# Talk:Injective cogenerator

User:Charles Matthews This rather isolated page is densely written, and really needs to be taken in hand.

The example about cogenerators in a category of topological spaces doesn't quite fit the definition, as the category doesn't have a zero object. AxelBoldt 23:35, 3 Feb 2004 (UTC)

OK - this is mentioned in the book of Barr and Wells as an example, but I was worrying about whether it was quite right.

Charles Matthews 07:17, 4 Feb 2004 (UTC)

I propose to join this page with Generator (category theory). Agreed? Tilmanbauer (talk) 11:48, 27 July 2010 (UTC)

## Abelian group case

The statement Assuming one has a category like that of abelian groups, one can in fact form direct sums of copies of G until the morphism f: Sum(G) →H is surjective makes no sense. What are G and H supposed to be here? Is it supposed to be true for any G and H (which it is not)? Is G supposed to be a generator? Do we know that the category has a generator? Most importantly, what is the reference? Lichfielder (talk) 08:43, 20 September 2012 (UTC)

The article does not actually provide a definition for the concept of injective cogenerator.

The definitions given for generator and cogenerator objects in categories having a zero object (as such an unnecessary restriction) are incorrect. They would imply that ${\displaystyle \mathbb {Z} /2}$ is both a generator and a cogenerator in the category of modules over the ring ${\displaystyle \mathbb {Z} /4}$, while in fact it is neither.

The formulation of the section titled "In general topology" is vague and incorrect. The closed interval [0,1] of the real line is indeed a cogenerator in the category of completely regular spaces, and the same goes for the open and half-open intervals. However, no interval is injective as an object of this category, since there exists a continuous map into it defined on the circle minus a point which cannot be extended continuously to the entire circle.

It seems best to replace the entire article.

Vdlee37 (talk) 15:07, 7 October 2020 (UTC)