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Categories for AGI

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Contents

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Front matter

  1. Prefacecfa-0001
  2. Why category theory?cfa-0002
  3. Why AGI?cfa-0003
  4. The path through the bookcfa-0004
  5. How to read the bookcfa-0005
  6. The boundary of the bookcfa-0006
  7. A Trail Map of Categories for AGIcfa-0007
  8. The landscape from abovecfa-0008
  9. Six regions of the bookcfa-0009
  10. The trails that cross the regionscfa-0010
  11. Five kinds of claimcfa-0011
  12. Choose a routecfa-0012
  13. Signposts for the journeycfa-0013

A Categorical Language

1 Category Theory for AGI5 sections
  1. 1.1 A Frontier Question: Consciousnesscfa-0015
  2. 1.2 Transformerscfa-0016
  3. 1.3 From Transformers to Function Categoriescfa-0017
  4. 1.4 Summarycfa-0018
  5. Suggested Readingcfa-0019
2 Functors for AGI8 sections
  1. 2.1 Generative AI with Universal Coalgebrascfa-0020
  2. 2.1.1 Reasoning about Circularity in AI and MLcfa-0021
  3. 2.1.2 Three Representations for Reasoning about Circularitycfa-0022
  4. 2.1.3 AI and ML Systems as Universal Coalgebrascfa-0023
  5. 2.1.4 Search Spaces as Coalgebrascfa-0024
  6. 2.1.5 Clustering as a Functorcfa-0025
  7. 2.2 Summarycfa-0026
  8. Suggested Readingcfa-0027
3 Representable Functors and the Yoneda Lemma5 sections
  1. 3.1 Natural Transformations and Universal Arrowscfa-0028
  2. 3.2 Yoneda Lemmacfa-0029
  3. 3.3 Yoneda Intuition via Visualizing Self-Attention in BERTsec-yoneda-bertviz
  4. 3.4 Summarycfa-0030
  5. Suggested Readingcfa-0031
4 Diagrams and Universal Constructions5 sections
  1. 4.1 Universality of Diagramscfa-0032
  2. 4.1.1 Universal Arrows and Elementscfa-0033
  3. 4.2 The Category of Elementscfa-0034
  4. 4.3 Summarycfa-0035
  5. Suggested Readingcfa-0036

Compositional Learning

5 Categorical Deep Learning11 sections
  1. 5.1 Symmetric Monoidal Categorycfa-0038
  2. 5.1.1 Symmetric Monoidal Categoriessmc
  3. 5.1.2 Attention in Transformersattention
  4. 5.1.3 Transformers as a Categorytranscat
  5. 5.1.4 LLM Next-token Distributions as Categoriesllmcat-defns
  6. 5.1.5 Synthetic Token Probabilities using Nearest-Neighbor Language Modelingknn-trans
  7. 5.2 Backpropagation as a Functor: Compositional Learningbackprop
  8. 5.2.1 Category of Supervised Learningcfa-0039
  9. 5.2.2 Backpropagation as a Functorcfa-0040
  10. 5.3 Summarycfa-0041
  11. Suggested Readingcfa-0042
6 Diagrammatic Backpropagation4 sections
  1. 6.1 Introductioncfa-0043
  2. 6.2 Minimizing Diagrammatic Curvature Energysec-diagrammatic-intro
  3. 6.3 Summarycfa-0044
  4. Suggested Readingcfa-0045

Geometric and Kan Extension Transformers

7 Geometric Transformers14 sections
  1. 7.1 A Yoneda–style view of the Geometric Transformerapp-yoneda-gt
  2. 7.1.1 Diagram functors for GT Lite and GT Fullcfa-0047
  3. 7.1.2 A diagrammatic Yoneda principlecfa-0048
  4. 7.2 Diagrammatic Backpropagation as Horn Filling in Simplicial Setshorn-filling
  5. 7.2.1 Simplicial Sets and Objectscfa-0049
  6. 7.2.2 Hierarchical Learning in DB and GT by solving Lifting Problemscfa-0050
  7. 7.2.3 Simplicial Subsets and Horns in DB and GTcfa-0051
