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.