On Left and Right Categories Zhaohua Luo (7/7/1998) The following note is based on my 1996 paper Unitary Categories. It was inspired by Vaughan Pratt's comment on "co-prefix" (7/6/1988).  Recall that an initial object 0 in a category is called strict if any map to 0 is an isomorphism. The dual notion is a strict terminal object. We say a map f: Y --> X is non-initial if Y is not an initial object. We say a category is trivial if any object is initial.  Definition 1. A left category is a category with a strict initial object such that for any non-initial map f: Y -->X there is a distinct pair of maps u, v: X -->Z such that uf ¹vf.  Remark 2. A mono f: Y --> X is called regular (in the general sense) if it generates the sieve {t: T--> X | ut = vt for any pair of maps u. v: X --> Z with uf = vf}. Note that the kernel of any pair of parallel maps is regular. Now Definition 1(b) can be restated as  (b') A left category is a category with a strict initial object 0 such that 0 --> X is regular for any object X.  Remark 3. Suppose C is a category with a strict initial object 0 and a terminal object 1.  (a) C is left iff 0 --> 1 is regular.  (b) If C is a non-trivial left category then 1 is not a strict terminal object.  A category is called right if its opposite is left.  Theorem 4. The opposite of any non-trivial left category is not left (i.e. any non-trivial right category is not left).  Proof. If C is a non-trivial left category then its strict initial object 0 is a terminal object in Cop, which is not a strict terminal object by Remark 3(b).  Remark 5. It follows from Theorem 4 that any non-trivial category belongs to only one of the following classes:  (a) Left categories;  (b) Right categories;  (c) Other categories.  Example 6. Most of the natural categories (at least from categorical-geometric point of view) are leftt or right categories:  (a) The category of sets is left; the category of complete atomic Boolean algebras is right.  (b) The category of locales is left; the category of frames is right.  (c) The category of coherent spaces (also called spectral spaces, see [Johnstone 1982, Stone Spaces, p.65, 3.4, and p.78]) is left; the category of distributive lattices is right.  (d) The category of Stone spaces is left; the category of Boolean algebras is right.  (e) The category of affine schemes is left; the category of commutative rings (with identity and identity-preserving homomorphisms) is right.  (f) The category of affine varieties is left; the category of finitely generated algebras over an algebraically closed field is right.  Remark 7. From Example 6 one concludes that left categories are geometric in nature, while right categories are algebraic in nature. Remark 8. Recall that a category with finite limits and stable disjoint finite sums is called a lextensive category. Any lextensive category is left, as it has a strict initial object and any map 0 ® 1 is regular (see [Carboni, Lack and Walters 1993], or [Luo 1998, (1.3.1)] ). In fact all the left categories in Example 6 are lextensive (more precisely, they are analytic geometries in the sense of [Luo 1998]). Thus it is natural to call the opposite of a lextensive category a right lextensive category, while a left lextensive category is simply a lextensive category in the usual sense.  Remark 9. All the right categories in Example 6 are regular categories (i.e. regular epis are stable). Thus it is natural to call the opposite of a regular category a left regular category, while a right regular category is simply a regular category in the usual sense.  Remark 10. In algebraic geometry a property and its dual are often referred by the same term. For instance, a flat morphism of affine schemes is really the dual of a flat homomorphism of commutative rings. Another example: integral ring and integral affine scheme. From categorical point of view one may wish to use "coflat" or "cointegral" in the category of affine schemes (or in any analytic category, see [Luo 1998]). The logic behind these practices is Theorem 4 (because all the major categories in algebraic geometry are unitary categories), and the trick of using different names for the objects (or morphisms) in the opposite situation. Thus if we agree to call an object in a left (resp. right) category a left (resp. right) object, then there is no confusion concerning an integral left object and an integral right object. Similarly if we agree to use map in a left category and morphism in a right category to denote arrow, then a flat map is exactly the dual of a flat morphism. This will eliminate the use of "co-" term (such as "coflat", "coreduced", "cointegral", or "cosimple") in categorical geometry.  (to be cont.)  [References][Notations][Home] 