https://thosgood.com/ Tim Hosgood's blog 2023-09-08T00:00:00Z Tim Hosgood https://thosgood.com tag:thosgood.com,2023-09-08:/blog/2023/09/08/cech-totalisation/ Čech totalisation 2023-09-08T00:00:00Z 2023-09-08T00:00:00Z <p>After a pretty long time (over two and half years or so), <a href="https://www.zeinalian.com/">Mahmoud Zeinalian</a> and I have finished our paper “Simplicial presheaves of Green complexes and twisting cochains” (arXiv:<a href="https://arxiv.org/abs/2308.09627">2308.09627</a>). In this blog post I want to give a brief overview of one of the main technical tools that we use, which we call <em>Čech totalisation</em>. The full story involves model categories and homotopy limits and all this sort of machinery, but the main part of this post will try to keep this to a minimum, and just talk about a surprisingly useful roundabout way of describing principal bundles. There might be a second part to this, but for those interested or wanting for more details I recommend just delving into the paper — I spent a lot of time trying to make it as readable as possible!</p> <!-- more --> <h1 id="the-motivating-example-principal-g-bundles">The motivating example: principal <span class="math inline">G</span>-bundles</h1> <p>Here’s a really over-the-top way of constructing/defining principal <span class="math inline">G</span>-bundles, whose usefulness we will soon come to justify. We’re first going to go very quickly, with an nLab-style approach, but we’ll come back and explain things more concretely afterwards.</p> <p>Let <span class="math inline">G</span> be a Lie group (i.e. a group object in the category <span class="math inline">\mathsf{Man}</span> of smooth manifolds). Then Yoneda gives us a presheaf <span class="math display"> y^G = \mathsf{Man}(-,G) \colon\mathsf{Man}^\mathrm{op}\to\mathsf{Set} </span> but we can in fact endow this presheaf with the structure of a Lie group, by using the Lie group structure of <span class="math inline">G</span> “pointwise”. This means that we can deloop <span class="math inline">y^G</span> to obtain a presheaf of one-element groupoids <span class="math display"> \mathbb{B}y^G = \mathbb{B}\mathsf{Man}(-,G) \colon\mathsf{Man}^\mathrm{op}\to\mathsf{Grpd}. </span> In other words, given any smooth manifold <span class="math inline">X</span>, we obtain a groupoid <span class="math inline">\mathbb{B}y^G(X)</span> with a single object <span class="math inline">*</span> and with endomorphism group <span class="math inline">\operatorname{Hom}(*,*)\cong\mathsf{Man}(X,G)</span>. Finally, for this step, we can take the categorical nerve <span class="math inline">\mathcal{N}\colon\mathsf{Cat}\to\mathsf{sSet}</span> to obtain a presheaf of simplicial sets <span class="math display"> \mathcal{N}\mathbb{B}y^G \colon\mathsf{Man}^\mathrm{op}\to[\Delta^\mathrm{op},\mathsf{Set}]. </span> So if we go back and forget about our choice of <span class="math inline">G</span>, instead leaving that open, we get a functor <span class="math display"> \mathcal{N}\mathbb{B}y^{(-)} \colon\mathsf{LieGroup}\to[\mathsf{Man}^\mathrm{op},[\Delta^\mathrm{op},\mathsf{Set}]]. </span></p> <p>Let’s now think about covers. Let <span class="math inline">\mathsf{Man}_\mathcal{U}</span> denote the category whose objects are pairs <span class="math inline">(X,\mathcal{U})</span> of a smooth manifold and a (good) cover, and whose morphisms <span class="math inline">(X,\mathcal{U})\to(Y,\mathcal{V})</span> are the morphisms <span class="math inline">f\colon X\to Y</span> in <span class="math inline">\mathsf{Man}</span> such that <span class="math inline">\mathcal{U}</span> is a refinement of <span class="math inline">f^{-1}(\mathcal{V})</span>. Then we can write the Čech nerve <span class="math inline">\check{\mathcal{N}}</span> as a functor <span class="math display"> \check{\mathcal{N}} \colon\mathsf{Man}_\mathcal{U}\to[\Delta^\mathrm{op},\mathsf{Man}] </span> and then the purely abstract fact that <span class="math inline">[\mathcal{C},\mathcal{D}]^\mathrm{op}\cong[\mathcal{C}^\mathrm{op},\mathcal{D}^\mathrm{op}]</span> for any categories <span class="math inline">\mathcal{C}</span> and <span class="math inline">\mathcal{D}</span> lets us apply an <span class="math inline">\mathrm{op}</span> to get a functor <span class="math display"> \check{\mathcal{N}}^\mathrm{op} \colon\mathsf{Man}_\mathcal{U}^\mathrm{op}\to[\Delta,\mathsf{Man}^\mathrm{op}]. </span></p> <p>We can precompose (which we write with as <span class="math inline">(-)^*</span>) the above <span class="math inline">\mathcal{N}\mathbb{B}y^{(-)}</span> with this <span class="math inline">\check{\mathcal{N}}^\mathrm{op}</span> to obtain a single functor <span class="math display"> (\check{\mathcal{N}}^\mathrm{op})^*\mathcal{N}\mathbb{B}y^{(-)} \colon\mathsf{LieGroup}\to[\mathsf{Man}_\mathcal{U}^\mathrm{op},\mathsf{csSet}] </span> where <span class="math inline">\mathsf{csSet}=[\Delta,[\Delta^\mathrm{op},\mathsf{Set}]]</span> is the category of cosimplicial simplicial sets. The very last step (for real this time) is to then <em>totalise</em> over the cosimplicial structure, i.e. to apply the functor <span class="math inline">\operatorname{Tot}\colon\mathsf{csSet}\to\mathsf{sSet}</span> that turns a cosimplicial simplicial set into a simplicial set in exactly the same way that taking the total complex turns a bicomplex into a complex (don’t worry, we’ll talk more about what this functor is later). All in all, we get a simplicial presheaf on the category of smooth manifolds with chosen cover, namely <span class="math display"> \operatorname{Tot}(\check{\mathcal{N}}^\mathrm{op})^*\mathcal{N}\mathbb{B}y^{(-)} \colon\mathsf{LieGroup}\to[\mathsf{Man}_\mathcal{U}^\mathrm{op},\mathsf{sSet}]. </span></p> <p>Now comes the punchline.</p> <div class="itenv" title="Theorem"> <p>For any Lie group <span class="math inline">G</span> and space with cover <span class="math inline">(X,\mathcal{U})</span>, the simplicial set <span class="math inline">\operatorname{Tot}(\check{\mathcal{N}}^\mathrm{op})^*\mathcal{N}\mathbb{B}y^{G}</span> is the <strong>space</strong> of principal <span class="math inline">G</span>-bundles.</p> </div> <p>What do we mean by this? Well, we are saying the following:</p> <ul> <li>the resulting simplicial set is actually a Kan complex, and thus earns the name of “space”;</li> <li>the points of this space are exactly principal <span class="math inline">G</span>-bundles;</li> <li>a path between two points in this space is exactly an isomorphism of principal <span class="math inline">G</span>-bundles;</li> <li>there is no higher homotopical information.</li> </ul> <p>We can actually even modify this construction to obtain a “space” where the paths are mere morphisms of principal <span class="math inline">G</span>-bundles, but this is then only a <em>quasi-category</em> instead of a true space.</p> <p>This story is very <span class="math inline">1</span>-categorical, and can even be seen as a <span class="math inline">1</span>-categorical version of §3.2.1 of “Čech cocycles for differential characteristic classes: an <span class="math inline">\infty</span>-Lie theoretic construction” by Fiorenza, Schreiber, and Stasheff (DOI:<a href="https://dx.doi.org/10.4310/ATMP.2012.v16.n1.a5">10.4310/ATMP.2012.v16.n1.a5</a>). The <span class="math inline">\infty</span>-categorical version turns up in the paper by me and Mahmoud in the following guise:</p> <blockquote> <p><em>The space of twisting cochains is given by the Čech totalisation of the maximal Kan complex of the dg-nerve of the category of finitely generated free complexes.</em></p> </blockquote> <p>This may mean nothing to you (twisting cochains à la Toledo and Tong seem to be rather underappreciated in modern literature, imho), but I promise that it’s a cool result (just not one that I have the space to delve into today). In fact, you can take this as a motto if you like:</p> <blockquote> <p><em>Twisting cochains are the <span class="math inline">\infty</span>-analogue of vector bundles, a sort of “homotopy-coherent bundles”.</em></p> </blockquote> <p>Now, I am sure that the <span class="math inline">1</span>-categorical statement (concerning principal <span class="math inline">G</span>-bundles) is a known result, even though I can’t find it written down anywhere (except for in our paper). Indeed, all it’s really saying is that <em>“bundles are homotopy limits of locally trivial stuff”</em>. Let’s take this rather glib summary and try to see if we can understand it in more detail.</p> <h1 id="the-čech-nerve-totalisation-and-reedy-fibrancy">The Čech nerve; totalisation and Reedy fibrancy</h1> <p>Given a cover <span class="math inline">\mathcal{U}</span> of a space <span class="math inline">X</span>, the <em>Čech nerve</em> is the simplicial space given in degree <span class="math inline">p</span> by the disjoint union of all non-empty <span class="math inline">p</span>-fold intersections of elements of <span class="math inline">\mathcal{U}</span>, i.e. <span class="math display"> \check{\mathcal{N}}\mathcal{U}_p = \coprod U_{\alpha_0\ldots\alpha_p} </span> and the face and degeneracy maps are given by dropping/repeating indices <span class="math inline">\alpha_i</span>. If the cover is “good”, then this gives a fibrant replacement of <span class="math inline">X</span>, and so can be used to compute cohomology (or more general mapping spaces). I love the Čech nerve — I think it turns up in almost everything I write — but I’d rather spend my little remaining energy describing totalisation instead.</p> <p>The prototypical cosimplicial simplicial set is <span class="math display"> \begin{aligned} \Delta[\star] \colon\Delta &amp;\to\mathsf{sSet} \\ [p] &amp;\mapsto\Delta[p]=\operatorname{Hom}_\Delta(-,[p]) \end{aligned} </span> which we can think of as “all of the simplices <span class="math inline">\Delta[p]</span> gathered together”. This lets us define a functor <span class="math display"> \begin{aligned} L \colon\mathsf{sSet} &amp;\to\mathsf{csSet} \\Y_\bullet &amp;\mapsto Y_\bullet\times\Delta[\star] \end{aligned} </span> and this admits a right adjoint, which we call <em>totalisation</em> <span class="math display"> \begin{aligned} \operatorname{Tot} \colon\mathsf{csSet} &amp;\to\mathsf{sSet} \\Y_\bullet^\star &amp;\mapsto \underline{\operatorname{Hom}}_\mathsf{csSet}(\Delta[\star],Y_\bullet^\star) \end{aligned} </span> where we make use of the fact that <span class="math inline">\mathsf{csSet}</span> is enriched over <span class="math inline">\mathsf{sSet}</span> via <span class="math display"> \left(\underline{\operatorname{Hom}}_\mathsf{csSet}(A_\bullet^\star,B_\bullet^\star)\right)_p = \operatorname{Hom}_\mathsf{csSet}(A_\bullet^\star\times\Delta[p],B_\bullet^\star). </span></p> <p>The particularly useful fact about totalisation is the following:</p> <blockquote> <p><em>If <span class="math inline">Y_\bullet^\star\in\mathsf{csSet}</span> is Reedy fibrant, then its totalisation and its homotopy limit are naturally weakly equivalent.</em></p> </blockquote> <p>There are lots of ways of thinking about totalisation, but there’s a very nice one that comes in useful for explicit calculations, namely that <em>a point in the totalisation <span class="math inline">\operatorname{Tot}Y_\bullet^\star</span> consists of <span class="math inline">(y^0,y^1,\ldots)</span> with <span class="math inline">y^p\in Y_p^p</span> such that “they all glue together”</em>. Of course, the end of this sentence is very vague, and I don’t really want to type out everything from the paper again, but I’ll include a nice picture here that sort of describes the idea.</p> <figure> <img src="totalisation-point.png" alt="A point in the totalisation of a cosimplicial simplicial set." /> <figcaption aria-hidden="true">A point in the totalisation of a cosimplicial simplicial set.</figcaption> </figure> <p>Trying to understand higher simplices in the totalisation is a bit of a mess, but it turns out that there is a really nice combinatorial approach involving paths of <span class="math inline">1</span>-simplices in products <span class="math inline">\Delta[p]\times\Delta[q]</span> (see Appendix B.2 in “Chern character for infinity vector bundles”, by Glass, Miller, Tradler, and Zeinalian (arXiv:<a href="https://arxiv.org/abs/2211.02549">2211.02549</a>)). In the paper by me and Mahmoud, we calculate some <span class="math inline">1</span>-simplices explicitly, showing that they recover pre-existing notions of weak equivalences between objects that some people really care about. Below is a nice picture from one of the proofs without any further explanation.</p> <figure> <img src="product-of-simplices.png" alt="The canonical simplicial structure of a product of simplices." /> <figcaption aria-hidden="true">The canonical simplicial structure of a product of simplices.</figcaption> </figure> <h1 id="the-čech-totalisation-functor">The Čech totalisation functor</h1> <p>The story in the previous section is a retelling of what Mahmoud explained to me almost three years ago now, and this kickstarted the project that became the paper now on the arXiv. In “Chern character for infinity vector bundles”, by Glass, Miller, Tradler, and Zeinalian (arXiv:<a href="https://arxiv.org/abs/2211.02549">2211.02549</a>), the same two-step fundamental construction turns up:</p> <ol type="1"> <li>apply the Čech nerve to a simplicial presheaf;</li> <li>totalise the resulting cosimplicial simplicial presheaf.</li> </ol> <p>This combination is what we call <em>Čech totalisation</em>, and it can be understood as a sort of partial sheafification. Indeed, if applied to a presheaf of mere sets, then it recovers the usual process sheafification given by taking sections of the espace étalé (though I must admit that this is not actually explicitly worked out anywhere in writing). There are actually some more general results about when this Čech totalisation does actually compute the sheafification, and you can find these in §5.1 of the Glass, Miller, Tradler, Zeinalian paper mentioned above. But it turns out to be an interesting construction even without thinking about sheafification.