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R. Street, R. Walters, The comprehensive factorization of a functor, Bull. Amer.
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C. Wells, A triple in Cat. Proc. Edinburgh Math. Soc. 23 (1980), 261-268.
G. C. Wraith, Algebras over theories. Colloq. Math. 23 (1971), 180-190.
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Springer Lecture Notes in Mathematics 871 (1981), 390-413.
Index

2-valued, 238 Beck™s Theorem, 115
S-separated, 224 Beck, Jon, 62
S-sheaf, 224 Birkhoªs Theorem, 288
U -contractible equivalence pair, 288 Boolean, 193, 237, 245
G-coalgebra, 105 Boolean algebra, 15
P, 170 Boolean topos, 240
j-closed, 196 Butler™s Theorem, 137
j-dense, 196
CABA, 15
j-separated, 201
canonical, 227
Cat(E ), 218
canonical underlying functor, 307
T-algebra, 103
cartesian closed, 180, 181
-divisible, 277
categorical theory, vii
Ab, 3, 24, 47 category, 1, 8, 162
absolute coequalizer, 110 category object, 217
absolutely closed, 202 category of Abelian groups, 3, 15, 62, 196
AC, 242 category of groups, 1, 3, 13, 16, 48, 49,
adjoint, 54, 56, 59, 99, 152 69, 104, 126
adjoint functor theorem, 59 category of models, 281
adjoint transpose, 54 category of sets, ix, 1, 15, 242, 302
adjoint triple, 138 category of topological spaces, 1
adjoint tripleable, 138 change of base space, 82
algebraic theory, 154 classifying topos, 280
application, 18 coalgebras, 105
arrow, 1 cocone induced by, 274
arrow category, 9 cocontinuous, 48
arrow congruence, 10 codomain, 1
arrows, 34, 35 coequalizer, 47
associated sheaf functor, 202 cogenerating set, 65
atom, 15 cograph, 8
atomic, 15 coherent, 253
Axiom of Choice, 242 coherent family, 241
coherent sketch, 280
Barr™s Embedding Theorem, 269 coherent theory, 280, 283
Beck conditions, 91 colimit, 47, 302
323
324 Index
comma category, 13 discrete op¬bration, 223
commutative cone, 36 distributive law, 296
commutative cone over D based on T , 29 domain, 1
commutative diagram, 3 domain of variation, 20
commute, 3 dual, 5
commute serially, 84, 111 Duskin™s Theorem, 289
compatible, 299
e¬ective descent type, 107
complete, 15, 37
e¬ective epimorphic family, 225
components, 16, 36
e¬ective equivalence, 89
composite, 1, 235
e¬ective equivalence relation, 51
Cone, 34
Ehresmann-Kennison Theorem, 147
congruence, 51
Eilenberg-Moore comparison functor, 105
connected, 10
element, 25, 29
connected diagrams, 44
element free, 3
connected graph, 44
element of, 19
conservative generator, 228
empty path, 66
continuous, 39
epi, 21
contractible, 109
epi-mono factorization, 183
contractible coequalizer, 109
epic, 21
contractible coequalizer pair, 110
epimorphic family, 224
contravariant functor, 11
epimorphism, 21, 53, 242
contravariant hom functors, 12
equalizer, 39, 84
coproduct, 47
equational theory, 67, 158
cotriple, 100
equivalence, 13
counit, 55
equivalence pair, 288
covariant, 12
equivalence relation, 40, 50, 89
covariant functor, 11
equivalent, 23
cover, 224
essentially small, 253
creates, 42
evaluation map, 75
crible, 223
exact, 141
Crude Tripleability Theorem, 126
exact category, 51, 237
CTT, 126
exact functor, 237
de¬ned over, 19 exponential, 180
Deligne™s Theorem, 253
factorization system, 187
dense, 150
factors through, 22
descent type, 107
faithful, 11
designated ¬nite limits, 39
¬ber, 81, 222
diagram, 35
¬ber product, 30
diagram of type, 35
¬bration, 218
direct image, 82
¬lter, 282
discrete, 37
¬ltered, 158
