Re: connected categories and epimorphisms.

[Robin Cockett <robin@cpsc.ucalgary.ca>]

Of course it depends rather heavily on what you mean by connected:

(1) if you mean that there is a way to get between any two objects via
arrows -- and one is allowed to go backwards along arrows -- then this
is not true.  Any category with products is necessarily connected in
this manner and the category of Sets provides a counter-example.
Any projection p_0: A x 0 -> A where 0 is the empty set and A is
non-empty is non-epic.

(2) if you mean that given any objects A and B there is always an arrow
f: A -> B (differs from (1) in that you are not allowed to go backwards
along arrows) -- that is homsets are non-empty -- then this IS true.
This is because every projection in such a category has a section as the
composite
          <1_A,f>          p_0
       A --------> A x B --------> A

is the identity. This makes the projection a retraction and thus epic.

(2) if you mean (stretching a bit) that every object has a (regular)
epic onto the final object (all objects have global support) then all
you need in addition is that the product functors _ x A preserves these
epics.  This will be the case, for example, if the category is cartesian
closed ... however, such a category better not have an initial object!

-robin

On 16 May, Flavio Leonardo Cavalcanti de Moura wrote:
> Hi,
>
> How can I show that, in a connected category, projections (of the
> product) are epimorphisms?
>
> Thank you,
>
>  Flavio Leonardo.