From Wikipedia:
A concrete category is 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 other sort of structure-preserving maps than standard homomorphisms, e.g. interpretations. (I understand that – basically – interpretations 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 cannot think of its morphisms as some equivalence class
of some sort of structure-preserving maps?
Best Answer
A theorem of Kučera asserts that every category arises as a quotient of a concrete category by suitable congruence: see Section 6B of Theory of Mathematical Structures by Adámek.