</p> <p>Let’s start with a proper definition.</p> <div class="rmenv" title="Definition"> <p>Let <span class="math inline">\mathcal{F}\colon\mathsf{Space}^\mathrm{op}\to\mathsf{sSet}</span> be a simplicial presheaf on the category of spaces, and let <span class="math inline">X\in\mathsf{Space}</span> be a space with cover <span class="math inline">\mathcal{U}</span>. We define the <em>Čech totalisation of <span class="math inline">\mathcal{F}</span> at <span class="math inline">\mathcal{U}</span></em> to be the simplicial set given by <span class="math inline">\operatorname{Tot}\mathcal{F}(\check{\mathcal{N}}\mathcal{U}_\bullet)</span>, i.e. the totalisation of the cosimplicial simplicial set given by evaluating <span class="math inline">\mathcal{F}</span> on the Čech nerve of <span class="math inline">\mathcal{U}</span>.</p> </div> <p>Here are some things that we show in our paper, <em>under the assumption that we have presheaves of Kan complexes</em> (i.e. “globally fibrant” simplicial presheaves).</p> <ol type="1"> <li>The Čech totalisation of <span class="math inline">\mathcal{F}</span> is a Kan complex.</li> <li>If <span class="math inline">\mathcal{F}</span> and <span class="math inline">\mathcal{G}</span> are weakly equivalent to one another, then their Čech totalisations are weakly equivalent to one another.</li> <li>The Čech totalisation computes the homotopy limit (this is due to the fact that evaluating on the Čech nerve gives a Reedy fibrant cosimplicial simplicial set).</li> </ol> <p>These facts turn out to be useful for our intended applications (maybe I haven’t made this very clear, but Čech totalisation is only really a tool in our paper, used to study a rather different problem!) but we can also use them to compare to the existing results in the literature concerning homotopy limits of presheaves of dg-categories. In “Explicit homotopy limits of dg-categories and twisted complexes”, by Block, Holstein, and Wei (arXiv:<a href="https://arxiv.org/abs/1511.08659">1511.08659</a>), it is explained how to calculate homotopy limits of presheaves of dg-categories by using the fact that totalisations of Reedy fibrant objects are weakly equivalent to their homotopy limits. The nice (but expected) result that we show is that one can switch from presheaves of dg-categories to simplicial presheaves by taking the dg-nerve (followed by the maximal Kan complex) without changing the resulting object. More precisely:</p> <div class="itenv" title="Theorem"> <p>Let <span class="math inline">\mathcal{F}\colon\mathsf{Space}^\mathrm{op}\to\mathsf{dgCat}</span> be a presheaf on the category of spaces that sends finite products to coproducts, and let <span class="math inline">X\in\mathsf{Space}</span> be a space with cover <span class="math inline">\mathcal{U}</span>. Then there is a weak equivalence of Kan complexes <span class="math display"> \operatorname{Tot}\langle\mathcal{N}^\mathrm{dg}\mathcal{F}(\check{\mathcal{N}}\mathcal{U})\rangle \simeq \langle\mathcal{N}^\mathrm{dg}\big(\operatorname{Tot}\mathcal{F}(\check{\mathcal{N}}\mathcal{U}\big))\rangle </span> where we write <span class="math inline">\langle-\rangle</span> to denote the maximal Kan complex, and on the left-hand side we are taking the totalisation of cosimplicial simplicial sets, and on the right-hand side we are taking the totalisation of cosimplicial dg-categories.</p> </div> <h1 id="applications">Applications</h1> <p>So what do Mahmoud and I actually use Čech totalisation for in our paper? Well, if you actually want to know the answer to that, then I recommend just having a look at either the introduction or the section named “Narrative” in the actual paper itself, since these are written with minimal technical details. However, here’s a very quick version of the story (and each sentence is justified with real details somewhere in the paper, I promise).</p> <blockquote> <p>Locally free sheaves (read: “vector bundles”) on ringed spaces (read: “schemes”, “manifolds”, or whatever geometric thing you like) are very interesting, but they’re very homotopically rigid. In fact, when we look at complex-analytic (i.e. holomorphic) manifolds, they become too rigid, and we can’t always break down problems about coherent sheaves (the things that geometers often really care about) into problems about locally free ones. There is a solution to this: the <strong>twisting cochains</strong> of Toledo and Tong, which are like homotopy-coherent locally free sheaves, and are very related to the dg-nerve. There is another solution though: the <strong>Green complexes</strong> of Green (a student of O’Brian, a co-author of Toledo and Tong), which are very related to simplicial objects. These two things share a common generalisation, namely <strong>simplicial twisting cochains</strong>.</p> <p>What is the relationship between these three things? Is there some sort of Dold–Kan correspondence? Can we place them all in a unified framework, and use the tools of homotopy theory to talk about them? What does this tell us about the geometry of complex-analytic manifolds? What does this tell us about geometry more generally? And do we learn anything about presheaves of dg-categories along the way?</p> </blockquote> <p>As always, I’d love to talk about this stuff with anybody interested — please do reach out if you like!</p> tag:thosgood.com,2023-07-04:/blog/2023/07/30/translations-part-4/ Translations 2023-07-04T00:00:00Z 2023-07-04T00:00:00Z <p>Hopefully one of my co-authors and I will be uploading a long-awaited preprint on dg-categories, twisting cochains, and homotopy limits to the arXiv “soon”. Until then, here are the small handful of translations that I’ve finished in the year since I last wrote</p> <!-- more --> <p>Unintentionally, all three translations here are Grothendieck related. Let’s start with the smallest one first: Grothendieck’s famous letter to Thomason on derivators, which makes for some interesting “light” reading.</p> <ul> <li>A Grothendieck, “Lettre d’Alexander Grothendieck sur les Dérivateurs, 02.04.91”. Edited by Matthias Künzer. <a href="https://labs.thosgood.com/translations/grothendieck-thomason-91-04-02.html">HTML</a></li> </ul> <p>Next, I <em>finally</em> finished EGA II. I have yet to proofread it, so it probably still has an awful amount of typos and possible errors (if you spot any the please do let me know, either by email or (preferably) by <a href="https://github.com/ryankeleti/ega/issues">submitting an issue on the repo</a>). Note that the web version is down at the moment due to some technical issues that neither Ryan nor I have had the time to fix, but you can still view and download the <a href="https://github.com/ryankeleti/ega#pdfs">PDF versions</a>.</p> <p>Lastly, I’ve been slowly working away at FGA, since it’s not inconceivable that I actually manage to finish this one off sometime by early 2024 (famous last words, I’m sure). I haven’t done much more, but have finished seminar IV (“Hilbert schemes”).</p> <ul> <li>Grothendieck, A. “Technique de descente et théorèmes d’existence en géométrie algébrique, IV”. <em>Séminaire Bourbaki</em> <strong>12</strong> and <strong>13</strong> (1959/60 and 1960/61), Talks no. 190, 195, and 212. <a href="https://thosgood.com/fga">HTML</a></li> </ul> <p>When I finish off the last two seminars (V and VI, both on Picard schemes), I will go back and fix all the little issues (weird bibliographies, no links to original scans, no PDF versions, etc. etc.) but I’m saving that as a nice little “treat” when all the typing is done.</p> <p>Hope you’re all having a lovely summer.</p> tag:thosgood.com,2022-12-03:/blog/2022/12/03/virtual-double-categories-workshop/ Virtual Double Categories Workshop 2022-12-03T00:00:00Z 2022-12-03T00:00:00Z <p>I now realise two things: firstly, I haven’t posted anything in really quite a while; secondly, I should have written this specific blog post a week ago. Over the past five days, <a href="https://bryceclarke.github.io">Bryce Clarke</a> and I have been running the Virtual Double Categories Workshop (a “fun” pun, since the workshop was entirely online, and there is a notion of a double category called “virtual”). The speakers were incredible, both in terms of the talks they gave, in their attitude and enthusiasm for the workshop, and in the variety of their interests. You can find abstracts, slides, and recordings of the talks on <a href="https://bryceclarke.github.io/virtual-double-categories-workshop/">the workshop webpage</a>, but if you want to hear a bit more about my personal interest in double categories and how this workshop came about, then read on.</p> <!-- more --> <p>Back in 2019 I wrote <a href="https://thosgood.com/blog/2019/07/15/more-than-one-less-than-three/">a very cursory post</a> about 2-categories and double categories because I’d heard so much about them at CT2019, and I figured I should start learning something about them. Originally I found them interesting just for their own sake: I think I’ve always found definitions particularly interesting in mathematics, and seeing a new structure that’s meant to generalise or build upon something you already know is pretty fascinating to me. But then thesis writing reared its head and I went back to the maths I was focusing on, which didn’t seem to have much relevance to 2-categorical stuff.</p> <p>Anyway, fast forward a bunch of years to this summer, when I was in Glasgow for ACT2022, talking about <a href="https://doi.org/10.3934/mine.2023036">some joint work</a> (summarised in two blog posts, <a href="https://topos.site/blog/2022/04/diagrammatic-equations-and-multiphysics-part-1/">here</a> and <a href="https://topos.site/blog/2022/04/diagrammatic-equations-and-multiphysics-part-2/">here</a>) with Evan Patterson, Andrew Baas, and James Fairbanks on a category-theoretic approach to (P)DEs using the language of diagram categories, and suddenly double categories seemed to turn up again. Firstly, Evan and I had been discussing a “higher” version of initial functors, which appealed to some inherent 2-categorical structure found in diagram categories, and which might resolve some of the more conceptually unsatisfying things we had come across in the theoretical aspect of this diagrammatic differential equations work. Secondly, it looked like moving up to double categories would be necessary if we wanted to extend the work to talk about multi-domain/mixed-dimensional problems. Thirdly, and somewhat unrelatedly, I saw Bryce again, having first spoken to him (very briefly) all the way back in CT2019.</p> <p>It would be exceedingly generous to describe my understanding of lenses as even “passable”, and yet I can quite honestly say that some of my favourite maths talks I’ve seen over the past few years have been Bryce’s on <a href="https://arxiv.org/abs/2009.06835">his work on internal lenses</a> and, more recently, on their links to <a href="https://www.youtube.com/watch?v=UyBz2uS7Ark">algebraic weak factorisation systems</a>. Bryce has always been very welcoming with the questions I’ve asked him after his talks, which have usually been basically the same: “why does this remind me so much of the story of derived categories and localisation?” (something I’ve written about a few times before — most recently <a href="https://topos.site/blog/2021/11/left-adjoints-lenses-and-localisation/">here</a>). Anyway, we got to talking over lunch one day, and he mentioned that he was thinking of organising an online workshop on double categories, and wanted to know a bit about how I’ve helped to run some online seminars in the past. Knowing that I would attend such a workshop anyway, I offered to help with the technical aspects, and then a few months later he emailed me a list of speakers he thought would be good to invite, and things started to fall into place. Organising an online event is, in many many ways, much simpler than an in-person one — no room bookings, catering, accommodation, etc. — and yet it’s somehow a bit stressful in an entirely different way (<em>will people turn up?</em> <em>what if my internet cuts out?</em> <em>what if a speaker’s internet cuts out?</em> <em>what time zone is best to do each talk in?</em>). But last week things ran pretty much as smoothly as possible, and it seemed like everybody involved got something out of the workshop. Once again, I’d really like to thank all of the speakers — I wish I had the time and energy to write about all of the talks individually, but let me just mention a few that were particularly relevant to the thing I’ve been thinking about recently. I’d also like to thank Bryce, for having the idea in the first place, and then for letting me get involved.</p> <p>Before I go any further, I really do recommend that you look at the abstracts on <a href="https://bryceclarke.github.io/virtual-double-categories-workshop/">the workshop webpage</a> yourself, because the variety of topics, given that all speakers were asked to talk about double categories, is really quite astonishing: formal category theory, size issues, cybernetics, higher homotopy theory, coloured symmetric sequences, monoidal things, rewriting theory, and operads, to name but a few.