Sheaves, Games, And Model Completions Book

1	Introduction	1
1	Motivating example	1
2	An overview of the book	4
3	How to read the book	6
4	Historical remarks	12
2	Preliminary Notions	15
1	Basic algebraic structures	15
2	Model theoretic background	25
3	Subobjects and regular subobjects	30
4	Finitely presented algebras	32
5	Principal congruences	36
6	The amalgamation property	40
3	Model Completions	49
1	r-Heyting categories	49
2	Model completions and fp algebras	60
4	Heyting Algebras	73
1	Basic definitions	73
2	Finitely presented Heyting algebras	78
3	Duality for Heyting algebras	82
4	A combinatorial result	88
5	Properties of M[subscript H] and [Psi][subscript H]	91
6	Second order extensions of IpC	95
7	Quantifier-elimination	98
8	M[subscript] H is not exact	101
9	Some applications	104
10	Projective Heyting algebras	106
5	Duality for Modal Algebras	115
1	Frames, evaluations and games	115
2	The category of finite S-frames	125
3	The category M[subscript S]	133
4	Finitely presented S-algebras	143
5	Duality	147
6	Regularity of monomorphisms	148
7	Combinatorial Conditions	153
6	Model Completions in Modal Logic	163
1	Negative results	163
2	Diagonalizable algebras	166
3	Varieties of interior algebras	169
7	Algebraically Closed Models	181
1	FLEA's	181
2	Grothendieck Topologies on algebras	182
3	Sh(T[superscript f], J[superscript f]) as a classifying topos	185
4	Models in Set	188
5	The case of Heyting algebras	193
6	Existentially closed algebras	198
8	Open Problems	203
9	Appendix	207
1	Glossary of basic categorical notions	207
2	Internal algebras in categories	222
3	Grothendieck topologies	224
4	Sheaves	225
5	Associated sheaf functor	226
6	Properties of Grothendieck toposes	227
7	Classifying toposes	229
	References	233
	Glossary of Notation	237
	Subject Index	241