- Email: [email protected]
The gleason cover of a topos, I
Journal
of Pure and Applied
;z North-Holland
Publishing
THE GLEASON
Algebra
19 (1980)
171-192
Company
COVER
OF A TOPOS,
I
P.-l-. JOHNSTONE Deparrmenr
of Pure .Marhetnarics.
Universir,v
of Cambridge,
England
0. Introduction Despite its title, this paper is the third in a series (the first two being [ 131 and [ 141) in which we investigate the consequences within topos theory of the logical principle known as De Morgan’s law:
In [13], it was observed that the topos Shv(X) of sheaves on a topological space X satisfies De Morgan’s law iff X is extremally disconnected, and it was claimed that most of the occurrences of extremally disconnected spaces in general topology and functional analysis could in fact be explained as appeals to De Morgan’s law in the internal logic of Shv(X). In this and a subsequent paper [17], we set out to justify this claim in the case of a particular topological construction involving extremally disconnected spaces: the Gleason cover. In [7], A.IM. Gleason 0.1. Theorem.
proved
the following
two results:
spaces, the projective objects maps) are precisely the extremally dis-
In the category of compact Hausdorff
(with respect to surjective connected spaces. q
continuous
0.2. Theorem. For any compact Hausdorff surjection e : yX+X, where yX is compact, connected, which is “minimal” in the sense that surjectively through e. Moreover, this property homeomorphism in the category of spaces over
space X, there is a continuous Hausdorff and extremally disevery other such surjection factors characterizes yX up to (unique) X. C
The space yX is called the Gleason cover of X. Subsequent authors [23, 5,9, 25, 19, 1, 31 extended both Gleason’s results, ultimately to the category of all Twspaces, but with the proviso that one must replace “surjective continuous map” by “surjective proper map” in both theorems when dealing with non-compact spaces. The object of the present paper is to provide a topos-theoretic analogue of Theorem 0.2: we shall construct for any topos 4 a surjective geometric morphism 171
172
P. T. Johnstone
e : y&*6, where yb satisfies De Morgan’s law, which is in some sense the “best approximation” to 8’ by a De Morgan topos. We shall also show that our construction coincides with the topological one in the case when W= Shv(X), at least for a regular space X, and investigate how it may be described in the case when 6 is a topos of presheaves. Finally, we shall investigate the connection between y& and the order-completion *lR of the Dedekind real numbers in b. The sequel [17] will investigate the analogue of Theorem 0.1: the extent to which De Morgan’s law is equivalent to projectivity for toposes.
1. The Gleason cover
In this section, we shall construct the Gleason cover y&’ of an arbitrary topos 8, and prove its basic properties. In doing this, we shall make use of certain properties of internal sites in a topos (in fact we shall consider only two particular cases,, that of an internal distributive lattice with its finite cover topology, and that of an internal locale (= complete Heyting algebra) with its canonical topology). These properties are simply the “internalizations” of various results which are well-known for sites in the topos of sets [8,11]; in certain cases, their proofs in a general topos involve mild technical complications, and we shall not give them here-the interested reader will find them in [16]. We recall [13, Theorem 1] that a topos R satisfies De Morgan’s law iff the internal Boolean algebra Q _ -, coincides with 2. In general, the Stone Representation Theorem [24] tells us that an arbitrary Boolean algebra may be represented as the global sections of the sheaf 2 over a suitable space. We therefore define the Gleason cover of d to be the internal “Stone space” of Q,,-except that, since Q-,, may fail to have any prime ideals [14], we must consider it as a locale rather than a space. Formally, we have: Definition. The Gleason cover yb of a topos ti is defined to be the topos of (Jvalued) sheaves for the finite cover topology on the internal Boolean algebra Q 7 ?, or equivalently the topos L[Idl(Q _ -)I of canonical sheaves on the internal locale of ideals of S27 -. (For the equivalence, see [ 16, Lemma 4.61.) We must first show that yB’ actually satisfies De Morgan’s law. In fact it suffices to prove that the locale Idl(Q,,) is an internal Stone algebra in 6, since it then follows easily that the same is true for the subobject classifier in y8 [16, Corollary 2.21. This follows from the internalization of a well-known result on Stone spaces: 1.1. Lemma. Let B be an internal Boolean algebra in a topos. Then the locale Idl(B) of ideals of B is a Stone algebra iff B is internally complete. Proof.
If I is an ideal of B, then the negation of I in Idl(B) is easily seen to be the
The Gleason cover of a
173
ropes.1
subobject J= (b E B / (Vi E f)(iAb = 0)} )-+B. Now if B is complete (and hence a locale), it is easy to see that J is closed under arbitrary joins, and so J=iseg(V@)); i.e. every 7 -t-closed ideal in B is principal and therefore complemented in Idl(B). So Idl(B) is a Stone algebra. Conversely if Idl(B) is a Stone algebra, then the principal ideal map lseg : B-+Idl(B) identifies B with the sublocale (Idl(B)) T7 of Idl(B), so B is complete. 1.2. Corollary. We global gated lattice
For any topos 8, yG satisfies De Morgan’s law.
0
0
have a canonical geometric morphism e : yG+G, whose direct image is the section functor for sheaves over Idl(Q--) (or over Q --). In [ 161, we investithe way in which surjectivity of this morphism is related to properties of the Q,,; in particular, we showed
1.3. Lemma. For a distributive lattice A in 6, the following are equivalent: (i) f[Idl(A)] -+ C; is surjective. (ii) The map ,I : Q-Idl(A) defined by A(p) = {a E A j (a = O)Vp} is a monomorphism. (iii) The map 2-A Proof.
defined by the global elements 0 and 1 is a monomorphism.
Combine Lemmas 2.8 and 4.4 of [16].
1.4. Corollary.
0
The canonical map e : yA + (! is surjective.
Cl
If t” satisfies De Morgan’s law, then Q -- E 2 and so Idl(SZ--) z Idl(2) 3 f2. (The map A of Lemma 1.3(ii) provides the latter isomorphism.) But Q is a terminal object in the category of internal locales in ci-[ 16, Lemma 2.41; so r; [Q] = 6, and we deduce 1.5. Corollary. Proof.
(-)
e : y6 --* 6 is an equivalence iff 6 satisfies De Morgan’s law.
follows from Corollary
1.2, (=) from the remarks above.
C
In [14], we observed that R,, has a unique proper ideal, namely false : 1 h n 7-7; so if Q : Idl(Q,7)+Q is the classifying map of the maximal element -a,TT : 1 * Idl(Q,,), then the diagram I
cl fdse
&/se) I Idl(Q,,)
I A52
171
P. T. Johnsrone
is a pullback-i.e. Idl(SZ,,) is a minimal locale in the 2.91. It follows that the surjection y i + i: is a minimal proper closed subtopos of y’_’ which maps surjectively being coherent over R, would map surjectively to t: iff also follows that there is a commutative diagram
terminology of [16, Lemma in the sense that there is no to J-for such a subtopos, its image were dense in 8. It
[16, Lemma 2.10]-indeed, this is obvious from the proof of Lemma 1.1. Translating this diagram of internal locales in A into a diagram of localic 6-toposes, we deduce 1.6. Lemma.
There is a commutative diagram sh,,(yB) I
A
sh,,(T’) I
in Top, where the vertical arrows are the canonical inclusions. 1.7. Corollary.
‘1
y*f is Boolean iff d is.
Proof. If rY is Boolean, then it satisfies De Morgan’s law and so y< is Boolean by Corollary 1.5. Conversely if yc! is Boolean, then the inclusion sh,,(G)*& is equivalent to the composite sh,,(yt) A Yd -A-..+ 6 and therefore surjective; so i is Boolean. 0
The next result characterizes y tl amongst localic C-toposes by a list of properties similar to those which characterize the Gleason cover of a space X amongst spaces over X. Let A be an internal locale in a topos h which is compact, regular, minimal and extremally disconnected (=a Stone algebra). Then A is isomorphic to Idl(R --,). 1.8. Proposition.
Proof. Since A is regular, it is semi-regular [15, Lemma 4.51, i.e. every element is a join of TY-stable elements. But since A is extremally disconnected, its TY-stable elements are complemented; and since A is compact, its complemented elements are finite ( = intranscessible in the terminology of [ 181). So A is coherent, i.e. A = Idl(B)
The Gleason cover of a
lopes, I
175
where B ti A is the sublattice of complemented elements. But by extremal discon0 nectedness and minimality we have B = A TT z f? --. Suppose we have a geometric morphism h : yC+yJ over 6. By the equivalence of [16, Theorem 2.71, h is uniquely isomorphic to the morphism induced by a locale map cp : Idl(Q,,)-, Idl(Q,,) in f; and since rp* preserves complemented elements, it must restrict to a Boolean algebra homomorphism rp’ : Sz__ -+Q -,-. Now @ must preserve the elements true andfalse; but since (true, false) : 2 * Q YT is l-r-dense and S2-- is a -~-sheaf, it follows that p’ must be the identity. Hence also cp is the identity, and h is (uniquely) isomorphic to the identity. So we have proved: 1.9. Lemma. The category Xop/& (~6, y(9) is equivalent to the trivial category with one object and one morphism. (In the terminology of [IO], Gleason covers are rigidly attached.) q All the above considerations follow directly from the definition of y:, without any reference to the particular nature of i. However, in order to identify ~(5 in particular cases, it will be convenient to have a means of converting a site of definition for /i into one for yti. We close this section by providing such a construction. Suppose (C,J) is a site (i.e. a small category with a Grothendieck topology), and suppose we have identified the sheaf a77 in the topos Shv(C,J). We define a new site (yC, yJ) as follows: the objects of yC are pairs (c, R) where c is an object of C and R E Q,,(c), and the morphisms (c’,R’)+(c,R) in yC are morphisms a : c’-c in C such that R’c a*(R). We write p : yC-C for the obvious projection functor, whose fibre over lc is the poset Q --(c). We define yJ to be the smallest topology on yC for which the following sieves are covering: (a) For every (c,R) E ob yC and every .SEJ(C), the sieve generated by {a : (c’,a*R)-(c,R)I(a : c’dc)~S}. (b) For every (c,R) E ob yC and every finite family {RI, . . . , R,) 2 Q,,(c) with joinR, the sievegenerated by {lc: (c,R;)+(c,R)ji=l,...,n}. The functor p : yC-+C mentioned above is not continuous (i.e. does not preserve covers), since the object (c,OJ of yC (where 0, is the least element of Q,,(C)) is covered by the empty sieve even if c is not. However, the right adjoint q of p, which sends c to (c,MJ where M, is the largest element of Q,,(C), does preserve covers; and so it defines a morphism of sites in the sense of [8, IV 4.9.11, i.e. it induces a geometric morphism Shv(yC, yJ)+Shv(C,
J)
whose direct image is given by restriction along q.
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P. T. Johnsrone
1.10. Theorem. With the notation introduced above, Shv(yC, yJ) is equivalent (as a topos over Shv(C, J)) to y(Shv(C, J)).
Let F be a presheaf on yC; then we know that F is a sheaf for the topology yJ iff it satisfies the sheaf axiom for the particular coverings described in (a) and (b) above. So suppose this is true, and define a presheaf G on C by Proof.
