I’ve spent the last week at CT2025, which has just come to a close. It was great getting to see so many old friends and meet so many new ones, and every time I go to a CT I’m reminded of just how much category theory there is in the world, as well as just how much I enjoy all of it! Right before this I was in Antwerp for some Noncommutative Geometry, where I learned a ton and met even more new friends! Then next week I go to Bonn for my third conference in a row. I’m trying to stay energetic, and thankfully I have a few days off between CT and QTMART to help me rest up! I want to write up a lot of things I learned over the last month, since I have a lot of new thoughts on noncommutative geometry, mirror symmetry, and deformation theory, all coming from just my time in Antwerp! I’ve also learned a lot at CT and talked to a lot of interesting people about interesting things, and I’m sure I’ll have even more to say after my time in Bonn. I think organizing all of those thoughts are going to take a while, though (if I end up writing them down at all), but today I have a quick observation inspired by a few lovely conversations I had with Clark Barwick at CT. One of the many questions I asked him was if there’s a conceptual reason the Sphere Spectrum (read: the homotopy groups of spheres) is so darn complicated. He gave me an answer that’s obvious in hindsight, but which totally rearranged the way I think about things: I think I internalized a while ago that “free” constructions are fairly concrete. After all, you look at the syntax of whatever object you’re interested in, quotient out by the relations you want to be true and you’re done! Plus, mapping out of a free thing is as simple as possible, since it’s a left adjoint! All you have to do is find a (usually simpler) map from your generating set to a structure of interest and let the magic of category theory build your (usually more complicated) map for you… Of course, this view is heavily influenced by the kind of free structures I have experience with, and the kinds of questions I was asking about them. I was thinking about free groups and monoids, which you can study with word combinatorics, free (dg-)algebras on (graded) vector spaces, which look like polynomials, free categories on graphs, free $k$-linear or dg-categories on categories, and free cauchy completions of these, all of which come from just looking at paths, linear combinations, concentrating things in degree $0$, or working with twisted closures to add shifts and cones and whatnot1. I was thinking about relatively free constructions like the universal enveloping algebra, with its PBW-basis, or the right angled artin group attached to a (reflexive, simple) graph… All of these constructions feel like friends to me, in part because I know how to compute with them. Why would the sphere spectrum – the free spectrum on a single point – be so different? The point is that I’ve internalized these constructions as being tractable because I’m usually mapping out of them, in the direction the category theory encourages. I’m also usually relying on serious “normal form” theorems that make computing with these things tractable, or I’m doing fairly simple combinatorics with my generating set before arguing that these extend in some obvious way to things defined on the whole free object. All of these constructions become much less friendly when you start mapping into them, or asking more difficult questions about their internal structure. In hindsight, I’ve even personally struggled with tons of questions about free structures in my research! Free groups are extremely interesting from basically any perspective, with deep questions about their first order theory (Tarski’s Problem), the combinatorics relating their generating sets (The Andrews-Curtis Conjecture), or the coarse geometry of their outer automorphisms. Free cauchy completions are obviously complicated when you want to understand them on their own terms! If you take an algebra $A$ and view it as a one-object dg-category, then its cauchy completion1 is its whole category of perfect complexes! An enormous chunk of representation theory is, in that lens, dedicated to nothing more than the study of a certain, complicated, free construction! I’ve personally given up2 on a problem about relatively free constructions in right angled artin groups! These interpolate between free and free-abelian groups, and geometric group theory is teeming with interesting open problems about raags. For instance, can you understand, at the level of the underlying graphs, when one raag will embed into another? I thought about this off and on for a year before I started working seriously with my current advisor, and I made almost no progress at all. I also spent some time working with the adjunctions It’s interesting to try and construct these explicitly, and to understand the essential images of the left adjoints. This amounts to understanding which essentially algebraic theories are actually algebraic, and which algebraic theories are actually props. One of the big difficulties here is that we have a relatively free construction which adds relations rather than just operations. Adding new operations tends to be a fairly mild thing to do – consider the free algebra on an abelian group, which sends $A$ to its tensor algebra $\bigoplus_n A^{\otimes n}$ where it’s easy to recover the $A$ you started with. If instead we want to add new relations or axioms, for instance by freely sending a group $G$ to its abelianization $G \big / [G,G]$, then we lose lots of information in this construction. After some conversations with John Baez and Todd Trimble I came quite close to characterizing the image of the left adjoint between finite product categories and symmetric monoidal categories by factoring it into a “lossy” construction adding new axioms forcing the monoidal unit to be terminal and a much simpler construction which freely adds new operations corresponding to the product projections. I’ve had to put that project on hold while I focus on my thesis work, but I really want to come back and finish it soon. Of course as soon as you’re interested in logic, you have to accept that free things are complicated! The freest version of any theory lives in its classifying category, where truth and provability coincide. Then proving anything at all about the free model gives immediate understanding about all other models of that theory! This is already true for groups, whose classifying finite-product category is just the category of finitely generated free groups and homomorphisms. We don’t usually think about it because the kinds of statements you prove in equational logic aren’t very deep. But if you look instead at the classifying topos for groups and ask geometric questions suddenly you’re able to do a lot more, and the game becomes much harder! Perhaps this is clearest in the semantics of programming languages, where the free model (often called the “term model” in this context3) is the programming language, and checking whether two terms are equal in this free model literally amounts to evaluating two programs and seeing if their respective values agree. Because of this, many important structural results about a programming language (such as canonicity) can be proven by building another model whose semantics you understand, and then producing a section of the unique map from the