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

Description

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