  8. 7.3 Language Modeling With Geometric Transformerscfa-0052
  9. 7.3.1 Experimental Results on WikiText-103sec-wikitext-103
  10. 7.4 Constructing Large Causal Modelsapp-democritus-relational
  11. 7.4.1 Geometric Refinement via DB and GTcfa-0053
  12. 7.5 A Global–Local Operator Perspectivesec-gt-theory-sketch
  13. 7.6 Summarycfa-0054
  14. Suggested Readingcfa-0055
8 Dynamic Compositionality136 sections
  1. 8.1 Probe, Scope, and Falsifiabilitycfa-0056
  2. 8.1.1 Čech-style obstruction proxycfa-0057
  3. 8.1.2 A Minimal Demonstration: Residual MLPs on Two Moonscfa-0058
  4. 8.1.3 Results: obstruction trajectories explain stabilitycfa-0059
  5. 8.2 Dynamic Compositionality and Order Sensitivity in Residual Learningsec-v2-dynamic-comp
  6. 8.2.1 From static composition to learned deformationssec-learned-deformations
  7. 8.2.2 Order sensitivity and commutator energysec-commutator-energy
  8. 8.2.3 A minimal residual examplesec-minimal-example
  9. 8.3 What the Minimal Residual Example Establishessec-v2-mlp-example
  10. 8.3.1 Two interacting residual blockssec-two-residual-blocks
  11. 8.3.2 Order sensitivity in residual updatessec-mlp-order-sensitivity
  12. 8.3.3 A conjectured connection to gradient interferencesec-mlp-gradient-interference
  13. 8.3.4 Implications for Diagrammatic Backpropagationsec-mlp-db-implications
  14. 8.4 Dynamic Compositionality in Transformer and Geometric Transformer Blockssec-v2-transformer-gt
  15. 8.4.1 Transformer encoder blocks as interacting sub-operatorssec-transformer-subops
  16. 8.4.2 Measuring order sensitivity inside a blocksec-transformer-order-sensitivity
  17. 8.4.3 Why vanilla Transformers exhibit high commutator energysec-baseline-transformer
  18. 8.5 Geometric Transformerssec-gt
  19. 8.5.1 GT-Lite: reducing order sensitivity via local smoothingsec-gt-lite
  20. 8.5.2 GT-Full: geometric transport and alignment of representation geometrysec-gt-full
  21. 8.5.3 GT-MoE: routing-induced order sensitivitysec-gt-moe
  22. 8.5.4 Summary: architectural control of dynamic compositionalitysec-summary-transformer
  23. 8.6 From Mechanism to Implementation: Explaining the DB+GT Code Pathsec-v2-code-explanation
  24. 8.6.1 The commutator-energy primitivesec-code-commutator
  25. 8.6.2 Layerwise instrumentation in Transformer blockssec-code-transformer
  26. 8.6.3 Extending the probe to Geometric Transformer variantssec-code-gt
  27. 8.6.4 Why commutator energy tracks training stabilitysec-code-stability
  28. 8.6.5 From instrumentation to controlsec-code-control
  29. 8.7 GT-Lite Under the Hood: Transformer Blocks with Local Geometric Smoothingsec-v2-gt-lite-code
  30. 8.7.1 GeomEncoderBlock: structure and intentcfa-0060
  31. 8.7.2 Sub-operators and dynamic compositionalitycfa-0061
  32. 8.7.3 Why local smoothing reduces commutator energycfa-0062
  33. 8.7.4 GeomDecoderBlock: a non-causal decoder cautioncfa-0063
  34. 8.8 GT-Full Under the Hood: Simplicial Transport as Coordinate Alignmentsec-v2-gt-full-code
  35. 8.8.1 GeomFullEncoderBlockSeq: code structurecfa-0064
  36. 8.8.2 Sub-operators in GT-Fullcfa-0065
  37. 8.8.3 What geometric transport actually doescfa-0066
  38. 8.8.4 Testing the transport hypothesiscfa-0067
  39. 8.8.5 Expressivity is a separate admission testcfa-0068
