TITLE: Why don't ideals and quotients work well for categories?
QUESTION [39 upvotes]: Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) category theory could be regarded as a common generalization of all these settings. Why is it that such important structures don't work well for categories?
I am aware that there is a categorical notion of congruence relation. However, this doesn't seem to take the spirit of multiple objects to heart: all it does is keep the same objects and relate morphisms within homsets. For one thing, the accompanying notion of quotient category doesn't correspond to coequalizers in $\mathbf{Cat}$ (of which there are many more).
It is not even clear how to define an ideal of a category. To allow for proper ideals, it probably shouldn't simply be a subcategory. Naively one thinks of a subset $I(X,Y)$ of each $\mathrm{Hom}(X,Y)$ that is invariant under composition with arbitrary morphisms, or just of subsets $I(X)$ of each $\mathrm{Hom}(X,X)$, or of $I(X)$ just for some objects; but this doesn't really take objects into account. Thinking of an appropriate definition is even more perplexing for higher categories.

Question: are there related notions of ideal and quotient for categories that have interesting consequences but are not trivial on the level of objects?

It is left open what roles left (postcomposition) or right (precomposition) ideals should play; a related question is if there is a notion of commutativity for categories with interesting consequences.
A convincing explanation why one shouldn't consider such questions would also be a good answer.

REPLY [5 votes]: I don't think I saw anyone give this construction above and if so I apologize for the repetition.
One could consider the orbit category by an endofunctor.  The example I have in mind is the cluster category $\mathcal{C}$ important in the categorification of cluster algebras.  One starts with the category of finite-dimensional representations $mod_\mathbb{C}~Q$ of a Dynkin quiver $Q$ over the field $\mathbb{C}$ of complex numbers.  Then the cluster category is the orbit category of the bounded derived category $D^b(mod_\mathbb{C}~Q)$ by the endofunctor $F=\tau^{-1}[1]$ where $\tau$ is the Auslander-Reiten translation of $mod_\mathbb{C}~Q$ and $[1]$ is the shift functor.  The paper at http://de.arxiv.org/pdf/math.RT/0402054.pdf was one of the first on cluster categories.
I guess to answer your main question: the objects in these categories are very different.  For example $D^b(mod_\mathbb{C}~Q)$ has infinitely many isomorphism classes of indecomposable representations, for example the shifts of indecomposable objects in $mod_\mathbb{C}~Q$.  However $\mathcal{C}$ has only finitely many indecomposable objects, namely the indecomposable representations of $Q$ considered as complexes concentrated in degree 0 and the shifts $P[1]$ of the projective representations of $Q$.  
I suppose this construction has a slightly different flavor than the quotient of a ring by an ideal.