Spacetime Groupoid in GR

Yet another post about equivalence of spacetime models. Apologies. I've been thinking about this since I learnt GR in 1985 (Part III Mathematics at the Other Place). In particular, learning that spacetimes related by "diffeomorphisms" are (are?) the same physical world. "But how can that be?", I thought. If they're different, how can they be the same? It just doesn't make sense ...

But part of the answer is quite simple: spacetime models are spacetime models, mathematical representations: and what they represent is different. So, we need to understand what "representation" means, and what the represented thing---i.e., a physical world---is. So, if $\mathcal{M}$ is a spacetime model and $w$ is a world, we need to understand what

(1) $\mathcal{M}$ represents $w$

means. And we wish to understand it in a way that explains why suitably related ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ represent the same $w$.

I think I know an answer to this, using the diagram conception of structure and worlds. (1) should first be rephrased to include the physical relations ${\mathsf{R}}_1,\dots$ which "interpret" the model, as

(2) $\mathcal{M}$ represents $w$ relative to ${\mathsf{R}}_1,\dots$.

This relationship can then be defined as,

(3) $w={\hat{\mathrm{\Phi }}}_{\mathcal{M}}[{\mathsf{R}}_1,\dots ]$

where ${\hat{\mathrm{\Phi }}}_{\mathcal{M}}$ is the propositional diagram of $\mathcal{M}$. It is an infinitary, categorical, propositional function which defines the isomorphism type of the model $\mathcal{M}$. It is the best candidate for the abstract structure of the model $\mathcal{M}$, since it is what all isomorphic copies "have in common". That is, the mapping

$\mathcal{M}\mapsto {\hat{\mathrm{\Phi }}}_{\mathcal{M}}$

satisfies Leibniz Abstraction:

Leibniz Abstraction
${\hat{\mathrm{\Phi }}}_{{\mathcal{M}}_1}={\hat{\mathrm{\Phi }}}_{{\mathcal{M}}_2}$ iff ${\mathcal{M}}_1\cong {\mathcal{M}}_2$.

And, from this framework, one can then prove Leibniz Equivalence. So, Leibniz Equivalence is not a peculiarity of a particular physical theory, Einstein's GR. Rather, GR helps to motivate the "propositional diagram conception" of what worlds are, and how they are "represented" by "models", and then Leibniz Equivalence comes out as consequence, not an assumption. And an interesting further consequence of this is that worlds do not have domains. For if one picks a domain, $D_w$, of "things existing at $w$", one can apply a "Hole argument" (an argument quite analogous to the one given in philosophy of physics literature: see, e.g, Norton 2011, "The Hole Argument", SEP) against any such choice, generating indiscernible duplicates, violating Leibniz Equivalence. The selection of a domain is something like a "gauge choice".

I keep wishing to see how this approach to abstract structure, Leibniz equivalence, anti-haecceitism, etc., fits together with the way that category theorists talk, as I don't know enough category theory beyond the basics and basic constructions. In model theory, models look like

$\mathcal{A}=(A,\dots )$, 

and one has some associated language $L$ whose signature matches the signature of $\mathcal{A}$. One can then define $\mathcal{A}\models \varphi$ and talk about definable sets in $\mathcal{A}$, etc.; in addition, we may consider the language $L(\mathcal{A})$, obtained by adding to $L$ a constant $\underset{―}{a}$ for each $a\in A$ (where $A$ is the domain of $\mathcal{A}$). This lets one do things a bit more easily: e.g., to prove results about $\mathcal{A}$ by considering the elementary diagram of $\mathcal{A}$.

A "model" is, in other contexts, sometimes called a "mathematical structure" or a "structured set": examples are orderings, graphs, groups, monoids, fields, ordered fields, algebras, lattices, etc. For example:

an ordering $(A,\le )$ with certain conditions on $\le$.
a graph $(A,E)$ with certain conditions on $E$ (the edge relation).
a monoid $(A,\circ )$ with certain conditions on $\circ$.
a group $(A,\circ )$ with certain conditions on $\circ$.
a field $(A,0,1,+,\times )$ with certain conditions on $0$,$1$,$+$, $\times$, etc.

