restriction of $f_1$ to $\Lambda X$ respects the trivialisations. This condition then ensures that the full $f_1$ is a map on the total spaces of $U(1)$ -bundles.that is $U(1)$ -equivariant, so we have a morphism of bundle gerbes in Murray’s sense. Now assume we have a pair of abelian $U(1)$ -bundle gerbes $X,Y$ , and a morphism between them in Murray’s sense. Then since $f_1\colon X_1 \to Y_1$ is a map of $U(1)$ -bundles covering $X_0\times_M X_0 \to Y_0\times_M Y_0$ , the resulting square is a pullback, and then the restriction of this square along the diagonal maps $X_0\to X_0\times_MX_0$ and $Y_0\times_MY_0$ (to get the map $\Lambda X\to \Lambda Y$ over $X_0\to Y_0$ ) is still a pullback, and then we have recovered my definition of morphism. This I have recovered the traditional definition up to equivalence.

The natural continuation of the above discussion is to examine what we should do with natural transformations (or rather, natural isomorphisms, as they all are). Of course, the data of a natural isomorphism $a\colon f\Rightarrow g\colon X\to Y$ over $M$ is a smooth function $X_0\to Y_1$ satisfying the usual naturality condition. To compare to the literature, we need to go to section 3.4 of Danny Stevenson’s PhD thesis (which builds a 2-category of bundle gerbes and morphisms à la Murray, in the lead up to Proposition 3.8). The definition of 2-arrow is slightly nontrivial and on inspection of Lemma 3.3 one can see that the definition of abelian bundle gerbe I have given is being implicitly relied on, where descent data is given by conjugation in the groupoid, to descend a certain bundle down to $M$ . The data of a 2-arrow $a\colon f\Rightarrow g\colon X\to Y$ in Stevenson’s sense gives a smooth map $X_0 \to Y_1$ that, I think by the construction in Lemma 3,3, is in fact the data of a natural isomorphism (implicit in the equation at the bottom of page numbered 27). The question is: what else do we get? Or, another tactic is to see if an arbitrary natural isomophism $f\Rightarrow g$ is enough to define a 2-arrow in Stevenson’s sense. Given the component map $a\colon X_0\to Y_1$ , it gives a section $s_a$ of Danny’s bundle $\hat{D}\to X_0$ (which is the pullback of $Y_1\to Y_0\times_M Y_0$ by $(f_0,g_0)\colon X_0\to Y_0\times_MY_0$ ). What we need to check is that $s_a$ descends to a section of the descended bundle $D_{\bar{f},\bar{g}}$ . However, we know that the descent data for $\hat{D}$ that is constructed in Lemma 3.3 is by conjugation, and the naturality condition on $a$ in fact tells us that $s_a$ is compatible with this descent data, and so descends to a section as needed.

So in fact we can define a 2-arrow between morphisms (in my sense) to just be a natural isomorphism satisfying no additional conditions. And so we can define a 2-category of bundle gerbes as a locally full—and not full—sub-2-category of the 2-category of Lie groupoids over $M$ . My discussion in the past two blog posts (this and the previous one) is much longer than the bare definition requires; this is all to motivate and explain. The actual definition could be at most a single paragraph long—cf the more than two pages including two lemmas with proofs needed to define what turns out to be an isomorphic 2-category in Danny’s thesis, in addition to the theory of bundle gerbes that requires.

It turns out this post is longer than I thought it would be, but that’s ok. The next step is to move on from this naive 2-category of bundle gerbes, because the above really isn’t the one that people want. But that will fall out from an existing construction in the literature. I’m not sure, though, if I should first treat the analog of the above definition stack where we add connections to things, before moving on to the “real” definitions for both.

Small fun observation about finite topological spaces, and some challenges

The start of Sid Morris' paper "Are finite topological spaces worth of study", including the introduction and the statement of the main theorem, described in the blog post.