  40. 8.9 GT-MoE Under the Hood: Routing-Induced Order Sensitivitysec-v2-gt-moe-code
  41. 8.9.1 GeomEncoderMoEBlock: code structurecfa-0069
  42. 8.9.2 Sub-operators in GT-MoEcfa-0070
  43. 8.9.3 Routing as state-dependent branchingcfa-0071
  44. 8.9.4 A routing hypothesis for intermediate energycfa-0072
  45. 8.9.5 Comparison with GT-Lite and GT-Fullcfa-0073
  46. 8.10 Instrumenting and Controlling Order Sensitivityalg-v2-commutator-probe
  47. 8.11 Interpreting the Experimentssec-v2-bridge
  48. 8.11.1 Reframing the obstruction proxycfa-0074
  49. 8.11.2 Baseline growthcfa-0075
  50. 8.11.3 GT-Lite: partial mitigation via local smoothingcfa-0076
  51. 8.11.4 GT-Full: transport as a candidate mechanismcfa-0077
  52. 8.11.5 GT-MoE: conditional computation and intermediate regimescfa-0078
  53. 8.12 Frontier Case Study: Kimi K3 as a Compositional Architecturecfa-0079
  54. 8.13 Conclusion: What the Probe Does and Does Not Establishsec-v2-conclusion
  55. Suggested Readingcfa-0080
  56. 9 Information Regimes in Geometric Transformerscfa-0081
  57. 9.1 Information Regimes as Typed Provenancecfa-0082
  58. 9.1.1 A Kan-Style Analogy—and Its Limitsec-kando-kan-transformer
  59. 9.2 The Geometric Transformersec-wikitext-gt
  60. 9.2.1 Categorical Interpretation of Auxiliary Channelssec-gt-categorical-sidechannels
  61. 9.2.2 Language-Model Instantiationsec-gt-lm-instantiation
  62. 9.2.3 Revised LM Interpretationcfa-0083
  63. 9.3 WikiText-103 and WikiText-2 Results by Regimecfa-0084
  64. 9.4 Interpretation and Scopecfa-0085
  65. Suggested Readingcfa-0086
  66. 10 Kan Extension and Topological Coend Transformerscfa-0087
  67. 10.1 Backgroundcfa-0088
  68. 10.1.1 Autoregressive Transformerscfa-0089
  69. 10.1.2 Geometric Mixing in Transformerscfa-0090
  70. 10.1.3 Why Kan Extensions Entercfa-0091
  71. 10.1.4 Information Regimescfa-0092
  72. 10.1.5 From GT to KET and TopoCoendcfa-0093
  73. 10.1.6 Empirical Questionscfa-0094
  74. 10.1.7 Related Workcfa-0095
  75. 10.2 Kan Extensions as Universal Structured Extensioncfa-0096
  76. 10.2.1 Informal Definitioncfa-0097
  77. 10.2.2 Pointwise Formulas: Colimits and Limitscfa-0098
  78. 10.2.3 Interpolation and Completion as Motivating Examplescfa-0099
  79. 10.2.4 Coends as Weighted Aggregationcfa-0100
  80. 10.2.5 Kan Extensions via Coendscfa-0101
  81. 10.2.6 Interpretation for Transformer Architecturescfa-0102
  82. 10.2.7 Why This Matters for the Chaptercfa-0103
  83. 10.3 Neighborhood Systems and Kan-Style Aggregationcfa-0104
  84. 10.3.1 A Common Setupcfa-0105
  85. 10.3.2 Token Neighborhoods: Attentioncfa-0106
  86. 10.3.3 Learned Latent Neighborhoods: TopoCoendcfa-0107
  87. 10.3.4 Simplicial Neighborhoods: KETcfa-0108
  88. 10.3.5 Quadratic and Incidence-Restricted KETcfa-0109
  89. 10.3.6 Relationship to Geometric Transformerscfa-0110
  90. 10.4 Causality and Information Regimescfa-0111
  91. 10.4.1 Three Information Regimescfa-0112
  92. 10.4.2 Causality Lives in the Valuescfa-0113
  93. 10.4.3 Gold Noncausal Regime (Invalid)cfa-0114
  94. 10.4.4 Strict Causal Regimecfa-0115
  95. 10.4.5 Predict-and-Detach Self-Conditioningcfa-0116
  96. 10.4.6 Prompt Repetition as an Internal Mechanismcfa-0117
