March 9, 2015

HITs and the Erlangen Program

Posted by Mike Shulman

Last week I had a minor epiphany, which I would like to share. If it holds up, it’ll probably end up somewhere in my chapter for Landry’s book.

I was fortunate during my formal education to have the opportunity to take a fair number of physics classes in addition to mathematics. Over that time I was introduced to tensors (more specifically, tensor fields on manifolds) several times in a row: first in special relativity, then in differential geometry, and then again in general relativity and Riemannian geometry.

During all those classes, I remember noticing that mathematicians and physicists tended to speak about tensors in different ways. To a mathematician, a tensor (field) was a global geometric object associated to a manifold; upon choosing a local chart it could be expressed using local coordinates, but fundamentally it was a geometric thing. By contrast, physicists tended to talk about a tensor as a collection of coordinates labeled by indices, with its “tensorial” nature encoded in how those coordinates transformed upon changing coordinates.

(To be sure, most physicists nowadays are aware of the mathematician’s perspective, and many even use it. But I still got the sense that they found it easier, or at least expected that we students would find it easier, to think in coordinates.)

Here is my epiphany: the symbiosis between these two viewpoints on tensors is closely related to the symbiosis between HITs and univalence in homotopy type theory. I will now explain…

So, last week I happened to be reading a paper by John Norton entitled General covariance and the foundations of general relativity: eight decades of dispute, and I ran across the following passage.

There is a presumption in much modern interpretation of Einstein… that much of what he says cannot be taken at face value. (Why does Einstein make such a fuss about introducing arbitrary spacetime coordinates? We have always been able to label spacetime events any way we please!)… [My] proposal… is that our modern difficulty in reading Einstein literally actually stems from a change… in the mathematical tools used…

In recent work… we begin with a very refined mathematical entity, an abstract differentiable manifold… We then judiciously add further geometric objects only as the physical content of the theory warrants….

In the 1910s, mathematical practices in physics were different…. one used number manifolds — $R^n$ or $C^n$ for example. Thus Minkowski’s ‘world’… was literally $R^4$, that is it was the set of all quadruples of real numbers.

Now anyone seeking to build a spacetime theory with these mathematical tools of the 1910s faces very different problems from the ones we see now. Modern differentiable manifolds have too little structure and we must add to them. Number manifolds have far too much structure…. the origin $\langle 0,0,0,0\rangle$ is quite different from any other point, for examples… The problem was not how to add structure to the manifolds, but how to deny physical significance to existing parts of the number manifolds. How do we rule out the idea that $\langle 0,0,0,0\rangle$ represents the preferred center of the universe…?

Felix Klein’s Erlangen program provided precisely the tool that was needed. One assigns a characteristic group to the theory… Only those aspects of the number manifold that remain invariant under this group are allowed physical significance…. As one increases the size of the group, one strips more and more physical significance out of the number manifold.

This passage places the two viewpoints on tensors in a larger context, both historically and mathematically. It’s not just about coordinates: it’s about whether we are adding structure to something excessively abstract, or subtracting structure from something excessively concrete. The “transformation rule” approach to tensors corresponds to the Erlangen subtractive approach: when we consider only structures invariant under our characteristic group, we can only use measurements that transform in some well-defined way under that group action.

Now, suppose that we are working in HoTT, and we want to define “the type of Minkowski spacetimes”. The modern mathematician who knows some type theory would go about this in a fairly straightforward way: a Minkowski spacetime is a 4-dimensional real affine space equipped with a certain kind of bilinear form, and we can express this in type theory in the usual way as an iterated $\Sigma$-type (in proof assistants, a “record”):

$Mink \coloneqq \sum_{M:Type} \sum_{v:RealAffineSpace(M)} \sum_{b:BilinearForm(M,v)} Dim(v,4) \times Nondeg(b) \times Symm(b) \times Sig(b,(+,+,+,-))$

Applying univalence, we conclude that for $M,N:Mink$, the identity type $M=N$ is equivalent to the type of affine isomorphisms $M\cong N$ preserving the metrics, as we would expect. In particular, $Mink$ is a 1-type.

Now, however, note that $Mink$ is also connected. That is, for any two $M,N:Mink$ there exists such an isomorphism; we can construct it by choosing origins and bases of $M$ and $N$ that put their metrics in canonical form. (Since $M$ and $N$ are not equipped with such bases, this is the truncated “there exists”, i.e. “there merely exists”; in symbols what we have is $\Vert M=N\Vert$.) Since $Mink$ is also clearly inhabited (by the standard $\mathbb{R}^{3,1}$), it is a $K(G,1)$. And what is $G$? It’s just the Poincare group $ISO(3,1)$.

