I've just started to wrap my head around category theory, and came across two (from my perspective not obviously equivalent) definitions of a (small) diagram in a category $\mathcal{C}$:
Definition 1 (Brandenburg, Definition 2.4.3):
Define a quiver $\Gamma$ as a pair of sets $(V,E)$ equipped with two functions $s,t: E \rightarrow V$. A diagram $D$ in a category $\mathcal{C}$ consists of the following data:
- for each $v \in V$ an object $D(v) \in \mathcal{C}$.
- for each $e \in E$ a morphism $D(e): D(s(v)) \rightarrow D(t(v))$.
Definition 2 (pretty much everywhere else, e.g. here):
A diagram $D$ in a category $\mathcal{C}$ is simply a functor from a small category $I$ to $\mathcal{C}$.
I am aware that a quiver is somewhat more general than a small category in the sense that it is a "small categories with identity morphisms and composition forgotten." To that end Definition 1 can be seen as a generalization of Definition 2.
On the other side, one can define the free diagram of a quiver by making $\Gamma$ to a free category first and then proceed with Definition 1.
Question
Is a diagram of a quiver (Definition 1) actually equivalent to a diagram of a small category (Definition 2) or is it a true generalization?
If yes, what do I gain by this generalization? Is this a generalization for the sake of generalizing things or are there truly situations where Definition 2 is needed?
Best Answer
As you noted, these definitions are more or less the same, by considering the underlying quiver and the generated free category.
However, commutativity conditions can't be posed as part of the 'diagram' with definition 1, whereas if e.g. $fg=uv$ in a small category $I$, it must hold for their images as well, under an arbitrary functor.
In this sense, definition 2 is the stronger definition, and it became the standard.
Nevertheless, Brandenburg might have had a reason to separate the notions of diagrams and functors (from a small category). One such reason can be to distinguish syntactics and semantics for the language of categories.
Consider this third, mixed definition:
Definition 3 An abstract diagram is a quiver $\Gamma$ with a set $P$ of 'commutativity conditions', i.e. pairs of paths with common start and end point.
A diagram of shape $(\Gamma,P)$ in category $\mathcal C$ is then a morphism of quivers $D:\Gamma\to\mathcal C$ such that $D(p)=D(q)$ for each $(p, q)\in P$.
So, for example, a 'square' and a 'commutative square' are two different abstract diagrams over the same quiver.
Though def. 3 could be more edaquate in certain sense, in practice it's just as effective as definition 2, which is much simpler to work with.