G(c)=
ll F(c,R) REfZ__(C)
and if a : c’+c and XE F(c, R) c G(c), then G(a)(x) = F(a)(x) E F(c’, a*R) c G(f) where on the right-hand side we regard a as a morphism (c’, a*R)d(c, R) in yC. First we claim that G is a sheaf for the topology J. Suppose given SEJ(C), and a compatible family of elements (x,~G(c3I(a:
c’+c)c.S);
then writing Ra for the unique element of Q,,(c’) such that X~E F(c’, Ra), it is clear that the elements (Ra j a E S) are compatible in the sheaf sZTT, and so define a unique R E Q,,(C) such that a*R = R, for all a. Since F satisfies the sheaf axiom for covers of type (a), it is now clear that there is a unique XE F(c, R) c G(c) with G(a)(x) =xa for all a. Next we observe that there is a map 6 : G-Q7 T in Shv(C, J) which sends x E G(c) to the unique R such that XE F(c, R). Moreover, we have a map j? : (Q-,7)1 xo,, G+G (where (a,,)~
denotes the order-relation
on Q,,)
defined by
P((R’, R), x) = F( 1c)(x) where R’ c R in Q,,(c), x~F(c, R) c G(c), and we regard lc as a morphism (c,R’)+(c,R) in yC. It is straightforward to verify that this map makes G into an internal presheaf on R TT in Shv(C, J). Moreover, since F satisfies the sheaf axiom for covers of type (b), F(c,Oc) = 1 for every c, and the diagram F(c, R IV&) -
I
FCC, R2)
-
F(c, RI)
I
F(c, R dR2)
is a pullback for every pair (RI, R2); so it follows from [16, Lemma 4.71 that G is a sheaf for the (internal) finite cover topology on J2 __. Conversely, suppose given such an internal sheaf (G, S,B). We define a presheaf F on yC as follows:
The Gleason cover o_fa ropes, I
and if (r : (c’,R?+(c,R) F(a)(x)
177
in yC, then
= B((R’,
ar*R),G(a)(x))
for XE F(c, R). Reversing the above arguments, it is easy to see that the two sheaf conditions (external and internal) on G imply that F satisfies the sheaf axiom for covers of types (a) and (b), and is thus a yJ-sheaf. It is clear that the two constructions given above are inverse up to natural isomorphism, and so Shv(yC, yJ) = y(Shv(C, J)). The fact that this is an equivalence of toposes over Shv(C, J) is easily seen by comparing the two direct image functors; for taking the sections of G over the maximal element M : l-Q,_ has the same 0 effect as taking the restriction of F along the functor 4 : C--yC. Note that if the category C has finite limits, then so does yC. Also, if every covering in J has a finite refinement, then the same is true of the covers which generate yJ, and hence of every covering in yJ. So we deduce:
If 6 is a coherent (Grothendieck)
1.11 Corollary.
topos, so is y”.
III
2. The spectrum of the real numbers In [13], we observed that De Morgan’s law in a topos 4 is equivalent to a number of interesting statements about the real numbers in I! : in particular, the statement that the object II? of Dedekind reals coincides with its order-completion *IR (which we now call the MacNeille reals). In this section we investigate the relation between the Gleason cover of a topos R and the object *YZin 6 ; we shall show that each may be defined in terms of the other. We begin with something [13, Proposition 1.31:
which appeared
as a throwaway
remark
in the proof of
In any topos, the Boolean algebra of idempotents of *R is isomorphic
2.1. Lemma.
to Q,,. Proof.
Let p be a variable L(p)=
of type R-,-,
{qEQl(q
and define l)AP)),
~(P)={qE~l(q>l)V((q>O)A~P)}. In the proof that the pair (L(p), U(p)) defines step comes in showing that
a MacNeille
real, the only nontrivial
(9 < @A l(Q E UP)) -) (9’ E U(P)) (and its dual),
where there is an apparent
use of De Morgan’s
law in proceeding
178
P. T. Johnsrone
l)V-p.
from -((q-C I)Ap) to (q? trichotomy, we have
But
since
the
order-relation
on
Q satisfies
(9 < l)v(q 2 l), so this deduction J-(P) = (UP),
is permissible.
Hence
we have a map f : R-,- - *R, defined
by
U(P)>. Now P-+(~(P) = 1)
and
~P+(J(P)
= O),
so that
l%-(P)2
-f(P)
= 0).
But *lR is -l-separated (by the same proof as for R, see [ll, 6.63]), so we deduce that the image off consists of idempotents. Conversely, since *IF?is a residue field (cf. [I 1, 6.65]), any idempotent x E *Z satisfies 1(x = l)--+ 1(x invertible in *?) +(x=0), and dually
1(x=0)-+(x
= 1). So it follows
easily that
x =f(Ox = 1II), and
hence
*iR.
that f maps
R -,-
isomorphically
onto
the object
of idempotents
of
0
2.2. Corollary. The Gleason cover yli of I< is (rhe underlying topos of) the Pierce representation [22] of the ringed topos (i, *R).. Classically, the Pierce representation of a ring lives in the topos of sheaves on the Stone space of the Boolean algebra of idempotents of the ring. When vve construct this representation for a ring in a general topos (cf. [12, pp. 257-g]), it is clear that we must replace this Stone space by the locale of ideals of the Boolean Proof.
algebra.
0
What is the sheaf of rings over Idl(R-,) which provides the Pierce representation of *R? To answer this question, note that Dedekind real numbers in y? correspond topos for the to geometric morphisms over G from y6 to the classifying or equivalently to locale morphisms propositional theory of real numbers, Idl(Qn,,)+L(R) in A, where L(R) is the locale of “formal real numbers” in 6 [6]. We recall that this locale is generated by the “formal rational intervals” (q,r) (q E Q u { - co}, r E Q u (a~}), subject to the relations (i) (-co,oo)=l, (ii) (q, r) = 0 if q 2 r, (iii)
(qh rdNq2,rz)
= (max(sl,q2),min(ri,rz)),
(iv) (st,ri)V(q2,rz)=(ql,r2) 09
if 91592
(q,r)=V{(q',r?Iq
The Gleason cover of u ropes. I
179
2.3. Proposition. There is a bijection between Dedekind real numbers in yr: and MacNeille reals in I:,
Given a Dedekind
Proof.
x in ye’, regard it as a locale map Idl(R - _)-
real number
L(lR) in E. By the definition of t(R), such a map is determined by the effect of its inverse image x* on the rational intervals (q, r); and x* must preserve the relations given above.
Define CI=IrE~I,~*(--,r)=R,_};
L={qEQ~x*(q,03)=n,7}, we shall show that (L, I/) is a MacNeille
real in ~5. (i) and (v), and compactness of Idl(Q,,),
From conditions 3q,rEQ
we deduce
(x*(q,r)=QT7),
whence 3qeQ
Again
(qeL)A3rEQ
from condition
(reU).
(v), we deduce
qeL-3q’>q
(q/EL)
its dual, the converse implications being an easy consequence of (iii). The disjointness of L and U follows from (ii) and (iii), since from (q E L)r\(r E I/) we deduce x*(q, r) = R -,_ and hence q < r. Finally, we observe that for ideals I, J of and
n 77 we have (IvJ=Q,-)A~(I=Q;~,,)-)(J=Q,,)
since ~(I=sZ7~)-(I=O), condition
and so (taking I=x*(q,m),
J=x*(-
a,r),
and
using
(iv)) we deduce q
0:
and dually q < rA I(r E U)“q Conversely, suppose x : Idl(Q,,)+L(R) by
given
EL. So (L, U) is a MacNeille a MacNeille
real
(L, U)
real. in A; then
we define
x*(q,r)={pEn,,i(3q’>q)(31’
First
we have
to show that x*(q, r) is indeed an ideal of R,,; Suppose PI, p2 are such that
it is obviously
downward-closed.
(3qi>q)(Pi-’
ll(qiE
L))
for i= 1, 2. Then since (qlELvqZEL)-)min(q,,qz)EL, ~?p~Vpz)-
13min(qt,
we have
q2) EL);
and clearly q < min(qI, 92). From this and the dual argument, we deduce p~Ex*(q,r)ApzEx*(q,r)-+(~-r(plVp~))~x*(q,r), so x*(q, r) is
an ideal of Q -_.
180
P. T. Johnsrone
Next we have to show that x* preserves the relations (i)-(v). The first is trivial, since the first axiom for a MacNeille real says (3q> -=)(3r
LAr‘E U)- ll(q’c
r?
+q‘c r’
and so q > r+x*(q, r) = {fake}.
For the third, we have to observe that
(q
and suppose p~x*(q~,r~). Then we have q’>ql,
LAr’E U).
Let q”= (2q2+ r1)/3, r”= (q2+2r1)/3,
Then clearly p~~x*(ql,n),
and define p2=pAll(q”EL).
pr=pA~~(r”EU),
11(pvVp2)=
r) f L).
pzcx*(q2,rz),
and
llpAll(q”~LVr”~
(/)
=pAl(l(q”EL)Al(r”EI/)) = PAT (‘&Ike) = p since q”< r”. So p E (x*(qI, rl)vx*(qz, r2)). The fifth condition is trivial from the form of x*(q, r). Lastly, we have to show that the two constructions we have defined are inverse to each other. One way round is easy; for the other, we have to show that if (L, U) is a MacNeille real, then qEL++(zJq’>q)ll(q’EL)* But this follows since ll(q’E
L) implies l(q’E I/).
0
2.4. Corollary. The object *I??of MacNeille reals in R is isomorphic to e*(lR,,,), where e : y& -+ G is the canonical geometric morphism. Proof. Proposition 2.3 establishes a bijection between the global elements of these two objects; to show that they are isomorphic, we have to extend this bijection (naturally in X) to their X-elements for an arbitrary object X of 6. But we may do
181
The Gleason cover of LIropes, I
this simply by repeating the Gleason
the argument
of 2.3 in the topos E/X, bearing
cover of b/X is (y&‘)/e*X.
2.5. Corollary.
in mind that
0
The Pierce representation of the ringed topos (8, *Rs) is (~8, iR,+).
Proof. We have to show that the Dedekind real a sheaf for the finite cover topology on Q7, in of *I?, i.e. the sheaf whose ring of sections over ideal I(p) generated by the idempotent 1 -f(p). be described as
number object in yR, considered as 4, is isomorphic to the Pierce sheaf p E Q7, Noting
is the quotient of *II?by the that Z(p) may alternatively
{XE *R/p-+(x=0)}, it is easy to see that this is precisely
the kernel
of the homomorphism
(where U is the 1 T-closed subobject of 1 corresponding to p) whose transpose is the *R&/U. Thus sections of the Pierce sheaf canonical isomorphism u*( *lRg) -? over p correspond to MacNeille reals in 6/U, which in turn correspond by the argument of 2.3 to Dedekind reals in y(e/u) = y6/e*U. But these are exactly the sections of ll?,g over p. 0 Since one of the conditions equivalent is a local ring [13, Proposition 1.3(r)], it be obtainable as the Zariski spectrum of *lR (and indeed I?) does not in general simple example to illustrate this. Let 6 be the topos Shv(alhl), where cation of the discrete space of natural
to De Morgan’s law is the assertion that *II? is tempting to conjecture that y6‘ might also (6, *II?). Unfortunately this is not true, since have Krull dimension zero; we now give a
aN = N U {a} numbers.
