1 Systems and the floor
A system is a pair \(S = (T, R)\) of a set of things \(T\) and a relation \(R \subseteq T \times T\).
Klir, Facets of Systems Science (2001), p. 5: “S, T, R denote, respectively, a system, a set of things distinguished within S, and a relation (or, possibly, a set of relations) defined on T. Clearly, the thinghood and systemhood properties of S reside in T and R, respectively.” His examples on the same page admit both an ordering, which runs one way, and a partition, which comes from an equivalence relation and runs both ways.
\(X\) depends on \(Y\) when what \(X\) is cannot be stated without \(Y\). For the parts of Definition 1.1 this is a matter of type: \(R\) is a subset of \(T \times T\), so the kind of object \(R\) is depends on \(T\). That \(R\) depends on \(T\), and \(T\) not on \(R\), holds by construction.
The shape \(\mathcal{I}_{\mathrm{Klir}}\) has two objects, the relation and the things, and one generating arrow
read “\(R\) is defined over \(T\)”. The arrow joins the parts of the definition, not two things.
Klir’s data form a graph. Each element of \(R\) is a pair with a first and a second thing, so \(R\) maps to \(T\) twice, \(R \rightrightarrows T\), and read as a schema for data, that two-arrow shape is the shape of graphs. The one-arrow shape is not a schema for data. It records dependency: collapsing both arrows to one forgets how \(R\) refers to \(T\) and keeps only that it does.
The comparison across definitions takes place at the dependency level. The two-arrow shape is not shared: the Myers shape has at most one arrow between any two positions, so two parallel arrows have nowhere to go. The collapse is an explanation and is not formalized.
A kernel is a system \((T, R)\) together with the constraint that every pair in \(R\) lies in \(T\). The constraint is the arrow of Definition 1.3 at the level of data.
For each of the definitions of Bunge, Mobus, Myers, Wymore, Mesarović, Joslyn, and Spivak, there is a functor from \(\mathcal{I}_{\mathrm{Klir}}\) to that definition’s shape that is injective on objects. With Klir’s own, the arrow sits inside all eight.
Each functor is given by a choice of positions in the target shape. Faithfulness is automatic because \(\mathcal{I}_{\mathrm{Klir}}\) is thin, so the content is injectivity on objects.
The functors place the arrow by position. They do not show that it means the same thing in each definition. Into Myers it lands on expose, by which state determines output, a functional dependence. Into Joslyn it lands on efferent, the effector acting on the controlled variables, a causal one. The shape files also declare different arrow conventions. Bunge’s draws from dependent to depended-on. Joslyn’s, Spivak’s, Wymore’s, Mesarović’s, and Willems’s draw in the functional direction. Mobus’s declares none, and Myers’s is inconsistent. See Problem 2.4.
No quiver that embeds in the Willems shape contains a path of two arrows.
The Willems shape has no composable pair of arrows.
Let \(V\) be a connected quiver that embeds in both the Joslyn and the Willems shapes, and let \(e : x \to y\) be an edge of \(V\). Then \(x \neq y\), \(V = \{ x, y\} \), and every edge of \(V\) runs from \(x\) to \(y\).
Joslyn’s shape forbids two arrows leaving one object, and Willems’s forbids two arrows entering one object and any two arrows in a row (Lemma 1.6). A connected quiver satisfying all three has one edge.
The result is relative to the encoded presentations. Willems serves as a witness and is not one of the eight. Among things, a relation may run both ways. Among the parts of a definition, the shared floor is one arrow.
Compared as free categories, the three-object fork embeds injectively on objects and faithfully into all eight shapes. So the claim that nothing larger than one arrow is shared is false at that level.
In a free category every path is a morphism, and the fork enters through composite paths that no definition asserts.