simulation

A simulation is a morphism of coalgebras for an endofunctor.

Let $C$ be a category, and let $F\colon C \to C$ be an endofunctor. Recall that a coalgebra of $F$ is an object $X$ of $C$ equipped with a morphism $\theta\colon X \to F(X)$.

Given two coalgebras $X, X'$ of $F$, a **simulation** of $X$ in $X'$ is a morphism $s\colon X \to X'$ in $C$ such that $F(s) \circ \theta = \theta' \circ s$. That is, this diagram commutes:

$\array {
X & \overset{\theta}\to & F(X) \\
\llap{s}\downarrow & & \downarrow\rlap{F(s)} \\
X' & \underset{\theta'}\to & F(X')
}$

The terminology comes from computer science, and we should probably explain what it’s about.

Consider the category of sets and the (covariant) power set functor $\mathcal{P}$. A coalgebra of $\mathcal{P}$ is a set $X$ equipped with a function $\theta\colon X \to \mathcal{P}(X)$, which is the same thing as a binary relation $\prec$ on $X$. To fix the notation, we write $a \prec b$ ($a$ **precedes** $b$) if $a \in \theta(b)$.

Then in concrete terms, a simulation of $X$ in $X'$ is a function $s\colon X \to X'$ such that

- $s(a) \prec' s(b)$ whenever $a \prec b$, and
- $t = s(a)$ for some $a \prec b$ whenever $t \prec' s(b)$.

Simulations are commonly used as morphisms between sets equipped with well-founded or extensional relations. In particular, between well-ordered sets (sets equipped with relations that are both well-founded and extensional, as well as transitive), a simulation (if it exists) is unique, and we recover the usual ordered class of ordinal numbers as a thin category.

In their book *Algebraic Set Theory*, Joyal and Moerdijk present the notion of a pretopos equipped with a class of open maps. As they remark, every presheaf category carries a canonical class of open maps. We reformulate their notion of canonical open map for an example of primary interest to them, the category of forests, in coalgebraic terms.

The category of forests? is by definition the category of presheaves over the first infinite ordinal $\omega$. Concretely, a forest $F$ is a diagram of sets and functions

$\ldots \to F_{n+1} \to F_n \to \ldots \to F_1 \to F_0.$

If $y \in F_{n+1}$ maps to $x \in F_n$, we call $y$ a **predecessor** of $x$.

The successor map $s: \omega \to \omega$ induces a pullback functor

$s^\ast = Set^{s^{op}}: Set^{\omega^{op}} \to Set^{\omega^{op}}.$

Let $P$ be the covariant power-object functor on the presheaf topos. Each forest $F$ carries a canonical coalgebra structure over the endofunctor $P s^\ast$. Namely, the structure is a presheaf map $\theta_F \to P s^\ast F$ that corresponds to the subobject

$R \hookrightarrow F \times s^\ast F$

where the subset $R_n$ is the set of pairs $(x, y) \in F_n \times F_{n+1}$ such that $y$ is an immediate predecessor of $x$.

If $\phi: F \to G$ is a map of forests, in other words if the functions $\phi_n: F_n \to G_n$ collectively preserve the predecessor relation, then we have an inclusion

$\array{
F & \stackrel{\phi}{\to} & G \\
^\mathllap{\theta_F} \downarrow & \leq & \downarrow^\mathrlap{\theta_G} \\
P s^\ast F & \underset{P s^\ast \phi}{\to} & P s^\ast G
}$

where $\leq$ comes from the internal poset structure of $P s^\ast G$. If this inclusion is an equality; in other words, if $\phi$ is a coalgebra map or simulation over $P s^\ast$, then we also say (following Joyal and Moerdijk) that $\phi$ is *open*. Alternatively, we could say that $\phi$ is open if the maps $\phi_n$ collectively preserve and reflect the predecessor relation.

Pursuing this more concretely: the composite $\theta_G \circ \phi$ corresponds to the subobject $S$ of $F \times s^\ast G$ occurring in the pullback

$\array{
S & \to & R_G \\
\downarrow & & \downarrow \\
F \times s^\ast G & \underset{\phi \times 1}{\to} & G \times s^\ast G
}$

so that $S_n \subseteq F_n \times G_{n+1}$ consists of pairs $(x, y)$ such that $y$ is a predecessor of $\phi_n(x)$ in $G$. The composite $P s^\ast(\phi) \circ \theta_F$ corresponds to the subobject $S'$ of $F \times s^\ast G$ occurring in an image factorization

$\array{
R_F & \twoheadrightarrow & S' \\
\downarrow & & \downarrow \\
F \times s^\ast F & \underset{1 \times s^\ast \phi}{\to} & F \times s^\ast G
}$

so that (taking advantage of the fact that image factorizations in a presheaf topos are computed objectwise) $S_{n}' \subseteq F_n \times G_{n+1}$ consists of pairs $(x', y')$ such that $y' = \phi_{n+1} y$ for some predecessor $y$ of $x$ in $F$. The equality of subobjects $S = S'$ matches the notion of open map given in *Algebraic Set Theory*.

**Remark:** Joyal and Moerdijk actually refine this idea as follows. Instead of taking forests to be set-valued presheaves, they take them to be presheaves over $\omega$ valued in a given pretopos $C$ equipped with a class of *small maps* (which are defined axiomatically). Then, a *small forest* is a diagram in $C$ of the form

$\ldots \to F_{n+1} \to F_n \to \ldots \to F_1 \to F_0$

where each map $F_{n+1} \to F_n$ and $F_0 \to 1$ is small. On the other hand, the small map axioms guarantee that small subobjects are representable by a functor $P_{small}: C \to C$. This induces a functor $P_{small}: C^{\omega^{op}} \to C^{\omega^{op}}$ on the category of forests, and analogous to what we described above, a small forest can be construed in terms of a coalgebra structure $F \to P_{small} (s^\ast F)$.