Definition talk:Isomorphism (Abstract Algebra)

Aren't there categories in which a mono-epi is not an iso? For example, monoid categories (but maybe it is provable in case we are dealing with a 'set-category' of sets and certain special functions). --Lord_Farin 18:46, 23 March 2012 (EDT)

If there are, I haven't heard of them. My understanding is: a mono is an injective homo, an epi is a surjective homo, a mono-epi is an inj-surj homo i.e. a bijective homo, i.e. an iso.

If there are categories out there which do not fit into this system, I haven't heard of them. If you have, please share. --prime mover 18:56, 23 March 2012 (EDT)

Well I am only just venturing into category theory with the arrival of two books; when time comes, I will think about this stuff. But one of them explicitly says:

'So it is clear that every iso is a mono-epi. However, the converse is, in general, false. For example, take a monoid category'.

But I'm not sure as of yet how to interpret this, so I'll leave it for now. --Lord_Farin 19:32, 23 March 2012 (EDT)

Bear in mind that the definitions are different for category theory. The disambiguation pages are in place: there are two versions, one for Abs Alg and one for Cat Thry. And epi and mono look a bit different in cat thry to in abs alg. --prime mover 20:16, 23 March 2012 (EDT)

Inverses

Methinks these should have links to show that the inverse of each sort of isomorphism is in fact a homomorphism, as that seems key to the concept. Since any significant change I make is greeted with harsh shrieking, I figured I should ask in the discussion page first. --Dfeuer (talk) 21:40, 24 January 2013 (UTC)

Perhaps a link to Inverse of Algebraic Structure Isomorphism is Isomorphism under "Also see" is sufficient? --Lord_Farin (talk) 21:58, 24 January 2013 (UTC)

Once that page is expanded (very slightly, I think) to cover algebraic structures with multiple operations, I think that will probably do it, although I'd have to read each definition to be entirely certain. --Dfeuer (talk) 22:05, 24 January 2013 (UTC)

Might as well go all the way and take on algebraic systems (though that rather easy generalisation lies almost completely in darkness as of yet). It'd be appreciated if any significant changes were published in "preprint" on e.g. your sandbox, so as to avoid the verbal fights I've grown tired of. --Lord_Farin (talk) 22:10, 24 January 2013 (UTC)

I'm in agreement with L_F: I don't see a great deal of point in raising a separate page to specify that the inverse of each kind of isomorphism is an isomorphism (see, we can be more precise than say "homomorphism") as we already have Inverse of Algebraic Structure Isomorphism is Isomorphism. If you believe that it is necessary to expand to multiple operations, then go for it, but I can't see the point. No published work I've read bothers to go into any such mind-sapping details. --prime mover (talk) 23:03, 24 January 2013 (UTC)

Just because it's trivial doesn't mean it's unimportant. Otherwise you wouldn't be wearing your fingertips off typing in propositional logic. --Dfeuer (talk) 23:26, 24 January 2013 (UTC)

I contend that it *is* unimportant. Once you've demonstrated that an isomorphism is an equivalence relation you've said it all. --prime mover (talk) 06:12, 25 January 2013 (UTC)

But I see your point. I'll get to it as and when, it's just that the thought of doing yet more tedious stuff does not fill me with enthusiasm.

Once I've done that, what do you want me to work on next? --prime mover (talk) 07:08, 25 January 2013 (UTC)

I will do it. I only came here to check on the appropriate manner, and got my answer from Lord_Farin. --Dfeuer (talk) 09:28, 25 January 2013 (UTC)