I'm looking for a categorial notion of a normal subgroup and an ideal (of a ring, and a non-associative algebra).

Basing on the following observations:

  • they are used to define quotient objects in the category of groups and commutative rings,
  • given a connected Lie group, there is a correspondence between its normal connected Lie subgroups and ideals of its Lie algebra,

I expect that there may exist a construction in category theory generalizing these notions.

  • $\begingroup$ Yes, $A\hookrightarrow B$ is (isomorphic to) an embedding of an ideal of $B$ iff it is a kernel of some $g:B\to C$ (that is, the equalizer of $g$ and the trivial morphism). $\endgroup$ – Berci Feb 26 at 20:58
  • 1
    $\begingroup$ You may find of interest my post on ideal-determined varieties. $\endgroup$ – Bill Dubuque Feb 26 at 23:26

What you want is the notion of a congruence. It's a convenient fact about groups resp. rings that congruences are equivalent to normal subgroups resp. (two-sided) ideals; in general, for example when dealing with monoids, you really need to work with congruences.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.