But, we may now notice, there is another way to construct $K(G,1)$s for any group $G$ in HoTT, using higher inductive types. (These two approaches to classifying spaces also came up in another recent post.) Specifically, the HIT $K(G,1)$ is generated by the following constructors: a point $x$, a path $g:x=x$ for each $g\in G$, 2-paths imposing the multiplication table of $G$, and a 1-truncation constructor to make it a 1-type. Since $K(G,1)$s are unique up to equivalence, if in this construction we take $G$ to be the Poincaré group, we obtain a very different-looking construction of a type that is nevertheless equivalent to $Mink$; let’s call it $Mink'$.

The point is that $Mink'$ can be regarded as the 1910s Erlangen version of the type of Minkowski spacetimes, where instead of starting from an abstract thing with too little structure (an affine space) and adding stuff (a metric), we start from a concrete thing with too much structure ($\mathbb{R}^4$) and impose invariance under some automorphism group (the Poincaré group). The basepoint $x:Mink'$ can be thought of as representing $\mathbb{R}^4$, and the added automorphisms $g:x=x$ “force everything we say about $x$ to be invariant under the Poincaré transformation group”.

Of course, the point $x$ is not literally $\mathbb{R}^4$. But from the fact that the Poincaré group acts on $\mathbb{R}^4$ (which is of course necessary for this approach to get off the ground), we get a canonical map $Mink' \to Type$ sending $x$ to $\mathbb{R}^4$. (This is basically immediate from the universal property of $Mink'$ as an HIT, plus univalence to identify paths $\mathbb{R}^4=\mathbb{R}^4$ with automorphisms.) We could even declare this map to be an implicit coercion, so that we can informally identify $x$ with $\mathbb{R}^4$. Finally, of course, this map also factors through our original type $Mink$ by the canonical equivalence $Mink' \simeq Mink$.

I’m still pondering what conclusion to draw from all of this. It seems that modern mathematics, perhaps driven by set theory’s axiom of extensionality, has gone to the extreme of defining everything so that the already-present notion-of-sameness becomes the desired one. For instance, this is how we define the quotient of an equivalence relation in set theory: we take the elements of the quotient to be the equivalence classes, so that by the axiom of extensionality, if $x\sim y$ then their equivalence classes are equal (for the already-present notion of “equal” in set theory). The same is true for higher sorts of “sameness” in category theory: we define things like “Minkowski space” using complicated notions like vector space, bilinear form, and signature so that the “natural” sort of sameness for such gadgets, namely isomorphism in the appropriate category (which in the presence of univalence can be identified with equality in the appropriate type), is what we want it to be.

But it doesn’t have to be that way. For instance, Bishop proposed that to specify a set we should be obligated to specify not only its elements but also what it means for two of those elements to be equal. While I believe that he (and the closely related literature on “setoids” in type theory) didn’t go far enough — with HITs we can make our “specified equality” be the actual equality, so that we don’t need to worry about proving or hypothesizing that it is preserved by functions — this is a valid step in the other direction. Our above example of $Mink'$ shows that we can also define higher types this way: we defined (something equivalent to) “Minkowski spacetimes” without making any reference to a metric!

So perhaps one way to state the conclusion is that HITs unify Bishop’s ideas on sets with the Erlangen approach to geometry. Bishop said that when we define a set, we are free to specify under what conditions two of its elements are the same as each other. And the Erlangen program says that when we define a type of structures, we are free to specify in what ways such a structure is the same as itself. Of course, to a higher category theorist this shouldn’t be at all surprising, since both are a quotient construction — the quotient of an equivalence relation and the homotopy quotient of a group action. But nevertheless, I feel like I know something that I didn’t know a week ago.

Posted at March 9, 2015 7:21 AM UTC

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Re: HITs and the Erlangen Program

Does HIT stand for higher inductive type?

Posted by: Tom Leinster on March 9, 2015 12:30 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Yes, HIT stands for higher inductive type. Sorry!

Posted by: Mike Shulman on March 9, 2015 5:40 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

The Two Cultures Problem: Mathematical Physics vs. Theoretical Physics

Theoretical Physicists still live only in Euler’s Paradise; while Mathematical Physicists also live in Cantor’s Paradise and even in Eilenberg’s Paradise!

Posted by: Prof. David A. Edwards on March 9, 2015 1:29 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

That’s very cool! So is Mink something like the classifying space of Lorentzian metrics on a manifold?

It confuses me a bit that you seem to treat the Poincaré group $ISO(3,1)$ as a discrete group. Shouldn’t the Lie group structure play a role here? I understand that Mink is a $K(G,1)$ where $G$ is the Poincaré group considered as a discrete group, but that seems a bit odd because the Poincaré group would generally not be considered discrete. Maybe cohesive HoTT helps here?

Posted by: Tobias Fritz on March 9, 2015 2:16 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Yes, this is exactly where cohesion comes to the rescue. If we construct $Mink$ inside cohesive HoTT, then it will end up with the correct topology/smoothness, and the same is true of $Mink'$ if we start from the correct version of the Poincaré group. Either version will then indeed serve as a classifying space for Lorentzian metrics.

