Definition:Kernel (Category Theory)/Uniqueness

On Uniqueness

Since the kernel is defined by a universal property it is only unique up to unique isomorphism.

While for example in group theory the kernel of a group homomorphism $f : G \to H$ is a subset of $G$, not all categorical kernels of $f$ in the category of groups are subsets of $G$.

Also see

• Kernel is Unique up to Unique Isomorphism

This article, or a section of it, needs explaining. In particular: "A" kernel? There nay be more than one? This needs explanation, as in most other contexts, there is only one kernel for any given object. Examples might be useful. Here is a first try, i can extend if needed. I usually write 'a kernel' if something is not strictly unique, although 'the kernel' makes sense if the uniqueness up to unique isomorphism is already proven. Note, that the definition of categorical kernel is not in a strict sense equivalent to the definition Definition:Kernel of Group Homomorphism. Only should also note, that everything defined in terms of a universal property comes equipped with one or several morphisms (in this case this is the inclusion of the kernel into the domain), that are relevant data to ensure uniqueness up to 'unique' isomorphism. In the modern understanding of these concepts these maps are geniunely part of the definition. It probably worth disussing how to deal with this on $\mathsf{Pr} \infty \mathsf{fWiki}$, as this seems to not have been relevant before. --Wandynsky (talk) 19:17, 28 July 2021 (UTC) One can also also say something like 'a kernel is a pair (K,i) ...'. It is clumsy but probably most precise. --Wandynsky (talk) 19:23, 28 July 2021 (UTC) And I just realized, that all of this should be a part of the main page, going to change this later. --Wandynsky (talk) 19:34, 28 July 2021 (UTC) Best would be to put it into a separate page and transclude into all three pages. Any statements made in the "on uniqueness" section needs to be backed up with proofs. If it's complicated then well-crafted theorems and proofs will of course be written separately and linked to. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.