Pierce Topologies of Extensive Categories
Zhaohua Luo
1. Introduction.  

An extensive category is a category with finite stable disjoint sums. In this note we show that each extensive category carries a natural subcanonical coherent Grothendieck topology defined by injections of sums. This Grothendieck topology is induced by a strict metric topology (in the sense of [Luo 1995]), which is a functor to the category of  Stone spaces. We call this metric topology the Pierce topology of the category, as it generalizes the classical Pierce spectrums of commutative rings. 
Recall that the Pierce spectrum of a commutative ring R is the spectrum of the Boolean algebra of idempotents of R, which is a Stone space. A theorem of R. S. Pierce states that R can be represented as the ring of global sections of a sheaf of commutative rings on its Pierce spectrum (called the Pierce sheaf or representation), whose stalks are indecomposable rings (with respect to product decomposations). Diers in [Diers 1986] showed that Pierce's theorem can be extended to any object in a locally finitely presentable category such that the opposite of the subcategory of finitely presentable objects is lextensive (called a locally indecomposable category). We shall see that a weak form of Pierce representation exists for any object in an extensive category. 

2. Boolean Algebras 

We recall some basic facts about a Boolean algebra. A Boolean algebra is a distributive lattice (with 0 and 1) such that any element a has a complement Øa (i.e. a Ù Øa = 0 and a Ú Øa = 1). Since the complement operation Ø is uniquely determined by the operations Ù and Ú, any lattice homomorphism (i.e. a function preserving Ù and Ú) between Boolean algebras is a Boolean algebra homomorphism (commutes with Ø). If A is a Boolean algebra then the set spec A of prime filters of A is naturally a compact Hausdorff space with clopen subsets as open base. Such a space is called a Stone (or Boolean) space. Conversely, the lattice of clopen subsets of a Stone space is a Boolean algebra. We obtain an equivalence spec: Boolop ---> Stone from the opposite of the category Bool of Boolean algebras to the category Stone of Stone spaces. Note that Boolop is a lextensive category with epi-reg-mono factorizations and the functor spec is simply the analytic topology of Boolop (in the sense of [Luo 1998 Categorical Geometry]). 


3. The Boolean algebra of direct subobjects 

Consider a category with an initial object 0 and finite sums. Two maps u: U ® X and v: V ® X are disjoint if 0 is the pullback of (u, v). Suppose X + Y is the sum of two objects with the injections x: X ® X + Y and y: Y ® X + Y. Then the sum X + Y is disjoint if the injections x and y are disjoint and monic. The sum X + Y is stable if for any map f: Z ® X + Y, the pullbacks ZX ® Z and ZY ® Z of x and y along f exist, and the induced map ZX + ZY ® Z is an isomorphism. A category with finite stable disjoint sums is an called an extensive category; a lextensive category is an extensive category with finite limits (see Carboni, Lack and Walters [1993]). 

If an object X = U + V is the sum of two objects U and V then we say that U together with the injection U --> X is a direct subobject of X + Y; the injection U --> X is called a direct mono; V is called the complement of U, denoted by Uc. Denote by Dir(X) the poset of direct subobjects of X

Theorem. Dir(X) is a Boolean algebra. 

Proof. (a) First we show that Dir(X) is a lattice. Suppose U and V are two direct subobjects of an object X. Since finite sums are stable, we have 

X = (U + Uc) Ç (V + Vc) = (U Ç V) + (Uc Ç V) + (U Ç Vc) + (Uc Ç Vc). 
(U Ç V) + (Uc Ç V) + (U Ç Vc)
are direct objects and 
(U Ç V) + (Uc Ç V) + (U Ç Vc) = (Uc Ç Vc)c.
(U Ç V) + (Uc Ç V) + (U Ç Vc
is contained in U Ú V. Since (U Ç V) + (U Ç Vc) = U  and  (U Ç V) + (Uc Ç V) = V, it also contains U and V. Thus 
(U Ç V) + (Uc Ç V) + (U Ç Vc) = U Ú V.