In each case, $A$ is the domain, or carrier set, of the model (structure). And $\le$, $E$, $+$, etc., are called distinguished relations and operations. A model (structure) is called an algebraic structure just when all of its distinguished parts are operations (except identity, which is taken as primitive). Not every relation/operation on the domain $A$ is distinguished or special, obviously. The special ones are singled out somehow. E.g., when we say,

"a set $A$ with (or equipped with) a binary operation $\circ :A\times A\to A$, etc., ...", 

we mean that this relation/operation $\circ$ is distinguished in the signature of the model. It follows from this that models are individuated as follows:

Individuation for Models/Structures
$(A,R_1,\dots )=(B,S_1,\dots )$ iff $A=B$, and $R_i=S_i$, for each $i$.

This individuation condition explains why models are represented as ordered n-tuples. Models are identical (literally identical: the same "thing") when they have the same carrier set and the same distinguished relations. I.e., extensionally identical. For example, for $n$-ary relations $R,S\subseteq A^n$, we have

$R=S$ iff for any $n$-tuple $\overrightarrow{a}\in A^n$, $\overrightarrow{a}\in R$ iff $\overrightarrow{a}\in S$.

With a bit more effort, one can think of vector spaces (over a field) as models or structured sets, but with modifications. A vector space $\mathbb{V}$ is a 2-sorted model:

$\mathbb{V}=((V,F);0,1,+,\times ,{+}_v,\cdot )$

where there are two carrier sets, $V$ and $F$, and the underlying field is $\mathbb{F}=(F,0,1,+,\times )$, and

$\cdot :F\times V\to V$

is scalar multiplication, and

${+}_v:V\times V\to V$

is vector addition. (Usually, in linear algebra, one just uses the same symbol for the field addition operation, and the vector addition operation.) Passing from the 2-sorted vector space $\mathbb{V}$ to the 1-sorted underlying field $\mathbb{F}$ would be what model theorists call a reduct. Category-theoretically this corresponds to a "forgetful functor". The "inverse" of this is called by model theorists an expansion: i.e., adding new distinguished relations and operations, usually to interpret new symbols in an extended language.

One can then write down axioms $\mathsf{Vect}$ in a 2-sorted language $L$ such that

$\mathbb{V}\models \mathsf{Vect}$ iff $\mathbb{V}$ is a vector space. 

[Here it's not so clear to me how to write down first-order axioms for a topological space $\mathbb{T}=(X,\mathcal{T})$ as a 2-sorted model, because $\mathcal{T}\subseteq \mathcal{P}(X)$ is required to be closed under arbitrary unions, and this requires quantification over sets of sets of points.]

A class of models isomorphic to some given one is called an isomorphism class or an isomorphism type. One asks questions, usually cardinality questions, about isomorphism types. E.g.,

• how many unique--up-to-isomorphism countable models does $PA$ have?

(The answer to this is $2^{{\mathrm{\aleph }}_0}$, by considering complete extensions of $PA$.)

Let $\mathsf{DLO}$ be the theory of Dense Linear Orders without endpoints. Then:

• how many unique-up-to-isomorphism countable models does $\mathsf{DLO}$ have?

(The answer to this is $1$. All such orders are isomorphic to $(\mathbb{Q},\le )$, a result due originally to Cantor and a proof is here.)

My interest in this, however, comes from General Relativity and Leibniz Equivalence. In GR, one considers spacetime models which are usually presented something like this:

$\mathcal{M}=(M,g,T,\psi ,\dots )$

where $M$ is understood to be a differentiable manifold, $g$ and $T$ symmetric $(0,2)$ tensors, and perhaps $\psi$ is some other tensorial field on $M$. We needn't worry about the conditions that these models have to satisfy, and when a model $\mathcal{M}$ is a solution to Einstein's equations. But, given the setup, one can then consider questions like:

"What are the properties of a massless scalar charged field $\varphi$ on a 26-dimensional curved space-time $(M,g)$?"

Is this mathematics? Or is it physics? The sense in which it is mathematics is obvious. For

• You know what a manifold $M$ is.
• You know that a symmetric $(0,2)$ tensor is.
• You know what a $\mathbb{C}$-valued function on $M$ is.
• You know what the differential equations involved are (Einstein's equation, the Klein-Gordon equation).

So, we can, if we work hard, work out properties of such models.

These models are not in fact given in the usual first-order (and possibly many-sorted) way. They involve further structure (definable, though usually second-order definable), related to various vector spaces which appear as values of tensorial fields on $M$. If one tried to make all this explicit---to make explicit everything that is being quantified over---one should probably start with the notion of a principle bundle.