  97. 10.5 Model Instantiationscfa-0118
  98. 10.5.1 Backbone Transformercfa-0119
  99. 10.5.2 Quadratic KETcfa-0120
  100. 10.5.3 Incidence-Restricted KETcfa-0121
  101. 10.5.4 TopoCoend Transformercfa-0122
  102. 10.5.5 GT as a Local Comparison Casecfa-0123
  103. 10.5.6 Summary of the Hierarchycfa-0124
  104. 10.6 Predict–Detach as Paired Operational Semanticssec-predict_detach_category
  105. 10.6.1 Forward and Backward Semanticscfa-0125
  106. 10.6.2 Detach as a Stop-Gradient Transformationcfa-0126
  107. 10.6.3 The Predict–Detach Paircfa-0127
  108. 10.6.4 Information Regimes Revisitedcfa-0128
  109. 10.6.5 What Predict–Detach Does and Does Not Guaranteecfa-0129
  110. 10.6.6 Diagrammatic Viewcfa-0130
  111. 10.6.7 Relation to KET and TopoCoendcfa-0131
  112. 10.6.8 Conceptual Summarycfa-0132
  113. 10.7 Datasets, Protocol, and Experimental Setupcfa-0133
  114. 10.7.1 Model Familiescfa-0134
  115. 10.7.2 Information Regimescfa-0135
  116. 10.8 Learning Dynamicscfa-0136
  117. 10.8.1 PTBcfa-0137
  118. 10.8.2 WikiText-2cfa-0138
  119. 10.8.3 WikiText-103cfa-0139
  120. 10.9 Main Resultscfa-0140
  121. 10.9.1 Strict-Causal Comparisoncfa-0141
  122. 10.9.2 Future Predicted-Carrier Diagnosticcfa-0142
  123. 10.9.3 Diagnostic Gold-Noncausal Regimescfa-0143
  124. 10.9.4 Hardware Effectscfa-0144
  125. 10.10 Discussion of the Experimental Findingscfa-0145
  126. 10.11 Limitations and Future Directionscfa-0146
  127. 10.12 Conclusioncfa-0147
  128. 10.13 Additional Detailssec-appendix
  129. 10.13.1 Quadratic Kan Extension Blockcfa-0148
  130. 10.13.2 Incidence-Restricted Kan Blockcfa-0149
  131. 10.13.3 Predict–Detach Carrier Constructioncfa-0150
  132. 10.13.4 Required Leakage Auditcfa-0151
  133. 10.13.5 Hyperparameters and Training Detailscfa-0152
  134. 10.13.6 Hardware and Runtime Measurementcfa-0153
  135. 10.14 Summarycfa-0154
  136. Suggested Readingcfa-0155
11 Structured Language Modeling9 sections
  1. 11.1 Two Tasks, Not One Leaderboardcfa-0156
  2. 11.2 The Product Target and Its Factorized Realizationcfa-0157
  3. 11.3 The Implemented Corruption Channelcfa-0158
  4. 11.4 Categorical Interpretation: What Is Establishedcfa-0159
  5. 11.5 Algorithms Realized by the Codecfa-0160
  6. 11.6 Experimental Snapshotcfa-0525
  7. 11.7 A Fair Completion Benchmarkcfa-0162
  8. 11.8 Conclusioncfa-0163
  9. Suggested Readingcfa-0164
12 Manifold Learning with Geometric Transformers34 sections
  1. 12.1 From Point Clouds to Witnessed Relationscfa-0165
  2. 12.2 Classical Manifold Learning Is Not One Operatorcfa-0166
  3. 12.3 Boundary Operators and Hodge Laplacianscfa-0167
  4. 12.4 From a Typed Instance to a Simplicial Complexcfa-0168
  5. 12.5 What the Geometric Transformer Realizescfa-0169
  6. 12.6 Controlled Relational Recoverycfa-0170
  7. 12.6.1 World and preregistered endpointcfa-0171
  8. 12.6.2 The decisive capacity diagnosticcfa-0172
  9. 12.7 MovieLens: A Registered Negative Resultcfa-0173
  10. 12.8 Categorical and Coalgebraic Claim Boundariescfa-0174
  11. 12.9 A Reproducible Comparison Contractcfa-0175
  12. 12.10 Conclusioncfa-0176
  13. Suggested Readingcfa-0177
  14. 13 Mean-Field Theory of Geometric Transformerscfa-0178
  15. 13.1 The Object Being Measuredcfa-0179