From
is the one-point compactifithe results which we shall
prove in the next section, we know that yA is Shv@N), where ph\l is the Stone-tech compactification of h\l. [RJ is the sheaf of continuous real-valued functions on c&J, i.e. of convergent sequences of real numbers; the global sections of *Re correspond to arbitrary
bounded
sequences
of real numbers.
We shall consider
the stalks
of
these sheaves at the point 03; clearly, points of the topos Spec(i;, *R) lying over the point 00 of i; correspond to prime ideals in the stalk *R,. For each point p ~/Ih\j - iN, the sets Mp = {If] E *R, j f converges
to 0 on the ultrafilter
p}
and
~~=~Lfl~*~;ol{nIf(n)=~~~~~ are prime ideals of *lR,. The ideals Mp are maximal (since *R-/M, is isomorphic to R by the map [f] -limp(f)), and every maximal ideal of *II?, is of this form; the ideals mp are minimal, being generated by idempotents ( = sequences with values in {0, I}), and hence define points of the Pierce spectrum of *R. The ideals Mp thus
182
P. T. Johnsfone
define points of the Zariski spectrum which do not belong to the Pierce spectrum. fact the ring *lRg is easily seen to have infinite Krull dimension; for any sequence positive reals x(n) converging to 0 on an ultrafilter p, we may define I,r,p= {[f] i 3A4, ar>O such that if(n)i
In of
for
and this is a prime ideal lying between mp and Mp. Similar remarks apply to the stalk of IRa at 00, with convergence on a particular ultrafilter replaced by convergence as n-w. On the other hand, it is possible to represent yr’ as a Zariski spectrum, by using the ring iR of “extended reals”. This is the ring whose elements are pairs (L, L/) satisfying the same axioms as MacNeille reals, except that the first axiom (L and lJ are inhabited) is weakened to the statement “L and U are nonempty”, i.e. lT( 3q(q E L)A 3r(r E u)). Mulvey has shown [20] that iR is the real number object in the topos sh,,(rl ); it is therefore a field (in the geometric sense) in sh,,(? ), and hence a (von Neumann) regular ring in ri [12, Proposition 5.61. So the Zariski and Pierce spectra of iR coincide. Also, since @ differs from *I??only “at infinity”, its algebra of idempotents coincides with that of *R, and so its Pierce spectrum is the same as that of *z; i.e. it is the topos y’. (The ring in yh which provides the Pierce/Zariski representation of II?J is of course pi,4.) Yet another way of representing y/C as a spectrum is suggested by the example I! =Shv(ah\i) which we considered above. We noted that the points of yft correspond not only to points of the Pierce spectrum (= minimal prime ideals) of *?, but also to maximal ideals of *R. Now Mulvey [21] has shown how the maximal spectrum of a commutative C*-algebra may be presented as a spectrum in the topostheoretic sense, i.e. as the classifying topos of a propositional geometric theory; and if we interpret this construction internally in 15 for the real normed algebra *?, we will again obtain a site of definition for ye!. We shall not give the details here.
3. The Gleason cover of a space Let ~5 be the topos of sheaves on a space X. What is the relation between ;J’ and the topos of sheaves on yX, where yX is the Gleason cover of X as constructed by topologists? In Section 1, we saw that y/i was characterized up to equivalence as the unique localic r: -topos generated by a minimal, compact, regular, extremally disconnected locale in fi; we also know that it has enough points when I( has [16, Corollary 4.91, and is therefore spatial when (5 is [ 11, Theorem 7.251. Similarly, the Gleason cover e : yX-X is characterized among spaces over X as the unique extremally disconnected space admitting a proper minimal surjection onto X [ 1,3]. Using the fact that topological spaces correspond proper maps f : Y-X between “reasonable” precisely to compact regular locales in Shv(X) [16,17], we may thus deduce that Shv(yX) is equivalent as a Shv(X)-topos to y(Shv(X)).
The Gleason cover of a
Nevertheless,
it is of interest
183
ropes.I
to give a direct
comparison
between
the site of
definition for y(Shv(X)) which we gave at the end of Section 1, and the site provided by the open-set locale of yX. We shall do this only in the case when X is regular; for more general spaces, it seems difficult to give a sufficiently explicit description of the open sets of yX. We recall that Gleason’s original construction [7] of yX, for a compact Hausdorff X, was as the (external) Stone space of the Boolean algebra Q,,(X) of regular open subsets of X-i.e. the space whose points are the prime filters of Q,-(X), and whose open sets correspond to ideals in Q TT (X) (a point being in a particular open set iff the corresponding filter meets the corresponding ideal). The map e : yX-X sends a filter F to its limit, i.e. the intersection of the closures of all the members of F (which can easily be shown to be a singleton), and e-i sends an open set U c X to the ideal of all regular open sets whose closures are contained in U. Subsequent authors [9,25, l] have worked with spaces of maximal filters in the open-set lattice Q(X) rather than in R TT(X); but this is merely a cosmetic change, as the following lemma shows: 3.1. Lemma.
Let H be a Heyting algebra. Then there is a natural bijection between maximal filters in H and maximal (=prime) filters in the Boolean algebra H,,, given by F-FnH,-. Proof. If F is a maximal filter in H, then an element h is in F iff 7 1 h is in F (since if 1 Ih E F, we can adjoin h to F and still have a proper filter). Thus F is determined by Fn H-7; moreover, Ffl H 7 7 is a prime filter in H-.,- if F is a prime filter in H. Conversely, if G is a prime filter in H,,, it must be contained in some maximal filter Fin H, whose intersection with H,, must be G (since G is maximal in H--) and which is therefore
uniquely
determined
by G.
a
Thus the space of prime filters of n,-(X) is homeomorphic to the space of maximal filters of Q(X). If X is not compact, however, it is necessary to work with the subspace of convergent maximal filters, i.e. those which contain the neighbourhood filter at some point of X. To identify following lemma:
the open sets of this subspace,
we use the
3.2. Lemma. Let I, J be ideals of Q,-,(X), and let SI, SJ denote the sets of convergent prime filters in R,,(X) which meet I, J respectively. Then SI c SJ r? for every UE I, there is a (set-theoretic) covering of X by regular open sets Va such that each V,n U is in J. Proof. Suppose the condition is satisfied; let F be a filter in SI, and let x be its limit point. Let U be a set in Ffl I; then by hypothesis we can cover X by regular opens V, with VA WE J. But one such V, must contain .Y and is therefore in F; so V,rl L/E Fn J and hence FE S_I.
184
P. T. Johnsrone
Conversely, suppose the condition is not satisfied; then we have an open set I/ in I for which no covering of X with the required properties exists. Let x be a point of X for which no regular open neighbourhood N of x satisfies Nfl U E J, and let F be the filter in Q-,-(X) based by the sets Nn U. Since this filter does not meet J, we can enlarge it to a maximal filter G not meeting J; but C does meet I (in the set U) and converges
t0 x,
so GESI-SJ.
0
3.3. Corollary. Let Y be the subspace of the Stone space of Q,-(X) consisting of convergent prime filters. Then the open subsets of Y may be identified with ideals I
in !2 7-,(X) satisfying the following condition: (*) For any UESZ~~(X), if there exists a set-theoretic covering { V,]~EA} with each V,II I/ in I, then (I E I. Proof.
Let /be
any ideal of Q,,(X).
of X
Then it is easy to verify that the set
I*={f_J~!Zn~(X)/thereisacovering{VO~ocuA}ofX with each V,n I/ in I} is an ideal of Q,,(X) which satisfies (*) and meets precisely the same filters in Y that Imeets. And if two ideals 1, J both satisfying (*) meet the same filters in Y, then Lemma 3.2 shows that they must be equal. Thus the ideals satisfying (*) provide a set of representatives for the equivalence classes of open sets in the Stone space of B,,(X) under the relation I-J iff Ifl Y=Jn Y. 0 Note that if X is compact, then every ideal of Q,,(X) satisfies (a), since we may take the covering { Va/ (YE A} to be finite, and I is closed under finite joins. Of course, this is just another way of saying that if X is compact then every prime filter in !ZT7(X) converges. Now if X is regular, then the map lim : Y-X, which sends a filter in Y to its limit point, is continuous [25, Lemma 21, and we may take it to be the Gleason cover of X. We thus wish to compare the site provided by the ideals of Q,,(X) satisfying (*) with the site (yC, yJ) which we constructed in Section 1. Note that in this case yC is the poset whose elements are pairs (U, V) with U open in X and V regular open in U, with (U, V)l (U’, V’) iff UC U’ and VC V’, and yJ is generated by covers of two types (a) and (b) as described earlier. (In the definition of yC, we may of course restrict U to lie in some base for the topology on X. For example if X is semiregular, we may require U to be regular open in X; this will have the advantage that the sets V will be regular open in X and not merely in U.) Let F denote the finite cover topology there is a left exact, continuous functor
on the Boolean
algebra
-Q,,(X).
Then
u : (Q~-(X>,FP(YC,YJ) which sends VE G--(X)
to (X, V); and in fact u has a left adjoint
v (necessarily
co-
continuous, geometric
by [8, III 2.51) which sends (U, V) to 71 V. It follows
that we have a
morphism g : y(Shv(X))
where g* is the functor
= Shv(yC, yJ)-Shv(R “compose
-.T(X),F)
with u”, and g* is the functor
and then sheafify”. Since both Shv(yC, rJ) and Shv(Q - _(X), F) are generated
“compose
by subobjects
with v of 1, we
shall investigate the effect of g* and g, on such subobjects (which may, of course, be identified with downward-closed subsets of the two posets which are closed with respect to coverings in the two topologies). 3.4. Lemma. Let I be an ideal in Q-,7(X), regarded as a subobject of 1 in Shv(_Q 7-(X), F). Then g*(i) is the set of all (U, V) E yC for which U admits a (set-
theoretic) covering by open sets U, such that 7 T( U,n V) E I. Proof. It is clear that the sub-presheaf of 1 defined by the specified set is a sheaf for coverings of type (a), and that the inclusion of lov in it is dense with respect to these coverings. So it suffices to prove that it is a sheaf for coverings of type (b). But if we are given (U, V) E yC, a finite cover { VI, . . . , V,,} of V in .Q 7_( I/), and for each i an open cover { CJ,,,I a E A,} of I/ such that each l~(U;,~n V,) is in 1, then on taking a common refinement { L/D~/~EB} of the given covers of I/ we deduce that each -~l(Uafl V,) is in I and hence (since I is an ideal) that each Tl(Upn V) is in 1. So (U, V) is in the given sub-presheaf of 1. C
An ideal I of QT7(X) (*) of Corollary 3.3.
3.5. Corollary.
Proof.
By Lemma VEg*g*I
satisfies g,g*(f) = I iff it satisfies condition
3.4, we have e (X, V)Eg*I b there is a cover of X by open sets U, such that l-qU& V)EZ.
Thus condition obvious. G
(*) is precisely
the assertion
g,g*Zc
I; the reverse
inclusion
is
3.6. Lemma. Suppose X is regular. Then every subobject K of 1 in Shv(yC,yJ) satisfies g*g,(K) = K. map g*g .-t 1, we clearly have g*g,K G K. So suppose we may cover U by open sets CJclwhose closures are contained in CJ; for convenience we shall assume that each Ua is regular open in X. Let V,= U,n V; then Va is regular open in U,, hence in X and a fortiori in U, so that (U, V,) is in K.