</p> <p>Lyne Moser gave <a href="https://www.youtube.com/watch?v=RBngz7WXaJw">a talk on representation theorems for enriched categories</a>, investigating the question of when universal properties could be stated in terms of existence of representable objects. This is something which works perfectly fine in the 1-categorical setting (a limit exists if and only if a certain functor is representable if and only if a certain terminal object exists), but becomes difficult the moment you move up to 2-categories. The thing that really stood out to me though was an explanation of how 2-categories relate to double categories, given by answering the even bigger, more general question that I had been wondering about: how do enriched categories relate to internal categories? To say that a category is <span class="math inline">\mathcal{V}</span>-enriched means that its hom-sets are actually objects of <span class="math inline">\mathcal{V}</span>, but to say that it is internal to <span class="math inline">\mathcal{V}</span> <em>also</em> means (in particular) that its hom-sets are actually objects of <span class="math inline">\mathcal{V}</span>. It turns out that <span class="math inline">\mathcal{V}</span>-enriched categories sit fully faithfully inside categories internal to <span class="math inline">\mathcal{V}</span>, and the inclusion has a right adjoint. Lyne explained how this could be used to give an alternative proof of these representation theorems.</p> <p>John Bourke <a href="https://www.youtube.com/watch?v=Zz8NXrDzac4">spoke about different flavours of factorisation systems via double categories</a>, and this reminded me of the left-adjoint/lenses/localisation story that I mentioned always asking Bryce about. I had wondered if there was something interesting to be gained from looking at a double category whose arrows in one direction were fibrations, and in the other direction were cofibrations, but John showed that one “should” instead look at <em>two</em> double categories: both with arbitrary arrows in one direction, and then arrows from the left (resp. right) part of the factorisation system in the other direction. Using this, he explained how to axiomatise various types of factorisation systems in the language of double categories. A particularly interesting question/suggestion was raised by somebody (whose name escapes me, I’m sorry) <a href="https://youtu.be/Zz8NXrDzac4?t=3300">after the talk</a>: maybe it would be worth looking at the <em>triple</em> category which has arbitrary morphisms in one direction, and then the left and right parts of the factorisation system in the other two directions. Something that I’ve been thinking about ever since that conversation is how, if you take two squares <span class="math display"> \begin{CD} X @&gt;{f}&gt;&gt; X&#39; \\@V{i}VV @VV{i&#39;}V \\Y @&gt;&gt;&gt; Y&#39; \end{CD} \qquad\qquad \begin{CD} X @&gt;{f}&gt;&gt; X&#39; \\@V{p}VV @VV{p&#39;}V \\Z @&gt;&gt;&gt; Z&#39; \end{CD} </span> where <span class="math inline">f</span> is arbitrary, <span class="math inline">i</span> and <span class="math inline">i&#39;</span> are in the left part of the factorisation system, and <span class="math inline">p</span> and <span class="math inline">p&#39;</span> in the right part, then you can think about “gluing them together along <span class="math inline">f</span>” to get the diagram <span class="math display"> \begin{CD} Y @&gt;&gt;&gt; Y&#39; \\@A{i}AA @AA{i&#39;}A \\X @&gt;&gt;&gt; X&#39; \\@V{p}VV @VV{p&#39;}V \\Z @&gt;&gt;&gt; Z&#39; \end{CD} </span> which looks exactly like a 2-morphism of spans, or a cell in <span class="math inline">\mathbb{S}\mathrm{pan}_{\mathrm{LR}}</span> (where spans have their left leg in the left part of the factorisation system, and their right leg in the right part). In the case where you don’t quite have a factorisation system, but something “close”, this category of spans turns up in geometry: a colleague (and friend) here at Stockholm recently wrote a paper (<a href="https://arxiv.org/abs/2204.08968">“A descent principle for compact support extensions of functors”</a>) about how such spans can be used to describe when arbitrary cohomology theories have compactly supported versions. I’d love to better understand what is actually going on here, even without talking about factorisation systems: if we glue together two double categories, we get some double category of spans; can we formalise this, and does it tell us something?</p> <p>One last talk I’d like to mention is Matteo Capucci’s <a href="https://www.youtube.com/watch?v=wtgfyjFIHBQ">talk on categorical systems theory and categorical cybernetics</a>, where he mentioned the specific example of lenses and how they form a double category. Something that’s already “well known” (see e.g. Example 3.8 in <a href="https://arxiv.org/abs/1908.02202">“Generalized Lens Categories via functors <span class="math inline">\mathcal{C}^\mathrm{op}\to\mathsf{Cat}</span>”</a>) is how the Grothendieck construction/category of elements/generalised lens construction (depending on your preferred nomenclature) can give you the category of ringed spaces: objects are pairs <span class="math inline">(X,\mathcal{O}_X)</span>, with <span class="math inline">X</span> a space and <span class="math inline">\mathcal{O}_X</span> a sheaf of rings on <span class="math inline">X</span>; morphisms <span class="math inline">(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)</span> are pairs <span class="math inline">(f_0,f^\sharp)</span>, with <span class="math inline">f_0\colon X\to Y</span> and <span class="math inline">f^\sharp\colon f_0^*\mathcal{O}_Y\to\mathcal{O}_X</span>. But sometimes we want to consider the category of sheaves on spaces as having different morphisms — if the space is the same, then we want the map of sheaves to go in the “forwards” direction: <span class="math inline">\varphi\colon(X,\mathcal{O}_X)\to(X,\mathcal{O}&#39;_X)</span> should be a morphism of sheaves <span class="math inline">\varphi\colon\mathcal{O}_X\to\mathcal{O}&#39;_X</span>. These two notions of morphisms of pairs consisting of a space and a sheaf on the space (one where we think of the pair as a ringed space, and the other where we think of the pair as an element of <span class="math inline">\mathsf{Sh}(X)</span> for some fixed <span class="math inline">X</span>) seem to be exactly the two notions that Matteo described. It seems like an interesting question for algebraic geometers: is this double category of “sheaves on spaces” useful? At the very least, it seems to be a nice and tidy way of expressing the two flavours of sheaf morphism geometers usually care about!</p> <p>Finally, I would like to thank Andrée Ehresmann, who surprised us at <a href="https://www.youtube.com/watch?v=zB_ifewP8Yk">the end of David Jaz Myers’ talk</a> with a few words, telling us a little bit about the history of double categories (since it was indeed her late husband who first wrote down a definition). This was a lovely touch to the end of the workshop.</p> <p>There are so many other things I would like to write about from all the other talks, but I’m rather tired from the week of late nights, and I still haven’t gotten back into the swing of blogging, so I’m going to leave it here for now. As always, I’d love to hear from any of you if you have any thoughts — I’m not on Twitter any longer (and I never got around to fixing comments on this blog), but you can always drop me an email, or I’m now semi-active (but rarely) <a href="https://mathstodon.xyz/@thosgood">on Mathstodon</a>.</p> tag:thosgood.com,2022-05-30:/blog/2022/05/30/various-notions-of-cosimplicial-presheaves/ Various notions of (co)simplicial (pre)sheaves 2022-05-30T00:00:00Z 2022-05-30T00:00:00Z <p>For the first time, I have released into the wild a preprint of which I am the sole author, and had no real supervision. This is a scary moment indeed — how do I know that I haven’t written complete made-up nonsense? It’s true that I talked with a couple of close colleagues about the results, and they nodded in vague agreement, but the responsibility of checking the actual formal details is all on me. Even worse, I wanted to include some results about something that I don’t really have any formal experience with. Anyway, I hope the resulting paper is at least mildly “good” (whatever that might mean). It’s called “Various notions of (co)simplicial (pre)sheaves”, and is now on the arXiv: <a href="https://arxiv.org/abs/2205.15185">2205.15185</a>.</p> <!-- more --> <h1 id="where-this-paper-came-from">Where this paper came from</h1> <p>Ever since my PhD thesis, I’ve been thinking on-and-off about these things called “sheaves on the Čech nerve”, which were introduced by Green<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>, and then later in a different way by Toledo and Tong<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a>, in order to do some nice homotopical construction in complex geometry. The vague idea is that a sheaf <span class="math inline">\mathcal{F}</span> on a space <span class="math inline">X</span> can be split up into finer parts, with an individual sheaf <span class="math inline">\mathcal{F}^p</span> for each simplicial level of the Čech nerve of (a cover of) <span class="math inline">X</span>, i.e. instead of just describing what <span class="math inline">\mathcal{F}</span> looks like on all of <span class="math inline">X</span>, we describe what it looks like on all the open sets in a cover of <span class="math inline">X</span> (this is <span class="math inline">\mathcal{F}^0</span>), and then what it looks like on each pairwise intersection <span class="math inline">U\cap V</span> of open sets from the cover (this is <span class="math inline">\mathcal{F}^1</span>), and then what it looks like on each triple-wise intersection <span class="math inline">U\cap V\cap W</span> of open sets from the cover (this is <span class="math inline">\mathcal{F}^2</span>), and… and so on. One reason for wanting to do this is that sometimes we want to endow our sheaf <span class="math inline">\mathcal{F}</span> with some extra structure or data (such as a <a href="https://en.wikipedia.org/wiki/Connection_(vector_bundle)#Formal_definition"><em>Koszul connection</em></a>, in the case of Green/Toledo–Tong) that might not exist globally (as is indeed the case for holomorphic connections), <em>but</em> that exists <em>locally</em> — decomposing our sheaf <span class="math inline">\mathcal{F}</span> into different simplicial levels over the Čech nerve lets us do exactly this!</p> <p>Anyway, what always annoyed me about this story (however petty this might be) was the terminology: a bunch of sheaves <span class="math inline">(\mathcal{F}^p)_{p\in\mathbb{N}}</span> living over a simplicial space <span class="math inline">(X_p)_{p\in\mathbb{N}}</span> were called <em>simplicial sheaves</em>, but they’re <em>not</em> simplicial objects in the category of sheaves on <span class="math inline">X</span>, and, even worse, they don’t actually really even look very simplicial (the morphisms all end up going the wrong way, making them look <em>co</em>simplicial). So is there any relation between these things (which I gave the uninventive but fairly descriptive name of <em>sheaves on a simplicial space</em>) and actual simplicial sheaves? Since the only reference that I know of which actually talks about these sheaves on simplicial spaces is Green’s aforementioned thesis (and Toledo–Tong’s aforementioned summary of it), I couldn’t find if anybody else actually even cared about these objects, let alone cared enough to figure out if they also deserved the name of “simplicial sheaves”.</p> <h1 id="an-unexpected-application">An unexpected application</h1> <p>When I finally got the motivation to sit down and write things down, I realised that there was also another pretty large gap in what I knew about these things: the second thing you normally learn about after learning about sheaves is <em>sections</em> of a sheaf. So what is a section of a sheaf on a simplicial space? In one of life’s little coincidences, when I started thinking about this specific question seriously, I was working in an office three floors above Vincent Wang-Maścianica, who patiently helped me to understand the basics of string diagrams as well as some other stuff that he was working on. The reason this is a coincidence is because, one afternoon (to mildly dramatise the story) he saw a diagram on my board and said “oh, I didn’t realise you were studying X”, to which I replied, “I’m not, I’m thinking about sections of sheaves on simplicial spaces”, to which we both replied “oh… interesting”.</p> <p>The upshot of the ensuing conversations with Vincent left me reasonably convinced that I could find an application for sheaves on simplicial spaces outside their one appearance in complex geometry: they could be used to describe “stuff, stuff relating that stuff, and the order in which those relations should evolve”. To give a more concrete example of this, consider the following scenario: we have some objects <span class="math inline">x_1,\ldots,x_n</span> in a category that has a nice notion of tensor product (e.g. a symmetric monoidal category), and we want to understand what endomorphisms of <span class="math inline">x_1\otimes x_2\otimes \ldots\otimes x_n</span> look like. One way of getting such endomorphisms is to build them up from smaller endomorphisms! For example, say we have objects <span class="math inline">A</span>, <span class="math inline">B</span>, <span class="math inline">C</span>, and <span class="math inline">D</span>, and an endomorphism <span class="math inline">f\in\operatorname{End}(A\otimes B)</span> and another endomorphism <span class="math inline">g\in\operatorname{End}(B\otimes C\otimes D)</span>. Then we can get an endomorphism on <span class="math inline">A\otimes B\otimes C\otimes D</span> by composing <span class="math inline">f</span> and <span class="math inline">g</span>, after extending them by identity morphisms: <span class="math inline">(\mathrm{id}_A\otimes g)\circ(f\otimes\mathrm{id}_C\otimes\mathrm{id}_D)</span>. But of course we could have also composed these in the other order, doing <span class="math inline">g</span> first and then <span class="math inline">f</span>, or we could have even picked different endomorphisms, or we could have had <em>more</em> endomorphisms on tensor products of <em>different</em> subsets of our objects.