# Groups in Categories

# Groups in Categories

### Learn more about this misleading term: algebraic groups.

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Join For FreeThe first time I saw a reference to a “group in a category” I misread it as something in the category of groups. But that’s not what it means. Due to an unfortunate choice of terminology, “in” is more subtle than just membership in a class.

This is related to another potentially misleading term, algebraic groups, mentioned in the previous post on isogenies. An algebraic group is a “group object” in the category of algebraic varieties. Note the mysterious use of the word “in.”

You may have heard the statement “A monad is just a monoid in the category of endofunctors.” While true, the statement is meant to be a joke because it abstracted so far from what most people want to know when they ask what a monad is in the context of functional programming. But notice this follows the pattern of an *X* in the category of *Y*‘s, even though here *X* stands for monoid rather than a group. The meaning of “in the category of” is the same.

If you want to go down this rabbit hole, you could start with the nLab article on group objects. A group object is a lot like a group, but everything is happening “in” a category.

Take a look at the list of examples. The first says that a group object in the category of sets is a group. That’s reassuring. The second example says that a group object in the category of topological spaces is a topological group. At this point, you may get the idea that an *X* in the category of *Y*‘s simply adds an *X* structure to a *Y* thing. But further down, you’ll see that a group object in the category of groups is an *Abelian* group, which is an indication that something more subtle is going on.

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