# What are examples of good toy models in mathematics?

This post is community wiki.

A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to understand it better, but I don't know of too many examples since I don't see too many people talk about them. What examples are common in your field, or what examples do you personally think are very revealing? Here's what I've got so far, starting with Terence Tao's example. Feel free to modify any of these examples if I'm not stating them correctly and to elaborate on them in answers if you want.

• $F_p[t]$ is a toy model for $\mathbb Z$.
• $F_p[[t]]$ is a toy model for $\mathbb Z_p$.
• Simplicial complexes are a toy model for topological spaces.
• $\mathbb Z/n\mathbb Z$ is a toy model for $\mathbb Z$ (for the purposes of additive number theory).
• The DFT is a toy model for the Fourier transform on the circle.

Which properties of the original objects carry over to your toy model, and which don't? As usual, stick to one example per post.

• Are simplicial complexes really toy models for topological spaces? While the category of simplicial complexes has numerous problems, they are mostly confined to the morphisms -- most spaces of interest are at least homotopy equivalent to simplicial complexes. Commented Oct 20, 2009 at 3:37
• Well, I wouldn't really know; feel free to edit that one or write an answer. Commented Oct 20, 2009 at 3:42
• Not sure if this is the right place to ask, but when you say F<sub>p</sub>[t] is a toy model for Z I guess you are referring to the general philosophy that problems over function fields are easier to deal with than those over number fields. Can someone actually elaborate on this analogy between number fields and function fields? I'm not sure where I can find information about this. Ring of integers being Dedekind, finite residue field, RH over function fields easier to deal with, anything else?
– user709
Commented Oct 20, 2009 at 4:07
• I have the same question; I think it should probably be asked separately. Commented Oct 20, 2009 at 4:14
• I think the category of CW complexes (c.f. Hatcher's book) is a toy model for topological spaces. Commented Dec 27, 2009 at 15:38

2x2 matrices are a toy model for general square matrices.

"If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this." -- Olga Tausky-Todd

• Similarly, the matrix algebras M_n(K) are nice toy models for general central simple algebras (along with the Hamiltonians in the Quaternionic case).
– user1073
Commented Jan 20, 2010 at 16:21

One can think of toric varieties as "toy examples" of algebraic varieties. A lot can be said about them via combinatorial data, but they definitely are special (they are always rational varieties, for example).

Topological quantum field theories are toy models for real quantum field theories.

This is a funny example though, as TQFTs later became "real" again, when it was discovered that they describe the physics of the Fractional Quantum Hall Effect. Toy models sometimes pay off big!

Shift spaces are examples of topological dynamical systems, but they also serve as a toy model for topological dynamical systems.

The best examples I've come up with come from rational homotopy theory--commutative differential graded Q-algebras as a toy model for spaces and chain complexes of Q-vector spaces as a toy model for spectra--though really this is an instance of "toy examples" because we can build actual spaces/spectra corresponding to these algebraic data. It feels a bit like a toy model, though, I guess because those spaces aren't very geometric.

• +1 for chain complexes being toy examples of spectra. Maybe taking a (slightly) broader perspective, we can say that categories of chain complexes are toy examples of stable $(\infty,1)$-categories, and their homotopy categories toy examples of triangulated categories. Commented Jul 16, 2016 at 19:03

One can think of Grothendieck ring of varieties as a toy model for motives.

• I think the opposite is true - motives are a toy model for the Grothendieck ring, which is much more complicated. Much recent work involves taking some simple property of the category of motives and asking if it still holds in the Grothendieck ring. Commented Jul 17, 2015 at 14:23

Your first example of Fp[t] being a toy model for Z is a facet of the well known (to number theorists) analogy between Q and Fp(t). This analogy extends fabulously to a more general connection between the arithmetic of number fields, i.e. finite extensions of Q, and of function fields, i.e. finite extensions of Fp(t).

Relations are a toy model for linear maps. In fact, they can be thought of as matrices over the boolean semi-ring.

