| Home | References | Dictionary | Library | Links A Note on Reduced Categories Zhaohua Luo (8/1998) (Revised 9/29/98) |
Let A be a category with a strict initial object (recall that an initial object is strict if any map to it is an isomorphism). A map is non-initial if its domain is non-initial.
Definition 1. (a) A pair of parallel maps f, g: X --> Z is disjointed if its kernel is the initial map to X (i.e., any map t: T --> X satisfying the condition tf = tg is initial).
(b) A pair of parallel maps f, g: X --> Z is nilpotent if any map t: T --> X such that (tf, tg) is disjointed is initial.
(c) An object X is reduced if any pair of distinct parallel maps with domain X is not nilpotent.
(d) A is reduced if any object is reduced.
(e) A is reducible if any non-initial object is the codomain of a map with a reduced non-initial domain.
Example 1.1. (a) Consider two homomorphisms u, v: R --> S of commutative rings with the cokernel h: S --> H. Then ker(h) = h-1(0) is a unit (resp. nilpotent) ideal of S iff (u, v) is disjointed (resp. nilpotent) in the opposite of the category CRing of commutative rings.
(b) A ring is reduced (i.e. with no non-zero nilpotent) iff it is a reduced object in CRingop.
(c) The opposite of the category of reduced rings is a reduced category.
Remark 2. Suppose f, g: X --> Z and t: T --> X are maps. Then
(a) If (f, g) is disjointed then (ft, gt) is disjointed.
(b) If (f, g) is nilpotent then (ft, gt) is nilpotent.
(c) A pair of parallel maps f, g: X --> Z is disjointed and nilpotent iff X is initial.
(d) If A is a disjunctable analytic category and t is epic then (ft, gt) is nilpotent implies that (f, g) is nilpotent..
(e) If s: Z --> S is a map and (f, g) is nilpotent then (sf, sg) is nilpotent.
(f) If J is a cogenerator of A then A is reduced iff any pair of distinct parallel maps with codomain J is not nilpotent (this follows from (g)).
Proposition 3. Any quotient of a reduced object is reduced, i.e., if t: T --> X is an epi and T is reduced then X is reduced.)
Proof. Consider a pair of distinct parallel maps f, g: X --> Z. Since t is epic, ft and gt are distinct. Since T is reduced, (ft, gt) is not nilpotent, then (f, g) is not nilpotent by Remark 2(b). Thus X is reduced.
Definition 4. (a) A non-initial object is simple if any non-initial map to it is epic.
(b) A non-initial object is strictly integral if it is a quotient of a simple object.
Proposition 5. (a) Any simple object is reduced.
(b) Any strictly integral object is reduced.
(c) Any quotient of strictly integral object is strictly integral (thus also reduced).
Proof. (a) If X is a simple object then any pair of distinct parallel maps f, g: X --> Z is disjointed, therefore is not nilpotent by Remark 2(c) (note that by definition any simple object is not initial).
(b) follows from Proposition 3 and Definition 4(b).
(c) follows from Definition 4(b).
Example 5.1. A simple (resp. strictly integral) object in CRingop is precisely a field (resp. integral object) in CRing.
Definition 6. A class D of objects of A is uni-dense if any non-initial object is the codomain of a map with a non-initial object in D as domain.
Remark 7. A is reducible iff the class of reduced objects of A is uni-dense.
Theorem 8. Any uni-dense class D of a reduced category A is a set of generators.
Proof. Consider a pair of distinct parallel maps f, g: X --> Z in A. Since X is reduced, (f, g) is not nilpotent. Thus there is a non-initial map w: W --> X such that (fw, gw) is disjointed. Thus fw and gw are distinct. Since D is uni-dense there is a non-initial map t: V --> W with V in D. Then fwt and gwt are distinct. Therefore D is a set of generators.
Proposition 9. Suppose G is a uni-dense object in A.
(a) If A is reduced then the functor homC(G, ~) is faithful (i.e. G is a generator of C).
(b) If G is a generator such that any map G --> G is an isomorphism then A is reduced (Note that in practice G is often the terminal object 1 of C).
Proof. (a) is a special case of Theorem 8.
(b) Consider a pair of distinct parallel maps f, g: Y --> X. Since G is a generator there is a non-initial map t: G --> Y such that (ft, gt) is distinct. We show that (ft, gt) is disjointed. Consider any non-initial map s: S --> G. Since G is uni-dense there is a map w: G --> S. Then tw: G --> G is an isomorphism by assumption, so (ftw, gtw) is disjointed. This shows that (ft, gt) is disjointed, therefore X is reduced.
Example 9.1. (a) The categories of sets, topological spaces, posets are reduced because they satisfy the condition of Proposition 9(b) with G = 1.
(b) In CRingop a ring is simple (resp. strictly integral) iff it is a field (resp. integral domain). The class of fields (resp. integral domains) is uni-dense in Ringop.
