In Conceptual Mathematics, Lawvere & Schanuel describe tell us how to create a type of structure (I'm paraphrasing here):
- A set of names for objects/structural components
- A set of names for crucial structural maps
- The specification of which structural component is required to be the domain and codomain of each structural map.
They say that an abstract structure arises from particular examples in this way:
Suppose $\mathcal{A}$ is a small family of objects and maps in a category $\mathcal{X}$, with the domain and codomain of any map in $\mathcal{A}$ being in $\mathcal{A}$. Let each object $A$ of $\mathcal{A}$ be considered as the name of '$A$-shaped figures' and each map $\alpha$ in $\mathcal{A}$ be considered as a name $\alpha^*$ of structural map. The domain of $\alpha^*$ is the codomain of $\alpha$ and the codomain of $\alpha^*$ is the domain of $\alpha$. Then every object $X$ of $\mathcal{X}$ gives rise to an $A$-structure whose $A$-th component set is the set of all $\mathcal{X}$-maps $A \to X$ and wherein for each $\alpha\colon B \to A$ the structural map on these figures has for all $x$: $$\alpha_X^*(x) = x \circ \alpha$$
Additionally, it was pretty trivial to prove that: Every map $f\colon X \to Y$ in $\mathcal{X}$ gives rise to a map in the category of $\mathcal{A}$-structures by the associative law, so in some sense Lawvere's definition seems natural.
Specific points I'm unclear on:
1) I'm just generally little lost and wondering if there's a name for this 'construction' so I can find another interpretation elsewhere?
2) Where does $B$ come from? Is it just a test object in $\mathcal{X}$? Is it just the assumed domain for any $\alpha$-map?
3) What is $\alpha$? For example, if we are trying to create a graph structure on the category of sets, we might have as objects two sets $(A,P)$ and two maps $s,t\colon A \to P$ where $A$ is a set of arrows, $s$ and $t$ determine the source and target of the arrows, and $P$ is a set of points. Is $\alpha$ supposed to represent a morphism of graphs? Or, is $\alpha = s, t$?
4) I guess somehow all the $\alpha^*$ become the concrete realization of a subcategory from the abstract $\mathcal{A}$-structure in $\mathcal{X}$, is that right? I guess the primary benefit of this view comes from giving us access to the morphisms $f$ and also any functors from $\mathcal{X}$.
Thanks