# The model theory of covers

The aim of this post is to briefly summarize some of what is known (and the many things that remain unknown) about the model theory of covers (which includes finite covers and affine covers). Though the motivation comes from questions in “pure model theory”, there are intriguing connections with algebra and a lot of nice but surprisingly resistant open questions which I hope will be interesting to a wide audience.

Covers to understand totally categorical theories

A countable first-order theory $T$ is totally categorical if for every infinite cardinal $\kappa$, $T$ has precisely one model of cardinality $\kappa$ up to isomorphism.

Early in the study of stable theories, some amazing results were proved about these theories, such as:

Theorem (Cherlin-Harrington-Lachlan) No totally categorical theory is finitely axiomatizable.

Theorem (Hrushovski) (a) Any totally categorical theory is “quasifinitely axiomatizable” (axiomatizable by a finite number of first-order sentences plus a finite number of sentences of the form “for every $\overline{a}, \varphi(\overline{x}; \overline{a})$ is infinite”).

(b) There are only countably many totally categorical theories (modulo bi-interpretability).

In fact, (a) and (b) above are true of $\omega$-stable, $\omega$-categorical theories in general.

In modern terminology (some of it explained below), here is what is going on in a totally categorical theory $T$: in $T^{eq}$, there is an $\emptyset$-definable strictly minimal set $D$ (that is, strongly minimal, plus there are no nontrivial definable equivalence relations with domain $D$). The universe of any model $M \models T$ can be constructed in finitely many stages as $D = D_0 \subseteq D_1 \subseteq \ldots D_n$, where each $D_{i+1}$ is an $\emptyset$-definable set in $T^{eq}$ which is a finite or affine cover of $D_i$ and $M$ is in $\textup{dcl}(M_n)$. Furthermore, the full definable structure on $D$ is either (i) completely trivial (the “disintegrated case”) or (ii) that of an infinite-dimensional projective space over a finite field (the “modular case”).

The definition of a cover

The dream is to develop a nice model theory for “covers of structures” $\pi: M \rightarrow N$, where $\pi$ is a map between two multi-sorted structures $M$ and $N$. The idea is that $M$ and $N$ may have completely different languages (indeed, that is the interesting case), but $\pi$ should still “preserve definable structure”.

The first challenge is just to give a sufficiently general and useful definition of “cover”, which I will now try to do (this is essentially a variation of the definition of Evans from (3)).

Definition: Call a function $\pi : M \rightarrow N$ a cover if it satisfies:

0. $N$ is in the definable closure of the image of $\pi$;

1. The equivalence relation on $M$ given by being in the same fiber of $\pi$ is $0$-definable in $M$;

2. The $0$-definable $n$-ary relations on $N$ are precisely the projections via $\pi$ of the $0$-definable relations on $M$; and

3. (“Stable embeddedness”) Every $0$-definable family of definable sets $\left\{R(\overline{x}, \overline{a}) : \overline{a} \in M\right\}$ in $M$ projects via $\pi$ onto a $0$-definable family of definable sets in $N$.

Question: Do properties 0-3 define a good, sufficiently general notion of “cover”? Condition 0 probably isn’t logically necessary, given condition 2.

Definition: Call a cover finite if the fibers $\pi^{-1}(a)$ are all finite. (“Locally finite” may be more accurate, but “finite cover” is now standard for this meaning.) Call a cover affine if there is a definable family $(G_a : \varphi(a))$ of vector spaces in $N$ over a fixed finite field and definable regular actions of each $G_a$ on the corresponding fiber $\pi^{-1}(a)$.

Now given a definable family $(G_a : \varphi(a))$ of groups in some structure $N$, we would like to know what are the various ways to build a cover $\pi: M \rightarrow N$ where the $G_a$ act on the fibers.

There is one fairly obvious way to do this which works for any $N$, which is to build a split cover of $N$ with structure groups $(G_a : \varphi(a))$ (sometimes in the work of Evans, this is called a principal cover). The idea is the following: $M$ will be a structure containing $N$ plus one new sort, call it $P$, which is a disjoint union of the groups $G_a$ for each $a \in \varphi(N)$, without the group structure. There are basic function symbols for the natural projection $\pi : P \rightarrow \varphi(N)$ and for the left-regular action of each $G_a$ on the corresponding fiber $\pi^{-1}(a)$. That’s it; we add no other basic definable relations between elements of the fibers fibers. (Note that this is basically like the construction of a wreath product in permutation group theory.)

More generally, suppose that $\pi: M \rightarrow N$ is a cover and that $M$ and $N$ are saturated models (a harmless assumption, if $\textup{Th}(M)$ is stable). Then we have an exact sequence

$1 \rightarrow K \rightarrow \textup{Aut}(M) \rightarrow \textup{Aut}(N) \rightarrow 1$

between the automorphism groups. (The existence of the arrow $\pi_*: \textup{Aut}(M) \rightarrow \textup{Aut}(N)$ induced by $\pi : M \rightarrow N$ comes from condition 2 in the definition of cover, and the fact that it is surjective follows from saturation plus stable embeddedness.) Call $K$ the kernel of the cover.

This cover is called split if there is a splitting $s: \textup{Aut}(N) \rightarrow \textup{Aut}(M)$ which is a topological group map (under the usual topology for automorphism groups, where an open basis for the identity map is the collection of all pointwise stabilizers of finite sets); that is, $\pi_* \circ s = \textup{id}$. (If $K$ is abelian, then there is always a continuous map $s$ with this property, but it may not be a group map.)