Proof.
Since there is a counit
(U, V) E K. Since X is regular,
186
P. T. Johnsrone
But (X, Vcl) is yJ-covered by (C-J,Vcl) and (X- oG, 0) since VGlc U,; and (X- U=, 0) is yJ-covered by the empty family, so (X, Va) is yJ-covered by (U, VU) and hence is in K. Hence k’,~g,K for each a; but since I/ is covered by the O;, and u,n v= v,, we deduce from Lemma 3.4 that (U, V) is in g*g,K. El 3.7. Theorem. Let X be a regular space. Then y(Shv(X)) topos to Shv(yX).
is equivalent as a Shv(X)-
Proof. Taking yX to be the space Y of convergent prime filters in n,,(X), in Corollaries 3.3 and 3.5 that the image of the geometric morphism
we saw
g : y(Shv(X))+Shv(spec(Q,,(X))) is precisely Shv(yX); and Lemma 3.6 shows inclusion, and so induces an equivalence y(Shv(X))
A
s
if X is regular
then
g is an
Shv(yX).
It remains to show that this equivalence the diagram y(Shv(X)) \
that
is a morphism
of Shv(X)-toposes,
i.e. that
Shv(yX) /
Y Yrn Shv(X)
commutes up to equivalence. But if F is a yJ-sheaf defined by U-F(U, U); and lim,(g,F) is the sheaf
on yC, then e.(F) is the sheaf
U-g,F(lim-‘(I/)) = -lim (F(X, V) j V regular
open,
PC U)
= A lim (F(U, V) 1V regular
open,
PC U)
=F(U, U) since (U, U) is yJ-covered by {(U, V) 1Pr U} and each (X, V) is yJ-covered G and (X- V, 0). So the result is established.
by (U, V)
The proof of Theorem 3.7 is not entirely constructive, since we made use of the maximal principle in the proof of Lemma 3.2. However, for compact regular spaces, we can eliminate 3.2 and 3.3, as indicated in the remark after 3.3; and the proofs of 3.4-3.6 are completely constructive. Thus we can interpret them internally in any topos R (replacing the compact regular space X by a compact regular locale A), to obtain
Let A be a compact regular locale in a topos 8. Then the Gleason cover of the localic G-topos &[A] is equivalent to #[[Id&A --)I.
3.8. Corollary.
The Gleason
cover of
a
ropos,I
187
4. The Gleason cover of a finite category In this section, our aim is to describe y6 in the case when ,: is the topos .pcop of presheaves on a small category C. We shall be particularly interested in the cases when C is a finite category or a monoid. If C is a finite category, then for every object c of C the Boolean algebra R--(c) is finite and therefore atomic, and we may identify its Stone space with the discrete set ,4(c) of its atoms. In fact, we do not need to determine Q,,(C) explicitly to find its Stone space:
Let H be a Heyting algebra with the property that every nonzero element of H contains a minimal nonzero element (e.g. any finite Heyting algebra). Then H,, is atomic, and its atoms are in l-l correspondence with the minimal noneero elements of H.
4.1. Lemma.
Proof. Let x be a minimal nonzero element of H. Then 1 lx is an atom of H _ _ ; for anyyEH7, satisfying 0
Cc@)= {~PERI is a minimal
sieve on d.
Theorem. Let C be a finite category, and let 6 : A-C be the discrete opfibration corresponding to the set-valued functor A defined above. Then the induced geometric morphism
4.2.
is (equivalent to) the Gleason cover of YcoD. We recall that the objects of A are pairs (c, R) where c is an object of C and R EA(c); and that morphisms (c, R)+(d,S) in A are morphisms (Y : c-d in C such Proof.
188
P. T. Johnstone
that A(a)(R)
=S.
But if A(a)(R)=&
then it is easy to see that a*(S) 1 R, and so
there is an obvious inclusion functor i : A-+, where (yC, yJ) is the site of definition for y(Shv(C,J)) constructed in Theorem 1.10 (with J taken to be the trivial Grothendieck topology on C). Moreover, in this case yJ contains only covers of type (b), so it is easy to see that every object of yC is yJ-covered by objects in the image of i, but the latter have no nontrivial Lemma of Grothendieck [8, III 4.11, i induces i* : Shv(yC, yJ) d It remains
commutes that Ed
..YAoP.
to check that this is an equivalence
up to isomorphism.
@covers. So by the Comparison an equivalence
of .j/Cop-toposes, i.e. that the diagram
But since 6 is a discrete
opfibration,
it is easy to see
i*F is the functor c - Rg,=,F(c, R), to c - F(c, MC) provided
which is clearly isomorphic
F is a y.J-sheaf.
EI
4.3. Remark. We may verify directly that the category A satisfies the “fill-in of [13, Proposition 1.11, i.e. that .T”*’ satisfies De ,Morgan’s law. condition” Suppose we are given a diagram Cc,RI
in A; then if y : b-c is any element of R, we deduce from the equalities C,(R) = T= Zp(.S) that we can write ay =pS for some 6 E S. Now if P is any minimal sieve on b, then we clearly have Z:,(P) c R and .Zd(P) c S; but by minimality these inclusions must be equalities, and so we have a commutative diagram (b, J-7 A
(c,R) I a
I
6 i
(4 8 p
(e, T)
in A, as required.
0
Next we consider poset or a monoid.
the special forms of Theorem
4.2 in the cases when C is either a
The Gleason cover of a
189
ropos,I
4.4. Corollary. Let P = (P, 5) be a finite poser. Then the Gleason cover of .YpO’is (equivalent to) 2 *Op,where A is the poset of pairs (m,p) of elements of P with m minimal and m up. ordered by (m,p)i(m’,p’)
* m=m’andpIp’.
Proof. After 4.2, we have only to observe that a minimal sieve on an element of P must be generated by a single morphism, and that (m-+p) generates a minimal sieve iff m is a minimal element of P. 0 Let M be a finite monoid. To identify the Gleason cover of the topos .li w of (left) M-sets, we need to identify the minimal sieves on the unique object of M considered as a category, i.e. the minimal left ideals of M. To do this, we employ a series of lemmas relating minimal left and right ideals of &I.
4.5. Lemma. Let R be a minimal right ideal of a monoid M. The set E of idempotents in R forms a right Q-semigroup, i.e. satisfies ef =f for every e, f E E. Proof. By minimality, R is generated by each of its elements: havef=exforsomexEiM.Butthenef=e~,~=ex=fsinceeisidempotent.
so if e, f E E, we must II
Let L, R be minimal left and right idea/s of a finite monoid ,tl. Then there is a unique idempotent in L 0 R.
4.6. Lemma.
Proof. If XE R, y E L, then xy E L rl R. But since M is finite, some power of xy must be idempotent, and this is again in L fI R. Uniqueness follows from Lemma 4.5; for if e, fare idempotents in L fl R, then ef =f since e, f E R, and ef = e since e, f E L. 0
4.1. Lemma. Let R be any minimal right ideal of a finite monoid M. Then the minimal left ideals of A4 are in 1-1 correspondence with the idempotents in R. Proof. Let e be an idempotent in R, and consider a minimal left ideal L contained in Me. By Lemma 4.6, this meets R in an idempotent f =xe say; but then fe=xe2= f = e by Lemma 4.5. So e E L, and hence L = Me, i.e. the ideals Me, e idempotent in R, are all minimal. It follows at once from Lemma 4.6 that they are all distinct, and 3 that they exhaust the minimal left ideals of M. The set of minimal
left ideals of A4 forms a right M-set under
multiplication,
i.e.
Lm= {xmIxE L}. Thus we obtain a right M-set structure on the set E of idempotents in a minimal right ideal R; explicitly, the multiplication is given by e*m = the unique
idempotent
in R 17Mem
= the unique
idempotent
power of em.
P. T. Johnsrone
190
Corresponding
to this M-set,
we have a discrete
opfibration
6 : E-M,
where the
category E may be described as follows: its objects are the elements of E, and morphisms e-f in E are elements m EM such that f is a power of em (the composition in E being given by multiplication in M).
4.8. Corollary. Let A4 be a finite monoid. Then the Gleason cover of the topos .? w of left M-sets is (equivalent to) YEa’, where E is the category described above. Z Let M be a finite monoid. Then the order-completion V? of the Dedekind reals in .YMopis the set of all functions E-R, where E is the set of idempotents in a minimal right ideal of M, with left M-action induced by the right lMaction on E (i.e. (ma f)(e) =f (e*m)). 4.9. Corollary.
Proof. It is well known that in any topos of presheaves, the object of Dedekind reals is simply the constant presheaf with value IR. So by Corollary 2.4 we have to compute the effect of the right Kan extension functor l&on this constant presheaf 0 on E, which gives the answer described above. How far can the results of this section be extended to infinite categories? Clearly, Theorem 4.2 requires only the hypothesis that Q(c) is finite for each object c of C, not that C itself is finite; so it extends, for example, to any poset in which the dounward segment of each element is finite, and to the category of nonempty finite sets. (In the latter case, since the minimal nonempty sieves on a finite set X are those generated by morphisms 1 -+X, we deduce that the Gleason cover of (presheaves on nonempty finite sets) is (presheaves on finite pointed sets).) If Q(c) is allowed to be infinite but still required to satisfy the hypothesis of Lemma 4.1, then the Boolean algebra R-,(c) will still be atomic; but if it is infinite then its Stone space will not be discrete, and so the Gleason cover of .fc” will not be a topos of presheaves. For simplicity, we now restrict ourselves to the case of a monoid. It is well known that if a monoid M has a single minimal left ideal L, then every other (nonempty) left ideal of M contains a minimal left ideal; for ifs is an element of a left ideal L’, then Lx is a minimal left ideal contained in L’. So Q( *) (where * is the unique object of the category M) satisfies the hypothesis of Lemma 4.1, and Q-,-(e) is atomic. The identification of minimal left ideals with idempotents in a minimal right ideal (Lemma 4.7) will remain valid provided some (and hence every) minimal left or right ideal of M is finite; but in general we shall have to work directly with the M-set E of minimal left ideals. Now the power-set of E may be identified with _Q-,(*), and hence with a (nonfull) subcategory of yM; so it is easy to see that yJ-sheaves on yM may be described as sheaves on the Stone-tech compactification /3E of the discrete space E, equipped structure. In particular, Dedekind real with a certain additional “A4-equivariant” numbers in Shv(yM,yJ) may be shown to correspond to continuous real-valued
cover of a
The Gleason
ropos,I
191
functions on /W (the extra structure being constant in this case), from which we deduce that Corollary 4.9 remains valid to the extent that *R is the M-set of bounded functions E-,R, with M-action induced by that on E. (It is of interest to note that this M-set appears in recent work of Banaschewski [2] as being the injective hull of R in the category of M-equivariant Banach spaces, at least when A4 has a finite minimal left ideal. Thus we can say that the Hahn-Banach theorem, in the sense of Burden [4], is valid in the topos of M-sets for any such M.)
Acknowledgements
I am indebted to Bernhard Banaschewski, Martin Hyland, John Isbell and Chris Mulvey for helpful discussions about the subject of this paper.
