Details
| Original language | English |
|---|---|
| Pages (from-to) | 541-572 |
| Number of pages | 32 |
| Journal | Applied categorical structures |
| Volume | 15 |
| Issue number | 5-6 |
| Publication status | Published - 26 Oct 2006 |
Abstract
We provide the appropriate common '(pre)framework' for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann-Lawson duality between continuous frames and locally compact sober spaces, the Hofmann-Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.
Keywords
- Domain, Duality, Locally compact, Open filter, Preframe, Quasiframe, Saturated, Sober, Spatial, Well-filtered
ASJC Scopus subject areas
- Mathematics(all)
- Theoretical Computer Science
- Computer Science(all)
- General Computer Science
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In: Applied categorical structures, Vol. 15, No. 5-6, 26.10.2006, p. 541-572.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Choiceless, pointless, but not useless
T2 - Dualities for preframes
AU - Erné, Marcel
PY - 2006/10/26
Y1 - 2006/10/26
N2 - We provide the appropriate common '(pre)framework' for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann-Lawson duality between continuous frames and locally compact sober spaces, the Hofmann-Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.
AB - We provide the appropriate common '(pre)framework' for various central results of domain theory and topology, like the Lawson duality of continuous domains, the Hofmann-Lawson duality between continuous frames and locally compact sober spaces, the Hofmann-Mislove theorems about continuous semilattices of compact saturated sets, or the theory of stably continuous frames and their topological manifestations. Suitable objects for the pointfree approach are quasiframes, i.e., up-complete meet-semilattices with top, and preframes, i.e., meet-continuous quasiframes. We introduce the pointfree notion of locally compact well-filtered preframes, show that they are just the continuous preframes (using a slightly modified definition of continuity) and establish several natural dualities for the involved categories. Moreover, we obtain various characterizations of preframes having duality. Our results hold in ZF set theory without any choice principles.
KW - Domain
KW - Duality
KW - Locally compact
KW - Open filter
KW - Preframe
KW - Quasiframe
KW - Saturated
KW - Sober
KW - Spatial
KW - Well-filtered
UR - http://www.scopus.com/inward/record.url?scp=36949013217&partnerID=8YFLogxK
U2 - 10.1007/s10485-006-9029-4
DO - 10.1007/s10485-006-9029-4
M3 - Article
AN - SCOPUS:36949013217
VL - 15
SP - 541
EP - 572
JO - Applied categorical structures
JF - Applied categorical structures
SN - 0927-2852
IS - 5-6
ER -