free model. Also coming from logic are various lattices, whose free models can be quite intricate. Famously the free modular lattice on $3$ generators has $28$ elements, while the free modular lattice on $4$ generators is infinite! Indeed, this lattice has an undecidable word problem4, so that no program can tell whether two descriptions of its elements are the same or not! Heyting algebras are extremely important for semantics of intuitionistic logic, yet already the free heyting algebra on one generator is infinite, and the free heyting algebra on two generators is famously complicated. The study of free heyting algebras is still ongoing and seems quite difficult (at least as an outsider). See, for instance, Almeida’s recent preprint Colimits and Free Constructions of Heyting Algebras through Esakia Duality. With all this in mind, it shouldn’t be surprising at all that the sphere spectrum is so complicated! It’s the free spectrum on a point, and as such the only “relations” it will have are those that hold in all spectra! But of course spectra should obviously be complicated – They control all possible (co)homology theories for all spaces! So in this sense one should expect the internal structure of the sphere spectrum to be quite complicated, since any simplification would persist to something true of all cohomology theories. Of course, it’s easy to be complicated without being interesting, and I still think it’s a bit of a miracle that the sphere spectrum should have all this intricate structure inside it. As I understand it, much of Chromatic Homotopy Theory came from trying to explain patterns in the homotopy groups of spheres, and this subject is now as famously intimidating to outsiders5 as it is famously fascinating once you put in the work to become an insider6. Thanks for reading all! This really was a quick one for once, since I already had a lot of these examples floating around in my head. I really had all of the tools to realize that free things are obviously complicated in general, especially their “internal structure” that doesn’t ride the coattails of the universal property, but for some reason I just didn’t put it together until my conversation with Clark. It’s always dangerous to say what I’m thinking about writing about, but I at least have one more short post planned from my time in Antwerp, and maybe a longer one too if I have the energy. I’m doing a ton of writing right now, since I have two half-finished papers that I want to submit by the end of the summer. I think I’ll be able to get it done, but between writing these and going to conferences it’s been a tiring month. It’s tiring in a fun way, though, and I really feel like I’ve been productive in a way that I haven’t felt in a little while. I’m excited to start crossing a lot of these projects off my long-term-todo-list, especially since I already have three more projects I want to start! Regardless, I hope you’re having a more restful summer than I am! Stay safe, all, and we’ll talk soon ^_^. Emily Roff, taken at the top of the main tower in Brno) Actually it’s not completely obvious to me that the cauchy completion of a dg-category should be its idempotent triangulated closure… The cauchy completion will certainly be idempotent complete and triangulated, but in the well-named paper Cauchy Completeness for DG-Categories, Nicolić, Street, and Tendas show that to be cauchy complete you also need to be closed under “cokernels of protosplit chain maps”… I think this is some kind of split idempotent condition? But I haven’t read the paper closely enough to know for sure. If you want to be guaranteed to be correct, instead of “cauchy completion” you can say “the idempotent closure of the twisted closure”. That’s still a free construction and it still gives you the derived category in the special case your dg-category is a ring. ↩ ↩2 At least for now ↩ Pun intended ↩ Which is made more interesting by the fact that if you look at the class of lattices coming from lattices of subgroups of abelian groups, the corresponding free lattice on $4$ generators does have solvable word problem! This is remarkable since (as I understand it) modular lattices are called that because they look like lattices of submodules (in particular, sublattices of a lattice of subgroups of an abelian group). This is apparently proven in Herrmann’s On the Equational Theory of Submodule Lattices, and I learned all this from Ralph Freese the comments of this n-Category Cafe post. ↩ Such as myself. ↩ Which it seems like I might start doing soon, for a project I’m not ready to talk about yet. ↩

Earlier this week my friend Shane and I took a day and just did a bunch of computations. In the morning we did some differential geometry, where he told me some things about what he’s doing with symplectic lie algebroids. We went to get lunch, and then in the afternoon we did some computations in derived algebraic geometry. I already wrote a blog post on the differential geometry, and now I want to write one on the derived stuff too! I’m faaaaar from an expert in this stuff, and I’m sure there’s lots of connections I could make to other subjects, or interesting general theorem statements which have these computations as special cases… Unfortunately, I don’t know enough to do that, so I’ll have to come back some day and write more blog posts once I know more! I’ve been interested in derived geometry for a long time now, and I’ve been sloooowly chipping away at the prerequisites – $\infty$-categories and model categories, especially via dg-things, “classical” algebraic geometry (via schemes), and of course commutative and homological algebra. I’m lucky that a lot of these topics have also been useful in my thesis work on fukaya categories and TQFTs, which has made the time spent on them easy to justify! I’ve just started reading a book which I hope will bring all these ideas together – Towards the Mathematics of Quantum Field Theory by Frédéric Paugam. It seems intense, but at least on paper I know a lot of the things he’s planning to talk about, and I’m hoping it makes good on its promise to apply its techniques to “numerous examples”. If it does, I’m sure it’ll help me understand things better so I can share them here ^_^. In this post we’ll do two simple computations. In both cases we have a family of curves where something weird happens at a point, and in the “clasical” case this weirdness manifests as a discontinuity in some invariant. But by working with a derived version of the invariant we’ll see that at most points the classical story and the derived story agree, but at the weird point the derived story contains ~bonus information~ that renders the invariant continuous after all! Ok, let’s actually see this in action! First let’s look at what happens when we intersect two lines through the origin. This is the example given in Paugam’s book that made me start thinking about this stuff again. Let’s intersect the $x$-axis (the line $y=0$) with the line $y=mx$ as we vary $m$. This amounts to looking at the schemes $\text{Spec}(k[x,y] \big / y)$ and $\text{Spec}(k[x,y] \big / y-mx)$. Their intersection is the pullback and so since everything in sight is affine, we can compute this pullback in $\text{Aff} = \text{CRing}^\text{op}$ as a pushout in $\text{CRing}$: Pushouts in $\text{CRing}$ are given by (relative) tensor products, and so we compute which is $k$ when $m \neq 