</p> <p>In the paper, I (timidly) make the argument that all of these choices can be bundled up into one sheaf on a simplicial space, and then that a single choice of all the data described above corresponds to a <em>section</em> of this sheaf. There are some caveats in this construction (for example, the method that I describe requires that we fix the number and type of endomorphisms beforehand), but I think that most of them could probably be done away with by somebody who actually knows something about these very restrictive sorts of string diagrams (e.g. if there are some “generating” shapes of these diagrams then you could maybe consider building a simplicial space for each one and then gluing them altogether with some limit-y process).</p> <figure> <img src="endomorphism-construction.png" alt="String diagrams of a certain form correspond to sections of a specific sheaf on a simplicial space." /> <figcaption aria-hidden="true">String diagrams of a certain form correspond to sections of a specific sheaf on a simplicial space.</figcaption> </figure> <p>Really, the only reason that I actually include this construction at all (beyond wanting to make people in another field aware of these objects that I find so fascinating) is in the hope that somebody comes along and makes it much better! I think it could be extended to describe much more general string diagrams, with copies and deletes and whatnot, but this is not something that I have the knowledge to do on my own.</p> <p>This application is, I think, not particularly useful, and might just be an example of “if all you have is a hammer, then everything looks like a nail”, or even just a tautology (“of <em>course</em> sections of a sheaf on a simplicial space describe such things — they’re just objects in the totalisation of the cosimplicial simplicial mapping space from a cofibrant replacement of a point, and the simplicial nature is exactly describing the notion of linear order”, or something like this, says some hypothetical expert), but “hey ho”, thought I, “why not just write it down anyway — maybe somebody else will also like to think about such things from this point of view”.</p> <h1 id="whats-actually-in-the-paper">What’s actually in the paper</h1> <p>The paper is mostly a survey, but contains some proofs of what I’m calling “pre-folklore results” — results which I think would be folklore if I’d ever actually heard anybody else care about them!</p> <p>The first two sections (after the introduction) define three (or really four) notions of “simplicial sheaf” and describes how they relate to one another; the next section recalls a technical construction that shows how to understand this story in the specific setting of (locally) ringed spaces via a lovely construction<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a> of lax homotopy limits of model categories by Bergner; the final section is split into two, and contains a summary of the application of sheaves on simplicial spaces to coherent sheaves in complex geometry, and then this semi-conjectural construction relating sections of a sheaf on a simplicial space to string diagrams describing endomorphisms generated by endomorphisms.</p> <p>If anybody does actually read this and have any comments, corrections, criticisms, or complaints, then please do let me know — you can leave a comment here, or email me, or tweet at me, or whatever you like. Happy reading!</p> <figure> <img src="three-notions.png" alt="The three main notions of simplicial sheaf." /> <figcaption aria-hidden="true">The three main notions of simplicial sheaf.</figcaption> </figure> <aside id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"> <hr /> <ol> <li id="fn1"><p>H.I. Green. “Chern classes for coherent sheaves”. PhD Thesis. University of Warwick, (1980). <a href="https://pugwash.lib.warwick.ac.uk/record=b1751746~S1">pugwash.lib.warwick.ac.uk/record=b1751746~S1</a><a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn2"><p>Domingo Toledo and Yue Lin L Tong. “Green’s theory of Chern classes and the Riemann-Roch formula”. In: <em>The Lefschetz Centennial Conference, Part 1</em>, Amer. Math. Soc. (1987). DOI:<a href="https://doi.org/10.1090/conm/058.1/860421">10.1090/conm/058.1/860421</a><a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn3"><p>Julia E. Bergner. “Homotopy limits of model categories and more general homotopy theories”. (2012). arXiv:<a href="https://arxiv.org/abs/1010.0717v2">1010.0717v2</a><a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li> </ol> </aside> tag:thosgood.com,2022-04-25:/blog/2022/04/25/some-research-questions-from-my-notesbooks/ Some research questions from my notebooks 2022-04-25T00:00:00Z 2022-04-25T00:00:00Z <p>One thing that the past few years have taught me is that I am not good at doing maths all by myself. In fact, I would go as far as to say I am completely useless and unmotivated. I do much better when I have co-authors to give me deadlines and friends to talk to, but, for obvious reasons, the past two years have not been good for this. Not really the ideal time for first postdocs, but alas, that’s life.</p> <p>I recently found an old notebook with some vague questions and research ideas in it, and then realised that I have had no motivation to work on any of these alone, so why not put them out there for other people to see?</p> <!-- more --> <p>A common fear amongst early-career mathematicians seems to be getting scooped: having somebody solve the questions you were working on before you do, maybe even by having read your work and using it as a launchpad. This is very understandable, since we live in a horrible academic world where publishing is everything, and trying to publish some results that you’ve arrived at <em>after</em> somebody else, no matter how shortly or how differently, suddenly becomes a <em>lot</em> harder, which makes getting a job next year <em>much</em> more difficult, and… Because of this, I’ve always been a bit cautious in writing what I’m working on. But, in the context of my life, the world around me, and some other things, I realise that these fears aren’t really very well founded for me right now. There’s no point in me worrying about other people using my ideas to solve problems before I do if <em>I’m not even working on applying my ideas myself</em>. Not only that, it’s not like I’m sitting on some goldmine of potential paths to solutions of big long-standing open problems — I just have a few small ideas about small things, none of them are particularly profound or ingenious (in fact, now that I read back what I’ve written, I see that they’re basically all just “<em>what if you did X, but with the Čech nerve instead?</em>”). If anything, I shouldn’t be worried about people scooping my ideas, but the opposite: trying to get anybody to listen at all!</p> <p>So I figured I might as well just open up some of<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> my “research” in the hopes that one of you might see one of the questions I have and think “oh, that’s interesting, I wonder if Tim would like to talk about this sometime” (to which I would almost certainly (time and obligations permitting) answer, “yes, that would be very lovely”).</p> <h1 id="some-simplicial-things">Some simplicial things</h1> <p><em>(These questions are about using the Čech nerve, and sometimes the notion of (pre)sheaves on the Čech nerve — cf. e.g. Definition 2.2.1 in “<a href="https://arxiv.org/abs/2003.10023">Simplicial Chern-Weil theory for coherent analytic sheaves, Part I</a>”.)</em></p> <h2 id="barycentric-global-sections-always-exist">Barycentric global sections always exist</h2> <p>Take a presheaf <span class="math inline">\mathscr{F}</span> (of <span class="math inline">\mathbb{R}</span>-modules, say) on some space <span class="math inline">X</span>, and pick a bunch of local sections <span class="math inline">\{s_i\in\Gamma(U_i,\mathscr{F})\}_{i\in I}</span>, where <span class="math inline">\{U_i\}_{i\in I}</span> is an open cover of <span class="math inline">X</span>. Since we only have a presheaf, there’s no reason for these local sections to glue to give a global section. <em>But</em> if we pull back our presheaf along the Čech nerve then we can consider the “barycentric global section” <span class="math inline">\sum_{j=0}^p t_j s_j</span> on any <span class="math inline">U_{\alpha_0\ldots\alpha_p}</span>. This is like the “uniform average” of all the sections: if we fibre integrate then we’d get e.g. <span class="math inline">\frac1{p+1}\sum_{i=0}^p s_{\alpha_j}</span> on each <span class="math inline">U_{\alpha_0\ldots\alpha_p}</span>.</p> <p>Is this ever useful at all, as a sort of “second best thing” for a global section that you know doesn’t exist?</p> <h2 id="čech-nerves-of-things">Čech nerves of things</h2> <p>Consider some complex-analytic “thing” (i.e. manifold, space, whatever) <span class="math inline">X</span> that is locally algebraic (whatever that might mean). If we take the Čech nerve then is the resulting simplicial “thing” an <em>algebraic</em> simplicial “thing” <span class="math inline">\widetilde{X}</span>? In the case where <span class="math inline">X</span> is <em>not</em> algebraic, can we measure how far away it is from being so by looking at the simplicial “thing” <span class="math inline">\widetilde{X}</span>?</p> <p>A simpler, similar question: is the Čech nerve of an affine cover of an (algebraic) scheme an “affine object” in the category of simplicial schemes?</p> <p><em>(This last question should be something already well known, but I just don’t know the answer myself.)</em></p> <h1 id="some-analytic-geometry">Some analytic geometry</h1> <p><em>(These questions are about taking things that we know how to do in the smooth or the algebraic world, and trying to do them in the complex-analytic world.)</em></p> <h2 id="chernweil-for-stacks">Chern–Weil for stacks</h2> <p>One way of cheekily summarising some of the results from <a href="https://tel.archives-ouvertes.fr/tel-02882140">my PhD thesis</a> would be that “sheaves on the Čech nerve sometimes allow you to apply smooth methods to complex-analytic things”, and the application of this that I considered was Chern–Weil theory via the Atiyah exact sequence. So is it possible to extend some results concerning Chern–Weil theory via the Atiyah exact sequence on differentiable stacks to the complex-analytic case?</p> <p><em>(This question was prompted by seeing two papers on the arXiv by Indranil Biswas, Saikat Chatterjee, Praphulla Koushik, and Frank Neumann: <a href="https://arxiv.org/abs/2012.08442"><code>2012.08442</code></a> and <a href="https://arxiv.org/abs/2012.08447"><code>2012.08447</code></a>.)</em></p> <h2 id="holomorphic-deligne-cohomology">Holomorphic Deligne cohomology</h2> <p>This is one that I’ve been thinking about ever since the middle of my PhD, and it was actually the original problem that I’d hoped to solve (but that turned out to be much more difficult than we’d first thought). Deligne cohomology in the <em>smooth</em> setting is really well understood — for example, Urs Schreiber has written SO much about this (and all very very lovely, albeit nearly entirely far over my head) under the name of <em>differential cohomology</em>. But at some point quite early on there is a partition of unity argument, which means that it fails in the holomorphic case. Indeed, there are lots of little worked examples you can do that show that Deligne cohomology in the complex-analytic world really is quite different.</p> <p>One “simple” concrete problem is the following: given a holomorphic vector bundle (not even an arbitrary coherent analytic sheaf!), write down <em>Čech representatives</em> for its Chern classes <em>in Deligne cohomology</em>. Just the first part (Čech representatives) was done<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a> in <a href="http://wrap.warwick.ac.uk/40592/">Green’s 1980 PhD thesis</a>; just the second part (holomorphic Deligne cohomology) was done in <a href="http://jgrivaux.perso.math.cnrs.fr/articles/Chern.pdf">Grivaux’s 2009 thesis</a>. There is a (beautiful and very good) paper by Brylinski and McLaughlin (“<a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-178/issue-1/%C4%8Cech-cocycles-for-characteristic-classes/cmp/1104286562.pdf">Cech Cocycles for Characteristic Classes</a>”) that seems to give an answer, but only gives a proof in the smooth setting, and when I sat down (many <em>many</em> times) and tried to work through it myself, the degrees in the holomorphic setting seemed to be exactly the wrong ones that you would get from trying to treat smooth Deligne cohomology like holomorphic Deligne cohomology (i.e. bidegree <span class="math inline">(2p,0)</span> instead of <span class="math inline">(p,p)</span>, if I remember correctly). It is also one of those answers where they say “ok, here are all the ingredients you need, so just put them together”, but then don’t just write down the explicit representatives (something that always frustrates me: if putting the pieces together is hard, then say it’s hard; if it’s easy, then why not just do it?!).</p> <p>To reduce this problem down to the very simplest form: given a rank-<span class="math inline">2</span> holomorphic vector bundle defined by transition functions <span class="math inline">g_{\alpha\beta}</span>, write down a Čech cocycle <em>in terms of the <span class="math inline">g_{\alpha\beta}</span></em> representing the second Chern class in <em>Deligne cohomology</em>.