0

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology?

I am mostly interested in categorical approach to Hausdorff compacts. In the category of Hausdorff compacts and continuous maps, i can define absolute retracts as objects such that every monomorphism of that object to another one splits, i.e. is a split monomorphism. Is this definition standard? Does it work well to generalize topological absolute retracts for non-compact spaces? If not, is there a better name for objects from which every monomorphism splits? Is there a better definition of "categorical absolute retracts"? Is there a standard name for the full subcategory generated by the class of all "absolute retracts"?

Alexey
  • 2,604
  • In order to do that, you should expressed "closed" and "normal" as pure categorical properties... I'm not sure it is possible. What the motivation behind your will of generalization ? – Pece Jan 26 '15 at 14:28
  • 1
    Then my question would be: is there a categorical term for the objects such that every monomorphism from them splits? I've found a purely topological definition of a closed interval, which i am satisfied with, and i was thinking about stating it in categorical terms. – Alexey Jan 26 '15 at 14:56
  • 1
    It sounds like you are interested in injective objects. – Zhen Lin Jan 26 '15 at 19:07
  • You might be interested in http://math.stackexchange.com/questions/450193/ where I gave a categorical description of open and closed embeddings. This way you should also be able to define normal spaces, and then absolute retracts. However, I don't know if anything of this is standard, and therefore it will just be a comment. (You explicitly asked for standard notions.) – Martin Brandenburg Apr 08 '15 at 01:09
  • 1
    See Definition 9.6. of this book, which I think partially answers your question at the beginning. –  Dec 17 '19 at 08:34

0 Answers0