  16. 13.2 Residual Updates and the First Obstructioncfa-0180
  17. 13.3 A Solvable Wide Random Modelcfa-0181
  18. 13.4 What Layer Normalization Changescfa-0182
  19. 13.5 Transport and Spectral Smoothingcfa-0183
  20. 13.6 Sheaf Energy Is a Different Obstructioncfa-0184
  21. 13.7 A Data-Limited Pilotcfa-0526
  22. 13.8 A Confirmatory Mean-Field Contractcfa-0186
  23. 13.9 Conclusioncfa-0187
  24. Suggested Readingcfa-0188
  25. 14 Depth Sweeps for Geometric Transformersgt-scaling
  26. 14.1 What Would Count as a Scaling Law?cfa-0189
  27. 14.2 Depth Changes More Than Depthcfa-0190
  28. 14.3 Averaging Is Not Accumulationcfa-0191
  29. 14.4 Evaluation of the Experiment Familycfa-0527
  30. 14.5 What the Surviving Curves Showcfa-0193
  31. 14.6 From a Depth Sweep to a Geometric Scaling Testcfa-0194
  32. 14.7 Three Testable Hypothesescfa-0195
  33. 14.8 Conclusioncfa-0196
  34. Suggested Readingcfa-0197

Categorical Models of Causality

15 Adjoint Functors9 sections
  1. 15.1 The Hom-Set Definitioncfa-0199
  2. 15.2 The Free-Monoid Examplecfa-0200
  3. 15.3 Unit, Counit, and Triangle Identitiescfa-0201
  4. 15.4 Galois Connectionscfa-0202
  5. 15.5 Limits and Colimits as Adjointscfa-0203
  6. 15.6 Adjunctions, Monads, and Equivalencescfa-0204
  7. 15.7 What an Adjoint Claim Would Require in Causalitycfa-0205
  8. 15.8 Summarycfa-0206
  9. Suggested Readingcfa-0207
16 Causal Claims from Language56 sections
  1. 16.1 From a Document to a Finite Causal Presentationcfa-0208
  2. 16.2 The Typed Presentationcfa-0209
  3. 16.3 Relational Geometry with the Geometric Transformercfa-0210
  4. 16.4 Ranking Local Neighborhoodscfa-0211
  5. 16.5 A Running Example: Dark Chocolate and Agingcfa-0212
  6. 16.6 An Empirical Audit across Four Domainscfa-0213
  7. 16.7 External Calibration against UniCausalcfa-0214
  8. 16.8 Geometry and Completion: A Boundary on the Claimscfa-0215
  9. 16.9 An Admission Protocol for Causal Usecfa-0216
  10. 16.10 Summarycfa-0217
  11. Suggested Readingcfa-0218
  12. 17 Temporal Diffusion over Causal Trajectoriescfa-0219
  13. 17.1 Modeling Corporate Geometrycfa-0220
  14. 17.1.1 Yearly Corporate States as Causal Objectscfa-0221
  15. 17.1.2 Corporate Trajectories as Functors Through Timecfa-0222
  16. 17.1.3 Why Geometry Matterscfa-0223
  17. 17.1.4 Temporal Diffusion as Structured Repaircfa-0224
  18. 17.1.5 From Causal Snapshots to Corporate Geometrycfa-0225
  19. 17.2 From Static Causal Snapshots to Temporal Structurecfa-0226
  20. 17.2.1 Yearly Snapshots Are Partial and Noisycfa-0227
  21. 17.2.2 Temporal Blocks as Local Diagramscfa-0228
  22. 17.2.3 Temporal Diffusion as Structured Repaircfa-0229
  23. 17.2.4 A Right-Kan Intuitioncfa-0230
  24. 17.2.5 From Individual Trajectories to a Company Metric Spacecfa-0231
  25. 17.3 Temporal Block Denoising Architecturescfa-0232
  26. 17.3.1 Temporal Block Representationcfa-0233
  27. 17.3.2 Temporal Corruption Modelcfa-0234
  28. 17.3.3 Denoising Objectivecfa-0235
  29. 17.3.4 Block Denoising Encodercfa-0236
  30. 17.3.5 Relation to Earlier Denoising Stagescfa-0237
  31. 17.3.6 Categorical Interpretationcfa-0238
  32. 17.3.7 Output Artifactscfa-0239
  33. 17.4 Temporal Diffusion as a Schrödinger-Bridge-Like Repair Processcfa-0240