</p> <p><em>(For example, we know how to do this in de Rham cohomology: you can take <span class="math inline">\operatorname{tr}(\omega_{\alpha\beta}g_{\alpha\beta}\omega_{\beta\gamma}g_{\alpha\beta}^{-1})</span> or, equivalently (via some algebraic manipulations), <span class="math inline">\operatorname{tr}(\omega_{\alpha\beta}(\omega_{\alpha\gamma}-\omega_{\alpha\beta}))</span>, where <span class="math inline">\omega_{\alpha\beta}=\operatorname{d}\log g_{\alpha\beta}=g_{\alpha\beta}^{-1}dg_{\alpha\beta}</span>, and you can then extend this to give a closed element of the Čech-de Rham bicomplex.)</em></p> <h1 id="some-analytic-sheaves">Some analytic sheaves</h1> <p><em>(More complex-analytic geometry, this time dealing with holomorphic vector bundles and their generalisations.)</em></p> <h2 id="complexes-of-coherent-sheaves">Complexes of coherent sheaves</h2> <p>If you’re dealing with sheaves in algebraic geometry, then you might care about whether or not they’re coherent. Given that we like to think of complexes of sheaves instead of just single sheaves (e.g. we like working in the derived category), we are interested in the derived category <span class="math inline">D^\mathrm{b}\mathsf{Coh}(X)</span> of (bounded) complexes of coherent sheaves. But there’s another category which arises quite often in practice, namely the category <span class="math inline">D^\mathrm{b}_{\mathsf{Coh}}(\mathsf{Sh}(X))</span> of complexes of sheaves that are not necessarily coherent, but whose (internal, i.e. “kernel of the differential of the complex modulo the image of the differential”) cohomology consists of coherent sheaves. The latter sounds like it should be more general, but in nice algebraic cases the two are actually equivalent! This follows from a result in SGA 6:</p> <ul> <li><strong>SGA 6, II, Corollaire 2.2.2.1.</strong> If <span class="math inline">X</span> is a Noetherian scheme, then the canonical fully faithful functor <span class="math inline">D^\mathrm{b}(\mathsf{Coh}(X))\hookrightarrow D(\mathsf{Sh}(X))</span> identifies the codomain with the full subcategory <span class="math inline">D^\mathrm{b}_{\mathsf{Coh}}(\mathsf{Sh}(X))</span> of the domain.</li> </ul> <p>Now there’s another related result, which follows from applying the above corollary to a specific case:</p> <ul> <li><strong>SGA 6, I, Exemples 5.11 (+ II, Corollaire 2.2.2.1).</strong> If <span class="math inline">X</span> is a smooth scheme, then there is a canonical equivalence of triangulated categories <span class="math inline">\mathsf{Perf}(X)\xrightarrow{\sim}D^{\mathrm{b}}(\mathsf{Coh}(X))</span>.</li> </ul> <p>This says that, for <em>smooth</em> schemes, pseudo-coherence is equivalent to perfectness (being <em>locally</em> resolved by locally free sheaves).</p> <p>Now, the analogue of this second statement is still true in the analytic case: coherent analytic sheaves always have <em>local</em> locally free resolutions. But the first statement is a corollary to Proposition 2.2.2, which uses the fact that “every quasi-coherent module is the filtrant colimit of its coherent submodules”, and this is <em>not</em> true in the analytic case (in fact, this raises a question I’ll talk about after this one).</p> <p>So are these two categories, “complexes of coherent sheaves” and “complexes of sheaves with coherent cohomology” equivalent in the analytic setting? In the specific case where <span class="math inline">X</span> is a smooth compact analytic <em>surface</em>, yes! This is Corollary 5.2.2 of Bondal and Van den Bergh’s “<a href="https://arxiv.org/abs/math/0204218">Generators and representability of functors in commutative and noncommutative geometry</a>”. But in higher dimensions, it is (as far as I can tell) still an open question.</p> <p>I don’t have any particular insight into this problem, except that I think (yet again) that sheaves on the nerve might have something to say about this. Furthermore, I haven’t read the details of Bondal and Van den Bergh’s proofs, nor those in SGA 6, so probably the following is obvious (or even tautological, actually), but it’s intriguing that these hypotheses (<span class="math inline">X</span> is smooth, compact, and a surface) are exactly those found in Schuster’s “<a href="https://www.degruyter.com/document/doi/10.1515/crll.1982.337.159/html">Locally free resolutions of coherent sheaves on surfaces</a>” which shows that, under these hypothesis, coherent analytic sheaves can be <em>globally</em> resolved by locally free sheaves (the <em>resolution property</em>).</p> <h2 id="the-resolution-property">The resolution property</h2> <p>Whether or not coherent analytic sheaves can be <em>globally</em> resolved by locally free sheaves (instead of just <em>locally</em> resolved) is controlled by the so-called <em>resolution property</em>, mentioned above. We say that something (e.g. a stack) <em>has the resolution property</em> if every coherent sheaf admits a surjection from a locally free sheaf — what happens if we change this to “… a surjection from a locally free sheaf <em>on the nerve</em>”? This could be useful for Riemann–Roch for Artin stacks: these have the resolution property when they are quotients of quasi-projective schemes by reductive groups, but not in general (and so this assumption appears in Toën’s Riemann–Roch paper). It could also be useful for a formal GAGA theorem, cf. Geraschenko and Zureick-Brown’s “<a href="https://arxiv.org/abs/1208.2882">Formal GAGA for good moduli spaces</a>”.</p> <h2 id="analogies-between-algebraic-and-analytic-geometry">Analogies between algebraic and analytic geometry</h2> <p>Here I’m just going to refer you to a previous blog post of mine: “<a href="https://thosgood.com/blog/2021/09/24/some-questions-about-analytic-geometry">Some questions about complex-analytic geometry</a>”. The main one here is “<em>how should we define the notion of quasi-coherence for an analytic sheaf?</em>”, followed immediately by “<em>why did we pick this definition, and not the others?</em>”. One possible answer is “read Scholze and Clausen’s lecture notes on condensed mathematics, Conrad’s paper on relative ampleness in rigid geometry, and Eschmeier and Putinar’s book <em>Spectral Decompositions and Analytic Sheaves</em>, and see if you can put all the pieces together”.</p> <p>Just for the sake of it, here are some possible contenders for the definition of quasi-coherence for an analytic sheaf <span class="math inline">\mathscr{F}</span> of <span class="math inline">\mathcal{O}_X</span>-modules:</p> <ol type="1"> <li>being <em>of local presentation</em>, i.e. for all <span class="math inline">x\in X</span> there exists an open <span class="math inline">U\subseteq X</span> on which there is an exact sequence <span class="math display">\mathcal{O}_X^{\oplus I}|U \to \mathcal{O}_X^{\oplus J}|U \to \mathscr{F}|U \to 0</span></li> <li>being the filtrant colimit of its coherent subsheaves</li> <li>being <em>Fréchet quasi-coherent</em>, or, equivalently, admitting a global “topologically free” resolution</li> <li>something analogous to arising from the right Kan extension of something like the pseudofunctor <span class="math inline">\mathsf{CRing}\to\mathsf{Cat}</span> defined by <span class="math inline">R\mapsto\mathsf{Mod}_R</span> (very vague, I know).</li> </ol> <p>I know that there are <a href="https://math.stackexchange.com/questions/2840594/gaga-and-quasicoherent-sheaf/2841087#2841087">examples of 1. that do not satisfy 2.</a>, but I actually don’t know how the other notions interact at all.</p> <aside id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"> <hr /> <ol> <li id="fn1"><p>Apart from one paper which I’m currently working on with a co-author, because that’s not just my story to share. But this will hopefully hit the arXiv… within a year? I dunno, I’ve been saying that about this specific paper for the past two years, so we’ll see what actually happens.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn2"><p>Well, the method was explained, but the actual computations were basically left as an exercise to the reader — I was one such reader, and I wrote them down in <em>my</em> PhD thesis.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li> </ol> </aside> tag:thosgood.com,2022-04-08:/blog/2022/04/08/diagrammatic-equations-and-multiphysics/ Diagrammatic equations and multiphysics 2022-04-08T00:00:00Z 2022-04-08T00:00:00Z <p>Just a very short post (you don’t even need to click “Continue reading” if you’re looking at this on my blog archive) — I finally managed to do some more maths (but only because I had some very hard working and very good coauthors), and I’ve written about it on the Topos blog as a two-part series: <a href="https://topos.site/blog/2022/04/diagrammatic-equations-and-multiphysics-part-1/">Part 1</a> and <a href="https://topos.site/blog/2022/04/diagrammatic-equations-and-multiphysics-part-2/">Part 2</a>.</p> <p>I would love to write more, but after having written these I’m afraid I’m all blogged out for the minute, so ciao for now!</p> <!-- more --> tag:thosgood.com,2022-02-11:/blog/2022/02/11/graded-commutative-and-graded-and-commutative/ Graded commutative rings and graded and commutative rings 2022-02-11T00:00:00Z 2022-02-11T00:00:00Z <p>One of the many reasons that teaching is fun is because you get to look back at things that you haven’t seen in a while and try to understand them in light of what you’ve learnt in the meantime. This means that you sometimes have the unexpected joy of having to teach something that always used to confuse you, but that now seems so much more straightforward! I experienced this last year when teaching an algebraic topology course: I remember being super lost when it came to the graded ring structure of cohomology and getting very annoyed at Hatcher’s book; now I look back and realise that it’s really neat! This post has a slightly different intended audience than normal: I’m just gonna assume that you know a bit about rings in the first half; the second half is aimed for somebody who’s a reasonable way through a first course on algebraic topology (e.g. knows what the cup product in cohomology is).</p> <!-- more --> <h1 id="graded-rings">Graded rings</h1> <p>Our starting point is the omnipresent <em>polynomial ring</em>: given a ring <span class="math inline">S</span>, we define <span class="math inline">S[x]</span> to be the set of polynomials in one variable (namely <span class="math inline">x</span>) with coefficients in <span class="math inline">S</span>, so elements of <span class="math inline">S[x]</span> are of the form <span class="math display"> f = s_n x^n + s_{n-1} x^{n-1} + \ldots + s_1 x + s_0 </span> where the <span class="math inline">s_i</span> are elements of <span class="math inline">R</span>. (More generally, we have the <em>polynomial ring in <span class="math inline">n</span> indeterminates</em>: <span class="math inline">S[x_1,x_2,\ldots,x_n]</span>, which is entirely analogous, but with more than one variable; this is important, but we don’t need to worry about it too much for now).</p> <p>These rings are <em>really</em> nice, in lots of ways that we won’t talk about today, but one <em>particularly</em> nice thing that they have is a <em>grading</em>, given by the <em>degree</em>. Recall that the degree of a <em>monomial</em> is defined to be the “total power”, i.e. <span class="math display"> \deg(x^n) = n </span> (we say <em>total</em> power to deal with the case of multiple variables, e.g. <span class="math inline">\deg(x^my^n)=m+n</span>). The degree behaves nicely under multiplication <span class="math display"> \deg(x^m\cdot x^n) = m+n </span> and we have a <em>unique</em> way of writing <em>any</em> polynomial <span class="math inline">f\in S[x]</span> as a sum of monomials, basically by definition of what it means to be a polynomial: <span class="math display"> f = \sum_{i=0}^n r_i x^i. </span></p> <p>So let’s do what we always do and generalise this to an abstract structure!</p> <div class="rmenv" title="Definition"> <p>A ring <span class="math inline">R</span> is a <em>graded ring</em> if there exist abelian groups <span class="math inline">(R_d)_{d\in\mathbb{N}}</span>, where <span class="math inline">R_d\subseteq R</span> for all <span class="math inline">d\in\mathbb{N}</span>, such that</p> <ol type="i"> <li><span class="math inline">R\cong\bigoplus_{d=0}^\infty R_d</span>;</li> <li><span class="math inline">R_d R_e\subseteq R_{d+e}</span>.</li> </ol> <p>Given <span class="math inline">r\in R\setminus\{0\}</span>, if there exists some <span class="math inline">d\in\mathbb{N}</span> such that <span class="math inline">r\in R_d</span>, then we say that <span class="math inline">r</span> is <em>homogeneous of degree <span class="math inline">d</span></em>, and we write <span class="math inline">|r|=d</span>.</p> </div> <p>The prototypical example is the thing we started with: the polynomial ring <span class="math inline">R=S[x]</span> is a graded ring with <span class="math inline">R_d\coloneqq\{sx^d \mid s\in S\}</span>; the more general polynomial ring <span class="math inline">R=S[x_1,\ldots,x_n]</span> in <span class="math inline">n</span> variables is a graded ring with <span class="math inline">R_d\coloneqq\{s x_1^{m_1}x_2^{m_2}\cdots x_n^{m_n}\mid s\in S, m_1+m_2+\ldots+m_n=d\}</span>.</p> <h1 id="commutativity">Commutativity</h1> <p>Now let’s talk about something confusing: what is a <em>graded commutative ring</em>? Or should we say <em>commutative graded ring</em>? Or should these two things be different?</p> <p>Well, it makes sense that a graded commutative ring would just be a commutative ring that is graded, i.e. we parse it as “graded (commutative ring)”. Annoyingly, however, this is <em>not</em> the way that most algebraists or geometers will parse this! If you want to talk about commutative rings that are graded, then your best bet is really to just say “a commutative ring that is graded”, but if you want to be snappier, then I would advise that you say <em>commutative graded ring</em>. Why am I making such a point out of this? What do people mean when they actually say “graded commutative ring” then?