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

[Mikael Vejdemo-Johansson]

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On Jun 26, 2009, at 8:07 AM, Jason Dusek wrote:
>  The notion of cancellable on the left or right are what define
>  monomorphisms and epimorphisms.
>
> .  Cancellable on the left is what makes a monomorphism.
>
> .  Cancellable on the right is what makes an epimorphism.
>
>  The notion of injection and surjection apply specifically to
>  sets; an injection is a monomorphism (or is monic) while a
>  surjection is an epimorphism (or is epic).
>

Furthermore, for categories that have a forgetful functor to Set, it  
is an interesting property of the category - that may or may not hold  
- - that epi's are structurepreserving surjections and mono's are  
structurepreserving injections.

One instance where it doesn't: the inclusion map of monoids N -> Z is  
an epi in the category of monoids. (caveat: I haven't actually worked  
this statement out myself; but Barr & Wells claim it, so I go on  
that :-)


Mikael Vejdemo-Johansson, Dr.rer.nat
Postdoctoral researcher
mik at math.stanford.edu






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