TITLE: Cogroup Objects
QUESTION [16 upvotes]: Pretty much anyone who does algebra is familiar with group objects in categories, but what about cogroup objects? Most of what I've been able to find about them is that they "arise naturally in algebraic topology" (from wikipedia) and that, somehow, the n-sphere is one (nLab's meager entry).  Is there a reference for more on the stuff? Specifically wondering if the spaces of pointed maps from a topological space X to a pointed sphere are cogroups, and if anything is known about these "cohomotopy groups."

REPLY [6 votes]: This isn't a homotopy theory answer, but the most widely used type of cogroup in mathematics is an algebraic group.  An algebraic group is a cogroup object in the category of commutative rings, or in particular commutative algebras over a field.  What is important in this case is that people care a lot about the opposite category, the category of affine schemes, or in the field case affine algebraic varieties.  People care just as much about schemes and algebraic varieties in general, but if you want the algebraic groups analogue of connected semisimple Lie groups, then all examples are affine.
In my view, any time you endorse a co(something fancy) object in a category, you are really endorsing the opposite category in general.  Of course, at the formal level a category and its opposite category have exactly the same information, but conceptually it's a big leap.  For example, non-commutative geometry is largely the study of the opposite category of non-commutative rings.  Although in this case, people (correctly) use Hopf algebras as if they were cogroup objects, but they aren't!  The tensor product operation on non-commutative algebras is not a coproduct.