Monoids, Acts and Categories, With Applications to Wreath Products and Graphs

Foreword
vii
Introduction
xii
I
Elementary properties of monoids, acts and categories
1
1
Sets and relations
1
2
Groupoids, semigroups and monoids
13
3
Some classes of semigroups
24
4
Acts over monoids (monoid automata)
42
5
Decompositions and components
62
6
Categories
78
7
Functors
89
II
Constructions
103
1
Products and coproducts
103
2
Pullbacks and pushouts
114
3
Free objects and generators
138
4
Cofree objects and cogenerators
149
5
Tensor products
155
6
Wreath products of monoids and acts
165
7
The wreath product of a monoid with a small category
. . . .
175
III Classes of acts
183
1
Injective acts
184
2
Divisible acts
195
3
Principally weakly injective acts
200
4
fg-weakly injective acts
204
5
Weakly injective acts
205
6
Absolutely pure acts
207
7
Cogenerators and overview
214
8
Torsion free acts
218
9
Flatness of acts and related properties
223
10
Principally weakly flat acts
225
11
Weakly flat acts
233
12
Flat acts
238
13
Acts satisfying Condition (Ρ)
249
14
Acts satisfying Condition (E)
257
15
Equalizer flat acts
262
16
Pullback flat acts and overview
267
17
Projective acts
274
18
Generators
291
19
Regular acts and overview
300
IV Homological classification of monoids
306
1
Principal weak injectivity
307
2
On fg-weak injectivity
311
3
Weak injectivity
319
4
Absolute purity
322
5
Injectivity and overview
326
6
Torsion freeness and principal weak
flatness
335
7
Weak
flatness
339
8
Flatness
344
9
Condition (P)
360
10
Strong
flatness
367
11
Projectivity
372
12
Projective generators
381
13
Freeness and overview
385
14
Regularity of acts
394
V
Equivalence and Duality
397
1
Adjoint functors
398
2
Categories equivalent to Act — S
427
3
Morita equivalence of monoids
437
4
Endomorphism monoids of generators
455
5
On Morita duality
467
Bibliography
483
Index of symbols
517
Index
521