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Projective
Kanpur
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THE GLEASON
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Company
COVER
OF A TOPOS,
I
P.-l-. JOHNSTONE Deparrmenr
of Pure .Marhetnarics.
Universir,v
of Cambridge,
England
0. Introduction Despite its title, this paper is the third in a series (the first two being [ 131 and [ 141) in which we investigate the consequences within topos theory of the logical principle known as De Morgan’s law:
In [13], it was observed that the topos Shv(X) of sheaves on a topological space X satisfies De Morgan’s law iff X is extremally disconnected, and it was claimed that most of the occurrences of extremally disconnected spaces in general topology and functional analysis could in fact be explained as appeals to De Morgan’s law in the internal logic of Shv(X). In this and a subsequent paper [17], we set out to justify this claim in the case of a particular topological construction involving extremally disconnected spaces: the Gleason cover. In [7], A.IM. Gleason 0.1. Theorem.
proved
the following
two results:
spaces, the projective objects maps) are precisely the extremally dis-
In the category of compact Hausdorff
(with respect to surjective connected spaces. q
continuous
0.2. Theorem. For any compact Hausdorff surjection e : yX+X, where yX is compact, connected, which is “minimal” in the sense that surjectively through e. Moreover, this property homeomorphism in the category of spaces over
space X, there is a continuous Hausdorff and extremally disevery other such surjection factors characterizes yX up to (unique) X. C
The space yX is called the Gleason cover of X. Subsequent authors [23, 5,9, 25, 19, 1, 31 extended both Gleason’s results, ultimately to the category of all Twspaces, but with the proviso that one must replace “surjective continuous map” by “surjective proper map” in both theorems when dealing with non-compact spaces. The object of the present paper is to provide a topos-theoretic analogue of Theorem 0.2: we shall construct for any topos 4 a surjective geometric morphism 171
172
P. T. Johnstone
e : y&*6, where yb satisfies De Morgan’s law, which is in some sense the “best approximation” to 8’ by a De Morgan topos. We shall also show that our construction coincides with the topological one in the case when W= Shv(X), at least for a regular space X, and investigate how it may be described in the case when 6 is a topos of presheaves. Finally, we shall investigate the connection between y& and the order-completion *lR of the Dedekind real numbers in b. The sequel [17] will investigate the analogue of Theorem 0.1: the extent to which De Morgan’s law is equivalent to projectivity for toposes.
1. The Gleason cover
In this section, we shall construct the Gleason cover y&’ of an arbitrary topos 8, and prove its basic properties. In doing this, we shall make use of certain properties of internal sites in a topos (in fact we shall consider only two particular cases,, that of an internal distributive lattice with its finite cover topology, and that of an internal locale (= complete Heyting algebra) with its canonical topology). These properties are simply the “internalizations” of various results which are well-known for sites in the topos of sets [8,11]; in certain cases, their proofs in a general topos involve mild technical complications, and we shall not give them here-the interested reader will find them in [16]. We recall [13, Theorem 1] that a topos R satisfies De Morgan’s law iff the internal Boolean algebra Q _ -, coincides with 2. In general, the Stone Representation Theorem [24] tells us that an arbitrary Boolean algebra may be represented as the global sections of the sheaf 2 over a suitable space. We therefore define the Gleason cover of d to be the internal “Stone space” of Q,,-except that, since Q-,, may fail to have any prime ideals [14], we must consider it as a locale rather than a space. Formally, we have: Definition. The Gleason cover yb of a topos ti is defined to be the topos of (Jvalued) sheaves for the finite cover topology on the internal Boolean algebra Q 7 ?, or equivalently the topos L[Idl(Q _ -)I of canonical sheaves on the internal locale of ideals of S27 -. (For the equivalence, see [ 16, Lemma 4.61.) We must first show that yB’ actually satisfies De Morgan’s law. In fact it suffices to prove that the locale Idl(Q,,) is an internal Stone algebra in 6, since it then follows easily that the same is true for the subobject classifier in y8 [16, Corollary 2.21. This follows from the internalization of a well-known result on Stone spaces: 1.1. Lemma. Let B be an internal Boolean algebra in a topos. Then the locale Idl(B) of ideals of B is a Stone algebra iff B is internally complete. Proof.
If I is an ideal of B, then the negation of I in Idl(B) is easily seen to be the
The Gleason cover of a
173
ropes.1
subobject J= (b E B / (Vi E f)(iAb = 0)} )-+B. Now if B is complete (and hence a locale), it is easy to see that J is closed under arbitrary joins, and so J=iseg(V@)); i.e. every 7 -t-closed ideal in B is principal and therefore complemented in Idl(B). So Idl(B) is a Stone algebra. Conversely if Idl(B) is a Stone algebra, then the principal ideal map lseg : B-+Idl(B) identifies B with the sublocale (Idl(B)) T7 of Idl(B), so B is complete. 1.2. Corollary. We global gated lattice
For any topos 8, yG satisfies De Morgan’s law.
0
0
have a canonical geometric morphism e : yG+G, whose direct image is the section functor for sheaves over Idl(Q--) (or over Q --). In [ 161, we investithe way in which surjectivity of this morphism is related to properties of the Q,,; in particular, we showed
1.3. Lemma. For a distributive lattice A in 6, the following are equivalent: (i) f[Idl(A)] -+ C; is surjective. (ii) The map ,I : Q-Idl(A) defined by A(p) = {a E A j (a = O)Vp} is a monomorphism. (iii) The map 2-A Proof.
defined by the global elements 0 and 1 is a monomorphism.
Combine Lemmas 2.8 and 4.4 of [16].
1.4. Corollary.
0
The canonical map e : yA + (! is surjective.
Cl
If t” satisfies De Morgan’s law, then Q -- E 2 and so Idl(SZ--) z Idl(2) 3 f2. (The map A of Lemma 1.3(ii) provides the latter isomorphism.) But Q is a terminal object in the category of internal locales in ci-[ 16, Lemma 2.41; so r; [Q] = 6, and we deduce 1.5. Corollary. Proof.
(-)
e : y6 --* 6 is an equivalence iff 6 satisfies De Morgan’s law.
follows from Corollary
1.2, (=) from the remarks above.
C
In [14], we observed that R,, has a unique proper ideal, namely false : 1 h n 7-7; so if Q : Idl(Q,7)+Q is the classifying map of the maximal element -a,TT : 1 * Idl(Q,,), then the diagram I
cl fdse
&/se) I Idl(Q,,)
I A52
171
P. T. Johnsrone
is a pullback-i.e. Idl(SZ,,) is a minimal locale in the 2.91. It follows that the surjection y i + i: is a minimal proper closed subtopos of y’_’ which maps surjectively being coherent over R, would map surjectively to t: iff also follows that there is a commutative diagram
terminology of [16, Lemma in the sense that there is no to J-for such a subtopos, its image were dense in 8. It
[16, Lemma 2.10]-indeed, this is obvious from the proof of Lemma 1.1. Translating this diagram of internal locales in A into a diagram of localic 6-toposes, we deduce 1.6. Lemma.
There is a commutative diagram sh,,(yB) I
A
sh,,(T’) I
in Top, where the vertical arrows are the canonical inclusions. 1.7. Corollary.
‘1
y*f is Boolean iff d is.
Proof. If rY is Boolean, then it satisfies De Morgan’s law and so y< is Boolean by Corollary 1.5. Conversely if yc! is Boolean, then the inclusion sh,,(G)*& is equivalent to the composite sh,,(yt) A Yd -A-..+ 6 and therefore surjective; so i is Boolean. 0
The next result characterizes y tl amongst localic C-toposes by a list of properties similar to those which characterize the Gleason cover of a space X amongst spaces over X. Let A be an internal locale in a topos h which is compact, regular, minimal and extremally disconnected (=a Stone algebra). Then A is isomorphic to Idl(R --,). 1.8. Proposition.
Proof. Since A is regular, it is semi-regular [15, Lemma 4.51, i.e. every element is a join of TY-stable elements. But since A is extremally disconnected, its TY-stable elements are complemented; and since A is compact, its complemented elements are finite ( = intranscessible in the terminology of [ 181). So A is coherent, i.e. A = Idl(B)
The Gleason cover of a
lopes, I
175
where B ti A is the sublattice of complemented elements. But by extremal discon0 nectedness and minimality we have B = A TT z f? --. Suppose we have a geometric morphism h : yC+yJ over 6. By the equivalence of [16, Theorem 2.71, h is uniquely isomorphic to the morphism induced by a locale map cp : Idl(Q,,)-, Idl(Q,,) in f; and since rp* preserves complemented elements, it must restrict to a Boolean algebra homomorphism rp’ : Sz__ -+Q -,-. Now @ must preserve the elements true andfalse; but since (true, false) : 2 * Q YT is l-r-dense and S2-- is a -~-sheaf, it follows that p’ must be the identity. Hence also cp is the identity, and h is (uniquely) isomorphic to the identity. So we have proved: 1.9. Lemma. The category Xop/& (~6, y(9) is equivalent to the trivial category with one object and one morphism. (In the terminology of [IO], Gleason covers are rigidly attached.) q All the above considerations follow directly from the definition of y:, without any reference to the particular nature of i. However, in order to identify ~(5 in particular cases, it will be convenient to have a means of converting a site of definition for /i into one for yti. We close this section by providing such a construction. Suppose (C,J) is a site (i.e. a small category with a Grothendieck topology), and suppose we have identified the sheaf a77 in the topos Shv(C,J). We define a new site (yC, yJ) as follows: the objects of yC are pairs (c, R) where c is an object of C and R E Q,,(c), and the morphisms (c’,R’)+(c,R) in yC are morphisms a : c’-c in C such that R’c a*(R). We write p : yC-C for the obvious projection functor, whose fibre over lc is the poset Q --(c). We define yJ to be the smallest topology on yC for which the following sieves are covering: (a) For every (c,R) E ob yC and every .SEJ(C), the sieve generated by {a : (c’,a*R)-(c,R)I(a : c’dc)~S}. (b) For every (c,R) E ob yC and every finite family {RI, . . . , R,) 2 Q,,(c) with joinR, the sievegenerated by {lc: (c,R;)+(c,R)ji=l,...,n}. The functor p : yC-+C mentioned above is not continuous (i.e. does not preserve covers), since the object (c,OJ of yC (where 0, is the least element of Q,,(C)) is covered by the empty sieve even if c is not. However, the right adjoint q of p, which sends c to (c,MJ where M, is the largest element of Q,,(C), does preserve covers; and so it defines a morphism of sites in the sense of [8, IV 4.9.11, i.e. it induces a geometric morphism Shv(yC, yJ)+Shv(C,
J)
whose direct image is given by restriction along q.
176
P. T. Johnsrone
1.10. Theorem. With the notation introduced above, Shv(yC, yJ) is equivalent (as a topos over Shv(C, J)) to y(Shv(C, J)).
Let F be a presheaf on yC; then we know that F is a sheaf for the topology yJ iff it satisfies the sheaf axiom for the particular coverings described in (a) and (b) above. So suppose this is true, and define a presheaf G on C by Proof.
