Ringoids
Posted by John Baez
Just as a “group with many objects” is a groupoid, a “ring with many objects” is called a ringoid.
Gregory Muller has emailed me a question about ringoids. I don’t want to get into the habit of posting emailed questions on this blog, because I already burnt myself out years ago helping moderate a newsgroup. But just this once, I will.
(Famous last words…)
Gregory Muller writes:
I was excited recently when I learned how to categorify the notion of
rings. I’ve long thought of groups as categories, and up to a few days ago,
it had bothered me that I lacked a parallel notion for rings, especially
given the significance of category-theoretic notions in algebraic geometry.
However, my current method of categorifying rings is ugly; specifically, it
relies on a definition that uses underlying sets and elements, which is
obviously distasteful. What is further frustrating is that this method
generalizes in a way that allows Lie groups and other common mathematical
objects to be categorified.The method I am using is as follows. Let X be a category with a forgetful
functor F to Set. Then define an “X-valued category” to be a category in
the usual sense, except that the Hom(A,B) is an object in X instead of a
set, and satisfying the following compatibility with composition:The composition function is
o: F(Hom(A,B)) × F(Hom(B,C)) → F(Hom(A,C)),
with the map from F(Hom(B,C)) → F(Hom(A,C)) given by ‘plugging in’
is the image of a morphism in X, and the similar statement for plugging in
on the right.This is a useful definition, since:
1) A ring (with unit) is the same as an AbGrp-valued category with one
object.2) A k-algebra (associative, with unit) is the same as a k-Vect-valued
category with one element.3) An R-algebra (associative, with unit) is the same as an R-Mod-valued
category with one element.4) A Lie Group is the same as a SmoothManifold-valued groupoid with one
element.…and likely others that I haven’t noticed yet.
The problem is the use of sets and elements in the defintion. I have tried
to clean things up and use a category-theoretic definition, but (at least in
the case of AbGrp-valued categories) it seems to be related to the problem
of expressing the formula “ab+cd” as a product of sums in an arbitrary ring,
which I think is about as hard and convoluted a question as any you are
likely to find in math.Another question is how to philosophically reconcile this notion of a
“category with arrows in category X”, with the other notion of such a
category coming from n-categories. Specifically, a 2- category can be
thought of as a category with Hom(A,B) taking its values in Cat, instead of
Set, subject to some very different notions of “compatibility with
compositions”.I would be very thankful for any insight into this stuff,
Greg
Your concept of an “X-valued category” is usually called an X-enriched category, or X-category for short. The idea is to fix a category X and define a X-category to have a set of objects and, for any pair of objects a and b, not a set but an object of X called hom(a,b). We can write down the whole definition of category this way as long as X is a monoidal category, that is, a category with tensor product. This allows us to say that for any objects a,b,c in our X-category, composition is a morphism in X:
o: hom(a,b) ⊗ hom(b,c) → hom(a,c)
By this means we can avoid referring to “elements” of hom(a,b).
Enriched categories were invented by Max Kelly, and you will enjoy reading his book, since he gives a very clean treatment:
- G.M. Kelly, Basic Concepts of Enriched Category Theory. Originally printed as Lecture Note Series 64, London Math. Soc., London, 1982.
Kelly was the first to bring category theory to Australia, and enriched category theory has been a mainstay of Australian category theory ever since he invented it. In addition to X-enriched categories, he defined X-enriched functors and X-enriched natural transformations. He then went ahead and redid all of category theory - well, lots of it anyway! - in this X-enriched setting. The basics are straightforward; things get more tricky when you reach the theory of limits and colimits.
One of Kelly’s most famous students is Ross Street, and you can read about the history of enriched category theory near the beginning of Street’s Australian conspectus of higher categories.
People usually denote the category of abelian groups by Ab instead of AbGrp. With the usual tensor product of abelian groups, Ab becomes a monoidal category, and an Ab-category is sometimes called a ringoid.
