So far, I’ve spent some time talking about groups and what they mean. I’ve also given a
brief look at the structures that can be built by adding properties and operations to groups –
specifically rings and fields.
Now, I’m going to start over, looking at things using category theory. Today, I’ll start
with a very quick refresher on category theory, and then I’ll give you a category theoretic
presentation of group theory. I did a whole series of articles about category theory right after I moved GM/BM to ScienceBlogs; if you want to read more about category theory than this brief introduction, you can look at the category theory archives.
Like set theory, category theory is another one of those attempts to form a fundamental
abstraction with which you can build essentially any mathematical abstraction. But where sets
treat the idea of grouping things together as the fundamental abstraction, category
theory makes the idea of mappings between things as the fundamental abstraction.
A category is thus a simple structure consisting of a collection of objects,
connecting by arrows (arrows are also called morphism). For intuition, you
can think of objects as sets, and arrows as functions: an arrow is a mapping from one object
to another, much like a function is a mapping from one set to another. (But that’s only a very
vague intuition: in category theory, we never consider the internal structure of the objects
or the arrows: the objects and arrows are the primitive atoms of category theory – and there
are some categories where that intuition will lead you astray.) If there’s an arrow f from
object a to object b, we’ll write that as “f:a→b”.
In some sense, the objects of a category are almost unnecessary; what we care about are
the arrows. The objects are only there to give us something to connect with arrows. It’s like
a connect-the-dots puzzle: the dots aren’t important parts of the picture: they’re just there
to show you where to draw the lines.
In addition to the arrows, there’s a single fundamental operation on arrows, called
composition. Given two arrows f:a→b and g:b→c, we can compose f and g,
getting a new arrow from a to c. We write the composition of f and g “gºf”. Arrow
composition has to meet two properties, which are going to look pretty familiar from group
theory: associativity, and identity. Associativity says that
aº(bºc)=(aºb)ºc; and identity says that that for any arrow f:a→b,
there are arrows 1a and 1b such that 1bºf = f =
fº1a. (In other words, for any arrow, there’s something you can compose it
with that won’t change it.)
Category theorists have a lot of jargon for describing arrows with additional properties.
There’s a rundown of the basic ones here.
For an example that’s important for looking at group theory related topics, we say than an
arrow is iso (i.e., it is a iso-morphism) if it’s reversable – which in category
theoretic terms means that if f is an iso-arrow, then there is an arrow f-1 such
In category theory, one useful technique is diagrams. You can draw a set of objects and arrows as a diagram. The diagram is said to commute if any two paths in the diagram that have the same start and end-points compose to the same arrow. It’s important to realize that a diagram is an explanatory tool – not a proof. (That’s a common error – I can’t count how many CS papers I’ve seen that show a diagram without proving that it actually commutes!) Just because you can draw it doesn’t mean that it commutes. You’ve got to show that.
For an example of a diagram, there’s a type of arrow between two objects A and B called a principle morphism. The idea of a principle morphism is that a morphism M from A to B is principle if and only if every self-arrow of A (or endomorphism, an arrow from A to A), composed with an arrow from A to B yields M. As a very visual thinker, I find that paragraph hard to follow. But in category theory, we can say that m is principle if and only if for every arrow x from A to A, and for every arrow y from A to B, the diagram to the right commutes.
The key to reading diagrams is to realize that you really need to look carefully at the labels. The same object can (and often will) appear multiple times in the same diagram. But any two paths between the same objects has to be equivalent. So in the example above, the object A appears two different times. That helps make clear that we’re saying that composing an arrow from A to A with an arrow from A to B always yields m; if we only drew A once, we’d have a looped arrow on A, and two arrows from A to B. It wouldn’t be clear what we were asserting about which arrows compose with which. By drawing A twice, we’ve done something which is equivalent, but easier for a human to read.
As I said, in category theory, we’re studying structures in terms of mappings. One of the
major tools of the field is to use higher-order mappings to study the structures of
categories. If you’ve got a category with arrows, you can define a higher-order mapping called
a functor, which is a mapping from arrows to arrows. (And it’s also an arrow in the
category of categories.) Above that, there’s something called a natural
transformation which, very loosely described, is a mapping from functors to functors. The
idea of all of these mappings is to provide a tool for talking about structures, how they
work, and what they mean. In every case, you can view a mapping as something that preserves
some kind of structure. An arrow in a category is a mapping between things that have some
structural similarity, and the mapping preserves that. A functor maps between arrows in a
category, preserving the structure described those arrows. A natural transformation maps
between functors, preserving the structure described by those functors.
Since from the viewpoint of abstract algebra, we’re building sets with a bit of structure
to them (provided by the operations), category theory is a great tool for studying groups,
rings, fields, and the other things in abstract algebra.