Definition talk:Homomorphism (Abstract Algebra)

Morphisms
I believe homomorphisms are morphisms. Certainly Wikipedia says so, and Quora. I know these are not definitive sources, but I have also heard such elsewhere. --Dfeuer (talk) 06:15, 18 January 2013 (UTC)


 * It's true. It's also true that $u \le v$ is a morphism in a poset category. A mapping is a morphism in the Definition:Category of Sets. Morphisms are everywhere. It's not helpful to add virtually all pages defining a mapping to the category Category:Definitions/Morphisms. There admittedly is currently a lack of application of CT to other areas covered on PW, but a more structured approach approved of by most or all of the maintenance team would be the way to go. I hope you see the distinction.
 * In general, maintenance in categories you wouldn't say to "master" to some extent (at the very least, to have consulted/worked through one or more source works) is not a good idea. Your work is appreciated, but currently it's best for all of us that you leave CT alone and work on areas we're convinced you are proficient in (or where the first few of your edits make us conclude you are). Thanks. --Lord_Farin (talk) 08:50, 18 January 2013 (UTC)
 * I was trying to organize things, not to pretend I know more than I do. If you don't think it's valuable to have a morphism category, then fine. But I feel rather strongly that an isomorphism category, including not only algebraic but also topological, etc., ones is quite reasonable. Certainly PM's move of making "isomorphism" a subcategory of abstract algebra and removing the abstract algebra categorization from pages defining isomorphisms goes the wrong way. --Dfeuer (talk) 17:24, 18 January 2013 (UTC)
 * Agreed on the part of isos - if I'm not mistaken, I didn't cancel your edits on those. In hindsight the current categorisation of "isomorphism" was unfortunate - OTOH, it is an understandable move for people not knowing about CT. Your efforts in organizing the site are appreciated; I just think that there are ample other areas you're (reasonably) knowledgeable about yearning for a similar treatment. Attacking those will be easier for everyone and prevents needless shouting and annoyance. --Lord_Farin (talk) 17:48, 18 January 2013 (UTC)


 * There is limited use in creating a subcategory based on nothing else but coincidence of language. True, the concept of "isomorphism" in general means the same thing wherever it is encountered: it means "same form". But whereas there is an "isomorphism" in abstract algebra, there is an "isomorphism" in graph theory and whatever else you can dream up, but that in no way makes it sensible to lump everything which relates to "isomorphism" in the same category.


 * If you do feel the need to create a category called "isomorphism" then make it a subcategory of "Definitions", rather than a subcategory of everything it is not a subcategory of. (If it's a subcategory of abstract algebraical homomorphisms, it is not a subcategory of Graph Theory, and so on.


 * Categories evolve. It is a questionable approach to generate a category just because there is a definition for it. Instead, you look at an existing category and see that there are many results inside it which all relate to the same concept (like e.g. Union in the category of Set Theory, or Conjunction in the category of Propositional Logic. When this happens, it makes sense to break the category down and thence to reduce the number of results in any given category.


 * So please lay off your organising of the categories until you've created some useful material to put in them. --prime mover (talk) 18:31, 18 January 2013 (UTC)

That is more or less what I tried to suggest. --Lord_Farin (talk) 22:01, 18 January 2013 (UTC)

Closedness
It doesn't make sense to say
 * $\forall x, y \in S: \phi \left({x \circ y}\right) = \phi \left({x}\right) * \phi \left({y}\right)$

if $S$ is not closed. --barto (talk) (contribs) 19:13, 13 January 2018 (EST)


 * Definition:Algebraic Structure has been replaced by Definition:Magma throughout this page. --prime mover (talk) 15:29, 25 October 2018 (EDT)