1. Unitary Categories 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.  (b) An object Z is called (right) null if the identity 1Z: Z --> Z is null.  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.  (b) An arrow with a null domain or null codomain is null by (a).  (c) An object T is null if and only if for any object Z there exists at most one arrow from Z to T.  (d) Any subobject of a null object is null.  (e) Any terminal object is null.  (f) Any subobject of a terminal object is null.  (g) Assume a terminal object exists. Then an object X is null if and only if the unique arrow from X to a terminal object is a monomorphism (i.e. X is a subobject of a terminal object).  (h) Any strict initial object is null.  (i) Any arrow with a strict initial domain 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.  (b) If f: Y --> X is both unitary and null then X is null.  (c) If f: Y --> X is unitary and Y is null then X is null.  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.  (b) If f: Y --> X is both unitary and null then f = 1X f: Y --> X is null by (1.2.a); thus 1X is null as f is unitary.  (c) Suppose f: Y --> X is unitary and Y is null. Then f is null by (1.2.b). Hence X is null by (b).  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 non-terminal 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.  (b) A unitary null object is a strict terminal object.  (c) A terminal object is unitary if and only if it is strict.  (d) If a category has a unitary object then any terminal object is unitary and strict.  (e) If a category has a unitary initial then any object is 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:  (i) tf: X --> T is null;  (ii) T is terminal.  (iii) t is null.  Now (i) implies (iii) implies that f is unitary by definition, and (iii) implies (ii) implies that Y is unitary.  (b) Any arrow started from a unitary null object P is a null arrow, thus with a terminal as codomain. Applying this to the identity arrow of P we see that P is terminal and any arrow with domain P is an isomorphism.  (c) - (e) follow from (a) and (b).  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.  (b) Any null object in a unitary category is terminal and any terminal object in a unitary category is strict.  (c) Any category with a unitary initial is unitary.  (d) Initial objects in a non-trivial unitary category are not strict.  Proof. (a) - (c) are direct consequences of (1.7).  (d) If a unitary category has a strict initial object, then it is also a strict terminal object by (b), which implies that any object is a zero object.  Proposition 1.10. A category is unitary if and only if the following two conditions are satisfied:  (a) Any arrow is unitary.  (b) Any null object is terminal.  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:  (a) There is a pair (r, s): Y --> X with r  s .  (b) Any arrow to a non-terminal object with domain X is monomorphic.  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:  An arrow with domain X is monomorphic if and only if its codomain is non-terminal.  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.  (b) A category such that any object is the codomain of an arrow with a simple domain is unitary.  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:  If g: X --> Z is an arrow such that gr = gs for any (r, s): W --> X with fr = fs, then g factors through f (uniquely).  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.  (a) X is unitary if and only if the unique arrow e: X --> 0 is a regular epimorphism.  (b) Suppose X --> 0 is a regular epimorphism and f: X --> Y is any arrow. Then Y --> 0 is a regular epimorphism.  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.  (b) If X --> 0 is a regular epimorphism then X is unitary by (a); thus Y is unitary by (1.7.b). It follows that Y --> 0 is also regular by (a).  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.  (b) A category with an initial and a terminal object is unitary if and only if the terminal object is strict and the arrow from an initial to a terminal object is a regular epimorphism.  Proof. (a) follows from (1.6.a).  (b) follows from (1.9.c) in view of (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:  (a) Any two objects have sum.  (b) Any arrow has a kernel pair.  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 .  First we assume (a) holds. Let X + W be the sum of X and W with the injections i: X --> X + W and j: W --> X + W. Denote by r': X + W --> U the arrow induced by u: X --> U and r: W --> U . Let s': X + W --> U be the arrow induced by u: X --> U and s: W --> U. Then r = r'j and s = s'j. Now r' and s' may be viewed as arrows from (X + W, i) to (U, u) in X/C. Since t: (U, u) --> (V, v) is a null arrow in X/C, we have tr' = ts'. It follows that tr = trj = tsj =ts. This shows that t is null in C.  Next assume any arrow has a kernel pair. Let (r',s'): Z --> U be the kernel pair of t: U --> V. Denote by i and j the arrows induced by r and s respectively. Using the argument similar to (a) we can show that t is null in C.  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.  (b) An object Z is called leftnull if the identity 1Z: Z --> Z is left null.  (c) An arrow f: Y --> X is called leftunitary if for any arrow t: T --> Y the composition ft is left null implies that t is left null.  (d) An object X is called leftunitary if the domain of any left null arrow with X as codomain is initial.  (e) An object X is called a leftsimple if there is a pair (r, s): X --> Y of arrows with r ¹ s and any arrow to X with a non-initial domain is epimorphic.  Proposition 1.21. The following conditions are equivalent for a category:  (a) The category is trivial.  (b) The category is both left and right unitary with an initial.  (c) The category is both left and right unitary with a terminal object.  (d) The category is right unitary with a strict initial.  (e) The category is left unitary with a strict terminal object.  (f) The category is right unitary with a zero object.  (g) The category is left unitary with a zero object.  Proof. These assertions follow easily from (1.9.b) and (d).  Example 1.21.1. (a) The category of groups is non-trivial with a zero object (the zero group). Thus it is neither left nor right unitary by (1.21). Similarly any non-trivial abelian category is neither left nor right unitary.  (b) The category of commutative rings with arbitrary homomorphism is neither left nor right unitary because it is non-trivial with a zero object (i.e. the zero ring).  Example 1.21.2. There exists non-trivial 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 unit-preserving 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.  (b) Similarly, the categories of reduced rings, integral domains, and algebras over a ring are right unitary.  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).  (b) Similarly, the categories of (bounded) lattices, distributive lattices, and frames are right unitary.  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 non-empty. 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 non-trivial right extensive category is not left extensive by (1.9.e).  (b) A category is left lextensive if it has finite limits and finite sums which are disjoint and universal. Any left lextensive category is left unitary.  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.