Counter-example for abelian category that is not concrete

Question

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work?

Thanks,

Answer 1

The term you want is concretizable (meaning admits a faithful functor to $\text{Set}$). The category of representations of a quiver does not work: if $Q$ is such a quiver with vertex set $Q_0$ and $V$ is a representation, then

$$V \mapsto \prod_{q \in Q_0} V(q)$$

is a concretization.

More generally, a category $C$ is concretizable if it has a small separating family, meaning a family of objects $c_i, i \in I$ indexed by a set $I$ such that the product $\prod \text{Hom}(c_i, -)$ is faithful. In particular, any essentially small category is concretizable. The category of representations of a quiver has a small separating family given by the free representations on each vertex.

On the other hand, any concretizable category must be locally small. An example of an abelian category which is not locally small is the category $\text{Vect}^{\text{Ord}}$ of ordinal-indexed families of vector spaces; this is the category of representations of a "large quiver." But arguably this is cheating: it's reasonable to require that all categories should be locally small (although this comes at the price of not being able to form all functor categories).

As Martin Brandenburg says in the answer to this math.SE question, Freyd showed that the stable homotopy category embeds into a (locally small) abelian category, and since Freyd also showed that the former category is not concretizable, neither is the latter. This is a variation on Freyd's famous result that the (locally small!) homotopy category of pointed spaces is not concretizable.