Talk:Group object

Are linearly ordered groups an example of the sort of thing this is about? Michael Hardy 01:04, 1 Dec 2003 (UTC)

Those would have to be group objects in the category of partially ordered sets with order-preserving functions as morphisms. Note that the category is that of partially ordered sets because the product of two linearly ordered sets is naturally partially ordered, but not linearly ordered. There is another glitch, though, as the inverse operation reverses the order, so we need to add the order-reversing morphisms to the category. It seems to be easier to just talk about ordered monoids. -- Miguel

Subgroup objects

Near the bottom of the article it talks about how most of group theory can be transfered over to group object theory. It uses subgroups and normal subgroups as examples. I do not understand how subgroups could be transfered over. The only way I could think of is to say a subgroup object is a subobject of a group object that follows the group object identities in the article but that won't work because a subobject isn't a object in its own right. It's a equivilence class of monomorphisms with codomain G! It just doesn't make sense. Possibly, someone out there can clarify this and put it in the article.--SurrealWarrior 22:56, 14 December 2006 (UTC)[reply]

I am not an expert, but can't you just do it with an individual representative of the subobject, which would be a monomorphism into the group object? You will probably have to work in the category of morphisms into the group object. ---- Hans Adler (talk) 19:55, 16 November 2007 (UTC)[reply]

Probably you are right, but it would be nice to see it worked out in a reference. For example, I suppose you would need a factorization system to define things like kernels, so the isomorphism theorems are perhaps not completely general... I don't know of a reference, though. Sam Staton (talk) 19:01, 18 December 2007 (UTC)[reply]

A good place to see it worked out for an example are the topological groups, Lie groups, and algebraic groups (here a great deal of elementary group theory is repeated nearly verbatim but with "continuous", "analytic", or "rational" sprinkled generously). Probably a good place to see how it doesn't just "work out" is in Hopf algebras where basic ideas like Lagrange's theorem become not so basic. Those two extremes might help to figure out what sort of reference for the general case might exist. I also have a dim idea that it might follow trivially if one looks at regular group actions rather than just plain groups, so one might look for a reference from that angle. JackSchmidt (talk) 19:43, 18 December 2007 (UTC)[reply]

Group objects and their morphisms in a category form a category. A subgroup of a group object is then a subobject in the category of group objects. Subobjects are defined as equivalence classes of monomorphisms, but then one normally works with a representative of the subobject since anything reasonable you want to do with a subobject is invariant under the equivalence relation.Wellsoberlin (talk) 20:59, 21 November 2008 (UTC)[reply]

Cogroup object

Someone requested an article cogroup be created. I have added the definition of cogroups as an example. I also created redirects from cogroup and cogroup object. It might be worthwhile to extend this and put it into a section of its own. ---- Hans Adler (talk) 20:01, 16 November 2007 (UTC)[reply]