What does it mean that "All diagrams commute in a posetal category"?

Question

I am quite familiar with posetal categories, however, I just randomly came accross the claim that "all diagrams commute in a posetal category" on Wikipedia.

I am confused, what does it even mean? Does it tell that if we compose two morphisms and get a composition, say from $A$ to $C$, it is the same as composing other two morphisms between these two? Or how to understand this?

Thank you for advice.

Answer 1

In a posetal category, there is at most one morphism $A \to B$ for any objects $A$ and $B$. Therefore, if you have two morphisms $f, g \colon A \to B$, it is necessarily the case that $f = g$, so that the diagram whose paths are $f$ and $g$ commutes.

(Repeating my comment as an answer as requested.)

Answer 2

"All diagrams commute" is one of those imprecise slogans that have different meanings in different contexts. In the first place, the standard definition of "diagram" is functor, so tautologically all diagrams commute (= preserve composition). So to make sense of the assertion that "all diagrams commute" we must first find a definition of "diagram" where this is not a tautology. Here is one:

Definition. A diagram shape consists of the following data:

• A set $K_0$.

• A set $K_1$ and maps $d_0, d_1 : K_0 \to K_1$.

Given such data:

• A vertex is an element of $K_0$.

• An edge is an element of $K_1$.

• The target of an edge $e$ is the vertex $d_0 (e)$, and the source of an edge $e$ is the vertex $d_1 (e)$.

• A path from a vertex $v$ to a vertex $w$ is a finite (but possibly empty) sequence of edges $(e_n, \ldots, e_1)$ such that $d_0 (e_i) = d_1 (e_{i+1})$ for $1 \le i < n$, $d_1 (e_1) = v$, and $d_0 (e_n) = w$. (If $n = 0$ we require $v = w$.)

A diagram of the above shape in a category $\mathcal{C}$ consists of the following data:

• A map $X_0 : K_0 \to \operatorname{ob} \mathcal{C}$.

• A map $X_1 : K_1 \to \operatorname{mor} \mathcal{C}$, such that for every edge $e$, the domain of the morphism $X_1 (e)$ is the object $X_0 (d_1 (e))$ corresponding to the source of $e$ and the codomain of the morphism $X_1 (e)$ is the object $X_0 (d_0 (e))$ corresponding to the target of $e$.

Definition. A commutative diagram shape is a diagram shape as above equipped with the following additional data:

• For every pair $(v, w)$ of vertices, a set of pairs of paths from $v$ to $w$.

A commutativity condition is such a pair of paths.

A diagram in $\mathcal{C}$ as above commutes if, for every commutativity condition, the composite of the images of the two paths in $\mathcal{C}$ are equal.

Remark. Commutativity conditions are often left unstated. In the case where there is at most one edge between any two vertices and no directed cycles, the maximal set of commutativity conditions is usually what is intended, i.e. every pair of paths between the same source and target is a commutativity condition.

Proposition. If $\mathcal{C}$ is a posetal category, or more generally any thin category, then every diagram in $\mathcal{C}$ commutes.

Proof. In a thin category, there is at most one morphism between any two objects. So if two paths have the same source and target, their image in $\mathcal{C}$ have the same composite.　◼