Let C be any category. Definition 1.1. (a) An arrow f:
X > Y is called (right) null
if for any pair of parallel arrows (r, s): Z > X
we have fr = fs.
Recall that a terminal (resp. an initial) object Z in a category is strict if any arrow Z > T (resp. T > Z ) is an isomorphism. Remark 1.2. (a) The composition of two
arrows is null if any one of them is null.
Definition 1.3. An arrow f: Y > X is called (right) unitary if for any arrow t: X > T the composition tf is null implies that t is null. Proposition 1.4. (a) Compositions of unitary
arrows are unitary.
Proof. (a) Suppose f: Y > X and g:
Z > Y are two unitary arrows. Consider the composition fg:
Z > X. Let t: X > T be any arrow such that t(fg)
is null. Then g is unitary implies that tf is null, and f
is unitary implies that t is null. This shows that fg is
unitary.
Definition 1.5. An object X is called (right) unitary if the codomain of any null arrow with domain X is terminal. Remark 1.6. An object X is unitary if and only if for any arrow f: X > Y such that Y is nonterminal there is a pair (r, s): Z > X with fr fs. Proposition 1.7. (a) An arrow with a unitary
domain is unitary; its codomain is also unitary.
Proof. Consider any arrow f: X > Y where X
is unitary. Suppose t: Y > T is an arrow. Then the following
assertions are equivalent:
Definition 1.8. A category is called (right) unitary if any object is unitary. Recall the a zero object in a category is an object which is both an initial and a terminate. We say a category is trivial if any object is a zero object. Proposition 1.9. (a) Any arrow in a unitary
category is unitary.
Proof. (a)  (c) are direct consequences of (1.7).
Proposition 1.10. A category is unitary
if and only if the following two conditions are satisfied:
Proof. One direction follows from (1.9.a) and (b). Suppose the conditions are satisfied. From (a) and (1.4.a) we know that the codomain of any null arrow is null, thus terminal by (b). It follows that any object is unitary. Definition 1.11. An object X in
a category is called (right) simple
if the following two conditions are satisfied:
Proposition 1.12. Suppose X is
an object in a category with a terminate 0. Then X is a simple
object if and only if the following condition is satisfied:
Proof. In this case the condition (1.11.a) is equivalent to that X > 0 is not monomorphic. Proposition 1.13. Any simple object is unitary. Proof. Any null arrow from a simple object is not monomorphic because of (1.11.a). Therefore its codomain must be terminal by (1.11.b). Corollary 1.14. (a) A category with a
simple initial is a unitary category.
Proof. These follow directly from (1.9.c) and (1.7.a). Definition 1.15. An epimorphism f:
X > U is called a regular epimorphism
if the following condition is satisfied:
Remark 1.16. Suppose f is an arrow having a kernel pair. Then it is regular if and only if it is the coequalizer of that kernel pair. The dual notion of a regular epimorphism is a regular monomorphism. Proposition 1.17. Suppose X is
an object in a category with a strict terminate 0.
Proof. (a) First suppose e: X > 0 is regular.
If t: X > Z is a null arrow, then t factors through
e:
X > 0 by (1.15). Since 0 is strict, Z
must be terminate. Thus X is unitary. Conversely suppose X
is unitary. We verify (1.15) for e. If g:
X > Z is an arrow such that gr = gs for any (r, s):
W > X (note that we always have er = es as 0 is terminal),
then g is null. Thus Z is terminal by assumption. It follows
that Z is isomorphic to 0. Thus g factors through
e: X > 0 uniquely. This proves that e is a regular
epimorphism.
Corollary 1.18. (a) A category with a
terminal object is unitary if and only if the terminal object is strict
and any arrow with a terminal codomain is a regular epimorphism.
Proof. (a) follows from (1.6.a).
Proposition 1.19. Suppose C is a unitary category. For
any object X the category X/C (consisting of arrows
with domain X ) is unitary if any of the following two conditions
is satisfied:
Proof. Suppose (U, u) and (V, v)
are two objects in X/C and t: (U, u)
> (V, v) is a null arrow in X/C. We need to
prove that (V, v) is a terminate object in X/C. It
suffices to prove that V is a terminate in C or equivalently,
that t is a null arrow in C as C is unitary. Consider
any pair (r, s): W > U such that tr = ts .
The notions introduced above are of righttype. The duals of these notions will be referred as of lefttype. The direct definitions for these duals are given below: Definition 1.20. (a) An arrow f:
X > Y is called left null
if for any pair of parallel arrows (r, s): Y > Z
we have rf = sf.
Proposition 1.21. The following conditions
are equivalent for a category:
Proof. These assertions follow easily from (1.9.b) and (d). Example 1.21.1. (a) The category of
groups is nontrivial with a zero object (the zero group). Thus it is neither
left nor right unitary by (1.21). Similarly any nontrivial
abelian category is neither left nor right unitary.
Example 1.21.2. There exists nontrivial categories which are both left and right unitary. Consider a group G with more than two elements viewed as a category with just one object G. Then any arrow in the category G is both monomorphic and epimorphic. Since G has more then two elements, (1.11.a) and its dual also hold for G. Thus the object G is both left and right simple. It follows that the category G is both a left and right unitary by (1.13). Example 1.21.3. (a) Consider the category
Ring
of commutative rings with unit and unitpreserving homomorphisms. The ring
Z of integers is an initial object and the zero ring 0 is a strict
terminate. A homomorphism of rings is a regular epimorphism if and only
if it is surjective. Thus Z > 0 is a regular epimorphism. Hence
Ring is a right unitary category by (1.18.b).
Note that in this case the initial Z is not right simple. It is
easy to see that a ring is right simple in Ring if and only if it
is a field.
Example 1.21.4. (a) A poset is called bounded if it has
a top element and a bottom element. Consider the category BPoset
of bounded posets with bounded maps (i.e. maps preserve top and bottom
elements). The poset 2 with exactly two elements is a right simple initial
in BPoset. Thus BPoset is a right unitary category by (1.14.a).
Example 1.21.5. Consider the category Set of sets. A map from a set S to a singleton P is surjective (i.e. epimorphism) if and only if S is nonempty. Since the empty set is an initial, P is left simple by the dual of (1.12). Since P is a terminal object, Set is left unitary by the dual of (1.14.a). Similarly, the categories of finite sets, topological spaces and posets (i.e. partially ordered sets) are left unitary. Example 1.21.6. An elementary topos has a strict initial and a terminate. Since any monomorphism in an elementary topos is regular, the unique arrow from an initial to a terminate is a regular monomorphism. Thus any elementary topos (hence also any Grothendieck topos) is left unitary. Example 1.21.7. (see [Carboni,
Lack and Walters 1993]) (a) A category with finite sums and pullbacks
along their injections is left extensive
if the sums are universal and disjoint. The opposite of a left extensive
category is called a right extensive category. Denote the initial of an
extensive category by 0. Then 0 is a strict initial object.
For any object X the arrow 0 > X is the equalizer of its
cokernel pair X > X + X consisting of injections because the sum
is disjoint. This shows that any left extensive category is left unitary.
Note that any nontrivial right extensive category is not left extensive
by (1.9.e).
Example 1.21.8. The underlying category
of any complete metric site (in the sense of [Luo
1995a]) is left extensive. Therefore it is left unitary by (1.21.7.a).
It follows that the categories of manifolds, schemes, and analytic spaces
are left unitary.
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