Page 7 - Volume 41
P. 7

Contents
Preface ix
1 Preliminaries 1
1.1 Posets ................................. 1 1.2 (Galois)adjunctions ......................... 1 1.3 Heytingalgebras ........................... 2 1.4 Pseudocomplementsandcomplements ............... 3 1.5 Frames................................. 4 1.6 Completely prime filters and meet-irreducibility . . . . . . . . . . 4 1.7 Sobriety ................................ 5 1.8 Whatwewillneedaboutcategories................. 6
2 Locales, frames and spaces 7
2.1 Spacesandframes........................... 7 2.2 LocalesandthecategoryLoc .................... 8 2.3 Pointsofalocaleandthespectrum................. 9 2.4 Localesassociatedwithspaces.ThefunctorLc. . . . . . . . . . . 10 2.5 Reconstructing spaces and continuous mappings . . . . . . . . . . 11
3 The spectrum adjunction. Spatiality 13
3.1 ThespecializationorderinPt(L) .................. 13 3.2 Thenaturaltransformationσ:LcPt→Id . . . . . . . . . . . . . 13 3.3 Thenaturaltransformationλ:Id→PtLc . . . . . . . . . . . . . 14 3.4 Thespectrumadjunction....................... 14 3.5 Spatiallocales............................. 15 3.6 Thesobrification ........................... 16
4 The basic structure of morphisms in Loc 19 4.1 Localicmapsversusframehomomorphisms . . . . . . . . . . . . 19 4.2 Localicmapsversuscontinuousmaps................ 19 4.3 Specialmonomorphismsandepimorphisms. . . . . . . . . . . . . 19 4.4 EpimorphismsinLoc......................... 21
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