first isomorphism theorem

first isomorphism theorem

[Andrei Prokopiw]

Is there a suitable first isomorphism theorem in category theoretic language? One barrier for me seems to be the correct notion of an image. What seems best right now would be that given f:X->Y, ker(coker f) = coker ( ker f). Here the left hand side would represent im f, and the right hand side X/kerf. If this is the right notion, what are the necessary conditions on the category for it to hold?

-Andrei Prokopiw

Re: first isomorphism theorem

[Toby Bartels]

Andrei Prokopiw wrote:

>Is there a suitable first isomorphism theorem in category theoretic language?
>One barrier for me seems to be the correct notion of an image. What seems best
>right now would be that given f:X->Y, ker(coker f) = coker ( ker f). Here the
>left hand side would represent im f, and the right hand side X/kerf. If this
>is the right notion, what are the necessary conditions on the category for it
>to hold?

I'm not really answering your question, instead addressing the "if".

One of the classic examples of the isomorphism theorems is for rings,
and the category of rings (with unit) has neither kernels nor cokernels.
There are ways around that limitation in this case, such as:
* Don't require rings to have units; or
* Analyse any specific f in a category of X-modules.

But more importantly, I think that much of the point of the theorem
is that the image is a purely set-theoretic notion,
definable without reference to the algebraic properties of f.
In the category of sets, the image is (of course)
not realisable as the kernel of the cokernel,
but it does exist as the equaliser of the cokernel pair.

It seems to me that the essence of the isomorphism theorem
is that the forgetful functor from algebras to sets
not only preserves limits (which we all know)
but also preserves images and coimages (defined as above),
even though it does *not* preserve coequalisers or cokernel pairs.
The classical form of the theorem is simply a reflection
of an inability to speak explicitly about
functors' preserving categorical constructions.


-- Toby Bartels

Re: first isomorphism theorem

[Michael Barr]

Let me just add to this reply (which is certainly one valid answer to
the question) the comment that I sent privately to the questioner that
exactness is sufficient for the FIT to be meaningful.

 On Thu, 19 Jun 2003, Toby Bartels wrote:

> Andrei Prokopiw wrote:
>
> >Is there a suitable first isomorphism theorem in category theoretic language?
> >One barrier for me seems to be the correct notion of an image. What seems best
> >right now would be that given f:X->Y, ker(coker f) = coker ( ker f). Here the
> >left hand side would represent im f, and the right hand side X/kerf. If this
> >is the right notion, what are the necessary conditions on the category for it
> >to hold?
>
> I'm not really answering your question, instead addressing the "if".
>
...