  34. 17.4.1 A Brief Reminder on Schrödinger Bridgescfa-0241
  35. 17.4.2 Temporal Blocks as Local Trajectory Fragmentscfa-0242
  36. 17.4.3 A Variational Interpretation of the Denoisercfa-0243
  37. 17.4.4 Why This Interpretation Is Usefulcfa-0244
  38. 17.4.5 Categorical View: Repair of Temporal Functorscfa-0245
  39. 17.4.6 Interpretive Consequences for Corporate Geometrycfa-0246
  40. 17.5 Multi-Company Metric Spaces over Causal Trajectoriescfa-0247
  41. 17.5.1 Company Trajectories as Functorscfa-0248
  42. 17.5.2 Alignments Between Corporate Trajectoriescfa-0249
  43. 17.5.3 Distances Between Corporate Trajectoriescfa-0250
  44. 17.5.4 A Composite Corporate-Trajectory Distancecfa-0251
  45. 17.5.5 Why Repair Comes Firstcfa-0252
  46. 17.5.6 Interpretationcfa-0253
  47. 17.5.7 Empirical Role in the Chaptercfa-0254
  48. 17.5.8 What the Structural-Indication Score Measurescfa-0255
  49. 17.6 Empirical Illustration: A Multi-Company Panel of Corporate Trajectoriescfa-0256
  50. 17.6.1 Middle-Year Reconstruction Benchmarkcfa-0528
  51. 17.6.2 Panel-Wide Structurecfa-0258
  52. 17.6.3 Cross-Sectional Regimes in Recent Yearscfa-0259
  53. 17.6.4 Temporal Diffusion and Multi-Company Geometrycfa-0260
  54. 17.6.5 Case Studies: Adobe and Nikecfa-0261
  55. 17.6.6 Summary of the Empirical Picturecfa-0262
  56. Suggested Readingcfa-0263
18 Building Agentic Systems using Kan Extension Transformers16 sections
  1. 18.1 Operational Plans as String Diagramscfa-0264
  2. 18.2 BASKET: Extracting Operational Plans from 10-K Filingscfa-0265
  3. 18.2.1 The Learned Action Vocabularycfa-0266
  4. 18.2.2 Company-Year Plan Graphscfa-0267
  5. 18.2.3 Kan Extension Interpretationcfa-0268
  6. 18.3 ROCKET: Financially Grounded Plan Selectiontab-rocket-financial-summary
  7. 18.3.1 Outcome Alignmentcfa-0269
  8. 18.3.2 Representative Editscfa-0270
  9. 18.3.3 Aggregate Plans Before and After ROCKETcfa-0271
  10. 18.4 From Workflow Extraction to Agentic Corporate Geometrycfa-0272
  11. 18.5 From Corporate Geometry to Topos Causal Modelscfa-0273
  12. 18.5.1 The Limitation of a Single Global Causal State Spacecfa-0274
  13. 18.5.2 From Global Geometry to Contextual Causal Semanticscfa-0275
  14. 18.5.3 Conceptual Continuitycfa-0276
  15. 18.5.4 Outlookcfa-0277
  16. Suggested Readingcfa-0278
19 Topos Causal Models11 sections
  1. 19.1 From Structured Completion to Topos Causal Modelscfa-0279
  2. 19.2 Introductioncfa-0280
  3. 19.3 Principles of Universal Causalitycfa-0281
  4. 19.4 Topos Causal Modelscfa-0282
  5. 19.5 Causal Models and the Arrow Toposcfa-0283
  6. 19.6 Causal Models Over a Topos of Sheavescfa-0284
  7. 19.6.1 Grothendieck Topology on Sitescfa-0285
  8. 19.6.2 Universal Property of TCM over Functor Categoriescfa-0286
  9. 19.7 Causal Mitchell-Bénabou Language and its Kripke-Joyal Semanticscfa-0287
  10. 19.8 Summarycfa-0288
  11. Suggested Readingcfa-0289
20 Judo Calculus124 sections
  1. 20.1 From Classical Do-Calculus to \(j\)-Do-Calculusscm-review
  2. 20.1.1 Classical Do-Calculuscfa-0290
  3. 20.1.2 \(j\)-do-Calculus: A Birds-Eye Viewcfa-0291
  4. 20.2 Causal Models Over a Topos of Sheavesgdc-stoch
  5. 20.2.1 Lawvere-Tierney Topologies on a Toposlawvere
  6. 20.2.2 Kripke-Joyal Semantics for Sheavescfa-0292