(Note, though, that they will not be what a classical algebraic topologist would call $B ISO(3,1)$: the cohesive structure of the Poincare group is kept separate from the $\infty$-groupoidal structure induced by delooping. In particular, $Mink$ is still a 1-type, which retains a topology/smoothness on its space of morphisms. If we want to pass from that topology to its homotopy-type, making topological paths in $ISO(3,1)$ into higher morphisms, then we can apply the shape modality $ʃ$.)

Posted by: Mike Shulman on March 9, 2015 5:50 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Perfect, you’ve even already answered my next question about $B$ISO(3,1) as well ;)

Posted by: Tobias Fritz on March 9, 2015 6:40 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Actually, the strategy of “defining everything so that the already-present notion-of-sameness becomes the desired one” goes back at least to Frege, who, in the Foundations of Arithmetic (1884) defines directions in terms of equivalence classes of lines (lines are equivalent iff they are parallel). So it goes back quite a long way.

Graham

Posted by: Graham White on March 9, 2015 4:40 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

One of my hobbies is finding modern abstract ideas instantiated in ancient texts, and so I present Euclid’s (borrowed) V.Def.iv-v:

Definition 4 Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.
Definition 5 Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

(That is, the ratio $a : b$ exists if there are natural numbers $m$ and $n$ with $a \lt m b$ and $n a \gt b$; ratios $a : b$ and $c : d$ are equal iff, for all natural numbers $m,n$ we have $m a \gt n b iff m c \gt n d$ $m a = n b iff m c = n d$ $m a \lt n b iff m c \lt n d$ …)

Note the “magnitude” type polymorphism! Euclid does not require that all magnitudes be mutually comparable! Neither does he exclude the possibility of nonarchimedean notions of magnitude: he simply says that pairs that would exhibit non-archimedean behaviour don’t have ratios.

Anyways, if that doesn’t mean Euclidean ratios are a Bishop setoid, …

Another fun example: gcd is a limit.

Posted by: Jesse C. McKeown on March 10, 2015 12:12 AM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Nice, thanks for sharing that! I may incorporate it in my chapter, if you don’t mind.

Posted by: Mike Shulman on March 10, 2015 3:34 AM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Interesting. Given some assumptions about the order, this is a way to assign a real-number ratio to two elements of an ordered monoid. For example, if we have $f \in O(g)$, then $f$ and $g$ have such a ratio, for instance (based on the “eventually less than” ordering).

Posted by: Ibrahim Tencer on March 11, 2015 8:59 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Mike said

Now, however, note that $\mathrm{Mink}$ is also connected… Since $\mathrm{Mink}$ is also clearly inhabited (by the standard $\mathbb{R}^{3,1}$), it is a K(G,1).

I think I’m missing something about the HoTT language here – why does the higher homotopy of $\mathrm{Mink}$ vanish?

There is something that strikes me as a little circular about the HIT approach , which could equally be said of the Erlangen program – namely, some of the work of defining $\mathrm{Mink}$ has simply been pushed off into how we choose to define the Poincaré group – we could define it as the automorphism group of a point of $\mathrm{Mink}$ or $\text{syntax error at token '}$, or else as certain matrices with a matrix composition law, or via generators and relations – the first two options seem circular, while the last two seem ad hoc to me.

Anyway, it’s cool to get this perspective of Bishop, Erlangen, higer inductive types!

Posted by: Tim Campion on March 10, 2015 3:06 AM | Permalink | Reply to this

Re: HITs and the Erlangen Program

[sorry, I messed up my latex and didn’t look closesly enough at the preview]

We could define the Poincaré group as the automorphisms of a point in $\mathrm{Mink'\mathrm{Mink}$ or $\mathrm{Mink}'$, or we could define it as certain matrices with a matrix composition law, or via generators and relations. The first two definitions seem circular, while the last two seem ad hoc to me.

Anyway, it’s cool to see this categorification of Bishop $\to$ Erlangen $\to$ higher inductive types!

Posted by: Tim Campion on March 10, 2015 3:10 AM | Permalink | Reply to this

Re: HITs and the Erlangen Program

why does the higher homotopy of $\mathrm{Mink}$ vanish?

Because every Minkowski spacetime is a set (a 0-type), the type of equivalences between any two such is again a set. Thus, the path-spaces of $Mink$ are sets, so $Mink$ is a 1-type. (If you’re expecting to see the higher homotopy that is to be found classically in the space called $B ISO(3,1)$, see my above response to Tobias.)

The first two definitions seem circular, while the last two seem ad hoc to me.