G(c)=
ll F(c,R) REfZ__(C)
and if a : c’+c and XE F(c, R) c G(c), then G(a)(x) = F(a)(x) E F(c’, a*R) c G(f) where on the right-hand side we regard a as a morphism (c’, a*R)d(c, R) in yC. First we claim that G is a sheaf for the topology J. Suppose given SEJ(C), and a compatible family of elements (x,~G(c3I(a:
c’+c)c.S);
then writing Ra for the unique element of Q,,(c’) such that X~E F(c’, Ra), it is clear that the elements (Ra j a E S) are compatible in the sheaf sZTT, and so define a unique R E Q,,(C) such that a*R = R, for all a. Since F satisfies the sheaf axiom for covers of type (a), it is now clear that there is a unique XE F(c, R) c G(c) with G(a)(x) =xa for all a. Next we observe that there is a map 6 : G-Q7 T in Shv(C, J) which sends x E G(c) to the unique R such that XE F(c, R). Moreover, we have a map j? : (Q-,7)1 xo,, G+G (where (a,,)~
denotes the order-relation
on Q,,)
defined by
P((R’, R), x) = F( 1c)(x) where R’ c R in Q,,(c), x~F(c, R) c G(c), and we regard lc as a morphism (c,R’)+(c,R) in yC. It is straightforward to verify that this map makes G into an internal presheaf on R TT in Shv(C, J). Moreover, since F satisfies the sheaf axiom for covers of type (b), F(c,Oc) = 1 for every c, and the diagram F(c, R IV&) -
I
FCC, R2)
-
F(c, RI)
I
F(c, R dR2)
is a pullback for every pair (RI, R2); so it follows from [16, Lemma 4.71 that G is a sheaf for the (internal) finite cover topology on J2 __. Conversely, suppose given such an internal sheaf (G, S,B). We define a presheaf F on yC as follows:
The Gleason cover o_fa ropes, I
and if (r : (c’,R?+(c,R) F(a)(x)
177
in yC, then
= B((R’,
ar*R),G(a)(x))
for XE F(c, R). Reversing the above arguments, it is easy to see that the two sheaf conditions (external and internal) on G imply that F satisfies the sheaf axiom for covers of types (a) and (b), and is thus a yJ-sheaf. It is clear that the two constructions given above are inverse up to natural isomorphism, and so Shv(yC, yJ) = y(Shv(C, J)). The fact that this is an equivalence of toposes over Shv(C, J) is easily seen by comparing the two direct image functors; for taking the sections of G over the maximal element M : l-Q,_ has the same 0 effect as taking the restriction of F along the functor 4 : C--yC. Note that if the category C has finite limits, then so does yC. Also, if every covering in J has a finite refinement, then the same is true of the covers which generate yJ, and hence of every covering in yJ. So we deduce:
If 6 is a coherent (Grothendieck)
1.11 Corollary.
topos, so is y”.
III
2. The spectrum of the real numbers In [13], we observed that De Morgan’s law in a topos 4 is equivalent to a number of interesting statements about the real numbers in I! : in particular, the statement that the object II? of Dedekind reals coincides with its order-completion *IR (which we now call the MacNeille reals). In this section we investigate the relation between the Gleason cover of a topos R and the object *YZin 6 ; we shall show that each may be defined in terms of the other. We begin with something [13, Proposition 1.31:
which appeared
as a throwaway
remark
in the proof of
In any topos, the Boolean algebra of idempotents of *R is isomorphic
2.1. Lemma.
to Q,,. Proof.
Let p be a variable L(p)=
of type R-,-,
{qEQl(q
and define l)AP)),
~(P)={qE~l(q>l)V((q>O)A~P)}. In the proof that the pair (L(p), U(p)) defines step comes in showing that
a MacNeille
real, the only nontrivial
(9 < @A l(Q E UP)) -) (9’ E U(P)) (and its dual),
where there is an apparent
use of De Morgan’s
law in proceeding
178
P. T. Johnsrone
l)V-p.
from -((q-C I)Ap) to (q? trichotomy, we have
But
since
the
order-relation
on
Q satisfies
(9 < l)v(q 2 l), so this deduction J-(P) = (UP),
is permissible.
Hence
we have a map f : R-,- - *R, defined
by
U(P)>. Now P-+(~(P) = 1)
and
~P+(J(P)
= O),
so that
l%-(P)2
-f(P)
= 0).
But *lR is -l-separated (by the same proof as for R, see [ll, 6.63]), so we deduce that the image off consists of idempotents. Conversely, since *IF?is a residue field (cf. [I 1, 6.65]), any idempotent x E *Z satisfies 1(x = l)--+ 1(x invertible in *?) +(x=0), and dually
1(x=0)-+(x
= 1). So it follows
easily that
x =f(Ox = 1II), and
hence
*iR.
that f maps
R -,-
isomorphically
onto
the object
of idempotents
of
0
2.2. Corollary. The Gleason cover yli of I< is (rhe underlying topos of) the Pierce representation [22] of the ringed topos (i, *R).. Classically, the Pierce representation of a ring lives in the topos of sheaves on the Stone space of the Boolean algebra of idempotents of the ring. When vve construct this representation for a ring in a general topos (cf. [12, pp. 257-g]), it is clear that we must replace this Stone space by the locale of ideals of the Boolean Proof.
algebra.
0
What is the sheaf of rings over Idl(R-,) which provides the Pierce representation of *R? To answer this question, note that Dedekind real numbers in y? correspond topos for the to geometric morphisms over G from y6 to the classifying or equivalently to locale morphisms propositional theory of real numbers, Idl(Qn,,)+L(R) in A, where L(R) is the locale of “formal real numbers” in 6 [6]. We recall that this locale is generated by the “formal rational intervals” (q,r) (q E Q u { - co}, r E Q u (a~}), subject to the relations (i) (-co,oo)=l, (ii) (q, r) = 0 if q 2 r, (iii)
(qh rdNq2,rz)
= (max(sl,q2),min(ri,rz)),
(iv) (st,ri)V(q2,rz)=(ql,r2) 09
if 91592
(q,r)=V{(q',r?Iq
The Gleason cover of u ropes. I
179
2.3. Proposition. There is a bijection between Dedekind real numbers in yr: and MacNeille reals in I:,
Given a Dedekind
Proof.
x in ye’, regard it as a locale map Idl(R - _)-
real number
L(lR) in E. By the definition of t(R), such a map is determined by the effect of its inverse image x* on the rational intervals (q, r); and x* must preserve the relations given above.
Define CI=IrE~I,~*(--,r)=R,_};
L={qEQ~x*(q,03)=n,7}, we shall show that (L, I/) is a MacNeille
real in ~5. (i) and (v), and compactness of Idl(Q,,),
From conditions 3q,rEQ
we deduce
(x*(q,r)=QT7),
whence 3qeQ
Again
(qeL)A3rEQ
from condition
(reU).
(v), we deduce
qeL-3q’>q
(q/EL)
its dual, the converse implications being an easy consequence of (iii). The disjointness of L and U follows from (ii) and (iii), since from (q E L)r\(r E I/) we deduce x*(q, r) = R -,_ and hence q < r. Finally, we observe that for ideals I, J of and
n 77 we have (IvJ=Q,-)A~(I=Q;~,,)-)(J=Q,,)
since ~(I=sZ7~)-(I=O), condition
and so (taking I=x*(q,m),
J=x*(-
a,r),
and
using
(iv)) we deduce q
0:
and dually q < rA I(r E U)“q Conversely, suppose x : Idl(Q,,)+L(R) by
given
EL. So (L, U) is a MacNeille a MacNeille
real
(L, U)
real. in A; then
we define
x*(q,r)={pEn,,i(3q’>q)(31’
First
we have
to show that x*(q, r) is indeed an ideal of R,,; Suppose PI, p2 are such that
it is obviously
downward-closed.
(3qi>q)(Pi-’
ll(qiE
L))
for i= 1, 2. Then since (qlELvqZEL)-)min(q,,qz)EL, ~?p~Vpz)-
13min(qt,
we have
q2) EL);
and clearly q < min(qI, 92). From this and the dual argument, we deduce p~Ex*(q,r)ApzEx*(q,r)-+(~-r(plVp~))~x*(q,r), so x*(q, r) is
an ideal of Q -_.
180
P. T. Johnsrone
Next we have to show that x* preserves the relations (i)-(v). The first is trivial, since the first axiom for a MacNeille real says (3q> -=)(3r
LAr‘E U)- ll(q’c
r?
+q‘c r’
and so q > r+x*(q, r) = {fake}.
For the third, we have to observe that
(q
and suppose p~x*(q~,r~). Then we have q’>ql,
LAr’E U).
Let q”= (2q2+ r1)/3, r”= (q2+2r1)/3,
Then clearly p~~x*(ql,n),
and define p2=pAll(q”EL).
pr=pA~~(r”EU),
11(pvVp2)=
r) f L).
pzcx*(q2,rz),
and
llpAll(q”~LVr”~
(/)
=pAl(l(q”EL)Al(r”EI/)) = PAT (‘&Ike) = p since q”< r”. So p E (x*(qI, rl)vx*(qz, r2)). The fifth condition is trivial from the form of x*(q, r). Lastly, we have to show that the two constructions we have defined are inverse to each other. One way round is easy; for the other, we have to show that if (L, U) is a MacNeille real, then qEL++(zJq’>q)ll(q’EL)* But this follows since ll(q’E
L) implies l(q’E I/).
0
2.4. Corollary. The object *I??of MacNeille reals in R is isomorphic to e*(lR,,,), where e : y& -+ G is the canonical geometric morphism. Proof. Proposition 2.3 establishes a bijection between the global elements of these two objects; to show that they are isomorphic, we have to extend this bijection (naturally in X) to their X-elements for an arbitrary object X of 6. But we may do
181
The Gleason cover of LIropes, I
this simply by repeating the Gleason
the argument
of 2.3 in the topos E/X, bearing
cover of b/X is (y&‘)/e*X.
2.5. Corollary.
in mind that
0
The Pierce representation of the ringed topos (8, *Rs) is (~8, iR,+).
Proof. We have to show that the Dedekind real a sheaf for the finite cover topology on Q7, in of *I?, i.e. the sheaf whose ring of sections over ideal I(p) generated by the idempotent 1 -f(p). be described as
number object in yR, considered as 4, is isomorphic to the Pierce sheaf p E Q7, Noting
is the quotient of *II?by the that Z(p) may alternatively
{XE *R/p-+(x=0)}, it is easy to see that this is precisely
the kernel
of the homomorphism
(where U is the 1 T-closed subobject of 1 corresponding to p) whose transpose is the *R&/U. Thus sections of the Pierce sheaf canonical isomorphism u*( *lRg) -? over p correspond to MacNeille reals in 6/U, which in turn correspond by the argument of 2.3 to Dedekind reals in y(e/u) = y6/e*U. But these are exactly the sections of ll?,g over p. 0 Since one of the conditions equivalent is a local ring [13, Proposition 1.3(r)], it be obtainable as the Zariski spectrum of *lR (and indeed I?) does not in general simple example to illustrate this. Let 6 be the topos Shv(alhl), where cation of the discrete space of natural
to De Morgan’s law is the assertion that *II? is tempting to conjecture that y6‘ might also (6, *II?). Unfortunately this is not true, since have Krull dimension zero; we now give a
aN = N U {a} numbers.