  7. 20.2.3 \(j\)-do-Calculus on Sitescfa-0293
  8. 20.3 Algorithms for Judo Calculuscfa-0294
  9. 20.3.1 Judo Calculus Model of Causal Inference under Interferencesec-min-interference
  10. 20.3.2 Computational and Statistical Efficiency of \(j\)-Stable Discoverysec-efficiency
  11. 20.3.3 Experimental Validation of Judo Calculus Efficiencycfa-0295
  12. 20.3.4 The \(j\)-stable do-operator (practical form)sec-jdo-practical
  13. 20.3.5 Relation to transportability (Pearl–Bareinboim)sec-transportability
  14. 20.4 Experimental Validation of \(j-\)Stable Causal Discoverysec-validation
  15. 20.4.1 Experimental Designsec-eval
  16. 20.4.2 Experimental Setupsec-exp_setup
  17. 20.4.3 Questionscfa-0296
  18. 20.4.4 Datasetscfa-0297
  19. 20.4.5 Methodscfa-0298
  20. 20.4.6 Metricscfa-0299
  21. 20.4.7 Experimental protocolcfa-0300
  22. 20.4.8 Computational efficiencycfa-0301
  23. 20.5 Experimental Resultscfa-0302
  24. 20.5.1 Why \(j\)-stable discovery works: an ensemble view (bagging & boosting)sec-bag-boost-intuition
  25. 20.5.2 Synthetic DAGs and data generationsec-synthetic-dag
  26. 20.5.3 Synthetic DAG: GES vs. \(j\)-stable GEScfa-0303
  27. 20.5.4 DCDI synthetic setups (perfect interventions)sec-dcdi-synth
  28. 20.5.5 Sachs protein signaling (11 nodes, multiintervention)cfa-0304
  29. 20.5.6 Empirical summary and limitationscfa-0305
  30. 20.5.7 LINCS L1000 perturbation signatures (cell line \(\times \) dose \(\times \) time)sec-lincs
  31. 20.5.8 OECD PISA ESCS Dataset (Countries as Regimes)sec-pisa-escs-dataset
  32. 20.6 Summarycfa-0306
  33. Suggested Readingcfa-0307
  34. 21 Csql : Mapping Documents into Topos Causal Model Databasescfa-0308
  35. 21.1 Introductioncfa-0309
  36. 21.2 A Formal View of TCM-DBsec-tcmdb_formal
  37. 21.2.1 Schema Categorycfa-0310
  38. 21.2.2 Definition of a TCM-DB Instancecfa-0311
  39. 21.2.3 Global Sections and the SQL Viewcfa-0312
  40. 21.2.4 Predicates, Subobjects, and \(\Omega \)cfa-0313
  41. 21.3 The Origin of Bipedal Walking: A Running Examplecfa-0314
  42. 21.4 Csql Data Modelsec-data_model
  43. 21.4.1 Overviewcfa-0315
  44. 21.4.2 Core Relationscfa-0316
  45. 21.4.3 Derived Relationscfa-0317
  46. 21.4.4 Query Semanticscfa-0318
  47. 21.5 Querying Causal Databases with Csqlsec-queries
  48. 21.5.1 Backbone Extractioncfa-0319
  49. 21.5.2 Causal Hubscfa-0320
  50. 21.5.3 Local Mechanism Explorationcfa-0321
  51. 21.5.4 Provenance and Auditabilitycfa-0322
  52. 21.5.5 Cycles and Feedback Structurescfa-0323
  53. 21.6 Example of a Csql Databasecfa-0324
  54. 21.6.1 Quantitative Properties of Csql Databasessec-quantitative_csql
  55. 21.6.2 Claim-graph ablation via SQL view rewritingcfa-0325
  56. 21.6.3 Quantitative Summary of a Csql Atlascfa-0326
  57. 21.6.4 Database Scale and Structurecfa-0327
  58. 21.6.5 Hub Dominance and Causal Centralizationcfa-0328
  59. 21.6.6 Heavy-Tailed Causal Strengthcfa-0329
  60. 21.6.7 Relation-Type Compositioncfa-0330
  61. 21.7 The Csql Data Modelsec-csql_data_model
  62. 21.7.1 Nodes: Canonical Causal Conceptscfa-0331
  63. 21.7.2 Edges: Aggregated Causal Relationscfa-0332
  64. 21.7.3 Edge Support and Provenancecfa-0333
  65. 21.7.4 Strongly Connected Componentscfa-0334