Thus Dir(X) is a lattice with 

U Ç V = U Ù V,
(U Ç V) + (Uc Ç V) + (U Ç Vc) = U Ú V. 
(b) Next we show that Dir(X) is distributive. If W is another direct subobject of X, then 
X = W Ç Uc + W Ç U + Wc 
implies that 
(W Ç U)c = W Ç Uc + Wc
(W Ç U)c Ç W = [(W Ç Uc) + Wc] Ç W = W Ç UcÇ W + Wc Ç W = W Ç Uc.
(W Ç V)c Ç W = W Ç Vc.
We have 
    W Ç (U Ú V
= W Ç [(U Ç V) + (Uc Ç V) + (U Ç Vc)]  
= W Ç U Ç V + W Ç Uc Ç V + W Ç U Ç Vc  
= W Ç U Ç V + (W Ç Uc) Ç V + W Ç (U Ç Vc).  
= (W Ç U) Ç (W Ç V) + ((W Ç U)c Ç W) Ç V + U Ç (W Ç (W Ç V)c)  
= (W Ç U) Ç (W Ç V) + (W Ç U)c Ç (W Ç V) + (W Ç U) Ç (W Ç V)c  
= (W Ç U) Ú (W Ç V). 
This shows that Dir(X) is a distributive lattice. Clearly Uc is the complement ØU of U in the lattice Dir(X). Thus Dir(X) is a Boolean algebra. (cf. [Luo 1998 Categorical Geometry, Section 4.3]) n 

4. Pierce Topologies 

Since finite sums are stable, if f: Y --> X is a map, the pullback of a direct mono is a direct mono, we obtain a function Dir(f):  Dir(X) --> Dir(Y), which preserves Ç and Ø, thus also Ú as (Uc Ç Vc)c U Ú V. So Dir(f) is a homomorphism of Boolean algebras. We obtain a functor DirA: A --> Boolop from A to the opposite of the category Bool of Boolean algebras. 

Denote by specA the composite of DirA: A --> Boolop with spec: Boolop --> Stone

Theorem. specA: A --> Stone is a strict metric topology. 

Proof. Clearly specA is a metric topology with direct  monos as effective open maps. Since Stone topology is compact, any open effective cover of an object X has a finite subcover. To see that specA is strict it suffices to consider a finite open effective cover {Ui}: i = 1, ..., n of X consisting of direct subobjects (i.e. the join of {Ui} is X). Write Vi for the complement (Ui)c of each Ui. Let W1 = U1, W2 = V1 Ç U2 , ..., Wi = V1 Ç V2 ... Ç Vi-1 Ç Ui. Then {Wi} is a direct cover of X, with Wi Ç Wj = 0 for i < j, and X = S Wi. Let z: Z ® X be the sum of ui: Ui ® X, and let s: X = S Wi ® Z be the map induced by the inclusion Wi ® Ui. Then z°s is the identity of X. Thus z is a retraction, hence a regular epi. This shows that {Ui} is a strict direct cover. Thus the specA is strict. n 

Definition. (a) The Boolean space specA X is called the Pierce Spectrum of an object X
(b) The strict metric topology specA is called the Pierce Topology of A

Remark. If U and V are two direct subobjects of X we define U +a V = U Ç Vc + V Ç Uc , then (Dir(X), +a, Ç, 0, 1) is a ring with U2 = U, thus is a Boolean ring. The spectrum of prime ideals of this Boolean ring is isomorphic to the spectrum specA X