• Nice answer. Is there a good example of a property that is not true for linear maps because it is obviously not true for relations? Commented Dec 16, 2019 at 7:41

Perhaps this doesn't count as a toy model, rather a toy example. A nice basic example for GIT is n points in CP1 under the action of SL(2,C). A lot of the usual elements of the theory look nice in this picture. For example, the Hilbert Mumford criterion shows that a collection of n points is semi-stable iff all points have multiplicity less or equal to n/2 whilst a collection of n points is stable iff all points have multiplicity strictly less than n/2.

It's also a nice example of the equivalence between symplectic reduction and the GIT quotient. If you fix a Fubini-Study metric on CP1 and look at the action of the corresponding SU(2) you can ask for a moment map. Thinking of CP1 as a coadjoint orbit in su(2)*, the moment map takes n points to their centre of mass. Now the equivalence of symplectic and GIT quotients says that, provided we don't have two points each of multiplicity n/2, you can move n points in CP1 by an element of SL(2,C) so that their centre of mass is zero if and only if all multiplicities are strictly less than n/2. (The case when two points each have multiplicity n/2 is special because of the additional C* stabiliser.) In one direction this is obvious, but in the other I think this is a neat non-trivial statement (at least when n is large).

2-bridge knots (rational knots) are a toy model for knots. If you think something should be right, first test it on 2-bridge knots. This happens so much in knot theory that it's not even funny.

Turing machines are in some sense toy models for real computers (which have parallel processors, networks, file systems, etc).

• But a "real computer," no matter how many networked machines it's made of, how many processors each machine has, or what kind of file system it's running, is still just a finite state automaton, so you could just as well say that "real computers" are examples from a class of toy models for Turing machines. Commented Dec 15, 2014 at 21:34
• Well, yes, but that would be inconvenient. Commented Dec 15, 2014 at 21:43

Derived categories of finite dimensional hereditary algebras for other derived categories. (Though under good circumstances other derived categories can turn out to be equivalent to derived categories of f.d. hereditary algebras.)

The Noncommutative torus serves as a toy model for noncommutative differentiable manifolds.

In Differential Geometry/Physics, Homogeneous spaces can be considered as nice examples of manifolds, where one can test conjectures.

This is not so much a new example of toy model but rather a complement to the excellent post by Tao on dyadic models. Like most good ideas, it has been rediscovered many times, in many different areas, and given different names. In quantum field theory and in particular in the framework of Wilson's renormalization group, many models have a Euclidean version (on $\mathbb{R}^d$) and a hierarchical version (for instance on $\mathbb{Q}_p^d$). The so-called hierarchical model in physics is another instance of what Tao calls "dyadic models". Wilson himself referred to it as "the approximate RG recursion" and it played a key role in his path to discovery of his RG theory (see quote by Wilson on page 8 of this paper). A good reference on the hierarchical model in physics is this review by Meurice. In probability theory, the dyadic model is the branching random walk or Brownian motion/Mandelbrot cascades used as a toy model for the Gaussian free field, see the references in this MO post.

Monounary algebras (that means, a set equipped with a single unary operation) are toy models for universal algebra.

Countable models of $ZFC$ (see, for example, Joel David Hamkins' blogpost "Upward closure in the toy multiverse of all countable models of set theory" on www.jdh.hamkins.org/upward-closure-in-the-toy-multiverse-of-all-countable-models-of-set-theory and his research on the set-theoretic multiverse).

There is a powerful analogy, developed over the last ~25 years, between graphs and algebraic curves (1-dimensional algebraic varieties, over $$\mathbb{C}$$ say, i.e., Riemann surfaces). The first paper to propose this analogy may be "The lattice of integral flows and the lattice of integral cuts on a finite graph" by R. Bacher, P. de la Harpe, and T. Nagnibeda, 1997. A big milestone was the Riemann-Roch theorem for graphs of M. Baker and S. Norine. This analogy lets us use graphs as a toy model for curves. Again, the idea is that we have something "combinatorial" in place of something more purely algebraic/algebro-geometric (so this answer is similar to my other one about Stanley-Reisner rings, and the one about toric varieties). Note that sitting between graphs and curves, and in some sense interpolating between them, are the metric graphs.

