Let C be a category. If (r_{1} , r_{2}): Y > X is a pair of parallel arrows then we say that r = (r_{1}, r_{2}) is a 2arrow in C; the common domain (resp. codomain) of r_{1} and r_{2} is called the domain (resp. codomain) of r. A 2arrow r = (r_{1}, r_{2}) is uniform if r_{1} = r_{2}. Any ordinary arrow t: Y > X (also called a 1arrow) determines a uniform arrow (t, t): Y > X . It is obvious how to define the compositions of 2arrows: If r = (r_{1}, r_{2}): Y > X and s = (s_{1}, s_{2}): Z > Y are two 2arrows we define rs to be the 2arrow A 2arrow with X as codomain is called a 2element of X. Denote by 1_{X} the set of 2elements of X. Any arrow f: Y > X determines a map 1_{Y} > 1_{X} sending each 2element r of Y to the 2element fr of X. We simply write f for this map. Denote by 0_{X} (or simply 0) the set of uniform 2elements of X. Note that 0_{X} is nonempty because it contains the uniform 2element (1_{X}, 1_{X}). If f: Y > X is an arrow then the subset f^{1}(0) is called the 2kernel of f, denoted by ker(f). Definition 2.1. (a) A set of 2elements
of X is called a 2kernel if
it is the 2kernel of some arrow t: X > Z.
Denote by I(X) the set of ideals of X. Remark 2.2. (a) A set a of 2elements of X is an ideal if and only if there is a collection {h_{i}: X > Z_{i}} of arrows such that (b) An ideal of X is a sieve on X as an object in C^{2}. The collection of ideals is closed under intersection in 1_{X}. Thus I(X) is a complete lattice with = . Any set T of 2elements of X generates an ideal (T) of X, which is the intersection of all the ideals (or 2kernels) containing T. An ideal of an object X is called a 2principal ideal if it is generated by a 2element of X. Remark 2.3. (a) Any ideal is an intersection
of 2kernels.
Example 2.3.1. (a) 0
is the smallest ideal of X.
Remark 2.4. Suppose f: Y
> X is an arrow.
(b) If S is any set of 2elements of X, the image f(S) generates an ideal of Y, denoted by (f(S)). We obtain a mapping f_{*}: I(Y) > I(X) sending each ideal a of Y to the ideal f_{*}(a) = (f(a)). (c) Since f_{*} is the left adjoint of f^{1}, it preserves joins. Thus if {a_{i}} is a set of ideals of X and a_{i} is the ideal generated by a_{i}, then f_{*}( a_{i}) = f_{*}(a_{i}). Remark 2.5. Suppose f: Y > X is an arrow and S is a set of 2elements of Y. Denote by (S) and (f(S)) the ideals generated by S and f(S) respectively. Then f_{*}((S)) = (f((S))) = (f(S)). Remark 2.6. Suppose f: X > Y
is an arrow. Then
Remark 2.7. The following are equivalent
for an arrow f: Y > X:
Next we shows that the notion of unitary arrow introduced in (1.3) can also be defined in terms of ideals (note that here by null or unitary we always mean right null and right unitary): Proposition 2.8. (a) An arrow t:
X > T is null if and only if t_{*}(1)
= 0 (i.e. ker(t) = 1).
Proof. (a) t is null if and only if t(1_{X})
0, i.e. t_{*}(1)
= 0, or equivalently, ker(t)
= 1.
Remark 2.9. (a) If f: Y
> X is not unitary then there is an arrow g: X
> Z such that Z is not null and (gf)_{*}(1)
= 0 (this follows from (2.6.d)
and (2.8.c)).
Proposition 2.10. An object is simple if and only if it is unitary with exactly two ideals. Proof. (a) Note that the condition (1.11.a)
in the definition of a simple object is equivalent to that 0
1 for X. If X is simple
then these two ideals are the only 2kernels of X. Since
any ideal is an intersection of 2kernels, these are the only ideals
of X. Also X is unitary by (1.13).
Corollary 2.11. An object in a unitary
category is simple if and only if it has exactly two ideals. n