0$ and is $k[x]$ when $m=0$, so we’ve learned that: When $m \neq 0$ the intersection of \(\{y=0\}\) and \(\{y=mx\}\) is $\text{Spec}(k)$ – a single point1. When $m = 0$ the intersection is $\text{Spec}(k[x])$ – the whole $x$-axis. This is, of course, not surprising at all! We didn’t really need any commutative algebra for this, since we can just look at it! The fact that the dimension of the intersection jumps suddenly is related to the lack of flatness in the family of intersections $k[x,y,m] \big /(y, y-mx) \to k[m]$. Indeed, this doesn’t look like a very flat family! We can also see it isn’t flat algebraically since tensoring with $k[x,y,m] \big / (y, y-mx)$ doesn’t preserve the exact sequence2 In the derived world, though, things are better. It’s my impression that here flatness is a condition guaranteeing the “naive” underived computation agrees with the “correct” derived computation. That is, flat modules $M$ are those for which $M \otimes^\mathbb{L} -$ and $M \otimes -$ agree for all modules $X$! I think that one of the benefits of the derived world is that we can pretend like “all families are flat”. I would love if someone who knows more about this could chime in, though, since I’m not confident enough to really stand by that. In our particular example, though, this is definitely true! To see this we need to compute the derived tensor product of $k[x,y] \big / (y)$ and $k[x,y] \big / (y-mx)$ as $k[x,y]$-algebras. To do this we need to know the right notion of “projective resolution” (it’s probably better to say cofibrant replacement), and we can build these from (retracts of) semifree commutative dg algebras in much the same way we build projective resolutions from free things! Here “semifree” means that our algebra is a free commutative graded algebra if we forget about the differential. Of course, “commutative” here is in the graded sense that $xy = (-1)^{\text{deg}(x) \text{deg}(y)}yx$. For example, if we work over the base field $k$, then the free commutative graded algebra on $x_0$ (by which I mean an element $x$ living in degree $0$) is just the polynomial algebra $k[x]$ all concentrated in degree $0$. Formally, we have elements $1, \ x_0, \ x_0 \otimes x_0, \ x_0 \otimes x_0 \otimes x_0, \ldots$, and the degree of a tensor is the sum of the degrees of the things we’re tensoring, so that for $x_0$ the whole algebra ends up concentrated in degree $0$. If we look at the free graded $k$-algebra on $x_1$, we again get an algebra generated by $x_1, \ x_1 \otimes x_1, \ x_1 \otimes x_1 \otimes x_1, \ldots$ except that now we have the anticommutativity relation $x_1 \otimes x_1 = (-1)^{1 \cdot 1} x_1 \otimes x_1$ so that $x_1 \otimes x_1 = 0$. This means the free graded $k$-algebra on $x_1$ is just the algebra with $k$ in degree $0$, the vector space generated by $x$ in degree $1$, and the stipulation that $x^2 = 0$. In general, elements in even degrees contribute symmetric algebras and elements in odd degrees contribute exterior algebras to the cga we’re freely generating. What does this mean for our example? We want to compute the derived tensor product of $k[x,y] \big / y$ and $k[x,y] \big / y-mx$. As is typical in homological algebra, all we need to do is “resolve” one of our algebras and then take the usual tensor product of chain complexes. Here a resolution means we want a semifree cdga which is quasi-equivalent to the algebra we started with, and it’s easy to find one! Consider the cdga $k[x,y,e]$ where $x,y$ live in degree $0$ and $e$ lives in degree $1$. The differential sends $de = y$, and must send everything else to $0$ by degree considerations (there’s nothing in degree $-1$). This cdga is semifree as a $k[x,y]$-algebra, since if you forget the differential it’s just the free graded $k[x,y]$ algebra on a degree 1 generator $e$! So this corresponds to the chain complex where $de = y$ is $k[x,y]$ linear so that more generally $d(pe) = p(de) = py$ for any polynomial $p \in k[x,y]$. If we tensor this (over $k[x,y]$) with $k[x,y] \big / y-mx$ (concentrated in degree $0$) we get a new complex where the interesting differential sends $pe \mapsto ey$ for any polynomial $p \in k[x,y] \big / y-mx$. Some simplification gives the complex whose homology is particularly easy to compute! $H_0 = k[x] \big / mx$ $H_1 = \text{Ker}(mx)$ We note that $H_0$ recovers our previous computation, where when $m \neq 0$ we have $H_0 = k$ is the coordinate ring of the origin3 and when $m=0$ we have $H_0 = k[x]$ is the coordinate ring of the $x$-axis. However, now there’s more information stored in $H_1$! In the generic case where $m \neq 0$, the differential $mx$ is injective so that $H_1$ vanishes, and our old “classical” computation saw everything there is to see. It’s not until we get to the singular case where $m=0$ that we see $H_1 = \text{Ker}(mx)$ becomes the kernel of the $0$-map, which is all of $k[x]$! The version of “dimension” for chain complexes which is invariant under quasi-isomorphism is the euler characteristic, and we see that now the euler characteristic is constantly $0$ for the whole family! Next let’s look at some kind of “hidden smoothness” by examining the singular curve $y^2 =x^3$. Just for fun, let’s look at another family of (affine) curves $y^2 = x^3 + tx$, which are smooth whenever $t \neq 0$. We’ll again show that in the derived world the singular fibre looks more like the smooth fibres. Smoothness away from the $t=0$ fibre is an easy computation, since we compute the jacobian of the defining equation $y^2 - x^3 - tx$ to be $\langle -3x^2 - t, 2y \rangle$, and for $t \neq 0$ this is never $\langle 0, 0 \rangle$ for any point on our curve4 (We’ll work in characteristic 0 for safety). Of course, when $t=0$ $\langle -3x^2, 2y \rangle$ vanishes at origin, so that it has a singular point there. To see the singularity, let’s compute the tangent space at $(0,0)$ for every curve in this family. We’ll do that by computing the space of maps from the “walking tangent vector” $\text{Spec}(k[\epsilon] \big / \epsilon^2)$ to our curve which deform the map from $\text{Spec}(k)$ to our curve representing our point of interest $(0,0)$. Since everything is affine, we turn the arrows around and see we want to compute the space of algebra homs so that the composition with the map $k[\epsilon] \big / \epsilon^2 \to k$ sending $\epsilon \mapsto 0$ becomes the map $k[x,y] \big / (y^2 - x^3 - tx) \to k$ sending $x$ and $y$ to $0$. Since $k[x,y] \big / (y^2 - x^3 - tx)$ is a quotient of a free algebra, this is easy to do! We just consult the universal property, and we find a hom $k[x,y] \big / (y^2 - x^3 - tx) \to k[\epsilon] \big / \epsilon^2$ is just a choice of image $a+b\epsilon$ for $x$ and $c+d\epsilon$ for $y$, so that the equation $y^2 - x^3 - tx$ is “preserved” in the sense that $(c+d\epsilon)^2 - (a+b\epsilon)^3 - t(a+b\epsilon)$ is $0$ in $k[\epsilon] \big / \epsilon^2$. Then the “deforming the origin” condition says that moreover when we set $\epsilon = 0$ our composite has to send $x$ and $y$ to $0$. Concretely that means we must choose $a=c=0$ in the above expression, so that finally: The tangent space at the origin of $k[x,y] \big / (y^2 - x^3 - tx)$ is the space of pairs $(b,d)$ so that $(d \epsilon)^2 - (b \epsilon)^3 - t(b \epsilon) = 0$ in $k[\epsilon] \big / \epsilon^2$. Of course, this condition holds if and only if $tb=0$, so that: When $t \neq 0$ the tangent space is the space of