Now, for the philosophy: what does "represent" mean? We have a model $\mathcal{M}=(M,g,\dots )$ and this somehow "represents" physical worlds. The models are themselves just models. The models are not spacetime! They "represent" spacetime. What is the relation between these models when they are to be considered as representing the same physical situation/world? That relationship, it turns out, is simply: isomorphism. That is,

Leibniz Equivalence (see Wald 1984: 428)
If ${\mathcal{M}}_1\cong {\mathcal{M}}_2$, then ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are physically equivalent.

Some authors suggest that this reflects a kind of "gauge freedom" in GR. Well, if "gauge freedom" is the right word, then gauge freedom in GR is simply selecting one model from an "isomorphism type" of isomorphic spacetime models. This is not a gauge freedom in the ordinary sense of the word, as it appears in, e.g., Yang-Mills gauge theories, where a gauge transformation is applied to a field $\varphi$, $A_{\mu }$, etc., living on $M$: e.g., with the vector potential for electromagnetism,

$A\mapsto A^{\prime }=A+d\mathrm{\Lambda }$. 

We then have two (mathematically) distinct fields $A$ and $A^{\prime }$, both on the same manifold $M$, but the "physical quantities" they correspond to are identical. One can easily see that the field strengths $F$ and $F^{\prime }$ are identical (because $F=dA$ and $d^2=0$). On the other hand, an isomorphism between a pair of different models is quite different from this!

Now, finally, the category theory connection: here I am very grateful to John Baez for a bit of help in the comments to the earlier M-Phi post. In algebra and model theory, we have isomorphism types; in category theory, an isomorphism type is a category, which is a groupoid. It is a connected groupoid, because for any pair objects $x,y$ of the type, there is an (iso)-morphism between them.

So, suppose that we pick some spacetime model $\mathcal{M}$. The collection of all isomorphic "copies" of $\mathcal{M}$ is the collection of all the physically equivalent representations of some (physical) world. This is a connected groupoid whose objects are these models, and whose morphisms are the isomorphisms. I want to call it "the spacetime groupoid" and denote it $\mathsf{Iso}(\mathcal{M})$. (In an earlier post I used a different notation.) That is,

$\mathsf{Iso}(\mathcal{M})$ is the groupoid of all spacetime models isomorphic to $\mathcal{M}$.

Such a groupoid is called a concrete category: all of its objects are "structured sets". But, they are all isomorphic. So, in a sense, what is being "abstracted away" are simply the irrelevant carrier sets.

For  GR, we may consider some solution $(M,g,T,\dots )$ of the relevant differential equations (e.g., famously the Schwarzchild solution one learns about in a GR course and Robertson-Walker models one learns about in a cosmology course). Then the spacetime groupoid $\mathsf{Iso}(\mathcal{M})$ somehow "represents" a single physical world. How does a whole groupoid of models represent a single world?

In the comments to the earlier M-Phi post, John mentions that the groupoid $\mathsf{Iso}(\mathcal{M})$ is equivalent (in the category theory sense) to $\text{Aut}(\mathcal{M})$, the automorphism group of $\mathcal{M}$.

I haven't quite got this clear in my mind. One puzzle is this. Suppose that ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are rigid spacetime models, but not isomorphic to each other. (One can consider, e.g., a pair of non-isomorphic models $(M,g)$ for which there are no isometries: i.e., no Killing fields $v$ such that ${\mathcal{L}}_{v}g=0$). In this case, the automorphism groups are trivial:

$\text{Aut}({\mathcal{M}}_1)=1=\text{Aut}({\mathcal{M}}_2)$.

where $1$ is the trivial group with one element.

So, though this is probably down to my own lack of knowledge of category theory, it seems that if ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$ are rigid but non-isomorphic spacetime models, the categories $\mathsf{Iso}({\mathcal{M}}_1)$ and $\mathsf{Iso}({\mathcal{M}}_2)$---which I'd expected to be distinct---are both equivalent to $1$. And this is very puzzling (the puzzlingness is probably due to my own confusion!).

[Update, 10 August: I changed the title from "Isomorphism Groupoid in GR" to "Spacetime Groupoid in GR", as "Isomorphism Groupoid" is kind of redundant.]