From
is the one-point compactifithe results which we shall
prove in the next section, we know that yA is Shv@N), where ph\l is the Stone-tech compactification of h\l. [RJ is the sheaf of continuous real-valued functions on c&J, i.e. of convergent sequences of real numbers; the global sections of *Re correspond to arbitrary
bounded
sequences
of real numbers.
We shall consider
the stalks
of
these sheaves at the point 03; clearly, points of the topos Spec(i;, *R) lying over the point 00 of i; correspond to prime ideals in the stalk *R,. For each point p ~/Ih\j - iN, the sets Mp = {If] E *R, j f converges
to 0 on the ultrafilter
p}
and
~~=~Lfl~*~;ol{nIf(n)=~~~~~ are prime ideals of *lR,. The ideals Mp are maximal (since *R-/M, is isomorphic to R by the map [f] -limp(f)), and every maximal ideal of *II?, is of this form; the ideals mp are minimal, being generated by idempotents ( = sequences with values in {0, I}), and hence define points of the Pierce spectrum of *R. The ideals Mp thus
182
P. T. Johnsfone
define points of the Zariski spectrum which do not belong to the Pierce spectrum. fact the ring *lRg is easily seen to have infinite Krull dimension; for any sequence positive reals x(n) converging to 0 on an ultrafilter p, we may define I,r,p= {[f] i 3A4, ar>O such that if(n)i
In of
for
and this is a prime ideal lying between mp and Mp. Similar remarks apply to the stalk of IRa at 00, with convergence on a particular ultrafilter replaced by convergence as n-w. On the other hand, it is possible to represent yr’ as a Zariski spectrum, by using the ring iR of “extended reals”. This is the ring whose elements are pairs (L, L/) satisfying the same axioms as MacNeille reals, except that the first axiom (L and lJ are inhabited) is weakened to the statement “L and U are nonempty”, i.e. lT( 3q(q E L)A 3r(r E u)). Mulvey has shown [20] that iR is the real number object in the topos sh,,(rl ); it is therefore a field (in the geometric sense) in sh,,(? ), and hence a (von Neumann) regular ring in ri [12, Proposition 5.61. So the Zariski and Pierce spectra of iR coincide. Also, since @ differs from *I??only “at infinity”, its algebra of idempotents coincides with that of *R, and so its Pierce spectrum is the same as that of *z; i.e. it is the topos y’. (The ring in yh which provides the Pierce/Zariski representation of II?J is of course pi,4.) Yet another way of representing y/C as a spectrum is suggested by the example I! =Shv(ah\i) which we considered above. We noted that the points of yft correspond not only to points of the Pierce spectrum (= minimal prime ideals) of *?, but also to maximal ideals of *R. Now Mulvey [21] has shown how the maximal spectrum of a commutative C*-algebra may be presented as a spectrum in the topostheoretic sense, i.e. as the classifying topos of a propositional geometric theory; and if we interpret this construction internally in 15 for the real normed algebra *?, we will again obtain a site of definition for ye!. We shall not give the details here.
3. The Gleason cover of a space Let ~5 be the topos of sheaves on a space X. What is the relation between ;J’ and the topos of sheaves on yX, where yX is the Gleason cover of X as constructed by topologists? In Section 1, we saw that y/i was characterized up to equivalence as the unique localic r: -topos generated by a minimal, compact, regular, extremally disconnected locale in fi; we also know that it has enough points when I( has [16, Corollary 4.91, and is therefore spatial when (5 is [ 11, Theorem 7.251. Similarly, the Gleason cover e : yX-X is characterized among spaces over X as the unique extremally disconnected space admitting a proper minimal surjection onto X [ 1,3]. Using the fact that topological spaces correspond proper maps f : Y-X between “reasonable” precisely to compact regular locales in Shv(X) [16,17], we may thus deduce that Shv(yX) is equivalent as a Shv(X)-topos to y(Shv(X)).
The Gleason cover of a
Nevertheless,
it is of interest
183
ropes.I
to give a direct
comparison
between
the site of
definition for y(Shv(X)) which we gave at the end of Section 1, and the site provided by the open-set locale of yX. We shall do this only in the case when X is regular; for more general spaces, it seems difficult to give a sufficiently explicit description of the open sets of yX. We recall that Gleason’s original construction [7] of yX, for a compact Hausdorff X, was as the (external) Stone space of the Boolean algebra Q,,(X) of regular open subsets of X-i.e. the space whose points are the prime filters of Q,-(X), and whose open sets correspond to ideals in Q TT (X) (a point being in a particular open set iff the corresponding filter meets the corresponding ideal). The map e : yX-X sends a filter F to its limit, i.e. the intersection of the closures of all the members of F (which can easily be shown to be a singleton), and e-i sends an open set U c X to the ideal of all regular open sets whose closures are contained in U. Subsequent authors [9,25, l] have worked with spaces of maximal filters in the open-set lattice Q(X) rather than in R TT(X); but this is merely a cosmetic change, as the following lemma shows: 3.1. Lemma.
Let H be a Heyting algebra. Then there is a natural bijection between maximal filters in H and maximal (=prime) filters in the Boolean algebra H,,, given by F-FnH,-. Proof. If F is a maximal filter in H, then an element h is in F iff 7 1 h is in F (since if 1 Ih E F, we can adjoin h to F and still have a proper filter). Thus F is determined by Fn H-7; moreover, Ffl H 7 7 is a prime filter in H-.,- if F is a prime filter in H. Conversely, if G is a prime filter in H,,, it must be contained in some maximal filter Fin H, whose intersection with H,, must be G (since G is maximal in H--) and which is therefore
uniquely
determined
by G.
a
Thus the space of prime filters of n,-(X) is homeomorphic to the space of maximal filters of Q(X). If X is not compact, however, it is necessary to work with the subspace of convergent maximal filters, i.e. those which contain the neighbourhood filter at some point of X. To identify following lemma:
the open sets of this subspace,
we use the
3.2. Lemma. Let I, J be ideals of Q,-,(X), and let SI, SJ denote the sets of convergent prime filters in R,,(X) which meet I, J respectively. Then SI c SJ r? for every UE I, there is a (set-theoretic) covering of X by regular open sets Va such that each V,n U is in J. Proof. Suppose the condition is satisfied; let F be a filter in SI, and let x be its limit point. Let U be a set in Ffl I; then by hypothesis we can cover X by regular opens V, with VA WE J. But one such V, must contain .Y and is therefore in F; so V,rl L/E Fn J and hence FE S_I.
184
P. T. Johnsrone
Conversely, suppose the condition is not satisfied; then we have an open set I/ in I for which no covering of X with the required properties exists. Let x be a point of X for which no regular open neighbourhood N of x satisfies Nfl U E J, and let F be the filter in Q-,-(X) based by the sets Nn U. Since this filter does not meet J, we can enlarge it to a maximal filter G not meeting J; but C does meet I (in the set U) and converges
t0 x,
so GESI-SJ.
0
3.3. Corollary. Let Y be the subspace of the Stone space of Q,-(X) consisting of convergent prime filters. Then the open subsets of Y may be identified with ideals I
in !2 7-,(X) satisfying the following condition: (*) For any UESZ~~(X), if there exists a set-theoretic covering { V,]~EA} with each V,II I/ in I, then (I E I. Proof.
Let /be
any ideal of Q,,(X).
of X
Then it is easy to verify that the set
I*={f_J~!Zn~(X)/thereisacovering{VO~ocuA}ofX with each V,n I/ in I} is an ideal of Q,,(X) which satisfies (*) and meets precisely the same filters in Y that Imeets. And if two ideals 1, J both satisfying (*) meet the same filters in Y, then Lemma 3.2 shows that they must be equal. Thus the ideals satisfying (*) provide a set of representatives for the equivalence classes of open sets in the Stone space of B,,(X) under the relation I-J iff Ifl Y=Jn Y. 0 Note that if X is compact, then every ideal of Q,,(X) satisfies (a), since we may take the covering { Va/ (YE A} to be finite, and I is closed under finite joins. Of course, this is just another way of saying that if X is compact then every prime filter in !ZT7(X) converges. Now if X is regular, then the map lim : Y-X, which sends a filter in Y to its limit point, is continuous [25, Lemma 21, and we may take it to be the Gleason cover of X. We thus wish to compare the site provided by the ideals of Q,,(X) satisfying (*) with the site (yC, yJ) which we constructed in Section 1. Note that in this case yC is the poset whose elements are pairs (U, V) with U open in X and V regular open in U, with (U, V)l (U’, V’) iff UC U’ and VC V’, and yJ is generated by covers of two types (a) and (b) as described earlier. (In the definition of yC, we may of course restrict U to lie in some base for the topology on X. For example if X is semiregular, we may require U to be regular open in X; this will have the advantage that the sets V will be regular open in X and not merely in U.) Let F denote the finite cover topology there is a left exact, continuous functor
on the Boolean
algebra
-Q,,(X).
Then
u : (Q~-(X>,FP(YC,YJ) which sends VE G--(X)
to (X, V); and in fact u has a left adjoint
v (necessarily
co-
continuous, geometric
by [8, III 2.51) which sends (U, V) to 71 V. It follows
that we have a
morphism g : y(Shv(X))
where g* is the functor
= Shv(yC, yJ)-Shv(R “compose
-.T(X),F)
with u”, and g* is the functor
and then sheafify”. Since both Shv(yC, rJ) and Shv(Q - _(X), F) are generated
“compose
by subobjects
with v of 1, we
shall investigate the effect of g* and g, on such subobjects (which may, of course, be identified with downward-closed subsets of the two posets which are closed with respect to coverings in the two topologies). 3.4. Lemma. Let I be an ideal in Q-,7(X), regarded as a subobject of 1 in Shv(_Q 7-(X), F). Then g*(i) is the set of all (U, V) E yC for which U admits a (set-
theoretic) covering by open sets U, such that 7 T( U,n V) E I. Proof. It is clear that the sub-presheaf of 1 defined by the specified set is a sheaf for coverings of type (a), and that the inclusion of lov in it is dense with respect to these coverings. So it suffices to prove that it is a sheaf for coverings of type (b). But if we are given (U, V) E yC, a finite cover { VI, . . . , V,,} of V in .Q 7_( I/), and for each i an open cover { CJ,,,I a E A,} of I/ such that each l~(U;,~n V,) is in 1, then on taking a common refinement { L/D~/~EB} of the given covers of I/ we deduce that each -~l(Uafl V,) is in I and hence (since I is an ideal) that each Tl(Upn V) is in 1. So (U, V) is in the given sub-presheaf of 1. C
An ideal I of QT7(X) (*) of Corollary 3.3.
3.5. Corollary.
Proof.
By Lemma VEg*g*I
satisfies g,g*(f) = I iff it satisfies condition
3.4, we have e (X, V)Eg*I b there is a cover of X by open sets U, such that l-qU& V)EZ.
Thus condition obvious. G
(*) is precisely
the assertion
g,g*Zc
I; the reverse
inclusion
is
3.6. Lemma. Suppose X is regular. Then every subobject K of 1 in Shv(yC,yJ) satisfies g*g,(K) = K. map g*g .-t 1, we clearly have g*g,K G K. So suppose we may cover U by open sets CJclwhose closures are contained in CJ; for convenience we shall assume that each Ua is regular open in X. Let V,= U,n V; then Va is regular open in U,, hence in X and a fortiori in U, so that (U, V,) is in K.