pairs $(b,d)$ with $b=0$, which is one dimensional. When $t = 0$ the tangent space is the space of pairs $(b,d)$ with no further conditions, which is two dimensional! Since we’re looking at curve, we expect the tangent space to be $1$-dimensional, and this is why we say there’s a singularity at the origin for the curve $y^2 = x^3$….. But what happens in the derived world? Now we want to compute the derived homspace. As before, a cofibrant replacement of our algebra is easy to find, it’s just $k[x,y,e]$ where $x$ and $y$ have degree $0$, $e$ has degree $1$ and and $de = y^2 - x^3 - tx$. Note that in our last example we were looking at quasifree $k[x,y]$-algebras, but now we just want $k$-algebras! So now this is the free graded $k$-algebra on 3 generators $x,y,e$, and our chain complex is: We want to compute the derived $\text{Hom}^\bullet(-,-)$ from this algebra to $k[\epsilon] \big / \epsilon^2$, concentrated in degree $0$. The degree $0$ maps are given by diagrams that don’t need to commute5! Of course, such maps are given by pairs $(a + b \epsilon, c + d \epsilon)$, which are the images of $x$ and $y$. As before, since we want the tangent space at $(0,0)$ we need to restrict to those pairs with $a=c=0$ so that $\text{Hom}^0(k[x,y] \big / y^2 - x^3 - tx, \ k[\epsilon] \big / \epsilon^2) = k^2$, generated by the pairs $(b,d)$. Next we look at degree $-1$ maps, which are given by diagrams which are given by a pair $r + s\epsilon$, the image of $e$. Again, these need to restrict to the $0$ map when we set $\epsilon=0$, so that $r=0$ and we compute $\text{Hom}^{-1}(k[x,y] \big / y^2 - x^3 - tx, \ k[\epsilon] \big / \epsilon^2) = k$, generated by $s$. So our hom complex is where the interesting differential sends degree $0$ to degree $-1$ and is given by $df = d_{k[\epsilon] \big / \epsilon^2} \circ f - f \circ d_{k[x,y] \big / y^2-x^3-tx}$. So if $f$ is the function sending $x \mapsto b \epsilon$ and $y \mapsto d \epsilon$ then we compute So phrased purely in terms of vector spaces we see our hom complex is (living in degrees $0$ and $-1$): So we compute $H^0 = \text{Ker} ((-t \ 0))$ $H^{-1} = \langle s \rangle \big / \text{Im}((-t \ 0))$ When $t \neq 0$, our map is full rank so that $H^0$ are the pairs $(b,d)$ with $b=0$ – agreeing with the classical computation. Then $H^{-1} = 0$, so again we learn nothing new in the smooth fibres. When $t=0$, however, our map is the $0$ map so that $H^0$ is the space of all pairs $(b,d)$ is two dimensional – again, agreeing with the classical computation! But now we see the $H^{-1}$ term, which is $1$ dimensional, spanned by $s$. Again, in the derived world, we see the euler characteristic is constantly $1$ along the whole family! There’s something a bit confusing here, since there seem to be two definitions of “homotopical smoothness”… On the one hand, in the noncommutative geometry literature, we say that a dga $A$ is “smooth” if it’s perfect as a bimodule over itself. On the other hand, though, I’ve also heard another notion of “homotopically smooth” where we say the cotangent complex is perfect. I guess it’s possible (likely?) that these should be closely related by some kind of HKR Theorem, but I don’t know the details. Anyways, I’m confused because we just computed that the curve $y^2 = x^3$ has a perfect tangent complex, which naively would make me think its cotangent complex is also perfect. But this shouldn’t be the case, since I also remember reading that a classical scheme is smooth in the sense of noncommutative geometry if and only if it’s smooth classically, which $y^2 = x^3$ obviously isn’t! Now that I’ve written these paragarphs and thought harder about things, I think I was too quick to move between perfectness of the tangent complex and perfectness of the cotangent complex, but I should probably compute the cotangent complex and the bimodule resolution to be sure… Unfortunately, that will have to wait for another day! I’ve spent probably too many hours over the last few days writing this and my other posts on lie algebroids. I have some kind of annoying hall algebra computations that are calling my name, and I have an idea about a new family of model categories which might be of interest… But checking that something is a model category is usually hard, so I’ve been dragging my feet a little bit. Plus, I need to start packing soon! I’m going to europe for a bunch of conferences in a row! First a noncommutative geometry summer school hosted by the institute formerly known as MSRI, then CT of course, and lastly a cute representation theory conference in Bonn. I’m sure I’ll learn a bunch of stuff I’ll want to talk about, so we’ll chat soon ^_^. Take care, all! In fact we know more! This $k$ is really $k[x,y] \big / (x=0,y=0)$, so we know the intersection point is $(0,0)$. ↩ Indeed after tensoring we get since here $k \cong k[m] \big / (m)$. But then we can simplify these to and indeed the leftmost map (multiplication by $m$) is not injective! The kernel is generated by $x$. ↩ Again, if you’re more careful with where this ring comes from, rather than just its isomorphism class, it’s $k[x,y] \big / (x,y)$, the quotient by the maximal ideal $(x,y)$ which represents the origin. ↩ The only place it could possibly be $\langle 0, 0 \rangle$ is when $y=0$, but the points on our curve with this property satisfy $x^3-tx=y^2=0$ so that when $t \neq 0$ the solutions are $(x,y) = (0,0), \ (\sqrt{t}, 0), \ (-\sqrt{t}, 0)$. But at all three of these points $\langle -3x^2 - t, 2y \rangle \neq \langle 0, 0 \rangle$. ↩ This is a misconception that I used to have, and which basically everyone I’ve talked to had at one point. Remember that dg-maps are all graded maps! Not just those which commute with the differential! The key point is that the differential on $\text{Hom}^\bullet(A,B)$ sends a degree $n$ map $f$ to so that $df = 0$ if and only if $d_B f = (-1)^n f d_A$ if and only if $f$ commutes with the differential (in the appropriate graded sense). This means that, for instance, $H^0$ of the hom complex recovers from all graded maps exactly $\text{Ker}(d) \big / \text{Im}(d)$, which are the maps commuting with $d$ modulo chain homotopy! ↩

This is going to be a very classic post, where we’ll chat about a computation my friend Shane did earlier today. His research is largely about symplectic lie algebroids, and recently we’ve been trying to understand the rich connections between poisson geometry, lie algebroids, lie groupoids, and eventually maybe fukaya categories of lie groupoids (following some ideas of Pascaleff). Shane knows much more about this stuff than I do, so earlier today he helped me compute a super concrete example. We got stuck at some interesting places along the way, and I think it’ll be fun to write this up, since I haven’t seen these kinds of examples written down in many places. Let’s get started! First, let’s recall what an action groupoid is. This is one of the main examples I have in my head for lie groupoids, which is why Shane and I started here. If $G$ is a group acting on a set $X$, then we get a groupoid: here we think of $X$ as the set of objects and $G \times X$ as the set of arrows, where the source of $(g,x)$ is just $x$ the target of $(g,x)$ is $g \cdot x$, using the action of $G$ on $X$ the identity arrow at $x$ is $(1,x)$ if $(g,x) : x \to y$ and $(h,y) : y \to z$, then the composite is $(hg,x) : x \to z$ if $(g,x) : x \to y$ then its