25:
629:) (hence the notation pt) with a suitable topology. But how can we recover the set of points just from the locale, though it is not given as a lattice of sets? It is certain that one cannot expect in general that pt can reproduce all of the original elements of a topological space just from its lattice of open sets – for example all sets with the
502:– the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras). Now an obvious question is: To what extent is a topological space characterized by its locale of open sets?
633:
yield (up to isomorphism) the same locale, such that the information on the specific set is no longer present. However, there is still a reasonable technique for obtaining "points" from a locale, which indeed gives an example of a central construction for Stone-type duality theorems.
736:) → Ω(1). The open set lattice of the one-element topological space Ω(1) is just (isomorphic to) the two-element locale 2 = { 0, 1 } with 0 < 1. After these observations it appears reasonable to define the set of points of a locale
469:
892:
Now that a set of points is available for any locale, it remains to equip this set with an appropriate topology in order to define the object part of the functor pt. This is done by defining the open sets of
592:
of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from
687:
clearly determines one point: the element that it "points" to. Therefore, the set of points of a topological space is equivalently characterized as the set of functions from 1 to
1344:
of spatial locales. This main result is completed by the observation that for the functor pt o Ω, sending each space to the points of its open set lattice is left adjoint to the
314:
1227:
265:
1214:. Using the characterization via meet-prime elements of the open set lattice, one sees that this is the case if and only if every meet-prime open set is of the form
771:
to 2. But these morphisms are characterized equivalently by the inverse images of the two elements of 2. From the properties of frame morphisms, one can derive that
299:
114:
228:
1151:
At this point we already have more than enough data to obtain the desired result: the functors Ω and pt define an adjunction between the categories
937:
as morphisms, but one can of course state a similar definition for all of the other equivalent characterizations. It can be shown that setting Ω(pt(
759:), it is worthwhile to clarify the concept of a point of a locale further. The perspective motivated above suggests to consider a point of a locale
54:
1128:
as described above. Another convenient description is given by viewing points of a locale as meet-prime elements. In this case we have ψ(
407:
844:(0) is a principal prime ideal. It turns out that all of these descriptions uniquely determine the initial frame morphism. We sum up:
702:, all set-theoretic elements of a space are lost, but – using a fundamental idea of category theory – one can as well work on the
1040:
to 2. Again, this can be formalized using the other descriptions of points of a locale as well – for example just calculate (
144:
Probably the most general duality that is classically referred to as "Stone duality" is the duality between the category
1451:
130:
136:
This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail.
1467:
1433:
1413:
1391:
1226:. Alternatively, every join-prime closed set is the closure of a unique point, where "join-prime" can be replaced by
235:(in fact, this statement is equivalent to that assumption). The significance of this result stems from the fact that
76:
884:
All of these descriptions have their place within the theory and it is convenient to switch between them as needed.
47:
394:
211:
Now one can easily obtain a number of other dualities by restricting to certain special classes of sober spaces:
1090:)) was applied. This mapping is indeed a frame morphism. Conversely, we can define a continuous function ψ from
334:
The starting point for the theory is the fact that every topological space is characterized by a set of points
313:
and totally order-disconnected). One obtains a representation of distributive lattices via ordered topologies:
247:
305:
Stone's representation for distributive lattices can be extended via an equivalence of coherent spaces and
126:
1368:. The case of the functor Ω o pt is symmetric but a special name for this operation is not commonly used.
295:
232:
1459:
1425:
1405:
1527:
181:
37:
568:(like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a
173:
161:
41:
33:
374:
are given by set unions and finite set intersections, respectively. Furthermore, it contains both
208:. The involved constructions are characteristic for this kind of duality, and are detailed below.
1191:). For this it is necessary that both ψ and φ are isomorphisms in their respective categories.
1164:
752:(consider again the indiscrete topology, for which the open set lattice has only one "point").
165:
58:
1522:
1306:
581:
106:
94:
837:
825:
808:
801:
533:
1477:
1443:
653:, but there is in fact a more useful description for our current investigation. Any point
8:
630:
153:
1056:
As noted several times before, pt and Ω usually are not inverses. In general neither is
1494:
1397:
780:
193:
122:
1249:)) is always surjective. It is additionally injective if and only if any two elements
666:
from the one element topological space 1 (all subsets of which are open) to the space
1517:
1463:
1429:
1422:
Categorical foundations. Special topics in order, topology, algebra, and sheaf theory
1409:
1387:
1345:
1071:
102:
1490:
1473:
1439:
1379:
602:
363:
205:
1301:
If this condition is satisfied for all elements of the locale, then the locale is
509:
of topological spaces has as morphisms the continuous functions, where a function
1383:
621:
that in a certain sense "inverts" the operation of Ω by assigning to each locale
383:
306:
272:
250:
98:
1207:
703:
220:
118:
1511:
1424:. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge:
1230:
since we are in a distributive lattice. Spaces with this property are called
1060:
514:
310:
1458:. Cambridge Tracts in Theoretical Computer Science. Vol. 5. Cambridge:
321:
Many other Stone-type dualities could be added to these basic dualities.