Remark. If A is a lextensive category, specA can be defined via the unique functor F: Finite --> A from the category Finite of finite sets to A, which preserves finite limits and finite sums (sending each finite set n to the n-sum of the terminal object 1). 
(a) If C is a small category denote by Pro-C the opposite category of functors from C to Set  preserving finite-limits. The dual of a theorem of Diers [Diers 1983] states that Pro-C is a lextensive category, which is called a coherent analytic category. For instance, Pro-Finite is a coherent analytic category which is equivalent to Boolop, thus Pro-Finite may be identified with the category Stone of Stone spaces. 
(b) Now consider the functor F: Finite --> A. If X is any object of A, the presentable functor homA(X, ~) preserves limits, so the composite F.homA(X, ~) is in Pro-Finite = Stone. Thus we obtain a functor from  A to Stone, which is precisely specA

5. Pierce representations.  

Suppose A is an extensive category with colimits. Consider an object X. For any open subset U of specA X let PX(U) be the colimits of the system of direct subobjects Ui whose spectrum is contained in U with  inclusions. Since specA is strict , the function PX sending each open subset U to the object PX(U) is a cosheaf on the Stone space specA X with value in the category A

Definition. PX is called the Pierce representation of X

Remark. Consider a coherent analytic category A (which is a locally finitely copresentable category whose subcategory of finitely copresentable objects is lextensive subcategory). The intersection of direct subobjects in a prime filter p of an object X is an indecomposable component of X, which is the stalk of the cosheaf PX at p. Thus specA X may be viewed as the space of indecomposable components with the open base defined by direct subobjects (cf. [Diers 1986] or [Luo 1998 Categorical Geometry, Section 4.3], and PX is a cosheaf with indecomposable objects as stalks. Note that this does not hold for a general extensive category. 

6. Examples 

Example 1. The opposite Ringop of the category Ring of commutative rings with unit is an extensive category. Let R be a commutative ring. The Boolean algebra Dir(R) is isomorphic to the Boolean algebra E(R) of idempotents of R because each idempotent e determine a  product decomposition R = R/(e) × R/(1 - e). Write [e] for R/(e). 
(a) The complement of [e] is [1 - e]. 
(b) If f is an idempotent and ef = 0 then [e] + [f] = [e + f]. 
(c) If e' is an idempotent then [e] Ù [e'] = [e] Ç [e'] = [ee']. 
(d) According to the proof of Theorem 3 and (b) we have (note that e(1 - e) = e'(1 - e') = 0): 

[e] Ú [e'] = [ee'] + [(1-e)e'] + [e(1-e')] = [ee' + (1-e)e' + e(1-e')] = [ee' + e' -ee' + e - ee'] = [e  + e' -ee'].
The space spec E(R) is called the Pierce spectrum of (cf. [Jonhstone 1982, p.181 - 187]). Thus the Boolean space specAR is homeomorphic to the Pierce spectrum spec E(R), and the cosheaf PR is precisely the classical Pierce sheaf of the ring R

Remark. In [Luo 1998 Categorical Geometry, Section 2.6] we introduced the notion of the extensive topology of an extensive category, which is also determined by the divisor of direct monos. The main difference between  the extensive topology and the Pierce topology is that any open cover for a Pierce topology (viewed as a cover for the corresponding Grothendieck topology) is always coherent (i.e., has a finite subcover) and subcanonical (i.e. any representable presheaf is a sheaf), while this is not true for an arbitrary extensive topology. In general the Pierce topology may be viewed as the " compactification " of the extensive topology of an extensive category. If the extensive topology is coherent (i.e. any open cover has a finite subcover) then these two topologies coincides. 

Example 2. The extensive topology of any  coherent analytic category is Stone, thus coherent. So the Pierce topology and extensive topology of a coherent analytic category are the same. 

Example 3. Consider a set X as an object in the category Set of sets. Any subset is a direct subset of X. Thus Dir(X) is the full power-set PX of X. The extensive topology of X is simply the discrete space X. On the other hand, the Pierce spectrum is given by the locale Idl(PX) of ideals PX of with the prime filters of PX as points, which is the Stone-Cech compactification of (cf. [Jonhstone 1982, p.93]). The two topologies are different.