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 a (i.e. a  a = 0 and a  a = 1). Since the complement operation  is uniquely determined by the operations  and , any lattice homomorphism (i.e. a function preserving  and ) between Boolean algebras is a Boolean algebra homomorphism (commutes with ). 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: Bool^op ---> Stone from the opposite of the category Bool of Boolean algebras to the category Stone of Stone spaces. Note that Bool^op is a lextensive category with epi-reg-mono factorizations and the functor spec is simply the analytic topology of Bool^op (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 Z_X --> Z and Z_Y --> Z of x and y along f exist, and the induced map Z_X + Z_Y --> 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 U^c. 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 + U^c)  (V + V^c) = (U  V) + (U^c  V) + (U  V^c) + (U^c  V^c). 

U  V 

(U  V) + (U^c  V) + (U  V^c)

(U  V) + (U^c  V) + (U  V^c) = (U^c  V^c)^c.

(U  V) + (U^c  V) + (U  V^c) 

U  V.

U  V

U  V^c

U  V

U^c  V

(U  V) + (U^c  V) + (U  V^c) = U  V.

Thus Dir(X) is a lattice with 

U  V = U  V,

(U  V) + (U^c  V) + (U  V^c) = U  V. 

X = W  U^c + W  U + W^c 

(W  U)^c = W  U^c + W^c. 

(W  U)^c  W = [(W  U^c) + W^c]  W = W  U^c W + W^c  W = W  U^c.

(W  V)^c  W = W  V^c.

    W 

U  V

= W 

U  V

U^c  V

U  V^c

= W  U  V + W  U^c  V + W  U  V^c 

= W  U  V +

W  U^c

  V + W 

U  V^c

W  U

 

W  V

W  U

^c  W

  V + U 

W 

W  V

W  U

 

W  V

W  U

^c 

W  V

W  U

 

W  V

W  U

 

W  V

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  and , thus also  as (U^c  V^c)^c =  U  V. So Dir(f) is a homomorphism of Boolean algebras. We obtain a functor Dir_A: A --> Bool^op from A to the opposite of the category Bool of Boolean algebras. 

Denote by spec_A the composite of Dir_A: A --> Bool^op with spec: Bool^op --> Stone. 

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

Proof. Clearly spec_A 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 spec_A is strict it suffices to consider a finite open effective cover {U_i}: i = 1, ..., n of X consisting of direct subobjects (i.e. the join of {U_i} is X). Write V_i for the complement (U_i)^c of each U_i. Let W₁ = U₁, W₂ = V₁  U₂ , ..., W_i = V₁  V₂ ...  V_i-1  U_i. Then {W_i} is a direct cover of X, with W_i  W_j = 0 for i < j, and X =  W_i. Let z: Z --> X be the sum of u_i: U_i --> X, and let s: X =  W_i --> Z be the map induced by the inclusion W_i --> U_i. Then z_°s is the identity of X. Thus z is a retraction, hence a regular epi. This shows that {U_i} is a strict direct cover. Thus the spec_A is strict. 

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

Remark. If U and V are two direct subobjects of X we define U +_a V = U  V^c + V  U^c , then (Dir(X), +_a, , 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 spec_A X. 

Remark. If A is a lextensive category, spec_A 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 Bool^op, 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 hom_A(X, ~) preserves limits, so the composite F.hom_A(X, ~) is in Pro-Finite = Stone. Thus we obtain a functor from  A to Stone, which is precisely spec_A. 
 

5. Pierce representations.  

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

Definition. P_X 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 P_X at p. Thus spec_A 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 P_X 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 Ring^op 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]  [e'] = [e]  [e'] = [ee']. 
(d) According to the proof of Theorem 3 and (b) we have (note that e(1 - e) = e'(1 - e') = 0): 

[e]  [e'] = [ee'] + [(1-e)e'] + [e(1-e')] = [ee' + (1-e)e' + e(1-e')] = [ee' + e' -ee' + e - ee'] = [e  + e' -ee'].

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 X  (cf. [Jonhstone 1982, p.93]). The two topologies are different.