Definition talk:Category of Sets

This definition does not define the composition, codomain, or domain operations directly, and this turns out to be problematic. Based on the definitions of mapping, relation, and of codomain, we can't actually define $\operatorname{cod}$, due to its reliance on the extrinsic information. A mapping $f : S \to T$ is not distinguishable, as an object, from the same mapping with a different codomain, since the mapping is ultimately just a subset of $S \times T$ and may as well be a subset of $S \times U$ for some other $U$. Scshunt (talk) 17:05, 27 May 2016 (UTC)


 * Thanks for being attentive. Your remark boils down to the most common definition of "relation" being imprecise. See Definition:Relation/Relation as Ordered Triple for a more rigorous definition of "relation" which takes care of encoding the codomain within the relation.
 * Maybe it would be good to emphasize this on Definition:Codomain because it is necessary to have uniqueness of codomain. &mdash; Lord_Farin (talk) 17:16, 29 May 2016 (UTC)


 * I have considered restructuring the pages defining mapping and relation so as to place the Definition:Relation/Relation as Ordered Triple and its equivalent on the mapping page as the primary definition. It's rare within conventional undergrad mathematics for this approach to be taken, and finding source works to give this definition is a challenge. It will also be a major undertaking, so I have been putting it off and mucking about with trivial stuff instead, I'm renowned for putting off things I've got to do.


 * In the process, I intend to emphasise the point that two mappings / relations with different codomains are, while otherwise identical, different mappings / relations. I will also add a warning that there are treatments which gloss over this, and sometimes completely ignore it. --prime mover (talk) 19:30, 29 May 2016 (UTC)


 * The above has now been done, to a certain extent. --prime mover (talk) 03:31, 8 September 2016 (EDT)