September 20, 2024 ~ David Roberts ~ Leave a comment

Here’s a fun thing: if you want to generate a random finite $T_0$ space, instead select a random subset from $\mathbb{S}^n$ , the $n$ -fold power of the Sierpinski space $\mathbb{S}$ , since every $T_0$ space embeds into some (arbitrary) product of copies of the Sierpinski space. (Recall that $\mathbb{S}$ has underlying set $\{0,1\}$ , and the only open subsets are $\emptyset, \{0\}, \{0,1\}$ . The point $0\in \mathbb{S}$ is open, which is true for all points in discrete spaces, but not for example in the reals with the standard topology, where they are closed. The point $1$ is closed in $\mathbb{S}$ .) And, in fact, a topological space is $T_0$ if and only if it can be embedded in some power of $\mathbb{S}$ : in one direction, $T_0$ spaces being closed under arbitrary products and subspaces are standard facts, and then one checks manually the Sierpinski spaces is $T_0$ ; in the other, we map the space $(X,\tau)$ to $\mathbb{S}^\tau$ via $x\mapsto (f_U(x))_{x\in U}$ , where $f_U\colon X\to \mathbb{S}$ is the classifying map for the open set $U\in \tau$ (this sends everything in $U$ to the open point $0\in \mathbb{S}$ , and everything else to $1$ ).

Challenge 1: Figure out what distribution on homeomorphism classes of $T_0$ spaces of size at most $2^n$ arises from the taking the uniform distribution on $P(\mathbb{S}^n)$ . It’s certainly not uniform, since (the homeomorphism classed of) $\mathbb{S}^n$ and $\emptyset$ appear once, but the one-point space appears many times. It feels like one should use some kind of approximation or asymptotics, rather than attempt to get an exact answer.

Challenge 2: Can one come up with a scheme to sample random $n$ -element topological spaces in this way (taking subspaces of products of Sierpinski spaces) that is “good”? Here “good” might be (approximately) uniform, or some other consideration.

If you want non- $T_0$ spaces, then it’s possible to do something similar with a 3-point space instead of the 2-point Sierpinski space, by a cute 1972 result of my fellow Australian Sid Morris, Are finite topological spaces worthy of study?. Here the space is the so-called “Davey space” $D = (\{0,1,2\},\delta)$ where one takes the codiscrete topology (with only two opens) and declares the point $\{0\}$ to be open: that is, the sets $\delta = \{ \emptyset, \{0\}, \{0,1,2\}\}$ are the only open subsets (note that now the subset $\{1,2\}$ is closed, and no points are closed). In particular, for a finite space $(X,\tau)$ , it embeds in $D^{X\sqcup \tau}$ , as shown in Sid’s note. One can now find a random arbitrary finite space of size at most $3^n$ by considering random subsets of $D^n$ .

Challenge 3: Repeat Challenge 1 for arbitrary spaces instead!

Abelian differentiable gerbes – recap

September 15, 2024 ~ David Roberts ~ 1 Comment

Bundle gerbes are a class of objects that are my bread and butter, but there are many different ways of thinking about them. In my paper with Raymond Vozzo we wrote a different take that starts from the point of view of plain Lie groupoids (which we allow to be infinite-dimensional) and slowly builds in the conditions so that we end up with something equivalent to Murray’s original definition.

I want to summarise our approach to bundle gerbes, because I want to do a series of posts building up to a new construction of the bicategory of bundle gerbes on a given manifold. As a first step, we need to define bundle gerbes, but also define morphisms between them as a certain class of internal functors. Actually, it’s going to develop a theory of a slightly more general object, which won’t impact the results. Ultimately this is because I need to fix The Paper where my coauthor and I used a somewhat naive approach to gauge transformations of curvings on a bundle gerbe, and have an idea I will slowly develop in public, since my time for pure research is now more constrained. And I miss the early days of the n-Category Café where half-developed research was posted all the time for public discussion.

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