</p> <p>The answer lies in “bracketing” the adjectives in a different way, namely: “(graded commutative) ring”. But this just prompts the question: what does it mean for a ring to be “graded commutative”?</p> <div class="rmenv" title="Definition"> <p>A ring <span class="math inline">R</span> is <em>graded commutative</em> if <span class="math inline">R</span> is a <em>graded</em> ring <span class="math inline">R=\bigoplus_{d=0}^\infty R_d</span> such that <span class="math display"> rs = (-1)^{|r||s|}sr </span> for all homogeneous elements <span class="math inline">r,s\in R</span>.</p> </div> <p>So this is a slightly odd definition: graded commutativity is like commutativity, but with a possible minus sign, depending on the degree of the (homogeneous) elements.<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> To understand this better, let’s turn back to our good old friend <span class="math inline">S[x]</span>. If <span class="math inline">S</span> is commutative, then <span class="math inline">S[x]</span> is clearly commutative, but is it graded commutative? Shockingly, no! Indeed, we are asking if the following equality holds <span class="math display"> x^2 = x\cdot x \overset{?}{=} (-1)^{|x||x|} x\cdot x = -x^2. </span> We see that this only happens in two cases:</p> <ol type="1"> <li>if <span class="math inline">2=0</span>; or</li> <li>if <span class="math inline">x^2=0</span>.</li> </ol> <p>The first one can happen (if <span class="math inline">S</span> is a field of characteristic <span class="math inline">2</span>, for example, e.g. <span class="math inline">\mathbb{Z}/2\mathbb{Z}</span>), but the second one cannot, by the very definition of <span class="math inline">S[x]</span>. <em>But</em> this second case does happen in the <em>ring of dual numbers</em> <span class="math inline">S[x]/(x^2)</span>.</p> <p>Going back to the polynomial ring <span class="math inline">S[x]</span>, we could also do something entirely different and <em>define</em> <span class="math inline">x</span> to be of <em>degree <span class="math inline">2</span></em>. That is, we are giving <span class="math inline">S[x]</span> a <em>different</em> graded ring structure: <span class="math inline">S[x]\cong\bigoplus_{d=0}^\infty R_d</span> where <span class="math display"> R_d = \begin{cases} \{sx^d \mid s\in S\} &amp;\text{if }d\text{ is even;} \\0 &amp;\text{if }d\text{ is odd.} \end{cases} </span> (After a little bit of thought, you can see that this is “the same as” simply looking at <span class="math inline">S[y^2]</span>, where <span class="math inline">y</span> is an indeterminate of degree <span class="math inline">1</span> again). Then we have that <span class="math display"> (-1)^{|x||x|} x\cdot x = x^2 </span> and so this graded ring <em>is</em> graded commutative.</p> <p>A nice little summary table would be helpful right about now (and, just in this table, we’re going to make the assumption that <strong><span class="math inline">S</span> is commutative</strong>).</p> <table> <thead> <tr class="header"> <th style="text-align: center;">Graded ring</th> <th style="text-align: center;">Commutative?</th> <th style="text-align: center;">Graded commutative?</th> </tr> </thead> <tbody> <tr class="odd"> <td style="text-align: center;"><span class="math inline">S[x]</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">❌ (<strong>unless</strong> <span class="math inline">2=0</span> in <span class="math inline">S</span>)</td> </tr> <tr class="even"> <td style="text-align: center;"><span class="math inline">S[x]/(x^2)</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">✅</td> </tr> <tr class="odd"> <td style="text-align: center;"><span class="math inline">S[x^2]</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">✅</td> </tr> </tbody> </table> <p>Let’s look at one more example that we’re going to need when it comes to doing some algebraic topology: the <em>exterior algebra</em> <span class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> of a ring <span class="math inline">S</span> can be defined as the graded ring <span class="math display"> \Lambda_S[\alpha_1,\ldots,\alpha_n] \coloneqq S[\alpha_1,\ldots,\alpha_n]/(\alpha_i^2,\alpha_i\alpha_j+\alpha_j\alpha_i)_{1\leq i,j\leq n} </span> where each <span class="math inline">\alpha_i</span> is of degree <span class="math inline">1</span> (although, again, we can modify this if we want to). Note that <span class="math inline">\Lambda_S[\alpha]\cong S[\alpha]/(\alpha^2)</span>, i.e. <em>the exterior algebra of <span class="math inline">S</span> in one variable is exactly the ring of dual numbers</em>.</p> <p>The exterior algebra is commutative (if <span class="math inline">S</span> is), but is it graded commutative? Well it’s enough to check the condition on the generators <span class="math inline">\alpha_i</span>, but we see that <span class="math display"> \alpha_i\alpha_j = -\alpha_j\alpha_i = (-1)^{|\alpha_i||\alpha_j|}\alpha_j\alpha_i </span> where the first equality is exactly by the definition of <span class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> (and note that, if <span class="math inline">i=j</span>, then everything is zero, and we definitely have that <span class="math inline">0=0</span>).</p> <p>Back to our table:</p> <table> <thead> <tr class="header"> <th style="text-align: center;">Graded ring</th> <th style="text-align: center;">Commutative?</th> <th style="text-align: center;">Graded commutative?</th> </tr> </thead> <tbody> <tr class="odd"> <td style="text-align: center;"><span class="math inline">S[x]</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">❌ (<strong>unless</strong> <span class="math inline">2=0</span> in <span class="math inline">S</span>)</td> </tr> <tr class="even"> <td style="text-align: center;"><span class="math inline">S[x^2]</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">✅</td> </tr> <tr class="odd"> <td style="text-align: center;"><span class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span></td> <td style="text-align: center;">❌ (unless <span class="math inline">n=1</span>)</td> <td style="text-align: center;">✅</td> </tr> </tbody> </table> <p>(we’ve removed the row for <span class="math inline">S[x]/(x^2)</span>, since this is just a specific example of the exterior algebra <span class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> where <span class="math inline">n=1</span>.)</p> <h1 id="freeness">Freeness</h1> <p>Unrelated to the property of (graded) commutativity is that of being <em>free</em>. For <em>commutative</em> rings, being free basically means being isomorphic to a polynomial ring in finitely many variables. So <span class="math inline">S[x]</span> is free (as a commutative ring), as is <span class="math inline">S[x^2]</span>, but <span class="math inline">S[x]/(x^2)</span> is <em>not</em> free (as a commutative ring) — the latter has a nilpotent element (i.e. some <span class="math inline">r</span> such that <span class="math inline">r^2=0</span>, namely <span class="math inline">r=x</span>) and free commutative rings never have nilpotent elements, so it cannot be isomorphic to a free commutative ring.<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a></p> <p>But note that I’ve been very careful to say “free <em>as a commutative ring</em>”, and this is important: <span class="math inline">\Lambda_S[\alpha]</span> is <strong>not</strong> a free <em>commutative ring</em>, but it <strong>is</strong> a free <em>graded commutative ring</em>. What do I mean by this? I mean that, if we just use the fact that <span class="math inline">\Lambda_S[\alpha]</span> is a graded commutative ring, <em>forgetting all about how it’s actually defined</em>, then we can recover the fact that <span class="math inline">\alpha_i^2=0</span> and <span class="math inline">\alpha_i\alpha_j=-\alpha_j\alpha_i</span> <em>without having to ask for it</em> (…most of the time).</p> <p>That is, if we know that <span class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span> is graded commutative, then we know that <span class="math display"> \alpha_i\cdot\alpha_i = (-1)^{|\alpha_i||\alpha_i|}\alpha_i\cdot\alpha_i = -\alpha_i\cdot\alpha_i </span> which rearranges to give <span class="math display"> 2\alpha_i^2=0. </span> But then, if we can <em>divide by <span class="math inline">2</span></em> (i.e. if <span class="math inline">2</span> is an invertible element) in our ring <span class="math inline">S</span>, then we see that <span class="math inline">\alpha^2</span> <em>must be</em> equal to zero, automatically! (The other relation that we need to impose, that <span class="math inline">\alpha_i\alpha_j=-\alpha_j\alpha_i</span> is exactly the definition of being graded commutative (since all our <span class="math inline">\alpha_i</span> are of degree <span class="math inline">1</span>), so we don’t even need to worry about enforcing that one separately!)</p> <p>Of course, I’m going to put this in the table (and I’ll write “gc-ring” to mean “graded commutative ring”).</p> <table> <colgroup> <col style="width: 18%" /> <col style="width: 19%" /> <col style="width: 31%" /> <col style="width: 31%" /> </colgroup> <thead> <tr class="header"> <th style="text-align: center;">Graded ring</th> <th style="text-align: center;">Commutative?</th> <th style="text-align: center;">Graded commutative?</th> <th style="text-align: center;">Free (as a gc-ring)</th> </tr> </thead> <tbody> <tr class="odd"> <td style="text-align: center;"><span class="math inline">S[x]</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">❌ (<strong>unless</strong> <span class="math inline">2=0</span> in <span class="math inline">S</span>)</td> <td style="text-align: center;">✅</td> </tr> <tr class="even"> <td style="text-align: center;"><span class="math inline">S[x^d]</span></td> <td style="text-align: center;">✅</td> <td style="text-align: center;">❌ (<strong>unless</strong> <span class="math inline">2</span> divides <span class="math inline">d</span>)</td> <td style="text-align: center;">✅</td> </tr> <tr class="odd"> <td style="text-align: center;"><span class="math inline">\Lambda_S[\alpha_1,\ldots,\alpha_n]</span></td> <td style="text-align: center;">❌ (unless <span class="math inline">n=1</span>)</td> <td style="text-align: center;">✅</td> <td style="text-align: center;">❌ (<strong>unless</strong> <span class="math inline">2</span> is invertible in <span class="math inline">S</span>)</td> </tr> </tbody> </table> <p>By now you have probably noticed that all of the difficulties and subtleties come from how the number <span class="math inline">2</span> behaves (except for the exterior algebra only being commutative when in one variable, but this isn’t a “graded” property, so we’ll ignore that one), and this is a very common thing to happen.<a href="#fn3" class="footnote-ref" id="fnref3" role="doc-noteref"><sup>3</sup></a></p> <h1 id="circles-and-spheres">Circles and spheres</h1> <p>Before diving into the applications, we just need one more abstract remark about graded rings: the tensor product <span class="math inline">R\otimes S</span> of two graded rings has a natural graded ring structure by setting <span class="math display"> \deg(r\otimes s) \coloneqq \deg(r)+\deg(s) </span> and defining the multiplication to have a sign that “makes things work nicely”: <span class="math display"> (r\otimes s)(r&#39;\otimes s&#39;) \coloneqq (-1)^{|r&#39;||s|}(rr&#39;)\otimes(ss&#39;) </span> (note that the exponent uses the two “inner” terms, <span class="math inline">r&#39;</span> and <span class="math inline">s</span>, <strong>not</strong> <span class="math inline">r</span> and <span class="math inline">s</span>, nor <span class="math inline">r&#39;</span> and <span class="math inline">s&#39;</span> (nor <span class="math inline">r</span> and <span class="math inline">s&#39;</span>, for that matter)).</p> <p>What does this all have to do with algebraic topology? Hopefully you’ve already seen the ring structure on cohomology (of a manifold), given by the cup product. In fact, the cup product lets us assemble the cohomology groups into a <em>commutative graded ring</em> (this is indeed one justification for there being a minus sign in the definition of the cup product: it ensures graded commutativity!).</p> <p><em>In what follows, we say “space” to mean “CW complex that is nice enough to satisfy whatever hypotheses the theorem in question might need”.</em></p> <p>Here’s a fun fact that we won’t prove:</p> <div class="itenv" title="Lemma"> <p>Let <span class="math inline">X</span> and <span class="math inline">Y</span> be spaces. If <span class="math inline">Y</span> is such that <span class="math inline">\operatorname{H}^\bullet(Y)</span> is <em>free</em> (as a graded commutative ring) and <em>finitely generated</em>, then <span class="math display"> \operatorname{H}^\bullet(X\times Y) \cong \operatorname{H}^\bullet(X)\otimes\operatorname{H}^\bullet(Y) </span> (where <span class="math inline">\cong</span> means “isomorphic <em>as graded rings</em>”).</p> </div> <p>Using this, we can look at two examples.</p> <div class="rmenv" title="Example 1"> <p>We know that <span class="math display"> \operatorname{H}^n(S^1) = \begin{cases} \mathbb{Z} &amp;\text{if }n=0,1\text{;} \\0 &amp;\text{otherwise.} \end{cases} </span> If we denote by <span class="math inline">\alpha</span> the generator of <span class="math inline">\operatorname{H}^1(S^1)</span>, then <span class="math display"> \alpha^2 \coloneqq \alpha\smile\alpha </span> is an element of <span class="math inline">\operatorname{H}^2(S^1)</span>, but this group is zero, and so it must be the case that <span class="math inline">\alpha^2=0</span>. Thus <span class="math display"> \operatorname{H}^\bullet(S^1) \cong \mathbb{Z}\langle1\rangle \oplus \mathbb{Z}\langle\alpha\rangle/(\alpha^2) = \Lambda_{\mathbb{Z}}[\alpha] </span> (where here the angled brackets mean “the abelian group generated by these elements”).