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

where $\hat{\Phi}_{\mathcal{M}}$ 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

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

satisfies Leibniz Abstraction:

**Leibniz Abstraction**

$\hat{\Phi}_{\mathcal{M}_1} = \hat{\Phi}_{\mathcal{M}_2}$ 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 $\mathcal{A} \models \phi$ 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 $\underline{a}$ 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 $(A, \leq)$ with certain conditions on $\leq$.

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 $\leq$, $E$, $+$, 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 $\vec{a} \in A^n$, $\vec{a} \in R$ iff $\vec{a} \in S$.

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 $2^{\aleph_0}$, 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 $(\mathbb{Q}, \leq)$, 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 $\phi$ 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 $\phi$, $A_{\mu}$, etc., living on $M$: e.g., with the vector potential for electromagnetism,

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

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 $\mathcal{L}_vg = 0$). 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.]