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

[Jason Dusek]

2009/06/27 d p chang <weasel at meer.net>:
> Crutcher Dunnavant <crutcher at gmail.com> writes:
>> 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.
>
> your last sentence has me really confused. for example, i think of
> surjection as (w/o the notation that i'm too lazy to check the emacs
> input mode for):
>
>  f is surjective iff, there exists x in X, for all y in Y, such that
>  f(x) = y
>
>
> are you saying that:
>
>  - we have a context in which surjective means something else?
>
>  - epimorphism isn't exactly surjection?
>
>  - something else that i didn't understand

  An epimorphism is an arrow that -- in any category what so
  ever -- can be cancelled on the right. If `e` is "epic" then
  we have:

    f . e  =  g . e    ----- implies ---->    f = g

  In the category set, "epimorphism" and "surjection" are
  synonymous; in other categories, they are not synonymous.

--
Jason Dusek