Apologies in advance for what is likely a very simple, pedantic question. My question is if there are concrete categories whose objects are not sets.
Here's my thinking: the definition I am following says that a category $\mathcal{C}$ is concrete provided there is a faithful functor ${\mathcal{C}}\xrightarrow{\sigma}\textsf{Set}$. I understand that one can realize any concrete category as a category whose objects are sets, but - on the face of it - I see no reason that the objects of ${\mathcal{C}}$ need to be sets themselves.
For example, let $\textsf{Pt}$ be the category with a single object and morphism. It can be realized by letting its object be a singleton and morphism be the identity map on that singleton. However, I see no reason why you could not let the only object of $\textsf{Pt}$ be a proper class, the identity morphism be a singleton, and let composition be the obvious one. Does this work? If it does, are there any less trivial examples?
Thanks for any insight/help!
By the usual definitions (working in ZFC, say), this is not allowed. A category is defined to have a class of objects, and by definition a class is a definable collection of sets. So, every object of a category is a set.