As you note, a one-object ringoid is a ring, just as a one-object groupoid is a group. These are not “categorifications” of the concept of ring and group, not in the technical sense anyway. A group is already a category; when we go to groupoids we are just letting it have more objects. Similarly for rings and ringoids. So, instead of categorification, one should call this process many-object-ification, or maybe oidization.
To categorify the concepts of group and ring, we need to go up to 2-categories. The resulting concepts are called “ring categories” and “categorical groups” (or “2-groups”). Ring categories were introduced by Kelly and Laplaza.
As you note, we can also many-object-ify the concept of algebra. The category of R-modules is called R-Mod, and it’s a monoidal category with the usual tensor product of R-modules whenever R is commutative. An R-Mod-category is called an R-algebroid or simply an algebroid. As you note, a one-object algebroid is an associative algebra with unit.
Note that when R = Z, R-Mod is just Ab, so a Z-algebroid is just a ringoid.
Similarly, as you note, a Cat-category is a 2-category.
Lately we’ve been talking about symmetric monoidal closed categories, for example cartesian closed categories. Any such category is enriched over itself! (Being symmetric is actually irrelevant here.)
There’s been a lot of work on all these subjects.
But don’t feel bad that you’re reinventing the wheel a bit here - it’s a very good wheel, and you can roll quite far with it.
One thing fans of category theory enjoy is how sufficiently general concepts can bend back, bite their own tails, and swallow themselves. This happens in the case of ringoids. Whenever R is any ring, R-Mod is a ringoid. But, this is also true when R is a ringoid! We define a module of a ringoid R to be an Ab-enriched functor
F: R → Ab
and define a homomorphism between these to be an Ab-enriched natural transformation. These notions reduce to the standard ones when R is a ring.
So, we get a category R-Mod of modules for any ringoid R… and R-Mod is again a ringoid!
For a very practical text on algebroids try this:
- P. Gabriel and A. V. Roiter, Representations of Finite-Dimensional Algebras, Enc. of Math. Sci., 73, Algebra VIII, Springer, Berlin 1992.
The terminology is a bit quirky, but there’s some amazing stuff in here.
enrichment
Gregory Muller described the concept of composition in $X$-enriched categories this way:
He noticed that various familiar algebraic concepts can hence nicely be understod as $X$-enriched categories of various sorts. In this context he also remarks that
I am not fully sure what this last comment on a “different notion” of compatibility is addressing. But I’ll say this:
Abstractly, the “compatibility of composition” is always the same, namely always given by the monoidal structure of the category $X$ that we enrich over. The only subtlety to be aware of is that there may be quite different monoidal structures on one and the same category $X$.
There is a standard monoidal structure $\times$ on $\mathrm{Cat}$. For categories enriched over $\mathrm{Cat}$ this implies that composition
is a functor from a product category, which implies that
does not depend on whether we first apply $F$ on $f_1, f'_1$ and on $f_2,f'_2$ seperately, and then compose the result “vertically”, or if we first compose vertically and apply $F$ (the “horizontal composition”) to the result.
This compatibility condition is called the exchange law in 2-categories. It is implied by the standard monoidal structure on $\mathrm{Cat}$.
I am assuming that what Gregory had in mind is that this looks different from the “distributive” compatibility condition which we have for $\mathrm{Ab}$-categories.
But the reason for that is just the choice of monoidal product in $\mathrm{Ab}$.
One choice would be the cartesian product of abelian groups. Using that for enriching over $\mathrm{Ab}$ does not produce the expected distributivity of composition over addition.
Instead, this can in fact be understood as a special case of the above “exchange law” in 2-categories, namely if we think of an abelian group as a special case of a category (with a single object) with addition being the composition of morphisms.
But there is another monoidal structure on $\mathrm{Ab}$, namely the tensor product obtained by regarding abelian groups as $\mathbb{Z}$-modules. Using this monoidal structure when enriching produces the expected distributive compatibility condition.
This is more or less obvious, but maybe it doesn’t hurt saying it. In fact, the only reason why I am making this comment is that I was myself mixed up about this at one point.