discrete category, 63
Index 325
¬ltered colimit, 67, 158 Grp, 1, 49
¬ltered diagram, 67
has ¬nite limits, 38
¬nite diagram, 35
has products, 38
¬nite products, 38
Heyting algebra, 191
¬nite-sum sketch, 278
homomorphism, 35
¬nitely complete, 37
homotopy types, 2
¬‚abby, 98
homset, 1
forgetful functor, 11
FP theory of groups, 143
IAC, 242
FP-category, 143
idempotent, 195
FP-sketch, 145, 283
idempotent triple, 217
FP-theory generated by S , 151
identity, 1
free, 105
identity property, 8
free group, 12, 28, 97
image, 183, 188
free triple, 306
index graph, 35
Freyd™s Embedding Theorem, 241
in¬‚ationary, 195
FS-sketch, 278
initial object, 5, 186
FS-theory, 278
injective, 77
full, 3, 11
internal axiom of choice, 242
function, 7
internal functor, 218
functor, 11
intersection, 184
functor category, 17, 74, 75
Intuitionistic logic, 190
inverse, 4
generator, 228, 293
isomorphic, 13
generic, 280
isomorphism, 4
generic element, 20
geometric morphism, 83, 208, 215, 233
jointly monic, 45, 288
geometric sketch, 279
geometric theory, 279, 283 Kan extension, 60
germ, 81 Kan extensions, 222
Giraud sieve, 234 kernel pair, 40, 89
Giraud topology, 234 Kleisli category, 103
Giraud™s Theorem, 228
global element, 23 LE-category, 156
global section, 242 LE-functor, 156
global support, 179, 242 LE-sketch, 156, 283
Godement™s rules, 19 LE-theory, 156
left C -object, 218
graph, 8, 34
Grothendieck topology, 224 left adjoint, 54
Grothendieck topos, 229, 233, 253, 268, left exact, 38, 42
269 left exact cotriple, 208
group, 38, 39, 143 left exact triple, 212
groupoid, 63 left Kan extension, 60
326 Index
lex total, 233 pointed endomorphism structure, 254
lifting, 297 points, 268
limit, 36 pointwise construction of adjoints, 56
limit slice, 244 poset, 2, 38
Linton™s Theorem, 302 power object, 73, 165, 170, 172
locally small, 1 powerful epi, 243
logical, 175 powerset, 12, 28
logical category, 242 pre-FS-theory, 278
pre-geometric theory, 279
map, 1 Precise Tripleability Theorem, 116
model, 145, 274 preorder, 10
model category, 281 preregular theory, 274
monic, 21 preserves, 11, 42, 75
mono, 21 presheaf, 77, 224
mono square, 85 pretopos, 237
monomorphism, 21 product, 37
morphism, 1, 145 product of categories, 3
morphism of sites, 224 product projection, 37
morphism of triples, 128 projections, 14, 37
projective, 293
natural equivalence, 16
projective generator, 293
natural number object, 255
projective sketch, 144
natural transformation, 16, 27, 35
PTT, 117
near exact, 237
pullback, 30, 84
NNO, 255
pullback diagram, 30
nodes, 35
pun, 141, 222
object, 1
quotient category, 10
objects, 34, 35
op¬bration, 222
re¬nes, 223
opposite category, 5
re¬‚ective, 64
order-preserving, 195
re¬‚ector, 64
orthodox semigroup, 287
re¬‚ects, 11, 42
over, 3
re¬‚ects isomorphisms, 108
re¬‚exive, 112
Par´™s Theorem, 170
e
re¬‚exive coequalizer, 112
parallel pair, 108
regular category, 48
path, 66
regular epimorphic, 282
path category, 67
regular epimorphic family, 225
PE-structure, 254
regular epimorphism, 48, 89
Peano property, 260
regular functor, 237
Peano system, 255
regular generator, 228, 293
point, 280
regular image, 303
pointed, 126
Index 327
regular monomorphism, 48 split monomorphism, 21
regular projective, 293 split op¬bration, 223
regular projective generating set, 282 square brackets, 31
regular projective generator, 282, 293 SS, 242
regular ring, 275 stable, 224, 225

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