CATS Are primes ever generators?

CATS Are primes ever generators?

[David Espinosa]

We can say that an object P in a category with coproducts is *prime*
if whenever f : P -> A+B, f factors through one of the injections into
A+B.

(1) I didn't find any reference to this (obvious) notion of
    primality in the standard texts.  Does it occur anywhere?

(2) Is there any condition on the category under which the set of
    primes is a generating family?  Since objects are decomposable
    into a "quotient of a coproduct of generators" (Borceux, volume 1,
    page 151), this would give a decomposition into primes.

Thanks,

David

Re: CATS Are primes ever generators?

[Steve Lack]

> X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f
> Date: Fri, 13 Feb 1998 09:12:18 -0800
> From: David Espinosa <espinosa@kestrel.edu>
> Cc: espinosa@kestrel.edu
> Precedence: bulk
> 
> 
> 
> We can say that an object P in a category with coproducts is *prime*
> if whenever f : P -> A+B, f factors through one of the injections into
> A+B.
> 
> (1) I didn't find any reference to this (obvious) notion of
>     primality in the standard texts.  Does it occur anywhere?
> 
> (2) Is there any condition on the category under which the set of
>     primes is a generating family?  Since objects are decomposable
>     into a "quotient of a coproduct of generators" (Borceux, volume 1,
>     page 151), this would give a decomposition into primes.
> 
> Thanks,
> 
> David
> 
> 
> 

Dear David,

One convenient setting for your question is provided by _extensive_
categories (see the paper ``Introduction to extensive and distributive
categories'' by Carboni, Lack, and Walters, appearing in JPAA 1993). A
category E with finite coproducts is said to be extensive if for all 
objects x,y of E, the ``coproduct functor'' E/x x E/y --> E/(x+y) is
an equivalence. For such a category E, an object p is prime in your
sense if and only if it is connected, i.e. if and only if it admits no
proper coproduct decomposition; this in turn is equivalent to the 
representable functor E(p,-):E-->Set preserving coproducts.

An example of an extensive category is given by Fam(C) for C a (small)
category. The objects of Fam(C) are finite families (C_i)_{i\in I}
and an arrow from (C_i)_I to (D_j)_J comprises a function f:I-->J and
a family of arrows C_i-->D_fi in C. Fam(C) is the free category with
finite coproducts on the category C. The connected (=prime) objects
are precisely the singleton families. One can characterize the 
categories of the form Fam(C) as those extensive categories with a 
small set of connected objects such that every object is a finite 
coproduct of connected objects. (It seems possible that you could 
replace ``extensive'' in the last sentence by ``category with finite
coproducts'' provided that you also replace ``connected'' by
``prime and connected'', but I haven't thought about this.)

Best wishes,

Steve.