</p> <p>But we know that <span class="math inline">\Lambda_{\mathbb{Z}}[\alpha]</span> is free as a graded commutative ring, and it has only one generator so it’s also finitely generated; we can apply the Lemma above to get that <span class="math display"> \operatorname{H}^\bullet(S^1\times S^1) \cong \operatorname{H}^\bullet(S^1)\otimes\operatorname{H}^\bullet(S^1) </span> but this is exactly <span class="math inline">\Lambda_{\mathbb{Z}}[\alpha_1,\alpha_2]</span> (where <span class="math inline">\alpha_i</span> is the generator of the first cohomology of the <span class="math inline">i</span>-th copy of <span class="math inline">S^1</span>).</p> <p>Finally, recall that <span class="math inline">\Lambda_{\mathbb{Z}}[\alpha_1,\alpha_2]</span> is graded commutative but <em>not</em> commutative (i.e. <span class="math inline">\alpha_1\alpha_2=-\alpha_2\alpha_1</span>).</p> </div> <div class="rmenv" title="Example 2"> <p>We know that <span class="math display"> \operatorname{H}^n(S^2) = \begin{cases} \mathbb{Z} &amp;\text{if }n=0,2\text{;} \\0 &amp;\text{otherwise.} \end{cases} </span> If we denote by <span class="math inline">\beta</span> the generator of <span class="math inline">\operatorname{H}^2(S^2)</span>, then <span class="math display"> \beta^2 \coloneqq \beta\smile\beta </span> is an element of <span class="math inline">\operatorname{H}^4(S^2)</span>, but this group is zero, and so it must be the case that <span class="math inline">\beta^2=0</span>. Thus <span class="math display"> \operatorname{H}^\bullet(S^2) \cong \mathbb{Z}\langle1\rangle \oplus \mathbb{Z}\langle\beta\rangle/(\beta^2) = \Lambda_{\mathbb{Z}}[\beta] </span> where <em><span class="math inline">\beta</span> is of degree <span class="math inline">2</span></em>.</p> <p>As in the previous example, the Lemma gives us that <span class="math display"> \operatorname{H}^\bullet(S^2\times S^2) \cong \operatorname{H}^\bullet(S^2)\otimes\operatorname{H}^\bullet(S^2) </span> which is exactly <span class="math inline">\Lambda_{\mathbb{Z}}[\beta_1,\beta_2]</span>, but <em>where <span class="math inline">\beta</span> is of degree <span class="math inline">2</span></em> (we’re repeating this because it’s important).</p> <p>Now, <span class="math inline">\Lambda_{\mathbb{Z}}[\beta_1,\beta_2]</span> is graded commutative (since it’s a cohomology ring, and recall that these are always graded commutative, by the sign in the definition of the cup product), but we see that <span class="math display"> \beta_1\beta_2 = (-1)^{|\beta_1||\beta_2|} \beta_2\beta_1 = (-1)^{4}\beta_2\beta_1 = \beta_2\beta_1 </span> and so this cohomology ring is actually commutative!</p> </div> <div class="itenv" title="Corollary"> <p>The cohomology rings of <span class="math inline">S^1\times S^1</span> and <span class="math inline">S^2\times S^2</span> are not isomorphic (because one is commutative and the other is not).</p> </div> <h1 id="the-künneth-theorem">The Künneth theorem</h1> <p>What happens to our useful lemma if the cohomology ring <span class="math inline">\operatorname{H}^\bullet(Y)</span> is finitely generated but <em>not</em> free? That is (by the classification of finite groups), what if it has <em>torsion</em>? The <em>Künneth theorem</em> tells us that this torsion is exactly the “correction term” needed to fix the Lemma.</p> <div class="itenv" title="Corollary (to the Künneth theorem)"> <p>Let <span class="math inline">X</span> and <span class="math inline">Y</span> be spaces. If <span class="math inline">Y</span> is such that <span class="math inline">\operatorname{H}^\bullet(Y)</span> is finitely generated, then <span class="math display"> \operatorname{H}^\bullet(X\times Y) \cong (\operatorname{H}^\bullet(X)\otimes\operatorname{H}^\bullet(Y)) \oplus \operatorname{Tor}_{\bullet+1}(\operatorname{H}^\bullet(X),\operatorname{H}^\bullet(Y)) </span> (note that this <span class="math inline">\operatorname{Tor}</span> term is zero if <span class="math inline">\operatorname{H}^\bullet(Y)</span> is free<a href="#fn4" class="footnote-ref" id="fnref4" role="doc-noteref"><sup>4</sup></a>, so this really is a generalisation of our previous lemma).</p> </div> <div class="itenv" title="Example"> <p>We know that <span class="math display"> \operatorname{H}^n(\mathbb{RP}^2) = \begin{cases} \mathbb{Z} &amp;\text{if }n=0\text{;} \\\mathbb{Z}/2\mathbb{Z} &amp;\text{if }n=2\text{;} \\0 &amp;\text{otherwise.} \end{cases} </span> Doing some <span class="math inline">\operatorname{Tor}</span> calculations (use the fact that <span class="math inline">\mathbb{Z}</span> is free and thus flat (or even just projective) to see that <span class="math inline">\operatorname{Tor}(\mathbb{Z},-)=\operatorname{Tor}(-,\mathbb{Z})=0</span>; use the fact that <span class="math inline">\operatorname{Tor}(A,\mathbb{Z}/m\mathbb{Z})\cong\{a\in A\mid ma=0\}</span> to see that <span class="math inline">\operatorname{Tor}(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}</span>), the above Corollary gives us that <span class="math display"> \operatorname{H}^n(\mathbb{RP}^2\times\mathbb{RP}^2) = \begin{cases} \mathbb{Z} &amp;\text{if }n=0\text{;} \$$\mathbb{Z}/2\mathbb{Z})^2 &amp;\text{if }n=2\text{;} \\\mathbb{Z}/2\mathbb{Z} &amp;\text{if }n=3{;} \\\mathbb{Z}/2\mathbb{Z} &amp;\text{if }n=4{;} \\0 &amp;\text{otherwise.} \end{cases} </span> We’ve written the case <span class="math inline">n=3</span> on a separate line because this is exactly the “correction term” given by the Künneth theorem, i.e. the previous Lemma would have given us everything else but would have said that <span class="math inline">\operatorname{H}^3=0</span>; Künneth tells us otherwise.</p> </div> <div class="rmenv" title="Exercise"> <p>Calculate <span class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^2\times\mathbb{RP}^2;\mathbb{Q}/\mathbb{Z})</span>.</p> <p><em>Hint: use the Universal Coefficient Theorem, or the long exact sequence associated to the short exact sequence <span class="math inline">0\to\mathbb{Z}\hookrightarrow\mathbb{Q}\twoheadrightarrow\mathbb{Q}/\mathbb{Z}\to0</span>, and recall that <span class="math inline">\operatorname{Tor}(A,\mathbb{Q}/\mathbb{Z})\cong\operatorname{tors}(A)</span>, the torsion part of <span class="math inline">A</span>.</em></p> </div> <h1 id="some-useful-cohomology-rings">Some useful cohomology rings</h1> <p>Real projective spaces have pretty nice cohomology rings, but they differ in presentation depending on the parity of the dimension:</p> <ul> <li><span class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^{2n})\cong\mathbb{Z}[\beta]/(2\beta,\beta^{n+1})</span>, where <span class="math inline">|\beta|=2</span>;</li> <li><span class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^{2n+1})\cong\mathbb{Z}[\beta,\varepsilon]/(2\beta,\beta^{n+1},\epsilon^2,\beta\epsilon)</span>, where <span class="math inline">|\beta|=2</span> and <span class="math inline">|\varepsilon|=2n+1</span> (here <span class="math inline">\epsilon</span> is the generator of <span class="math inline">\operatorname{H}^{2n+1}</span>).</li> </ul> <p>What’s nice is that, if we work with <span class="math inline">\mathbb{Z}/2\mathbb{Z}</span> coefficients (instead of <span class="math inline">\mathbb{Z}</span>), then we can write both cases together as one thing:</p> <ul> <li><span class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^n;\mathbb{Z}/2\mathbb{Z})\cong(\mathbb{Z}/2\mathbb{Z})[\alpha]/(\alpha^{n+1})</span>, where <span class="math inline">|\alpha|=1</span>.</li> </ul> <p>Given that <span class="math inline">\mathbb{C}</span> is <span class="math inline">2</span>-dimensional over <span class="math inline">\mathbb{R}</span>, complex projective space behaves much nicer, since we don’t need to worry about parity:</p> <ul> <li><span class="math inline">\operatorname{H}^\bullet(\mathbb{CP}^n)\cong\mathbb{Z}[\beta]/(\beta^{n+1})</span>, where <span class="math inline">|\beta|=2</span>.</li> </ul> <p>Finally, if you know how to define infinite dimensional projective spaces, then it’s a very lovely cheeky little fact that the cohomology rings are just given by “taking the limit <span class="math inline">n\to\infty</span>”, i.e.</p> <ul> <li><span class="math inline">\operatorname{H}^\bullet(\mathbb{RP}^\infty)\cong\mathbb{Z}[\alpha]</span>, where <span class="math inline">|\alpha|=1</span>;</li> <li><span class="math inline">\operatorname{H}^\bullet(\mathbb{CP}^\infty)\cong\mathbb{Z}[\beta]</span>, where <span class="math inline">|\beta|=2</span>.</li> </ul> <p>Nice!</p> <aside id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"> <hr /> <ol> <li id="fn1"><p>Note that this condition extends to give some condition on the multiplication of arbitrary elements of <span class="math inline">R</span>, since every element can be written (in a unique way) as a sum of homogeneous elements, by the fact that <span class="math inline">R\cong\bigoplus_{d=0}^\infty R_d</span>.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn2"><p>This is very similar to the fact that a group homomorphism must send an element of order <span class="math inline">n</span> to an element of order <span class="math inline">m</span> <em>such that <span class="math inline">m</span> divides <span class="math inline">n</span></em>.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn3"><p>Both number theorists and algebraic geometers get very tired of having to write “let <span class="math inline">k</span> be a field of characteristic not equal to <span class="math inline">2</span>”, since <span class="math inline">2</span> “behaves badly” for them; some algebraic topologists, on the other hand, get very excited when you say “calculate (co)homology with coefficients in a field of characteristic <span class="math inline">2</span>”, since <span class="math inline">2</span> “behaves nicely” for them. Basically, <span class="math inline">2</span> is a tricky number.<a href="#fnref3" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn4"><p>Or if we work over a field instead of over <span class="math inline">\mathbb{Z}</span>…<a href="#fnref4" class="footnote-back" role="doc-backlink">↩︎</a></p></li> </ol> </aside> tag:thosgood.com,2022-02-05:/blog/2022/02/05/every-bundle-is-flat-from-infinity-pov/ Every principal bundle is flat, in the infinity world 2022-02-05T00:00:00Z 2022-02-05T00:00:00Z <p>Earlier today, Mahmoud Zeinalian explained something to me that Dennis Sullivan once explained to him, and it’s been sitting in my brain ever since then. In an attempt to empty out my thoughts, and also preserve what little understanding I currently believe to have of the story, I thought I’d write a little blog post about it. It’s going to move quite quickly, because I don’t want to spend time developing the prerequisites — the main purpose is for this to jog my brain two weeks down the line when I forget all the details!</p> <!-- more --> <p>The rather “provocative” statement that I’m going to try to justify today is the following:</p> <blockquote> <p><em>Every principal bundle is flat, from the <span class="math inline">(\infty,1)</span>-point-of-view.</em></p> </blockquote> <p>Let’s give some setup and explain why, first of all, this should sound rather bizarre, if not completely incorrect.</p> <h1 id="the-setup">The setup</h1> <p>Pick some principal <span class="math inline">G</span>-bundle <span class="math inline">\pi\colon E\to B</span>, endowed with a connection <span class="math inline">\nabla</span>. We’re going to look at what sort of information we can associated to singular cells in the base space <span class="math inline">B</span>, using the things that we have available to us.</p> <p>We know that the fibre <span class="math inline">E_b=\pi^{-1}(b)</span> of <span class="math inline">E</span> at any point <span class="math inline">b\in B</span> “looks like” the group <span class="math inline">G</span>, so, using the axiom of choice if we have to (or whatever really — we’re playing fast and loose here!), pick an element in each fibre to be the identity element of <span class="math inline">G</span>. That is, <em>very</em> discontinuously think of all the fibres as copies of <span class="math inline">G</span>.</p> <p>Now let’s look at what a <span class="math inline">1</span>-simplex (i.e. a line) in <span class="math inline">B</span> gives us. Say the line goes from the point <span class="math inline">b</span> to the point <span class="math inline">b&#39;</span>. We get a <span class="math inline">G</span>-equivariant map <span class="math inline">\Gamma_b^{b&#39;}\colon E_b\to E_{b&#39;}</span> by using the parallel transport offered to us from the connection <span class="math inline">\nabla</span>. But, using the fact that we have already identified our fibres with <span class="math inline">G</span>, and a nice technical lemma about Lie groups (saying (roughly) that, if a map commutes with all left actions, then it must be given by right action, i.e. by an element of the Lie group itself), we see that this is exactly the data of an element of <span class="math inline">G</span>, say <span class="math inline">g_{b,b&#39;}</span>.</p> <p>The next step is to think about what happens with <span class="math inline">2</span>-simplices (i.e. triangles) in <span class="math inline">B</span>. If we call the three vertices <span class="math inline">0</span>, <span class="math inline">1</span>, and <span class="math inline">2</span>, and label the edge between <span class="math inline">i</span> and <span class="math inline">j</span> by <span class="math inline">ij</span>, then, by the above, the three edges will give us three elements <span class="math inline">g_{01},g_{12},g_{02}\in G</span>. But here is where things “break”: it is <em>not</em> necessarily the case that <span class="math inline">g_{12}g_{01}=g_{02}</span>; this identity that the notation suggestively implies is only true if <span class="math inline">\nabla</span> is flat! (Recall: flatness of the connections corresponds to path-independence of the associated parallel transport).