Intrinsic kernels

In a pointed category, such as the category of groups, the kernel of a morphism f: A -> B can be defined as the inverse image of the distinguished point of B (which, in the case of groups, is the unit of B).

In a non-pointed category, the kernel of a morphism f: A -> B can still be defined as the inverse image of some suitably defined substructure of B. For example, the kernel of a morphism f: A -> B of inverse semigroups is defined as the inverse image of the subsemigroup E(B) of idempotents of the inverse semigroup B.

However, in this more general context, the choice of this substructure of B that we use to take kernels becomes a free parameter of the kernel construction ? we get different notions of kernel if we choose different substructures of B. Ideally, the notion of kernel would be something that is intrinsic to the category, meaning it is something that depends on the structure of the category, and not on some arbitrary choice.

We try to solve that problem by specifying a property that the choice of substructure should satisfy, and the intrinsic notion of kernel will be the one defined in terms of the smallest such substructure (under the assumption that the smallest such substructure exists). For example, in the case of inverse monoids, the submonoid of idempotents is a suitable choice of substructure, while the one element monoid isn't.

As a further example, in the category of sets, the smallest suitable substructure of a set will be the set itself, while for groups the substructure will be the one element group, so that intrinsic kernels reduce to the classical notion.