Let $S$ be a set equipped with a binary relation$\prec$. Then a bisimulation is a binary relation $\sim$ on $S$ such that for all $x, y \in S$ with $x \sim y$, the following conditions hold:

for all $a \in S$ such that $a \prec x$, there exists a $b \in S$ such that $b \prec y$ and $a \sim b$

for all $b \in S$ such that $b \prec y$, there exists a $a \in S$ such that $a \prec x$ and $a \sim b$

On a system with points $P,Q,\ldots$ and transition relations $\{\overset{\alpha}{\rightarrow} \colon \alpha\in L\}$, a bisimulation is a symmetric relation$\sim$ such that whenever $P\sim Q$, $\alpha\in L$ and $P\overset{\alpha}{\rightarrow}P'$, then $Q\overset{\alpha}{\rightarrow}Q'$ with $P'\sim Q'$.

Definition

The union of all bisimulation relations, as subsets of $S \times S$, is a relation called bisimilarity.

Let $\mathrm{traces}(s)$ be the set of all possible (finite or infinite) transition sequences starting as $s$. Trace equivalence, defined as $s\sim t$ if and only if $\mathrm{traces}(s)=\mathrm{traces}(t)$, is a bisimulation.

In category theory

In Joyal-Nielsen-Winskel (p.13) is given the following definition. For what a “path category in a category of models” is, see there.

Definition

Let $P$ be a path category in a category of models $M$. Two objects $X_1,X_2$ are called to be $P$-bisimilar if there is a span of $P$-open maps $X_1\leftarrow X\to X_2$.

The relation of $P$-open maps and open maps is given by Proposition 11, p.32:

Proposition

If $P$ is a dense full subcategory of $M$, then $f$-is $P$-open iff $M(-,f)$ is an open map.