From Wikipedia:

A

concrete categoryis a category that is equipped with a faithful functor to the category of sets.This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions.

This definition gives rise to the well-established dichotomy between concrete and abstract categories.

The examples of abstract (= non-concrete) categories I've heard of come in two flavours:

categories with structured sets as objects, and

**some**of structure-preserving maps than standard homomorphisms, e.g. interpretations. (I understand that - basically - interpretations*other*sort*are*structure-preserving maps, just involving re-definitions of individuals (as*n*-tupels) and relations.)categories with structured set as objects, and

**equivalence classes**of structure-preserving maps as morphisms, e.g. the homotopy category of topological spaces or much simpler: the category $C'$ which collapses all arrows $X \rightarrow Y$ of a (concrete) category $C$ into one (what's its name?)

[*Side remark:* The trivial equivalence relation on morphisms: $f \sim g :\equiv f = g$ leaves a given category $C$ unchanged.]

Given such a vast variety of possible definitions of structure-preserving maps and equivalence relations between them, I wonder why only classical homomorphisms and the trivial equivalence relation give rise to so-called **concrete** categories?

The other way around:

What is an example of an abstract category (in the standard sense) with structured sets as objects, such that we

cannotthink of its morphisms assomeequivalence class ofsomesort of structure-preserving maps?