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.
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: 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 Thus Dir(X) is a lattice with W Ç (U Ú V) = W Ç [(U Ç V) + (U^{c} Ç V) + (U Ç V^{c})] = W Ç U Ç V + W Ç U^{c} Ç V + W Ç U Ç V^{c} = W Ç U Ç V + (W Ç U^{c}) Ç V + W Ç (U Ç V^{c}). = (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 U^{c} 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 (U^{c} Ç V^{c})^{c }= U Ú 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 W_{1} = U_{1}, W_{2} = V_{1} Ç U_{2 }, ..., W_{i} = V_{1} Ç V_{2} ... Ç V_{i-1} Ç U_{i}. Then {W_{i}} is a direct cover of X, with W_{i} Ç W_{j} = 0 for i < j, and X = S W_{i}. Let z: Z ® X be the sum of u_{i}: U_{i} ® X, and let s: X = S 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. n Definition. (a) The Boolean space spec_{A
}X is called the Pierce Spectrum of an object X.
Remark. If U and V are two direct subobjects of X we define U +_{a} V = U Ç V^{c} + V Ç U^{c} , then (Dir(X), +_{a}, Ç, 0, 1) is a ring with U^{2} = 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).
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).
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. |