inverse is $(g^{-1},y) : y \to x$. Action groupoids are interesting and important because they allow us to work with “stacky” nonhausdorff quotient spaces like orbifolds in a very fluent way. See, for instance, Moerdijk’s Orbifolds as Groupoids: An Introduction, which shows how you can easily define covering spaces, vector bundles, principal bundles, etc. on orbifolds using the framework of groupoids. The point is that a groupoid $E \rightrightarrows X$ is a “proof relevant equivalence relation” in the sense that $E$ keeps track of all the “proofs” or “witnesses” that two points in $X$ are identified, rather than just the statement that two points are identified. Indeed, we think of $e \in E$ as a witness identifying $s(e)$ and $t(e)$ in $X$. Then reflexivity comes from the identity arrow, symmetry comes from the inverse, and transitivty comes from composition. The “proof relevance” is just the statement that there might be multiple elements $e_1$ and $e_2$ which both identify $x$ and $y$ (that is, $s(e_1) = s(e_2) = x$ and $t(e_1) = t(e_2) = y$). By keeping track of this extra information that “$x$ and $y$ are related in multiple ways” we’re able to work with a smooth object (in the sense that both $E$ and $X$ are smooth) instead of the nonsmooth, or even nonhausdorff quotient $X/E$. Next, we know that a central part of the study of any lie group $G$ is its lie algebra $\mathfrak{g}$. This is a “linearization” of $G$ in the literal sense that it’s a vector space instead of a manifold, which makes it much easier to study. But through the lie bracket $[-,-]$ it remembers enough of the structure of $G$ to make it an indispensable tool for understanding $G$. See, for instance, Bump’s excellent book Lie Groups or Stillwell’s excellent Naive Lie Theory. With this in mind, just playing linguistic games we might ask if there’s a similarly central notion of “lie algebroid” attached to any “lie groupoid”. The answer turns out to be yes, but unlike the classical case, not every lie algebroid comes from a lie groupoid! We say that not every lie algebroid is integrable. This, finally, brings us to the computation that Shane and I did together: As explicitly as possible, let’s compute the lie algebroid coming from the action groupoid of $SL_2(\mathbb{R}) \curvearrowright \mathbb{R}^2$. Let’s start with a few definitions coming from Crainic, Fernandes, and Mărcuț’s Lectures on Poisson Geometry. A Lie Algebroid on a manifold $M$ is a vector bundle $A \to M$ whose space of global sections $\Gamma(A)$ has a lie bracket $[-,-]_A$, equipped with an anchor map $\rho : A \to TM$ to the tangent bundle of $M$ that’s compatible with the lie bracket in the sense that for $\alpha, \beta \in \Gamma(A)$ and $f \in C^\infty(M)$ we have \([\alpha, f \cdot \beta]_A = f \cdot [\alpha,\beta]_A + (\rho(\alpha) f) \cdot \beta\) Then, given a lie groupoid $E \rightrightarrows M$ with source and target maps $s,t : E \to M$ and identity map $r : M \to E$, its lie algebroid is explicitly given by letting $A = r^* \text{Ker}(dt)$, which is a vector bundle over $M$ $[-,-]_A$ coming from the usual bracket on $TE$ (since $\text{Ker}(dt)$ is a subbundle of $TE$) $\rho : A \to TM$ given by $ds$ (which is a map $TE \to TM$, thus restricts to a map on $\text{Ker}(dt)$, and so gives a map on the pullback $A = r^* \text{Ker}(dt)$) If this doesn’t make perfect sense, that’s totally fine! It didn’t make sense to me either, which is why I wanted to work through an example slowly with Shane. For us the action groupoid is where First let’s make sense of $A = r^* \text{Ker}(dt)$. We know that $dt : T(SL_2(\mathbb{R}) \times \mathbb{R}^2) \to T\mathbb{R}^2$, that is, $dt : TSL_2(\mathbb{R}) \times T\mathbb{R}^2 \to \mathbb{R}2$, and is the derivative of the map $t : (M,v) \mapsto v$. If we perturb a particular point $(M_0, v_0)$ to first order, say $(M_0 + \delta M, v_0 + \delta v)$, then projecting gives $v_0 + \delta v$, which gives $\delta v$ to first order. So the kernel of this map are all the pairs $(\delta M, \delta v)$ with $\delta v = 0$, and we learn where we’re identifying the zero section of $T\mathbb{R}^2$ with $\mathbb{R}^2$ itself. Now to get $A$ we’re supposed to apply $r^*$ to this. By definition, this means the fibre of $r^* \text{Ker}(dt)$ above the point $v$ is supposed to be the fibre of $\text{Ker}(dt)$ over $r(v) = (\text{id},v)$. But this fibre is \(T_{\text{id}}SL_2(\mathbb{R}) \times \{v\}\), so that the pullback is a trivial bundle with fibre $\mathfrak{sl}_2(\mathbb{R})$: viewed as a trivial fibre bundle over $\mathbb{R}^2$. (Recall that the lie algebra $\mathfrak{sl}_2(\mathbb{R})$ is defined to be the tangent space of $SL_2(\mathbb{R})$ at the identity). Next we want to compute the bracket \([-,-]_A\). Thankfully this is easy, since it’s the restriction of the bracket on $TSL_2(\mathbb{R}) \times T\mathbb{R}^2$. Of course, an element of $A$ comes from $T_\text{id}SL_2(\mathbb{R}) = \mathfrak{sl}_2(\mathbb{R})$ and the zero section of $T\mathbb{R}^2$, so we get the usual lie bracket on $\mathfrak{sl}_2(\mathbb{R})$ in the first component and the restriction of the bracket on $T\mathbb{R}^2$ to \(\{0\}\) in the second slot. This zero section piece isn’t doing anything interesting, and so after identifying this bundle with the trivial bundle $\mathfrak{sl}_2(\mathbb{R}) \times \mathbb{R}^2$ we see the bracket is just the usual bracket on $\mathfrak{sl}_2(\mathbb{R})$ taken fibrewise. Lastly, we want to compute the anchor map. This caused me and Shane a bit of trouble, since we need to compute $ds$ at the identity, and we realized neither of us knew a general approach for computing a chart for $SL_2(\mathbb{R})$ near $\text{id}$! In hindsight for a $k$ dimensional submanifold of $\mathbb{R}^n$ the idea is obvious: Just project onto some well-chosen $k$-subset of the usual coordinate directions! I especially wish it were easier to find some examples of this by googling, so I’m breaking it off into a sister blog post which will hopefully show up earlier in search results than this one will. The punchline for us was that $SL_2(\mathbb{R})$ is defined to be So a chart near a point $(a_0, b_0, c_0, d_0)$ can be computed by looking at the jacobian of $f : \mathbb{R}^4 \to \mathbb{R}$ with $f(a,b,c,d) = ad-bc$ evaluated at the matrix of interest $(a_0, b_0, c_0, d_0)$. Since $1$ is a regular value for $f$, the jacobian will have at least one nonzero entry, and by locally inverting that coordinate we’ll get our desired chart! Since we want a chart near the identity, we compute the jacobian of $ad-bc$ at the identity to be We see that $\frac{\partial f}{\partial a} \neq 0$, so that locally near \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) the manifold looks like and this is our desired local chart! Now… Why were we doing this? We wanted to compute the anchor map from $A = r^* \text{Ker}(dt)$ to the tangent bundle $T\mathbb{R}^2$. This is supposed to be $ds$ (restricted to this subbundle). So how can we compute this? In the main body, I’ll make some identifications that make the presentation cleaner, and still show what’s going on. If you want a very very explicit version of this computation, take a look at this footnote1. Well $s : SL_2(\mathbb{R}) \times \mathbb{R}^2 \to \mathbb{R}^2$ is the map sending $(M,v) \mapsto Mv$. Explicitly, if we fix an $(x,y)$, this is the map From the previous discussion, we can write this as a map