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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.


My question now is

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?

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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.

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It's true that the term "diagram" is polysemantic in mathematics. Both definitions you mentioned are valid (and there are many others). All these definitions are connected, but I wouldn't say that one of them is a strict generalization of another.

Definition 1 is simply a morphism of quivers, i.e. a mapping, which preserves the quiver structure. Quivers with such morphisms constitute a category, which we may denote by $\mathbf{Quiv}$. Definition 1 is actual when we speak about quivers.

Definition 2 is simply a functor. The smallness of the domain category is required to avoid set-theoretical issues (namely, we want the category of all such diagrams "to be a category"). Definition 2 is actual when we speak about categories and categorical constructions, especially using functor categories (or "diagram categories") and universal constructions.

The connection between these definitions may be expressed in the following way: the forgetful functor $U\colon\mathbf{Cat}\to\mathbf{Quiv}$ has a left adjoint $F\colon\mathbf{Quiv}\to\mathbf{Cat}$, which sends every quiver $Q$ to the free category $F[Q]$, generated by it. Therefore, there exists a canonical isomorphism $$ \hom_{\mathbf{Cat}}(F[Q],C)\cong\hom_{\mathbf{Quiv}}(Q,U(C)), $$ which means that functors $F[Q]\to C$ naturally correspond to morphisms of quivers $Q\to U(C)$. So, in some sense, Def.1 is equivalent to Def.2 in case when the codomain quiver is an underlying quiver of a category. This equivalence may be used to easily define functors to $C$ when we only have morphisms of quivers to $U(C)$.

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