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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.

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You have serre subcategories of abelian categories, and thick subcategories of triangulated categories. They allow the definition of quotients, which is what ideals are good for. These quotients kill the objects in the subcategory, as it happens with quotient rings. You can look this up in wikipedia. –  Fernando Muro Mar 31 '11 at 22:58
For me an ideal is the kernel of a homomorphism between rings. What is the kernel of a functor? –  Qiaochu Yuan Mar 31 '11 at 23:04
I am with Qiaochu here. There is no good notion of a "normal subset" or an ideal of sets either. The only reason the kernel is so important for a homomorphism of groups (resp. rings) is that the additive structure lets you translate f(x) = f(y) into f(x-y) = 0. This is why fibers of a homomorphism are all cosets of the kernel. You might be interested in my post mathoverflow.net/questions/41955/… –  Steven Gubkin Apr 1 '11 at 0:12
@Qiaochu: the kernel could be the subcategory of morphisms that map to some identity morphism. That notion is actually used for groupoids. I don't think it is useful in general, though. –  Omar Antolín-Camarena Apr 1 '11 at 0:34
@Chris: The quotient (of an abelian category by a Serre subcategory, or a triangulated category by a thick subcategory) does have a universal property, and it is universal in the class of all functors. –  t3suji Apr 1 '11 at 16:19
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7 Answers

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Here is a shortish answer that relates to several of the above replies: Yes! There is such a theory.

Ideals correspond to a particular type of internal category or groupoid in the category of rings, normal subgroups correspond to `dittos' in the category of groups. Quotients by an ideal/normal subgroup are the coequalisers of the source and target maps of those internal categories (in many settings, quotients may not exist, e.g. this is especially important in geometric cases, hence the theory of Lie groupoids etc.) Both of these, internal categories and coequalisers still work perfectly well in $Cat$, so your impression is not quite right.

For a much lengthier gloss on that:

(i) The peculiarity of the categories of rings and groups (and other similar categories) is that congruences can be replaced by such `normal subobjects'. The congruence is the important thing here not the normal subobject, and in very many situations analogues of lattices of ideals work well. Of course, they are lattices of congruences, and so on.

(ii) For 'quotient', there seems a lot of confusion in terminology. From what you say in the question, I presume that you mean 'quotient ring', for instance, rather than 'ring of quotients'. (Several of the comments seem more in tune with the latter situation.) If that is right then the comment that I made above is relevant. quotients are `really' just coequalisers in a category.

(iii) Turning to $Cat$ itself, congruence relations make perfect sense in $Cat$ and correspond to a particular type of double category. In the category, $Cat/\Sigma$ , of categories with a fixed set, $\Sigma$ of objects, congruences are as you state. In other words, it depends on where one is as to which type of congruence and which type of quotient is appropriate.

(iv) Finally, to answer your specific question: yes, one can look at quotients with respect to congruences of categories or groupoids, that crush objects together. This sort of setting was thoroughly explored by Phil Higgins in his book which was reprinted in TAC a few years ago. (See [[http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html]]) That looks at the algebra of groupoids as algebraic objects of a fairly classical nature.

(edit: PS. I had not glanced at the linked Wikipedia article. That is way too restrictive in how it defines the notion of a quotient category, IMHO.)

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Excellent, Tim! Chapter 12 of Higgins is precisely what I was looking for without knowing it, and what the question tried to express I was missing. I don't quite understand the internal perspective yet, though. How does an ideal $I$ of a ring $R$ induce a groupoid in the category of rings? Do you take the objects of objects and morphisms to be $I$ and $R$ -- are you considering nonunital rings? –  Chris Heunen Apr 4 '11 at 1:20
Take $I\oplus R$ (as a sort of semi-direct product, so the action of $R$ on $I$ is used) as the object of 'arrows'. That has two morphisms to $R$ (which is the object of 'objects'). I gave a discussion of this for (commutative) rings in Internal Categories and Crossed Modules, Proc. Inter. Conf. of Category Theory, Gummersbach, 1981, in Springer Lecture Notes in Mathematics, Vol. 962, 1982. but it is 'well known' and can be found in other places. I can provide more detail if you want, (but probably not in MO as the comments are more tricky to use and I have only a few characters left!) –  Tim Porter Apr 4 '11 at 6:56
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An operation that insists on making certain objects equal breaks the spirit of category theory -- we should only insist that they be equal... up to isomorphism. To me, it seems more natural to add more morphisms to accomplish this effect, rather than trying modding objects out by an equivalence relation.

There are operations that can do this. Localization, for example, formally inverts a class of morphisms, making the corresponding objects isomorphic. I once saw a construction that formally added isomorphisms to create a natural isomorphism from the identity functor to a given endofunctor.

But, even modding out by an equivalence relation can have this effect. If the composites $A \xrightarrow{f} B \xrightarrow{g} A$ and $B \xrightarrow{g} A \xrightarrow{f} B$ are congruent to the corresponding identity morphisms, then $A \cong B$ in the quotient category.