So the cover splits just in case there is a closed subgroup $H$ of $\textup{Aut}(M)$ such that $H \cap K = 1$ and $H K = \textup{Aut}(M)$. Thus the map $\pi_* : \textup{Aut}(M) \rightarrow \textup{Aut}(N)$ restricts to an isomorphism from $H$ onto $\textup{Aut}(N)$.

Assuming the structures $M$, $N$ are $\aleph_0$-categorical, the definable structure can be recovered (up to bi-interpretability) from the automorphism group, and we have a more enlightening way to think about splitting: the cover $\pi: M \rightarrow N$ splits if and only if there is an expansion $M'$ of $M$ such that $N$ and $M'$ are bi-interpretable via $\pi$.

Evans in (4) found many cases of structures $N$ where every finite cover of $N$ with finite kernel splits, including basic examples such as the infinite structure in an empty language and a dense linear ordering, and also gave an example of an $\aleph_0$-categorical $N$ where this is not the case. This should be compared with Hrushovski’s work in (7) where he related the splitting of finite covers with finite kernels in stable theories with definable groupoids: these covers will always split (modulo adding finitely many algebraic constants) if and only if all connected groupoids definable in the base structure with finite fundamental groups are definably equivalent to groups.

Question: How can one decide, in general, if all finite covers with finite kernel will split? Given a base structure $N$ and a potential kernel $K$, when can one classify all finite covers of $N$ with kernel $K$?

Old open questions on totally categorical theories

Question: What, exactly, do the disintegrated (geometrically “trivial”) totally categorical structures look like?

At first, this question sounds like it should be easy, and at least some people think that Hrushovski settled the question in his paper (6) by showing that these are essentially just Grassmanians over infinite indiscernible sets, obtained from the trivial structure in the empty language by gluing copies of finite structures above finite sets. However, what Hrushovski showed was only that this holds after naming a finite set of constants. See the groupoids in (7) and the structures in (5) for examples of nontrivial phenomena in this context.

Without naming constants, what Hrushovski proved was that these disintegrated, totally categorical structures $M$ with strictly minimal set $D$ can be analyzed by a finite chain of $\emptyset$-definable sets

$M_0 \subseteq M_{1,0} \subseteq M_1 \subseteq M_{2,0} \subseteq M_2 \subseteq \ldots$

where $M_i$ is the intersection of the universe $M$ with algebraic closures of $i$-element sets from $D$, $(M_i / M_{i,0})$ is a Grassmanian constructed by gluing independent copies of a finite structure over $i$-element subsets of $D$, and $\textup{Aut}(M_{i,0} / M_{i-1})$ is nilpotent of class $i$. Note that all of the extensions in this chain are certain sorts of finite covers (which generally will be non-split).

Question: Is $\textup{Aut}(M_{i,0}/M_{i-1})$ abelian?

For $i = 2$, the group $\textup{Aut}(M_{2,0}/M_1)$ is connected with the definable groupoids studied in (7). For higher values of $i$, I suspect these will be connected with “higher-dimensional groupoids” of some sort.

Question: Can we say anything useful in general about the affine totally categorical theories?

Answering this last question would require understanding what affine covers can look like. Note that this case includes structures such as $(\mathbb{Z} / {4 \mathbb{Z}})^{\omega}$, which can be thought of as a nonsplit affine cover of an infinite-dimensional space over $\mathbb{F}_2$. As for results, I only know of some special cases proved by Ahlbrandt and Zielger in (1) which appear to only cover the case of spaces over $\mathbb{F}_2$ (!).

Hrushovski in (6) conjectures: “It seems clear that every $\aleph_0$-categorical, $\aleph_0$-stable structure of modular type is interpretable in a finite disjoint union of universal locally lexicographically ordered vector spaces over finite fields.” I have know idea if anybody else has pursued this. Having some kind of classification of modular, totally categorical structures would be quite interesting.

References

1. G. Ahlbrandt and M. Zielger, “What’s so special about $(\mathbb{Z}/{4 \mathbb{Z}})^{\omega}$?”, Archive for Mathematical Logic 31 (1991), 115-132.
2. Greg Cherlin, Leo Harrington, and Alistair Lachlan, “$\aleph_0$-categorical, $\aleph_0$-stable structures”, Ann. Pure Appl. Logic 28 (1985), 103-135.
3. David Evans, “Finite Covers,” talk at the ESF Conference on Model Theory, Bedlewo, Poland, August 2009 (http://www.uea.ac.uk/~h120/Bedlewo09.pdf).
4. David Evans, “Splitting of finite covers of $\aleph_0$-categorical structures”, J. London Math. Soc. (2) 54 (1996), 210-226.
5. David Evans and Elisabetta Pastori, “Second cohomology groups and finite covers of infinite symmetric groups”, Journal of Algebra 330 (2011), 221–233, http://arxiv.org/abs/0909.0366.
6. Ehud Hrushovski, “Totally categorical structures”, Trans. Am. Math. Soc. 313 (1989), 131-159.
7. Ehud Hrushovski, “Groupoids, imaginaries and internal covers”, Turkish J. of Math. (2012), http://arxiv.org/abs/math/0603413
8. W. M. Kantor, Martin W. Liebeck, and H. D. Macpherson, “$\aleph_0$-categorical structures smoothly approximated by finite substructures”, Proc. London Math. Soc. (3) 59 (1989), 439-463.