The following question is taken from "Theory of Categories" by Barry Mitchell and "Abstract and Concrete Categories The Joy of Cats" by Adamek, Herrlich, and Strecker.
$\color{Green}{Background:}$
[From Mitchell]
$\textbf{Definition 1:}$ A $\textbf{diagram scheme } \Sigma$ is a triple $(I,M,d)$ where $I$ is a set whose elements are called $\textbf{vertices,}$ $M$ is a set whose elements are called $\textbf{arrows,}$ and $d$ is a function from $M$ to $I\times I.$ If $m\in M$ and $d(m)=(i,j),$ then we call $i$ the $\textbf{origin}$ of $m$ and $j$ the $\textbf{extremity}.$ A $\textbf{diagram}$ in a category $\mathcal{A}$ over the scheme $\Sigma$ is a function $D$ which assigns to each vertex $i\in I$ an object $D_i\in \mathcal{A},$ and to each arrow $m$ with origin $i$ and extremity $j$ a morphism $D(m)\in [D_i,D_j].$ If $I$ and $M$ are finite sets then we call $\Sigma$ a $\textbf{finite}$ scheme and $D$ a $\textbf{finite}$ diagram.
[From Adamek, Herrlich, and Strecker]
$\textbf{Definition 2:}$
A $\textbf{diagram}$ in a category $\textbf{A}$ is a functor $D:\textbf{I}\to \textbf{A}$ with codmain $\textbf{A}.$ The domain, $\textbf{I},$ is called the $\textbf{scheme}$ of the diagram.
$\color{Red}{Questions:}$
For the first definition, it seems that the word $\textbf{scheme}$ are used in defining $\textbf{diagram scheme},$ which is a diagram/graph for depicting commutative diagrams. But does that mean the same as how it is used in the second definition, where a $\textbf{scheme}$ is used to mean the domain of a functor? Thank you in advavnce.
Best Answer
What Mitchell calls a diagram scheme is more commonly called a directed graph. A directed graph consists of vertices and edges, and each edge has a specified source and target. Every category $C$ has an underlying directed graph, whose vertices are the objects and whose edges are the morphisms in the same way that every ring e.g. has an underlying set.
There is also a category of directed graph, let me write $\operatorname{Grph}$ for it. The morphisms are morphismsm of directed graphs. We get a 1-functor $U:\operatorname{Cat}\to \operatorname{Grph}$.
When we speak of a diagram $D$ in a category $C$ of a specific shape $\mathscr J$ (here $\mathscr J$ is a directed graph) then we normally mean a morphism of graphs $D:\mathscr J \to UC$. For example, when your shape is the one drawn here
then a graph morphism $D:\mathscr J\to UC$ will choose exactly a composable pair of arrows in the category $C$. The problem now is that the theory flows more smoothly when we only speak about categories and functors, and not also about directed graphs and graph morphisms. In order to specify certain configurations of arrows and objects in $C$ we would instead like to use small categories $I$ and functors $D:I\to C$. What kind of category can we take so that functors $I\to C$ contain the same data as graph morphisms $\mathscr J\to C$?
The answer is that we take the "free category generated from the graph $\mathscr J$. This is a functor $F:\operatorname{Grph}\to \operatorname{Cat}$ which has the following universal property: \begin{align} \operatorname{Cat}(F\mathscr J,C) \cong \operatorname{Grph}(\mathscr J,UC) \end{align} The universal property is exactly what we need. Functors $F\mathscr J\to C$ correspond exactly to graph maps $\mathscr J\to UC$ so that in our example a pair of composable arrows in $C$ will be the same thing as a graph map $D:\mathscr J\to UC$ which in turn is the same thing as a functor $F\mathscr J\to C$.
How does the category $F\mathscr J$ look like? Well, we have the three vertices, call them $a,b,c$ and the two edges, call them $e:a\to b$ and $w:b\to c$. We need identity maps for each of the objects, so we just add them formally and introduce new arrows $1_a$, $1_b$ and $1_c$. Next we also need a composite of $e$ and $w$ which we also introduce formally and denote it for example by the string $"we"$. That is we just add new edges to the directed graph, just as you do when you formally define a free algebraic structure generated from some set. Next we need to define a composition operation on $F\mathscr J$. We do this by setting for example $1_b\circ e = e$ and $w\circ e = "we"$ where $"we"$ is the extra edge which we just introduced to be a formal composite.
Now we see that a functor $D:F\mathscr J\to C$ must send $D1_a$ to $1_{Da}$ and $D("we") = D(w\circ e) = D(e)\circ D(w)$, so that $D$ is completely determined by its action on the objects $a,b,c$ and the two arrows $e$ and $w$. This is how we see that a functor $D:F\mathscr J\to C$ is exactly the same thing as a composable pair of arrows in $C$, and this is also how your first and your second definition of a scheme relate.
If you feel uncertain about this, then you can try to come up with a graph $\mathscr J$ such that a graph map $D:\mathscr J\to UC$ is exactly an endomorphism in $C$. How does $F\mathscr J$ look like? (hint: $\mathscr J$ should have only one edge, but $F\mathscr J$ should have infinitely many morphisms)
Hence I think it is fair to say that the "scheme" of your second definition is $F\mathscr J$ of the scheme in your first definition, at least in intension. But note that taking categories as schemes for diagrams and defininig diagrams to be functors is also more general than working with graphs, because it allows you to enforce enforce equations in your diagrams which are not enforcable when you only use graphs and graph morphisms. For example, try to come up with a diagram $\mathscr J$ such that graph maps $\mathscr J\to UC$ correspond exactly to isomorphisms in $C$. It is impossible. But you can find a small category $I$ such that functors $I\to C$ are exactly isomorphisms in $C$ (the category $I$ is called the walking isomorphism).