in local charts \(\{(a,b,c) \in \mathbb{R}^3 \mid a \neq 0 \} \to \mathbb{R}^2\) given by and now it’s very easy to compute $ds$. It’s just the jacobian but we only care about the value at the identity, since $A$ comes from pulling back this bundle along $r : v \mapsto (\text{id},v)$. So evaluating at $(a,b,c) = (1,0,0)$ we find our anchor map is Moreover, by differentiating \(\begin{pmatrix} a & b \\ c & \frac{1+bc}{a} \end{pmatrix}\) in the $a$, $b$, and $c$ directions and evaluating at $(1,0,0)$ we see that the basis $(\partial_a, \partial_b, \partial_c)$ for the tangent space to $(1,0,0)$ in our chart gets carried to the following basis for the tangent space at the identity matrix in $SL_2(\mathbb{R})$: which we recognize as $H$, $E$, and $F$ respectively. But this is great! Now we know that the lie algebroid of the action groupoid of $SL_2(\mathbb{R}) \curvearrowright \mathbb{R}^2$ is given by the trivial bundle $\mathfrak{sl}_2(\mathbb{R}) \times \mathbb{R}^2 \to \mathbb{R}^2$ with the usual bracket on $\mathfrak{sl}_2$ taken fibrewise, and the anchor map $\rho : \mathfrak{sl}_2 \times \mathbb{R}^2 \to T\mathbb{R}^2$ sending (in the fibre over $(x,y)$) $\rho_{(x,y)}(H) = (x, -y)$ $\rho_{(x,y)}(E) = (y, 0)$ $\rho_{(x,y)}(F) = (0, x)$ (which is the standard representation of $\mathfrak{sl}_2(\mathbb{R})$ on $\mathbb{R}^2$, viewed in a kind of bundle-y way.) Thanks for hanging out, all! It was fun to go back to my roots and write a post that’s “just” a computation. This felt tricky while Shane and I were doing it together, but writing it up now it’s starting to feel a lot simpler. There’s still some details I don’t totally understand, which I think will be cleared up by just doing more computations like this, haha. Also, sorry for not spending a lot of time motivating lie algebroids or actually doing something with the result of this computation… I actually don’t totally know what we can do with lie algebroids either! This was just a fun computation I did with a friend, trusting that he has good reasons to care. I’ve been meaning to pester him into guest-writing a blog post (or very confidently holding my hand while I write the blog post) about lie algebroids and why you should care. As I understand it, they give you tools for studying PDEs on manifolds which have certain mild singularities. This is super interesting, and obviously useful, and so I’d love to spend the time to better understand what’s going on. Stay safe, and if you’re anywhere like Riverside try to stay cool! If you want to be super duper explicit, our function $s$ sends $SL_2(\mathbb{R}) \times \mathbb{R}^2 \to \mathbb{R}^2$, which in a chart around the identity looks like the function given by now we differentiate to get $ds : TSL_2(\mathbb{R}) \times T\mathbb{R}^2 \to T\mathbb{R}^2$ sending $(a,b,c,\delta a,\delta b,\delta c, x, y, \delta x, \delta y)$ to $(ax+by, cx+\frac{1+bc}{a}y, ?, ?)$ Where the two $?$s are the output of matrix multiplication against the jacobian of $s$: Then we’re supposed to restrict this to $\text{Ker}(dt)$, which are the points $(a,b,c, \delta a, \delta b, \delta c, x, y, 0, 0)$. Since $\delta x = \delta y = 0$, we don’t even bother writing those entries of the matrix, and that’s how we get $ds$ as written in the main body. Now, as in the main body, we pull this bundle back along $r : v \mapsto (\text{id},v)$, which in our chart is $(x,y) \mapsto (1,0,0,x,y)$ so that our bundle $A$ (with its structure map to $\mathbb{R}^2$) is which, in the main text, we identify with \(\mathfrak{sl}_2 \times \mathbb{R}^2 = \{(\delta a, \delta b, \delta c, x, y)\} \to \{(x,y)\} = \mathbb{R}^2\) so we learn that our anchor map $ds$ is the restriction of the above map $ds$ to this pulled back subbundle, and sends where, again the $?$s are the result of the matrix multiplication which brings us back to the result of the main body. Of course, most working differential geometers wouldn’t write out this much detail to do this computation! I think it might be helpful to some newcomers to the field, and I certainly found it clarifying to write down exactly what happened, even if Shane and I weren’t nearly this careful when we were doing this together at a whiteboard. ↩

While doing a computation with my friend Shane the other day, we realized we needed to explicitly compute a local chart near the identity of $SL_2(\mathbb{R})$. It took us longer than I’d like to admit to figure out how to do this (especially since it’s so geometrically obvious in hindsight), and so I want to write down the process for future grad students looking to just do a computation! If you want to see what Shane and I were actually interested in, you can check out the main post here. Ok, let’s hop right in! Say you have a $2$-manifold in $\mathbb{R}^3$ to start1: If we think of the purple manifold $M$ as being an open disk, representing a small neighborhood of some possibly larger 2-manifold, then we can see the projection onto the $xy$-plane is a diffeomorphism onto its image (an open disk in $\mathbb{R}^2$) while the projections onto the $yz$ and $xz$ planes are not open in $\mathbb{R}^2$! This is because the normal to $M$ is parallel to the $z$ axis inside $M$, (indeed, at the “top of the hill”) so the tangent plane at that point degenerates and projects to a line whenever we project onto a coordinate plane containing the $z$-direction. For a more computational example, let’s try the hyperboloid $x^2 + y^2 - z^2 = 1$, and let’s see what happens near a few points. The tangent plane at a point is controlled by the jacobian of the defining equation, which for us is $\langle 2x, 2y, -2z \rangle$. This gives us three (disconnected) charts: \(\{2x \neq 0\}\), \(\{2y \neq 0\}\), and \(\{-2z \neq 0\}\), which we can see visually here (and we also drop the unnecessary scalars): These turn into 6 honest-to-goodness charts where we turn the disconnected condition \(\{x \neq 0\}\) into the pair of connected conditions \(\{x \gt 0\}\) and \(\{x \lt 0\}\). Indeed it’s easy to see that the 6 connected components in the above pictures are all diffeomorphic to an open subset of $\mathbb{R}^2$, and we can see this algebraically by projecting onto the plane avoiding the nonzero coordinate. On \(\{x \gt 0 \}\), for example, we have an open set of the $yz$-plane, shown here in orange: Algebraically, we compute this chart by noting on \(\{x \gt 0 \}\), we can solve for $x$ and (using the positive square root, since $x \gt 0$) write our surface locally as which is diffeomorphic in the obvious way to its projection onto the $yz$-plane so this is one of our charts! Similarly, we can look at \(\{z \gt 0\}\), solve for $z$ and locally write our surface as which is diffeomorphic to \(\{ (x,y) \mid x^2 + y^2 \gt 1 \}\) – another chart. On the intersection of these charts, \(\{x, z \gt 0 \}\), it’s now easy to write down our transition maps (if one is so inclined): Here our charts are the diffeomorphisms so it’s easy to compose them to see our transition maps between these charts are As a (fun?) exercise, compute the \(\{y \gt 0 \}\) chart, and the other two transition maps. For another example, let’s take a look at $SL_2(\mathbb{R})$, which is defined to be \(\left \{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid ad-bc = 1 \right \} \subseteq \mathbb{R}^4\). Then the jacobian of our defining map is $\langle d, -c, -b, a \rangle$, and we get charts corresponding to \(\{d \neq 