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Profinite Algebraic Homotopy (Shortened Version)

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Profinite Algebraic Homotopy (Shortened Version) Timothy Porter Profinite Algebraic Homotopy (shortened version) April 18, 2009 Contents 1 Algebraic Preliminaries .................................... 5 1.1 Pro-objects . 5 1.2 Profinite topological spaces and profinite spaces . 7 1.3 Profinite and pro-C groups. 9 1.4 Free profinite groups . 13 1.5 Group and groupoid objects in a category. 15 1.6 Enriched categories - just a taster . 20 1.7 Profinite completions for groupoids . 23 1.8 Free profinite groupoids . 26 1.9 Pseudocompact Algebras and Modules. 28 1.10 The completed group algebra of a profinite group, G. 29 1.11 Some generalities on Pseudocompact modules . 30 2 Algebraic homotopy theory: preliminaries. 33 2.1 Algebraic Homotopy and Algebraic Models for Homotopy Types.................................................. 33 2.2 What is a Homotopy Theory? . 36 2.3 Simplicial Groups . 41 2.3.1 Simplicial Objects in Categories other than Sets. 41 2.3.2 Homotopies for simplicial groups . 42 2.3.3 The Moore complex and the homotopy groups of a simplicial group . 43 2.4 Kan complexes and Kan fibrations. 44 2.5 Again: ` What is a homotopy theory?' . 46 2.6 Simplicial background . 48 2.6.1 Simplicially enriched categories and groupoids. 48 2.6.2 The Dwyer-Kan S-groupoid functor. 50 2.6.3 The W construction. 53 2.6.4 Simplicial Automorphisms and Regular Representations. 54 2.6.5 W , W and twisted cartesian products. 55 2.6.6 n-types of spaces, simplicial sets and S-group(oid)s . 57 4 Contents 2.7 Profinite homotopy . 59 2.7.1 Homotopy procategories . 59 2.7.2 Profinite homotopy types . 61 2.7.3 Profinite completion of homotopy types . 63 3 Pro-C Crossed Modules. ................................... 65 3.1 Crossed modules and Pro-C crossed modules. 65 3.1.1 Crossed Modules . 65 3.1.2 Pro−C crossed modules . 68 3.2 Elementary Properties. 70 3.2.1 Images are normal. 71 3.2.2 Kernels are central . 71 3.2.3 Kernels are modules . 71 3.2.4 Categorical gloss on `extreme cases' . 72 3.2.5 Abelianisation. 73 3.2.6 The intersection A \ [C; C]. 73 3.2.7 Free pro-C groups: some difficulties. 74 3.3 Induced and restricted pro-C crossed modules. 74 3.3.1 Restriction along a homomorphism φ. 75 3.3.2 The extreme cases: normal subgroups and modules. 75 3.3.3 Extension along φ. ................................ 76 3.3.4 The extreme cases revisited. 78 3.3.5 Extension along an epimorphism. 78 3.3.6 Induction from final crossed modules. 79 3.3.7 Various forgetful functors and their right adjoints. 81 3.4 Free pro-C crossed modules. 81 3.4.1 A simplified construction. 82 3.4.2 A special case: when f(X) topologically generates G . 83 3.4.3 A generalisation. 84 3.4.4 Projective profinite crossed modules. 85 4 Identities amongst relations, simplicial groups and other connections ................................................ 87 4.1 Identities among relations and crossed modules . 87 4.1.1 The complex of a presentation . 87 4.1.2 Group presentations, identities and 2-syzyzgies: the discrete case . 88 4.2 Profinite analogues of identities . 96 4.2.1 The module of identities. 96 4.2.2 The Abelianisation of the `top' group in the free case. 97 4.2.3 Identifying problems and problems of identification.. 97 4.2.4 Comparisons. 98 4.3 Profinite simplicial groups and crossed modules. 98 4.3.1 Replacing topological by simplicial constructions. 98 4.3.2 A crossed module from a simplicial group. 99 Contents 5 4.3.3 The kernel and cokernel. 99 4.3.4 Ker @. ...........................................100 4.3.5 \ . and back again". 100 4.3.6 Back yet again! . 101 4.3.7 Simplicial normal subgroups.. 101 4.3.8 The Brown-Loday lemma . 102 5 Pro-C completions of crossed modules. 107 5.1 Cat1-groups, and their pro-C analogues . 107 5.1.1 Cat1-groups. 107 5.1.2 Cat1-groups and crossed modules . 108 5.1.3 The pro−C versions . 108 5.2 Pro−C completions of cat1-groups and crossed modules . 110 5.2.1 Pro−C completions of cat1-groups . 110 5.2.2 Pro−C completions for crossed modules. 110 5.3 Pro−C completions of groups and crossed modules. 111 5.3.1 The two completions agree on the base. 111 5.3.2 The cofinality condition . 112 5.3.3 Completions and the cofinality condition. 112 5.3.4 Completions of crossed modules with nilpotent actions.. 114 5.4 Pro-C completions of free crossed modules. 115 5.5 Remarks on the non exactness of pro-C completions . 116 5.6 Completions of simplicial groups and crossed modules. 117 6 Pro-C crossed complexes and chain complexes . 121 6.1 Continuous Derivations and derived pseudocompact modules. 121 6.1.1 Definitions . 121 6.1.2 Existence . 122 6.1.3 Derivation modules and augmentation ideals: the case of finite groups . 122 6.1.4 Derivation modules and augmentation ideals: the general case. 124 6.1.5 Topological generation of I^C(G). 126 6.1.6 When G is free pro-C...............................128 6.1.7 (Dφ; dφ); the general case. 128 6.1.8 Dφ for φ, the counit FC(X) ! G. 129 6.2 Associated module sequences . 129 6.2.1 Homological background . 129 6.2.2 The exact sequence in the pro-C case. 130 6.2.3 Reidemeister-Fox derivatives and Jacobian matrices . 133 6.3 Profinite crossed complexes. 137 6.3.1 Profinite crossed complexes: the Definition. 138 6.3.2 Example: pro-C crossed resolutions . 138 6.3.3 The standard crossed pro-C resolution . 139 6.3.4 G-augmented pro−C crossed complexes. 140 6 Contents 6.4 From crossed complexes to chain complexes. 140 6.4.1 From chain complexes to crossed complexes, ... 140 6.4.2 ... and back again. 141 6.4.3 Pro−C crossed resolutions and chain resolutions . 142 6.4.4 Standard pro−C crossed resolutions and bar resolutions 144 6.4.5 The pseudocompact module of identities. 144 6.4.6 Example: . 145 7 Pseudocompact coefficients ................................147 7.1 Introduction. 147 7.2 Continuous cohomology of profinite groups. 149 7.2.1 The definition . 149 7.2.2 Interpretation in low dimensions . 150 7.2.3 The bar resolution . 150 7.2.4 Induced and Coinduced Modules . 151 7.2.5 Limits of finite coefficients. 151 7.3 Relative cohomology groups. 152 7.3.1 What are they? . 152 7.3.2 The Auslander-Ribes theory . 153 7.3.3 The Adamson-Hochschild theory. 156 7.4 Extensions and long exact sequences. 158 7.4.1 Long exact sequences in cohomology . 158 7.4.2 The Lyndon-Hochschild-Serre spectral sequence. 159 7.4.3 Long exact sequences and relative cohomology . 160 7.5 The Profinite Schur Multiplier . ..
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