Stanley-Reisner rings have a combinatorial structure which makes them amenable to computation and experimentation. Because of their simple structure, they can serve as toy models of general quotients of polynomial rings (or even more generally finitely generated modules over commutative rings). This is sort of similar to the way that toric varieties can be toy models of general algebraic varieties, as was mentioned in another answer to this question. Hailong Dao has a nice talk explaining this perspective: https://www.msri.org/seminars/25063.

The Theory of Metric Spaces is a toy model for the General Topology.

Quantum graphs have become popular in the 1990s as toy models for quantum chaos, and they have been quite successful ever since, as the BGS-conjecture has been settled in the graph context.

Trees (resp. labelled trees) are a toy example for Young diagrams (resp. Young tableaux). Feel free to elaborate (I don't have time to).

Galois representations as toy models for varieties (more generally, motives). Conjecturally this isn't even a toy model, but for the moment it's a lot easier to work on the Galois side.

Boolean algebras are toy models for distributive lattices, which in turn are toy models for lattices in general (and partially ordered sets).

Semigroups of shifts on some function space are toy models for one-parameter $$C_0$$-semigroups.

On the one hand they are easy: Whereas in the general theory one often thinks of an operator $$A$$ as given ("the generator") and wants to study the unknown $$C_0$$-semigroup $$(\mathrm{e}^{tA})_{t\geq 0}$$, in the case of shift semigroups both objects are given by explicit formulas (the generator being the "first derivative"). On the other hand semigroups of shifts are interesting enough: By varying the underlying function space one can already produce many (counter-)examples which highlight crucial parts of the general theory.

The example that comes to mind is the "Toric Code" topological quantum field theory. It is in a strong sense the simplest non-trivial TQFT. The toric code is a testing ground both mathematically for people trying to understand quantum topology, and also physically for people who try to make these things in the lab.

In a Simon's Foundation interview on the topic, Kitaev (the inventor of the toric code) said this:

"Throughout my career I have been successful inventing toy models, some simple models that capture important features of a more complex problem" - Alexei Kitaev

Often homogeneous trees (connected acyclic graphs such that each vertex has the same valence) are considered a good toy model for real hyperbolic space.

Consider for example the behaviour of the heat semigroup https://www.ams.org/journals/tran/2000-352-09/S0002-9947-00-02460-0/S0002-9947-00-02460-0.pdf in comparison to the hyperbolic space.

Furthermore the Dirichlet space on the Poincarè disc has a lot of similarities with the discrete Dirichlet space on a tree. See for example http://www.dm.unibo.it/~arcozzi/rafrot_nyjm.pdf

Sequences of real numbers, and their sums and differences, as a toy model for real-valued functions, integrals and derivatives.

In the field of KPZ universality, the particular model of TASEP has led to a number of formulas and in particular the KPZ-fixed point formula that served as guide for finding analogous formulas for other models in the KPZ universality class.

It originates from the approach of using automata to model turbulence, interface models and biological models.

finite sets of points and the line segments connecting pairs of them are toy model for graphs and very popular for developing and testing heuristics for combinatorial optimization, most prominently the TSP.

• I'm confused by this — a finite set of points and a set of edges connecting them is a (finite) graph. What's toy about this model? Commented Jun 3, 2023 at 20:42
• A (simple symmetric) graph in the mathematical sense is A pair $(V,E),\,E\subseteq \lbrace\lbrace u,v\rbrace, u,v\in V\rbrace$ and that, to my understanding, isn't the same as geometric ponts (which have coordinates) or their connecting line segments (which are continuous point sets). Apart from that, a straight line embedding doesn't exist for every every graph. Commented Jun 4, 2023 at 4:18