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Good point! But is there what I called an interesting consequence, e.g. in the form of an appropriate first isomorphism theorem? Sure, a universal property is satisfied, but that doesn't involve arbitrary functors that may act nontrivially on objects, does it? –  Chris Heunen Apr 1 '11 at 4:49
Localizing a category $C$ at a set of morphisms $S$ gives a new category $C[S^{-1}]$ whose objects are the same as $C$ and that has more morphisms. It satisfies a universal property that does involve arbitrary functors: Any functor $C \to D$ that sends morphisms in $S$ to invertible morphisms in $D$ factors through $C[S^{-1}]$. Of course, localizations for an arbitrary $S$ are hard to work with, just like localizations of noncommutative rings, so usually people localize at an $S$ that has nice properties, like the morphisms of a Serre subcategory. –  Peter Samuelson Apr 1 '11 at 16:35
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Hurkyl has made some very good points, which I may be getting back to. On occasion though we might wish to break the spirit of modern category theory (which would prefer to view $Cat$ as a 2-category), and consider $Cat$ as a good old-fashioned 1-category of models of an essentially algebraic theory (or of a finite limit sketch). For example, the monoid of natural numbers could be defined as the (strict) coequalizer in $Cat$ of the pair of maps

$$\mathbf{1} \stackrel{\to}{\to} \mathbf{2}$$

which name the objects of the generic arrow category $\mathbf{2}$. Here objects get identified and we break the "taboo". (But notice how "odd" this example is, as the coequalizer from $\mathbf{2}$ to $\mathbb{N}$ is not a surjection between the sets of morphisms. This example shows for example that regular epis in $Cat$ don't compose: consider the composite $\mathbf{2} \to \mathbb{N} \to \mathbb{Z}/(3)$.)

Thus, the category of small categories $Cat$ is a beast quite unlike traditional algebraic categories like $Grp$ and $Ring$. Categories which are finitary algebraic (finitary monadic) over $Set$ are effective regular (Barr exact) categories, where there are nice stable Galois correspondences between quotients of objects and congruence relations on objects. (The cases of groups and rings is even better, since congruence relations can be morally identified with kernels.) The (1-)category $Cat$ is not effective regular -- it's not even regular.

This sort of harks back to Hurkyl's point: evidently a more sophisticated point of view needs to be applied to get a good theory of quotients and kernels in enriched category theory. I only just found out about this myself, but I think I can recommend to your attention the nLab article on generalized kernels, which may give you something along the lines that you are considering. See in particular examples 4, 5, and 6, all relating to 2-categorical analogues of coequalizers (co-inverters, coequifiers, and coinserters) in $Cat$ seen from the perspective of generalized kernels.

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Thank you Todd! Those are valuable pointers. This is more in the direction I was angling in. It is especially interesting to see factorization systems come in, but I suppose that was really only to be expected when one thinks about kernels and coequalizers in $\mathbf{Cat}$. –  Chris Heunen Apr 1 '11 at 4:58
I'm glad you find value in them! At least the generalized kernels are a positive direction in which to proceed; seems to me they ought to be better known and appreciated. –  Todd Trimble Apr 1 '11 at 16:25
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Hurkyl's point is a good one. It might also be worth mentioning that in your motivating examples for quotients, the Hom sets are modified and the objects are not. More precisely, a group is the same thing as a category with one object and all morphisms invertible, and a $k$-algebra is the same thing as a $k$-linear category with one object, so quotients of these have the same set of objects but have smaller Hom-sets.

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You're saying, that the definition of an Ideal $I(X,Y)\leq Hom(X,Y)$ does not affect objects. That is not true, because in the quotient category there can be more objects isomorphic to one another than before which is exactly the same behaviour in groups and ring where elements can become identical by passing to a quotient.

Consider this: If C is the category of finitely generated modules over an artinian ring (more general: modules of finite length over an aribitrary ring) and A is a full, isomorphism closed subcategory that is also closed w.r.t. taking direct summands, then define the following ideal: $I_A(X,Y):=\lbrace f:X\to Y | \exists P\in A: f \text{factors through} P\rbrace$

Then one can show that in the quotient category $C/A:=C/I_A$ the following charaterizations hold:

$X\cong^{C/A} 0 \iff X\in A$ and more general

$X\cong^{C/A} Y \iff \exists P_0, Q_0\in A: X=X_0\oplus P_0, Y=Y_0\oplus Q_0, X_0 \cong^C Y_0$ (i.e. "isomorphic in C/A" means nothing else than "isomorphic in C if direct summands from A are neglected".

This construction with A:={projective modules} gives you the stable module category of a ring. If you're dealing with group rings then things like the Green-correspondence become equivalences of such quotient categories where certain subcategories of relative projective modules are chosen as A.

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In quantum topology quotients of (tensor) categories by (tensor) ideals are quite important. For example, see Theorem 2.9 of Barrett and Westbury's "Spherical Categories." You just set a certain collection of morphisms equal to the zero moprhism. Note that this is a bit different from setting morphisms equal to the identity.

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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.

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