[formal-methods] Injective, Surjective, and Bijective functions

[d p chang]

Mikael Vejdemo-Johansson <mik at stanford.edu> writes:

> On Jun 28, 2009, at 4:35 AM, Crutcher Dunnavant wrote:
>> I find un-cited definitions un-compelling evidence.
>>
>> I'm perfectly willing to listen to argument, and read proofs. But  
>> for appeals to authority, I would like citation.
>>
>> As to the issue at hand, it seems trivial to say that injection  
>> abstracts to monomorphism and surjection abstracts to epimorphism.  
>> Awody's book (Category Theory) makes this claim on page 25. What I  
>> haven't seen anywhere, and I've looked, is a claim that it is  
>> incorrect to say that injection and surjection are synonyms for  
>> their more general concepts.
>>
> Actually, checking that, the idea that monomorphisms generalize  
> injections but that the generalization is strictly larger than the  
> initial concept, does get discussed on
> http://en.wikipedia.org/wiki/Monomorphism
> where among other things, it is stated:
>> It is not true in general, however, that all monomorphisms must be  
>> injective in other categories. For example, in the category Div of  
>> divisible abelian groups and group homomorphisms between them there  
>> are monomorphisms that are not injective: consider the quotient map  
>> q : Q → Q/Z. This is clearly not an injective map; nevertheless, it  
>> is a monomorphism in this category.

as i started poking around for stuff during this discussion i also found
some terminology clarification at

  http://en.wikipedia.org/wiki/Epimorphism#Terminology

>> The companion terms epimorphism and monomorphism were first
>> introduced by Bourbaki. Bourbaki uses epimorphism as shorthand for a
>> surjective function. Early category theorists believed that
>> epimorphisms were the correct analogue of surjections in an arbitrary
>> category, similar to how monomorphisms are very nearly an exact
>> analogue of injections. Unfortunately this is incorrect; strong or
>> regular epimorphisms behave much more closely to surjections than
>> ordinary epimorphisms.
[ ... deletia ... ]
>> It is a common mistake to believe that epimorphisms are either
>> identical to surjections or that they are a better
>> concept. Unfortunately this is rarely the case; epimorphisms can be
>> very mysterious and have unexpected behavior. It is very difficult,
>> for example, to classify all the epimorphisms of rings. In general,
>> epimorphisms are their own unique concept, related to surjections but
>> fundamentally different.

this at least clarifies the origins of the terminology and gives a
little background to the divergence.

\p
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