(d) Denote by RedCRing the category of reduced rings. Then RedCRingop is a reduced category and
the class of fields (resp. integral domains, reduced rings, resp. reduced local rings) is uni-dense in RedRingop, thus is a set of generators by Theorem 8.
(e) A coherent analytic category is a locally finitely generated category whose subcategory of finitely generated objects is lextensive. In a coherent analytic category the class of simple (resp. integral, resp. reduced) objects is uni-dense. Thus in a reduced coherent analytic category any of these class of objects is a set of generator. This generalizes (d) as RedRingop is a reduced analytic coherent category.
Two maps u: U ® X and v: V ® X are disjoint if 0 is the pullback of (u, v). A map f: Y --> X is called unipotent if any non-initial map to X is not disjoint with f.
A collection of maps S to X is called a unipotent cover if the only map to X that is disjoint with any map in S is an initial map.
Remark 10. Suppose f, g: X --> Z and t: T --> X are maps. Then
(a) (f, g) is nilpotent iff the collection of maps t such that ft = gt is unipotent.
(b) Suppose (f, g) has a kernel k: K --> X. Then (f, g) is nilpotent iff the map k is unipotent.
Proposition 11. Suppose X is a reduced object.
(a) Any proper strong subobject of X is not unipotent.
(b) Any unipotent map to X is epic.
If any pair of parallel maps in A has a kernel then conditions (a) or (b) is also sufficient for that X is reduced.
Proof. (a) Any proper strong subobject is contained in a proper regular subobject U of X. Thus we only need to consider a regular mono u: U --> X, which is a kernel of two distinct parallel maps f, g: X --> Z. Since X is reduced, (f, g) is not nilpotent, so by Remark 10(b) u is not unipotent. If any two parallel maps has a kernel then the above arguments can be reversed, so the condition is also sufficient.
(b) Consider a non-epic map t: T --> X. We can find two distinct parallel maps f, g: X --> Z such that ft = gt. Since X is reduced, (f, g) is not nilpotent. Thus there is a non-initial map w: W --> X such that (fw, gw) is a unit. Clearly u is disjoint with w. Thus t is not unipotent.
Note that (b) implies (a) as any epic map is not proper regular. Thus if A has kernels then condition (b) is also sufficient as (a) is so.
Remark 12. In Chapter 3, Section 3.1 of Categorical Geometry we used the condition of Proposition 11(b) as the definition of a reduced object in an analytic category. Since any analytic category has finite limits (by definition), these two definitions of reduced objects agree.
In the following we assume A has finite limits with a terminal object 1.
A mono u: U --> X is called normal if any of its pullbacks is not proper (i.e. non-isomorphic) unipotent.
Proposition 13. A is reduced iff any strong (or regular) mono is normal.
Proof. If A is reduced then by Proposition 11 any proper strong mono is not unipotent. Since the pullback of any strong mono is strong, this implies that any strong mono is normal. Thus the condition is necessary. It is also sufficient as it implies Proposition 11(a) (as any proper normal mono is not unipotent). This is true also for regular monos as any proper strong mono is contained in a proper regular mono.
Definition 14. (a) A generic mono for A (or any category with finite limits) is a regular mono t: K --> J such that any regular mono in A is an intersection of pullback of t; if furthermore K = 1 is a terminal object then we say that t is a standard mono (the dual notions are generic epi and standard epi).
(b) Suppose t: K --> J is a generic mono. A map X --> J is called a zero map (resp. nilpotent) (with respective to t) if the pullback of t along it is an isomorphism (resp. unipotent).
Proposition 15. Suppose t: K --> J is a generic mono.
(a) An object X is reduced iff any non-zero map from X to J is not nilpotent.
(b) A is reduced iff t is normal.
Proof. (a) follows from Proposition 11.
(b) The condition is sufficient by Proposition 13 as t is a regular mono. Conversely, assume t is normal. Since any regular mono is an intersection of pullbacks of t, and normal monos are closed under pullbacks and intersections, any regular mono is normal. Thus A is reduced by Proposition 13.
Remark 16. Suppose J is a cogenerator of A.
(a) The diagonal map : J --> J J is a generic mono. Thus A is reduced iff is normal.
(b) If t: 1 --> J is a map such that is a pullback of t then t is a standard mono.
Example 16.1. In CRing the zero-homomorphism Z[X] --> Z is a standard epi, where X is a variable over the ring Z of integers. Thus its dual Z --> Z[X] is a standard mono in CRingop. Now Proposition 15(a) states that a ring R is reduced iff it has no non-zero nilpotent element (corresponding to a homomorphism Z[X] --> R).
Example 16.2. (a) In a topos A the true arrow: 1 --> W is a standard mono. Thus A is reduced iff 1 --> W is normal, which is equivalent to that A is boolean (cf. Categorical Geometry, Section 3.7). Home | References | Dictionary | Library | Links
(b) The category of locales is reduced by Remark 16(a) (the topological space with two points and three open subsets is a cogenerator whose diagonal map is normal).