</p> <p>So we’ve come to the conclusion that a bundle is flat if and only if… it is flat. Great!</p> <p>So how about this claim that <em>every</em> bundle is flat “from the <span class="math inline">(\infty,1)</span>-point-of-view”?</p> <h1 id="some-cubical-wizardry">Some cubical wizardry</h1> <p>Trying to be ever so slightly more precise, we can formalise what we were doing in the above construction: for <span class="math inline">p=1,2</span>, to each singular <span class="math inline">p</span>-chain (i.e. each <span class="math inline">p</span>-simplex) in <span class="math inline">B</span> we were associating a <span class="math inline">(p-1)</span>-cube with values in <span class="math inline">G</span>. That is, were were constructing an element of the singular cochain complex on <span class="math inline">B</span> with values in “cubical chains with values in <span class="math inline">G</span>”, which we could write as <span class="math display"> \varphi_\nabla \in \mathrm{C}^{\bullet,\mathrm{sing}}(B,\mathrm{C}_{\bullet-1}^\mathrm{cube}(B,G)). </span> The fact that our connection is not flat is reflected in the fact that this element does not satisfy the Maurer–Cartan equation: <span class="math display"> \mathrm{d}\varphi_\nabla+\varphi_\nabla^2 \neq 0 </span> where defining the product turns out to be really easy, because cubes, <em>unlike simplices</em>, actually satisfy “the product of two cubes is a cube”.</p> <p>So here’s the question: we’ve constructed <span class="math inline">\varphi_\nabla</span> to be concentrated only in two (low) degrees; can we “extend” it to give an element that <em>does</em> satisfy Maurer–Cartan? The answer is <strong>YES</strong>, and this is exactly what we mean by our opening claim that “every bundle is flat in the <span class="math inline">(\infty,1)</span>-sense”. I won’t spell out the details on how <em>exactly</em> this works (since I’ve already been very informal up until now anyway), but I’ll show you some very lovely pictures, and then afterwards tell you where you can go to read about this in proper detail.</p> <p>Thinking about what we need to “construct” in order to extend our element, we can guess that we need some sort of “homotopy” between <span class="math inline">g_{12}g_{01}</span> and <span class="math inline">g_{02}</span>. But how exactly can we do this? And, further, how can we do it in a cubical way?</p> <p>Well we know that every path in <span class="math inline">B</span> gives us an element of <span class="math inline">G</span>, and we know that the path <span class="math inline">02</span> gives us <span class="math inline">g_{02}</span>, and the composite path <span class="math inline">01;12</span> gives us the element <span class="math inline">g_{12}g_{01}</span>, so let’s draw a bunch of intermediate paths! We can do this by starting at <span class="math inline">0</span>, heading along the line <span class="math inline">01</span> <em>for some time <span class="math inline">0\leq t_0\leq1</span></em>, and then heading towards the point <span class="math inline">2</span> for the rest of the time <span class="math inline">t_0-1</span>. When <span class="math inline">t_0=0</span> we recover exactly the line <span class="math inline">02</span>, and when <span class="math inline">t_0=1</span> we recover exactly the composite <span class="math inline">01;12</span>; for intermediate values, we get something like in the (very hastily drawn) picture below:</p> <figure> <img src="paths-in-delta-2.png" alt="Paths in the 2-simplex." /> <figcaption aria-hidden="true">Paths in the <span class="math inline">2</span>-simplex.</figcaption> </figure> <p>What’s wonderful is that, when we look at doing something similar for the <span class="math inline">3</span>-simplex, we see that all our paths are parametrised by variables <span class="math inline">0\leq t_0,t_1\leq 1</span>, i.e. by the <span class="math inline">2</span>-cube!</p> <figure> <img src="paths-in-delta-3.png" alt="Paths in the 3-simplex." /> <figcaption aria-hidden="true">Paths in the <span class="math inline">3</span>-simplex.</figcaption> </figure> <h1 id="behind-the-scenes-and-some-history">Behind the scenes, and some history</h1> <p>This whole story is really an explicit example of the <em>universal twisting cochain construction</em> for the path space bundle. What do I mean by that? Well, given any based topological space <span class="math inline">(X,x_0)</span>, we have the (based) path space bundle <span class="math inline">\mathcal{P}_{x_0}X\hookrightarrow X</span> given by sending a path to its endpoint; the fibre at the point <span class="math inline">x_0</span> is exactly the (based) loop space. Although this bundle doesn’t have a connection, we can use the homotopy lifting property to get something akin to parallel transport. We can make a nice simplification too: instead of looking at singular chains in <span class="math inline">X</span>, we can look at singular chains in <span class="math inline">X</span> whose <span class="math inline">0</span>-cells are all sent to the point <span class="math inline">x_0</span>; somehow this carries the same “homological information” as the usual singular chain complex. Then a <span class="math inline">1</span>-simplex is exactly a loop based at <span class="math inline">x_0</span>, which is exactly a <span class="math inline">0</span>-simplex in the loop space — this corresponds to the same “dimension drop” that we saw in the example above (where e.g. a line was sent to an element of <span class="math inline">G</span>).</p> <p>You know what? At this point, I’m just going to give up trying to explain things any better. Hopefully you’re confused enough by my meandering and poorly-written exposé that you want to read some proper references. This construction was described abstractly in J.F. Adams’ “On the Cobar Construction”<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a>; the concrete description (using these paths parametrised by the <span class="math inline">(p-1)</span>-cube) is found in Edgar H. Brown, Jr.’s “Twisted Tensor Products, I”<a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a>.</p> <p>Enjoy!</p> <aside id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"> <hr /> <ol> <li id="fn1"><p>J. F. Adams, “On the Cobar Construction”. <em>PNAS</em> <strong>42</strong> (1956), 409–412. <a href="https://www.jstor.org/stable/89694">JSTOR:89694</a>.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li> <li id="fn2"><p>Edgar H. Brown, Jr., “Twisted Tensor Products, I”. <em>Annals of Mathematics</em> <strong>69</strong> (1959), 223–246. <a href="https://doi.org/10.2307/1970101">DOI:10.2307/1970101</a>.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li> </ol> </aside> tag:thosgood.com,2022-01-23:/blog/2022/01/23/translations-part-3/ Translations 2022-01-23T00:00:00Z 2022-01-23T00:00:00Z <p>I haven’t blogged about it in a while, but I’ve been working on just making <a href="https://thosgood.com/translations">my translations</a> a bit better, both in terms of content and accessibility. Let’s have a look at what I’ve done, shall we?</p> <!-- more --> <p>The main change is that <em>nearly</em> all of the translations are now viewable as web versions, <em>as well as</em> PDFs. This hopefully makes them much more accessible to people who use screen readers, but is also nice even for people who don’t: you can change the font, the font size, switch to dark mode, use the table of contents which scrolls down the page with you, fit the whole page onto your phone, etc. etc. I also took this opportunity to refresh the CSS a bit, so hopefully things look slick and nice now.</p> <figure> <img src="new-css.png" title="A screenshot of the new style for the web versions of my translations" alt="New style for web versions of translations: the buttons in the top left are for toggling the table of contents, changing font options, and viewing the PDF version. Thanks Bookdown!" /> <figcaption aria-hidden="true">New style for web versions of translations: the buttons in the top left are for toggling the table of contents, changing font options, and viewing the PDF version. Thanks <a href="https://bookdown.org/">Bookdown</a>!</figcaption> </figure> <p>There are some translations which I probably won’t switch over to HTML, simply because they have commutative diagrams which cannot be rendered in the <a href="https://ctan.org/pkg/amscd?lang=en"><code>CD</code> environment</a> (which is now supported by KaTeX), and I just haven’t gotten around to sorting out image generation for these web versions yet.</p> <p>I’ve also finished some new translations since I last blogged. Here are the articles:</p> <ul> <li>M Balazard, E Saias, M Yor. “Notes sur la fonction ζ de Riemann, 2”. <em>Adv. in Math.</em> <strong>143</strong> (1999), 284–287. <a href="https://labs.thosgood.com/translations/AIM-143-1999-284.html">HTML</a></li> <li>P Deligne. “Variétés abéliennes ordinaires sur un corps fini”. <em>Inv. Math.</em> <strong>8</strong> (1969), 238–243. <a href="https://labs.thosgood.com/translations/IM-8-1969-238.html">HTML</a></li> <li>P Deligne. “Théorie de Hodge I”. <em>Actes du Congrès intern. math.</em> <strong>1</strong> (1970), 425–430. <a href="https://labs.thosgood.com/translations/ACIM-1-1970-425.html">HTML</a></li> <li>Y Diers. “Catégories Multialgébriques”. <em>Archiv der Math.</em> <strong>34</strong> (1980), 193–209. <a href="https://labs.thosgood.com/translations/ADM-34-1980-193.pdf">PDF</a></li> <li>A Grothendieck. “Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires”. <em>Annales de l’Institut Fourier</em> <strong>4</strong> (1952), 73–112. <a href="https://labs.thosgood.com/translations/AIF-4-1952-73.html">HTML</a></li> </ul> <p>And here are the seminars:</p> <ul> <li>H Cartan. “Les travaux de Koszul, I, II, and III”. <em>Séminaire Bourbaki</em> <strong>1</strong> (1952), Talks no. 1, 8, and 12. <a href="https://labs.thosgood.com/translations/SB-1-1%2B8%2B12.html">HTML</a></li> <li>A Douady. “Variétés et espaces mixtes; Déformations régulières; Obstruction primaire à la déformation”. <em>Séminaire Henri Cartan</em> <strong>13(1)</strong> (1960/61), Talks no. 2, 3, and 4. <a href="https://labs.thosgood.com/translations/SHC-13(1)-2+3+4.html">HTML</a></li> <li>Grothendieck, A. “Technique de descente et théorèmes d’existence en géométrie algébrique, I, II, and III”. <em>Séminaire Bourbaki</em> <strong>12</strong> and <strong>13</strong> (1959/60 and 1960/61), Talks no. 190, 195, and 212. <a href="https://thosgood.com/fga">HTML</a></li> </ul> <p>You might notice that this last one is Grothendieck’s FGA, and I’ve already talked a lot about how much progress has been made (although, admittedly, not for a while, due to general life chaos) about the <a href="https://github.com/ryankeleti/ega">EGA translation</a>, so the natural question to ask is <em>what about SGA?</em> Although I said I wasn’t going to work on this until I’d at least finished the FGA translation (of which there are only two and a half chapters left), I ended up needing to read bits of SGA 6 for my research anyway, so I just gave in and translated some: the homepage of the SGA translation is <a href="https://thosgood.com/sga/">thosgood.com/sga</a> (to accompany <a href="https://thosgood.com/fga">thosgood.com/fga</a>), and here you can see what I’ve done so far (just the first section of SGA 1, and a little bit of the introduction from SGA 6), but <em>do not expect any updates to this any time soon</em> — I really have to finish up some old projects before I get started on any news ones! (Although, as always, you can always contribute anything yourself if the fancy ever takes you: <a href="https://github.com/thosgood/sga">github.com/thosgood/sga</a>.)</p> <p>Anyways, that’s enough for now. Belated happy new year to you all!</p> tag:thosgood.com,2021-12-22:/blog/2021/12/22/blog-comments/ Comments on blog posts 2021-12-22T00:00:00Z 2021-12-22T00:00:00Z <p><strong>Update: comments have been “temporarily” disabled (I had some server issues).</strong> This blog now has support for comments! I haven’t had a chance to properly test things yet, and there are still some kinks left to iron out, but please do use this as an excuse to browse back through old posts and say nice things.</p> <!-- more --> <p>Using the lovely <a href="https://github.com/souramoo/commentoplusplus#how-is-this-different-to-the-original-commento">Commento++</a>, I’ve now enabled comments at the bottom of blog posts. Some things to point out:</p> <ul> <li>I haven’t yet made LaTeX rendering work, but it’s next on my to-do list. For now, feel free to write stuff surrounded by <code></code> or <code></code> (or wrap it in <code>\($$</code> or <code></code>), and at some point in the future this should actually render properly.</li> <li>I also haven’t yet made it so that people can post comments via their Twitter/GitHub/whatever accounts, but I also don’t expect that this will really be something that people particularly want to do? I assume I get around six readers per month on this blog, so there’s not particularly high demand for anything…</li> <li>If you want to comment a lot, you can create an account (using the “Login” button) which will let you set a profile picture and link to your website, but this isn’t necessary! You can always just leave a name, or even comment anonymously.</li> <li>Things might generally be a bit buggy, but if that’s the case, then please be patient, and also just <a href="mailto:tim.hosgood@posteo.net">let me know what went wrong</a>.</li> </ul> <p>Hopefully see you all in the comments soon!</p>