280:
149:
90:
748:) is in one-to-one correspondence to a point of the topological space
1211:
379:
525:
464:{\displaystyle x\wedge \bigvee S=\bigvee \{\,x\wedge s:s\in S\,\},}
367:
351:
343:
1499:
744:
to 2. Yet, there is no guarantee that every point of the locale Ω(
505:
As already hinted at above, one can go even further. The category
371:
109:. Today, these dualities are usually collected under the label
1493:(2011). "A topos-theoretic approach to Stone-type dualities".
324:
1175:
The above adjunction is not an equivalence of the categories
1484:
1167:ψ and φ provide the required unit and counit, respectively.
315:
Priestley's representation theorem for distributive lattices
1328:) is sober. Hence, it follows that the above adjunction of
1420:
Pedicchio, Maria
Cristina; Tholen, Walter, eds. (2004).
613:
The goal of this section is to define a functor pt from
266:
Stone's representation theorem for distributive lattices
637:
Let us first look at the points of a topological space
121:. Stone-type dualities also provide the foundation for
1336:
restricts to an equivalence of the full subcategories
1309:
for a similar condition in more general categories.)
972:
rather canonically by defining, for a frame morphism
410:
271:
When restricting further to coherent spaces that are
1269:
can be separated by points of the locale, formally:
300:
Stone's representation theorem for
Boolean algebras
139:
115:
Stone's representation theorem for
Boolean algebras
1394:. (available free online at the website mentioned)
1120:is just the characteristic function for the point
463:
223:(and coherent maps) is equivalent to the category
1377:Stanley N. Burris and H. P. Sankappanavar, 1981.
1078:)). However, when introducing the topology of pt(
961:))). It is common to abbreviate this space as pt(
1509:
1051:
641:. One is usually tempted to consider a point of
46:but its sources remain unclear because it lacks
1419:
1404:, Cambridge Studies in Advanced Mathematics 3,
815:preserves finite infima and thus the principal
800:preserves arbitrary suprema). In addition, the
1144:} denotes the topological closure of the set {
231:(and coherent maps), on the assumption of the
113:, since they form a natural generalization of
1312:Finally, one can verify that for every space
490:) is not an arbitrary complete lattice but a
401:) inherits the following distributivity law:
953:} does really yield a topological space (pt(
755:Before defining the required topology on pt(
455:
429:
1305:, or said to have enough points. (See also
329:
325:Duality of sober spaces and spatial locales
968:Finally pt can be defined on morphisms of
362:) has certain special properties: it is a
192:is the basis for the mathematical area of
164:with appropriate frame homomorphisms. The
1498:
1489:
1163:, where pt is right adjoint to Ω and the
564:). Furthermore, it is easy to check that
454:
432:
290:, the restriction yields the subcategory
77:Learn how and when to remove this message
1450:
397:finite infima and arbitrary suprema, Ω(
200:—the category of all locales, of which
117:. These concepts are named in honor of
16:Relationship between certain categories
1510:
1170:
1012:. In words, we obtain a morphism from
740:to be the set of frame morphisms from
694:When using the functor Ω to pass from
309:(ordered topological spaces, that are
783:), which contains a greatest element
683:. Conversely, any function from 1 to
608:
657:gives rise to a continuous function
18:
196:, which is devoted to the study of
13:
1320:) is spatial and for every locale
14:
1539:
887:
1183:(or, equivalently, a duality of
1148:} and \ is just set-difference.
239:in turn is dual to the category
140:Overview of Stone-type dualities
23:
933:. Here we viewed the points of
540:. Thus any continuous function
390:) into the powerset lattice of
1380:A Course in Universal Algebra.