I agree there is a question there. What about as the subgroup of $Aut(\mathbb{R}^4)$ generated by translations, rotations, and Lorentz boosts? That frees us from having to specify any relations explicitly, and I don’t think it is very ad hoc — it corresponds exactly to saying that we’re interested in properties of $\mathbb{R}^4$ that are invariant under a certain specific collection of automorphisms.

Posted by: Mike Shulman on March 10, 2015 3:31 AM | Permalink | Reply to this

Re: HITs and the Erlangen Program

That is very interesting. The general dichotomy of approaches between adding structure from something too abstract or subtracting structure from something too concrete is still prevalent in mathematics now. A construction in set theory is the latter, and sometimes we need to do such a thing to give an existence proof. (Or maybe to give an easy existence proof.) Also sometimes the latter makes the mathematics easier.

In model theory there is the tendency to work in a “monster model”, i.e. a universal domain, and adopt the convention that a model of a theory means a submodel of the monster model. It has some advantages, chiefly that various different notions all become subsets of the monster model and then when you want to do the same things with the different notions it is easy. On the other hand geometers abandoned this approach a long time ago, and it makes communication between model theorists and geometers rather difficult.

A more concrete example: a number field is either an abstract field extension of Q of finite rank, or a finitely generated subfield of Q^{alg}. With the second approach, a Galois extension of Q is just a number field fixed setwise by the automorphism group of Q^{alg}. I find it an easier concept than a (separable and) normal extension of Q.

Posted by: Jonathan Kirby on March 12, 2015 4:16 PM | Permalink | Reply to this

Re: HITs and the Erlangen Program

Actually, I believe that pure Bishop-style equality is a better approach than HoTT, and I find it strange that you describe setoids as “[not] go far enough” compared to HoTT when I don’t see the two approaches as going the same direction. Saying that ‘we can make our “specified equality” be the actual equality’ is misleading, because with Bishop too the specified equality is the same as the actually equality – it’s the definition of equality. (The fact the under certain circumstances it is possible to define some notion of “intensional equality” does not mean we should assign this concept any weight!)

One problem with HoTT-style truncation is that it “traps” critical information about an object when it is not equality-relevant. In contrast, with setoids this information may be used in arbitrary ways, as long as the final result of a function preserves equality. An analogy may be made with Java-style enforced private fields and Python-style private-by-convention, though it’s not clear whether this helps my point. To be more precise, there are many constructions that use the Axiom of Choice in HoTT where it is not necessary with the Bishop-style approach. Conversely this problem essentially disappears (and perhaps completely disappears, though I don’t know how to prove that) if you freely assume the Axiom of Choice. Nonetheless it seems significant that whether a theorem assuming the Axiom of Choice can be translated to a theorem about setoids without using the Axiom of Choice is a very good test for whether the usage of the Axiom of Choice is ultimately just an innocent technicality.

Posted by: Itai Bar-Natan on April 1, 2015 1:35 PM | Permalink | PGP Sig | Reply to this

Re: HITs and the Erlangen Program

As I see it, Bishop-style equality is strictly less general than theories with “intrinsic equality”, so I don’t see any reason to use it when developing mathematics. The category of setoids is the free exact completion of the base category (whose objects are “pre-sets” or types or whatever); so when you work with setoids, you’re essentially limiting your models to categories that are free exact completions. By contrast, a theory with intrinsic equality can in particular be interpreted into a free exact completion (and thereby yield conclusions about setoids), but also into other models that are not a free exact completion of anything. So if you can prove something in the latter theory, it yields a more general result.

Of course, free exact completions satisfy certain principles that not all categories do, notably the presentation axiom (and hence countable choice). So it may be easier to prove certain things there than in the general case. But I think it’s still better style to simply assume whatever principles you need, rather than explicitly restricting yourself to a certain class of models.

I also don’t think that it’s correct to say

In contrast, with setoids this information may be used in arbitrary ways, as long as the final result of a function preserves equality.

If the final result of a function really does preserve equality, then you can define it with HoTT-style quotients: that’s exactly what the induction principle for quotients says. Setoids do let you construct things that actual quotients don’t, but I think that is more correctly attributed to a partial blurring of the line between $\Sigma$ and $\exists$, and hence to a choice principle such as the presentation axiom.

Philosophically, I object to the idea of calling a relation “equality” when it doesn’t satisfy Indiscernibility Of Identicals. If two things really are equal, then you shouldn’t be able to do anything that distinguishes them. If we want to do constructions on Cauchy sequences that don’t respect equivalence, that’s fine, but we should be honest that that’s what we’re doing and not claim to be performing constructions on “real numbers”.

Finally, setoids are an enormous pain to deal with when formalizing mathematics in a proof assistant, since you have to explicitly prove all the time that everything you do respects the setoid equality. HoTT is much nicer.

Posted by: Mike Shulman on April 1, 2015 9:11 PM | Permalink | Reply to this

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