connected categories and epimorphisms.

connected categories and epimorphisms.

[Flavio Leonardo Cavalcanti de Moura]

Hi,

How can I show that, in a connected category, projections (of the
product) are epimorphisms?

Thank you,

 Flavio Leonardo.

Re: connected categories and epimorphisms.

[Robin Cockett]

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.

Re: connected categories and epimorphisms.

[Peter Freyd]

The quickest natural example I know of a connected category in which
projections from products needn't be epi is the category of
commutative rings. Well, actually, the opposite category.  The
coproduct of  Z_2  and  Z_3  is the terminal ring. The two
co-projections fail to be monic (the coproduct of a pair of objects in
this category is their tensor product).

Re: connected categories and epimorphisms.

[Nikita Danilov]

Peter Freyd writes:
 > The quickest natural example I know of a connected category in which
 > projections from products needn't be epi is the category of
 > commutative rings. Well, actually, the opposite category.  The
 > coproduct of  Z_2  and  Z_3  is the terminal ring. The two
 > co-projections fail to be monic (the coproduct of a pair of objects in
 > this category is their tensor product).
 >

It is interesting that construction of the coproduct in the category of
"just" rings (associative, unitary, but not necessary commutative) is
not easy to find in the literature.

It seems, however, that it is relatively easy to construct: for given
rings R1 and R2, let M1 and M2 be their multiplicative monoids and M be
coproduct of M1 and M2 in the category of monoids. Form monoid ring of M
over Z. That is, form free abelian group generated by underlying set of
M and extend multiplication from generators by distributivity. Let's
call resulting ring G. Let K be ideal in G generated by all elements of
the form

  (A + B) - A - B,

where both A and B belong to the same ring: either R1 or R2. That is, in
((A + B) - A - B) "+" is addition in R1 or R2, and "-" is subtraction in
G.

Now, quotient G/K is a coproduct of R1 and R2 in the category of rings, which
is easy to prove using universal properties of coproduct M, monoid ring G, and
epicness of the projection G ->> G/K.

Is there more "direct" description of coproduct in the category of rings?

 >
 >

Nikita.