852:is equivalently described as:
229:coherent (or spectral) locales
93:, there is an ample supply of
1:
1371:
1052:The adjunction of Top and Loc
1032:before applying the morphism
874:a completely prime filter of
474:for every element (open set)
127:theoretical computer science
7:
1020:) by applying the morphism
862:a principal prime ideal of
552:defines an inverse mapping
275:, one obtains the category
233:Boolean prime ideal theorem
172:is the category of spatial
10:
1544:
1460:Cambridge University Press
1426:Cambridge University Press
1406:Cambridge University Press
1082:) above, a mapping φ from
775:(0) is a lower set (since
723:is mapped to a morphism Ω(
1237:Conversely, for a locale
828:. Now the set-inverse of
524:) of any open set in the
1281:, then there is a point
1265:is not less-or-equal to
868:a meet-prime element of
492:complete Heyting algebra
330:The lattice of open sets
32:This article includes a
1165:natural transformations
838:completely prime filter
350:, i.e. a subset of the
182:categorical equivalence
61:more precise citations.
1485:Abstract Stone Duality
1206:)) is a homeomorphism
855:a frame morphism from
706:. Indeed, any "point"
465:
107:partially ordered sets
1307:well-pointed category
582:contravariant functor
513:is continuous if the
466:
358:. It is known that Ω(
125:and are exploited in
95:categorical dualities
1340:of sober spaces and
848:A point of a locale
763:as a frame morphism
408:
154:continuous functions
1408:, Cambridge, 1982.
1228:(join-) irreducible
1171:The duality theorem
631:indiscrete topology
625:a set of points pt(
298:. Thus one obtains
1456:Topology via logic
925:for every element
826:meet-prime element
609:Points of a locale
584:from the category
570:morphism of frames
461:
194:pointless topology
123:pointless topology
105:and categories of
103:topological spaces
34:list of references
1491:Caramello, Olivia
1364:)) is called its
1346:inclusion functor
1016:to 2 (a point of
580:then Ω becomes a
572:. If we define Ω(
478:and every subset
346:of elements from
283:. On the side of
156:and the category
129:for the study of
87:
86:
79:
1535:
1528:Duality theories
1504:
1502:
1481:
1447:
1098:)) by setting ψ(
1072:order-isomorphic
588:to the category
470:
468:
467:
462:
364:complete lattice
307:Priestley spaces
296:Boolean algebras
206:full subcategory
131:formal semantics
97:between certain
82:
75:
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68:
62:
57:this article by
48:inline citations
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1452:Vickers, Steven
1436:
1398:P. T. Johnstone
1384:Springer-Verlag
1374:
1222:} for a unique
1173:
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802:principal ideal
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338:and a system Ω(
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264:—one obtains
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166:dual category
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111:Stone duality
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1523:Order theory
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1402:Stone Spaces
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1300:
1294:
1293:) = 1 and p(
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1246:
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1199:
1195:
1194:For a space
1193:
1188:
1184:
1180:
1176:
1174:
1160:
1156:
1152:
1150:
1145:
1141:
1140:}, where Cl{
1137:
1133:
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1103:
1099:
1095:
1091:
1087:
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1068:
1064:
1061:homeomorphic
1057:
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1045:
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1024:to get from
1021:
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670:by defining
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398:
391:
387:
382:. Since the
375:
359:
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339:
335:
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281:Stone spaces
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248:distributive
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150:sober spaces
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53:Please help
45:
809:prime ideal
796:(0) (since
649:of the set
548:to a space
486:). Hence Ω(
370:and finite
260:is dual to
246:of bounded
176:denoted by
160:of spatial
91:mathematics
59:introducing
1512:Categories
1478:0668.54001
1444:1034.18001
1372:References
1261:for which
1124:from 1 to
1067:)) nor is
99:categories
67:March 2018
1500:1103.3493
1212:bijective
1111:), where
836:(1) is a
807:(0) is a
601:which is
449:∈
437:∧
427:⋁
418:⋁
415:∧
395:preserves
384:embedding
380:empty set
344:open sets
273:Hausdorff
253:. Hence,
1518:Topology
1454:(1989).
1094:to pt(Ω(
1086:to Ω(pt(
1074:to Ω(pt(
1063:to pt(Ω(
996:) as pt(
957:), Ω(pt(
941:)) = {φ(
921:) = 1 },
840:because
781:monotone
526:codomain
378:and the
352:powerset
251:lattices
1360:, pt(Ω(
1303:spatial
1273:if not
1245:→ Ω(pt(
1202:→ pt(Ω(
992:) → pt(
603:adjoint
560:) to Ω(
556:from Ω(
368:suprema
311:compact
174:locales
55:improve
1476:
1466:
1442:
1432:
1412:
1390:
1297:) = 0.
1285:in pt(
1210:it is
1102:) = Ω(
1048:)(0).
988:): pt(
905:) = {
811:since
715:: 1 →
679:(1) =
605:to Ω.
534:domain
500:locale
372:infima
237:CohLoc
225:CohLoc
180:. The
162:frames
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