Proof.
Since there is a counit
(U, V) E K. Since X is regular,
186
P. T. Johnsrone
But (X, Vcl) is yJ-covered by (C-J,Vcl) and (X- oG, 0) since VGlc U,; and (X- U=, 0) is yJ-covered by the empty family, so (X, Va) is yJ-covered by (U, VU) and hence is in K. Hence k’,~g,K for each a; but since I/ is covered by the O;, and u,n v= v,, we deduce from Lemma 3.4 that (U, V) is in g*g,K. El 3.7. Theorem. Let X be a regular space. Then y(Shv(X)) topos to Shv(yX).
is equivalent as a Shv(X)-
Proof. Taking yX to be the space Y of convergent prime filters in n,,(X), in Corollaries 3.3 and 3.5 that the image of the geometric morphism
we saw
g : y(Shv(X))+Shv(spec(Q,,(X))) is precisely Shv(yX); and Lemma 3.6 shows inclusion, and so induces an equivalence y(Shv(X))
A
s
if X is regular
then
g is an
Shv(yX).
It remains to show that this equivalence the diagram y(Shv(X)) \
that
is a morphism
of Shv(X)-toposes,
i.e. that
Shv(yX) /
Y Yrn Shv(X)
commutes up to equivalence. But if F is a yJ-sheaf defined by U-F(U, U); and lim,(g,F) is the sheaf
on yC, then e.(F) is the sheaf
U-g,F(lim-‘(I/)) = -lim (F(X, V) j V regular
open,
PC U)
= A lim (F(U, V) 1V regular
open,
PC U)
=F(U, U) since (U, U) is yJ-covered by {(U, V) 1Pr U} and each (X, V) is yJ-covered G and (X- V, 0). So the result is established.
by (U, V)
The proof of Theorem 3.7 is not entirely constructive, since we made use of the maximal principle in the proof of Lemma 3.2. However, for compact regular spaces, we can eliminate 3.2 and 3.3, as indicated in the remark after 3.3; and the proofs of 3.4-3.6 are completely constructive. Thus we can interpret them internally in any topos R (replacing the compact regular space X by a compact regular locale A), to obtain
Let A be a compact regular locale in a topos 8. Then the Gleason cover of the localic G-topos &[A] is equivalent to #[[Id&A --)I.
3.8. Corollary.
The Gleason
cover of
a
ropos,I
187
4. The Gleason cover of a finite category In this section, our aim is to describe y6 in the case when ,: is the topos .pcop of presheaves on a small category C. We shall be particularly interested in the cases when C is a finite category or a monoid. If C is a finite category, then for every object c of C the Boolean algebra R--(c) is finite and therefore atomic, and we may identify its Stone space with the discrete set ,4(c) of its atoms. In fact, we do not need to determine Q,,(C) explicitly to find its Stone space:
Let H be a Heyting algebra with the property that every nonzero element of H contains a minimal nonzero element (e.g. any finite Heyting algebra). Then H,, is atomic, and its atoms are in l-l correspondence with the minimal noneero elements of H.
4.1. Lemma.
Proof. Let x be a minimal nonzero element of H. Then 1 lx is an atom of H _ _ ; for anyyEH7, satisfying 0
Cc@)= {~PERI is a minimal
sieve on d.
Theorem. Let C be a finite category, and let 6 : A-C be the discrete opfibration corresponding to the set-valued functor A defined above. Then the induced geometric morphism
4.2.
is (equivalent to) the Gleason cover of YcoD. We recall that the objects of A are pairs (c, R) where c is an object of C and R EA(c); and that morphisms (c, R)+(d,S) in A are morphisms (Y : c-d in C such Proof.
188
P. T. Johnstone
that A(a)(R)
=S.
But if A(a)(R)=&
then it is easy to see that a*(S) 1 R, and so
there is an obvious inclusion functor i : A-+, where (yC, yJ) is the site of definition for y(Shv(C,J)) constructed in Theorem 1.10 (with J taken to be the trivial Grothendieck topology on C). Moreover, in this case yJ contains only covers of type (b), so it is easy to see that every object of yC is yJ-covered by objects in the image of i, but the latter have no nontrivial Lemma of Grothendieck [8, III 4.11, i induces i* : Shv(yC, yJ) d It remains
commutes that Ed
..YAoP.
to check that this is an equivalence
up to isomorphism.
@covers. So by the Comparison an equivalence
of .j/Cop-toposes, i.e. that the diagram
But since 6 is a discrete
opfibration,
it is easy to see
i*F is the functor c - Rg,=,F(c, R), to c - F(c, MC) provided
which is clearly isomorphic
F is a y.J-sheaf.
EI
4.3. Remark. We may verify directly that the category A satisfies the “fill-in of [13, Proposition 1.11, i.e. that .T”*’ satisfies De ,Morgan’s law. condition” Suppose we are given a diagram Cc,RI
in A; then if y : b-c is any element of R, we deduce from the equalities C,(R) = T= Zp(.S) that we can write ay =pS for some 6 E S. Now if P is any minimal sieve on b, then we clearly have Z:,(P) c R and .Zd(P) c S; but by minimality these inclusions must be equalities, and so we have a commutative diagram (b, J-7 A
(c,R) I a
I
6 i
(4 8 p
(e, T)
in A, as required.
0
Next we consider poset or a monoid.
the special forms of Theorem
4.2 in the cases when C is either a
The Gleason cover of a
189
ropos,I
4.4. Corollary. Let P = (P, 5) be a finite poser. Then the Gleason cover of .YpO’is (equivalent to) 2 *Op,where A is the poset of pairs (m,p) of elements of P with m minimal and m up. ordered by (m,p)i(m’,p’)
* m=m’andpIp’.
Proof. After 4.2, we have only to observe that a minimal sieve on an element of P must be generated by a single morphism, and that (m-+p) generates a minimal sieve iff m is a minimal element of P. 0 Let M be a finite monoid. To identify the Gleason cover of the topos .li w of (left) M-sets, we need to identify the minimal sieves on the unique object of M considered as a category, i.e. the minimal left ideals of M. To do this, we employ a series of lemmas relating minimal left and right ideals of &I.
4.5. Lemma. Let R be a minimal right ideal of a monoid M. The set E of idempotents in R forms a right Q-semigroup, i.e. satisfies ef =f for every e, f E E. Proof. By minimality, R is generated by each of its elements: havef=exforsomexEiM.Butthenef=e~,~=ex=fsinceeisidempotent.
so if e, f E E, we must II
Let L, R be minimal left and right idea/s of a finite monoid ,tl. Then there is a unique idempotent in L 0 R.
4.6. Lemma.
Proof. If XE R, y E L, then xy E L rl R. But since M is finite, some power of xy must be idempotent, and this is again in L fI R. Uniqueness follows from Lemma 4.5; for if e, fare idempotents in L fl R, then ef =f since e, f E R, and ef = e since e, f E L. 0
4.1. Lemma. Let R be any minimal right ideal of a finite monoid M. Then the minimal left ideals of A4 are in 1-1 correspondence with the idempotents in R. Proof. Let e be an idempotent in R, and consider a minimal left ideal L contained in Me. By Lemma 4.6, this meets R in an idempotent f =xe say; but then fe=xe2= f = e by Lemma 4.5. So e E L, and hence L = Me, i.e. the ideals Me, e idempotent in R, are all minimal. It follows at once from Lemma 4.6 that they are all distinct, and 3 that they exhaust the minimal left ideals of M. The set of minimal
left ideals of A4 forms a right M-set under
multiplication,
i.e.
Lm= {xmIxE L}. Thus we obtain a right M-set structure on the set E of idempotents in a minimal right ideal R; explicitly, the multiplication is given by e*m = the unique
idempotent
in R 17Mem
= the unique
idempotent
power of em.
P. T. Johnsrone
190
Corresponding
to this M-set,
we have a discrete
opfibration
6 : E-M,
where the
category E may be described as follows: its objects are the elements of E, and morphisms e-f in E are elements m EM such that f is a power of em (the composition in E being given by multiplication in M).
4.8. Corollary. Let A4 be a finite monoid. Then the Gleason cover of the topos .? w of left M-sets is (equivalent to) YEa’, where E is the category described above. Z Let M be a finite monoid. Then the order-completion V? of the Dedekind reals in .YMopis the set of all functions E-R, where E is the set of idempotents in a minimal right ideal of M, with left M-action induced by the right lMaction on E (i.e. (ma f)(e) =f (e*m)). 4.9. Corollary.
Proof. It is well known that in any topos of presheaves, the object of Dedekind reals is simply the constant presheaf with value IR. So by Corollary 2.4 we have to compute the effect of the right Kan extension functor l&on this constant presheaf 0 on E, which gives the answer described above. How far can the results of this section be extended to infinite categories? Clearly, Theorem 4.2 requires only the hypothesis that Q(c) is finite for each object c of C, not that C itself is finite; so it extends, for example, to any poset in which the dounward segment of each element is finite, and to the category of nonempty finite sets. (In the latter case, since the minimal nonempty sieves on a finite set X are those generated by morphisms 1 -+X, we deduce that the Gleason cover of (presheaves on nonempty finite sets) is (presheaves on finite pointed sets).) If Q(c) is allowed to be infinite but still required to satisfy the hypothesis of Lemma 4.1, then the Boolean algebra R-,(c) will still be atomic; but if it is infinite then its Stone space will not be discrete, and so the Gleason cover of .fc” will not be a topos of presheaves. For simplicity, we now restrict ourselves to the case of a monoid. It is well known that if a monoid M has a single minimal left ideal L, then every other (nonempty) left ideal of M contains a minimal left ideal; for ifs is an element of a left ideal L’, then Lx is a minimal left ideal contained in L’. So Q( *) (where * is the unique object of the category M) satisfies the hypothesis of Lemma 4.1, and Q-,-(e) is atomic. The identification of minimal left ideals with idempotents in a minimal right ideal (Lemma 4.7) will remain valid provided some (and hence every) minimal left or right ideal of M is finite; but in general we shall have to work directly with the M-set E of minimal left ideals. Now the power-set of E may be identified with _Q-,(*), and hence with a (nonfull) subcategory of yM; so it is easy to see that yJ-sheaves on yM may be described as sheaves on the Stone-tech compactification /3E of the discrete space E, equipped structure. In particular, Dedekind real with a certain additional “A4-equivariant” numbers in Shv(yM,yJ) may be shown to correspond to continuous real-valued
cover of a
The Gleason
ropos,I
191
functions on /W (the extra structure being constant in this case), from which we deduce that Corollary 4.9 remains valid to the extent that *R is the M-set of bounded functions E-,R, with M-action induced by that on E. (It is of interest to note that this M-set appears in recent work of Banaschewski [2] as being the injective hull of R in the category of M-equivariant Banach spaces, at least when A4 has a finite minimal left ideal. Thus we can say that the Hahn-Banach theorem, in the sense of Burden [4], is valid in the topos of M-sets for any such M.)
Acknowledgements
I am indebted to Bernhard Banaschewski, Martin Hyland, John Isbell and Chris Mulvey for helpful discussions about the subject of this paper.
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