0 \}\), \(\{-c \neq 0 \}\), \(\{ -b \neq 0 \}\), and \(\{ a \neq 0 \}\). In the \(\{d \neq 0\}\) chart, for instance, our defining equation looks like $a = \frac{1+bc}{d}$, so that $SL_2(\mathbb{R})$ looks locally like In the main post you can see how my friend Shane and I used this to compute the anchor map for a certain lie algebroid. Again, it makes a nice exercise to explicitly write out the various charts and transition maps What about a codimension 2 example? Let’s go back to our happy little hyperboloid, and intersect it with the surface $xyz = 1$. That is, we want to consider the manifold This is the levelset of the map $\mathbb{R}^3 \to \mathbb{R}^2$ sending $(x,y,z) \mapsto (x^2 + y^2 - z^2, \ xyz)$ taking value $(1,1)$. So we compute the jacobian and our charts are all the ways this matrix can have full rank. These conditions are: $2x \neq 0$ and $xz \neq 0$ $2x \neq 0$ and $xy \neq 0$ $2y \neq 0$ and $yz \neq 0$ $2y \neq 0$ and $xy \neq 0$ $-2z \neq 0$ and $yz \neq 0$ $-2z \neq 0$ and $xz \neq 0$ If we look at the \(\{2x \neq 0, \ xz \neq 0\}\) chart, we can ask sage to solve for $x$ and $z$ as functions of $y$: So as in the previous hyperboloid example, we need to break this into four charts, depending on whether $x$ and $z$ are positive or negative. Following the sage computation, in the \(\{x \gt 0, z \gt 0\}\) chart, we can write our curve as which, by projecting out the $y$ coordinate, is diffeomorphic to the open subset of $\mathbb{R}$ where all these square roots are defined. Ok, thanks for reading, all! This was extremely instructive for me, and hopefully it’s helpful to some of you as well! Sometimes it’s nice to just do some computations. Talk soon! That’s right. I finally bought an ipad. This has already dramatically improved my paper-reading experience (I love you Zotero) and my remote teaching experience, and I’m so glad that the example-drawing experience is shaping up to be everything I wanted it to be as well! ↩

The other day my friend Lucas Salim was asking me some questions about categorical logic and constructive math, and he mentioned he’d never seen a proof that there’s no constructive proof of the intermediate value theorem before. I showed him the usual counterexample, and since my recent blog post about choice was so quick to write I decided to quickly write up a post about this too, since I remember being confused by it back when I was first learning it. The key fact is Soundness and Completeness of the topos semantics of constructive logic. This says that there is a way of interpreting the usual syntax of mathematics into a topos in such a way that (Soundness) If you can constructively prove a statement, then its interpretation in every topos is true (Completeness) If a statement is interpreted as true in every (elementary1) topos, then there must exist a constructive proof For a proof, see Chapter II of Lambek and Scott’s Introduction to Higher Order Categorical Logic or Section D4.3 of The Elephant. This means that as long as we’re careful to avoid choice and excluded middle, anything we prove will be true when interpreted in any topos we like! Then there’s a mechanical procedure that lets us convert this interpretation into a corresponding statement in the “real world2”, and this gives us lots of “theorems for free” for each individual constructive theorem! A nice case study is given by the Wierstrass Approximation Theorem, which I gave a talk on years ago3. Since this theorem is constructively provable4… by interpreting it in the effective topos we learn there’s a computer program $\mathtt{Approx}$ which takes as input a function $f : [0,1] \to \mathbb{R}$5 and an $\epsilon \gt 0$ and outputs the coefficients of a polynomial approximating $f$. by interpreting it in a sheaf topos $\text{Sh}(\Theta)$ we learn that for any continuous family of functions $f_\theta(x) : \Theta \times [0,1] \to \mathbb{R}$, there’s locally6 a polynomial $p_\theta(x)$ whose coefficients are continuous functions of $\theta$ which approximates each $f_\theta(x)$ etc. If you’re interested in learning how to externalize a statement in a topos to a statement about the real world, I highly recommend Ingo Blechschmidt’s excellent paper Exploring mathematical objects from custom-tailored mathematical universes which gives a high level overview of the topic, while still giving enough details to let you externalize a few statements of your own! But where were we? The usual proofs of the intermediate value theorem aren’t constructive. See, for instance, Bauer’s Five Stages of Accepting Constructive Mathematics or Section 1 of Taylor’s A Lambda Calculus for Real Analysis for a discussion of how some common proofs fail (as well as great lists of constructively provable alternatives). Since the usual proofs seem to fail, we might guess that IVT is not provable constructively… but how could we prove this? Say, towards a contradiction, that there were a constructive proof of the intermediate value theorem. Then it would be true in every topos, and thus its various externalizations would all be true in the real world. So to show that there isn’t a constructive proof, all we have to do is find a topos which doesn’t think it’s true! Following many who came before me7, we’re going to use the topos of sheaves on $(-1,1)$. It’s been a minute since we’ve externalized a statement together, so let’s do it now! In full symbolic glory, the IVT says So now using the forcing language for $\text{Sh}((-1,1))$ (check out Theorem $1$ in Chapter VI.7 of Mac Lane and Moerdijk’s Sheaves in Geometry and Logic if you’re not sure what this means), we compute: We cash out the universal quantifiers to get “for every open $U \subseteq (-1,1)$, and for every continuous $f : U \times \mathbb{R} \to \mathbb{R}$, $a : U \to \mathbb{R}$, $b : U \to \mathbb{R}$, we have…” Next, cashing out the implication gives “for every open $U \subseteq (-1,1)$, and for every continuous $f : U \times \mathbb{R} \to \mathbb{R}$, $a : U \to \mathbb{R}$, $b : U \to \mathbb{R}$, so that for all $t \in U$ we know $a(t) \lt b(t)$, $f(t,a(t)) \lt 0$, and $f(t,b(t)) \gt 0$, we have…” Finally, cashing out the existential quantifer and the stuff inside it we get the external statement: The IVT is true inside $\text{Sh}((-1,1))$ if and only if the following is true externally: For every open $U \subseteq (-1,1)$ What a mouthful! Of course, we’re trying to prove this fails, so all we have to do is find an open set $U$ and functions $f$, $a$, and $b$ satisfying the assumptions so that the conclusion fails. We’ll choose $U = (-1,1)$ to be the whole set, $a(t) = -2$ and $b(t) = 2$ to be constant functions, and $f(t,x) : (-1,1) \times \mathbb{R} \to \mathbb{R}$ to be Then we see that, indeed, $f(t,a) \lt 0$ and $f(t,b) \gt 0$ for all $t \in (-1,1)$, so to prove the IVT fails in this topos we just need to show there’s no open cover on which $x(t)$ with $f(t,x(t)) = 0$ can be chosen continuously. The idea is that no matter how hard we try, $x(t)$ cannot be continuous in a neighborhood of $t=0$. Indeed, here’s an animation showing how $x(t)$ changes as we change $t$: You can see that when $t=0$ the root $x(t)$ jumps between $\pm 1$! Indeed, if we plot $x(t)$ we get: and it’s obvious that this is not the graph of a continuous function in any neighborhood of $t=0$. As long as we’re showing pretty graphics, you can also visualize this whole function $f$ as a surface over the strip $(-1,1) \times \mathbb{R}$. Then choosing a $t$ amounts to choosing a “slice” of the surface, and we can see that where that slice intersects the $(t,x)$-plane jumps suddenly as we cross $t=0$. In this example the axes are labeled $x$ and $y$ rather than $x$ and $t$: So where did we start, and where did we end? If the IVT were constructively provable, it would be true inside $\mathsf{Sh}((-1,1))$ and thus for our $f(t,x)$ we could find an open cover on which the zero $x(t)$ varies continuously in $t$. But this can’t possibly happen in a neighborhood of $t=0$, so we learn there is no constructive proof! Buuuuut, all is not lost! Usually classical theorems do have constructive analogues, either by adding new assumptions, weakening the conclusion, or by finding a different statement of the theorem that’s more positive. Andrej Bauer’s paper Five Stages of Accepting Constructive Mathematics lists many possibilities. For instance, one way to weaken the conclusion is to prove that for any $\epsilon$ you like, there’s an $x$ with $|f(x)| \lt \epsilon$. In our example, if we plot those $x$ so that $|f(t,x)| \lt \epsilon$ we get and it’s easy to fit the graph of a continuous selection function $x(t)$ inside this thickened region. Another approach is to recognize that the problem comes from $f$ “hovering” at $0$ when $t=0$. If we forbid this hovering, for instance by assuming $f$ is strictly monotone, then we can constructively prove the IVT (See Bauer’s Five Stages paper again). There’s yet another version, coming from Abstract Stone Duality, where we say that whenever $f(a) \lt 0 \lt f(b)$, the compact subspace $Z_f = { x \in [a,b] \mid f(x) = 0 }$ is occupied (Cor 13.11 in A Lambda Calculus for Real Analysis). This is a condition that’s weaker than inhabited but stronger than nonempty, which you can read about in Section 8 of the same paper. I don’t understand this condition very well, because I haven’t spent as much time thinking about ASD as I would like. Hopefully sometime soon I’ll find some time to work through some examples! Ok, thanks for reading, all! It’s nice to get a few quick posts up while I’m working on some longer stuff. I’m still thinking a lot about a cool circle of ideas involving Fukaya Categories, Skein Theory and T(Q)FTs, and Hall Algebras, and I’m slowly making progress on writing posts about all these fun things. Now, though, I have to go run a review session for a calculus class, haha. I’ll try to resist telling them about the fascinating subtleties that show up when you try to do everything constructively. Stay safe, and we’ll talk soon 💖 Usually on this blog when I talk about topoi I mean grothendieck topoi, but for this completeness result we really do need to allow more general elementary topoi with NNO. Indeed there are statements true in all grothendieck topoi that are not constructively provable (since they fail in some elementary topos). See here for a partial list. I would actually love to know if there’s a reference for what one has to add to IHOL to get something sound and complete for grothendieck topoi… I spent some time looking, but the only thing I found was Topological Completeness for Higher-Order Logic by Awodey and Butz, but it seems like they use $1+1$ in place of $\Omega$, so this isn’t the usual interpretation of logic in a grothendieck topos (which is also where the classical completeness comes from). ↩ Maybe “base topos” would be less philosophically charged ↩ I was really fumbling around with topos theory back then, haha. I’m much more confident now, and in the last few years I’ve just worked through a lot more examples and done more computations and read more papers and generally just learned a lot. Rereading that post was surprisingly… nostalgic isn’t the right word… but it’s fun to see how much I’ve grown! ↩ The usual proof with bernstein polynomials works if we’re careful to check some constructively relevant details. I’ll copy it here in case the wikipedia article changes someday: Fix $\epsilon > 0$. We write $b_{k,n}(x) = \binom{n}{k} x^k (1-x)^{n-k}$, and note that: $\sum_{k=0}^n b_{k,n} = 1$ $\sum_{k=0}^n \left ( x - \frac{k}{n} \right )^2 b_{k,n} = \frac{x(1-x)}{n}$ These are all provable by just expanding the left hand side, which is constructive. We also fix a $\delta$ so that whenever $|x-y| \lt \delta$ we have $|f(x) - f(y)| \lt \epsilon$. This is because, constructively, every continuous function on a compact sublocale of $\mathbb{R}$ is uniformly continuous. Note that here we crucially need to be working with locales! (see, eg, Thm 10.7 in Taylor’s A Lambda Calculus for Real Analysis) Lastly, we fix $M$ an upper bound for $\lvert f \rvert$. This is possible since $[0,1]$ is compact, overt, and inhabited (see Rmk 10.4 in Bauer and Taylor’s The Dedekind Reals in Abstract Stone Duality) thus the continuous $\lvert f \rvert$ admits a maximum (Thm 12.9 in A Lambda Calculus for Real Analysis). Now let $B_n f = \sum_{k=0}^n f(k/n) b_{k,n}$. We compute: In step (a) we use (1), and in step (b) we use the fact that $\forall x \in \mathbb{R} . x \lt \delta \lor x \gt \frac{\delta}{2}$ is constructively true (since the intervals overlap we don’t need excluded middle here!) But we can bound the first sum by noticing in this region $|x-k/n| \lt \epsilon$ so that (using (1) again) And since $\lvert f(x) - f(k/n)\rvert \leq \lvert f(x) \rvert + \lvert f(k/n) \rvert \leq 2M$, we compute: In step (c) we use $|x - k/n| \gt \frac{\delta}{2}$ to say that $\left ( \frac{\delta}{2} \right )^{-2} \left ( x - \frac{k}{k} \right )^2 \geq 1$, so that we can multiply through by it and make our sum bigger. In step (d) we use (2), and at the end we use the fact that $x(1-x) \leq \frac{1}{4}$ on $[0,1]$. Now since $\mathbb{R}$ is constructively archimedian (Defn. 1.1 in The Dedekind Reals in Abstract Stone Duality) we see $\exists n \in \mathbb{N} . \frac{2M}{\delta^2n} \lt \epsilon$. Since these bounds were uniform in $x$, we learn that as desired. ↩ A real number $x$ in this topos is a program that eats a natural number $n$ and outputs a rational number $x(n)$. We think about this as a sequence of rational approximations $x(n)$ converging to some real number $x$. So a function $[0,1] \to \mathbb{R}$ in this topos is a program that takes as input a program $x$ (outputting rational approximations between $0$ and $1$) and outputs a new program $f(x)$ which outputs rational approximations. ↩ Because our statement of Weierstrass $\forall \epsilon . \forall f . \exists p . \lVert f - p \rVert_\infty \lt \epsilon$ includes an existential quantifier there’s no way to get around the fact that the $p$ we build is only defined locally on an open cover. If you were more careful and gave a type theoretic proof that $\prod_\epsilon \prod_f \sum_p \lVert f - p \rVert_\infty \lt \epsilon$ then you could take $p$ to be a single polynomial defined on the whole of $\Theta$… I haven’t thought very hard about how possible this is (mainly because I haven’t spent much time thinking about what theorems about locales are provable in type theory), but I’m sure a talented undergraduate could figure it out. ↩ The earliest version of this example which I’ve seen comes from Stout’s Topological Properties of the Real Numbers Object in a Topos back in 1976. I think basically every other paper I’ve cited in this post gives some version of this example too, so it’s very well trodden ground! ↩