  66. 21.8 Causal Reasoning as SQLsec-csql_queries
  67. 21.8.1 Identifying Causal Backbonescfa-0335
  68. 21.8.2 Causal Hubs and Downstream Influencecfa-0336
  69. 21.8.3 Causal Compositioncfa-0337
  70. 21.8.4 Cycles and Feedbackcfa-0338
  71. 21.8.5 Quantitative Summary of the Csql Corpus Databasesec-quant-csql
  72. 21.9 Algorithmic Construction of Csql Databasessec-csql_algorithms
  73. 21.9.1 Input Contractsec-csql_input
  74. 21.9.2 Csql Schemasec-csql_schema
  75. 21.9.3 Canonicalization and Keyingsec-canonicalization
  76. 21.9.4 Atlas Builder: From LCMs to Parquetalg-build_atlas
  77. 21.9.5 Corpus Merge: Union of Csql Databasesalg-merge_atlas
  78. 21.9.6 Practical Notes for Userssec-csql_practical
  79. 21.10 Csql from RAG-Compiled Causal Corporasec-csql_from_rag
  80. 21.10.1 Csql from a RAG-Compiled Causal Corpus: Testing Causal Claims (TCC)sec-tcc_csql_results
  81. 21.10.2 Topos Theoretic cSQL : Red-Wine Pullbacks, Pushouts, and \(\Omega \)sec-redwine_functorflow_results
  82. 21.10.3 Applying Topos Causal Models in the TCC Domain: Pullback and Method-Conflict Subobjectssec-tcc_functorflow_pullback
  83. 21.11 Csql over Other Domainscfa-0339
  84. 21.12 Discussion and Implicationscfa-0340
  85. 21.12.1 Relation to Knowledge Graphs and RAGcfa-0341
  86. 21.12.2 Relation to Causal Inferencecfa-0342
  87. 21.12.3 Csql as a Causal Compilercfa-0343
  88. 21.13 Limitations and Future Worksec-limitations_future
  89. 21.13.1 Limitationscfa-0344
  90. 21.13.2 Future Workcfa-0345
  91. Suggested Readingcfa-0346
  92. 22 Homotopy in Language and Causal Inferencecfa-0347
  93. 22.1 Introductioncfa-0348
  94. 22.1.1 Why Homotopy Matters for Democrituscfa-0349
  95. 22.2 From Language to Causal Contentcfa-0350
  96. 22.3 Localization and the Homotopy Categorycfa-0351
  97. 22.4 Homotopical Repair of a Causal-Discourse Sketchcfa-0352
  98. 22.5 LLMs as a Special Casecfa-0353
  99. 22.6 Causal Semantics and Markov Categoriescfa-0354
  100. 22.7 Paraphrase Groupoids and Simplicial Structurecfa-0355
  101. 22.7.1 Weighted and Fuzzy Simplicial Structurecfa-0356
  102. 22.8 Homotopy, Obstructions, and Aggregation in Democrituscfa-0357
  103. 22.8.1 Evidence poolingcfa-0358
  104. 22.8.2 Graph consolidationcfa-0359
  105. 22.8.3 Ambiguity detectioncfa-0360
  106. 22.8.4 Contradiction and polarity sensitivitycfa-0361
  107. 22.8.5 A motivating examplecfa-0362
  108. 22.9 Empirical Boundary of the Current Systemcfa-0363
  109. 22.10 On Model Structurescfa-0364
  110. 22.11 Outlook: Homology and Higher Invariantscfa-0365
  111. 22.12 Bridge to Markov Categories, \(j\)-Stable Semantics, and Topos Causalitycfa-0366
  112. 22.13 Summarycfa-0367
  113. Suggested Readingcfa-0368
  114. 23 Model Categories for Causality and Languagecfa-0369
  115. 23.1 Why Language and Causality Need Weak Equivalencescfa-0370
  116. 23.2 Syntax, Causal Semantics, and Extractioncfa-0371
  117. 23.3 Localization and Homotopy Categoriescfa-0372
  118. 23.4 Lifting Problemscfa-0373
  119. 23.5 Simplicial Models of Paraphrase Coherencecfa-0374
  120. 23.6 Model Categoriescfa-0375
  121. 23.7 Axiomatic Homotopy for LLMs and Causal Inferencecfa-0376
  122. 23.8 Bridge to Markov Semantics and Interventionscfa-0377
  123. 23.9 Summarycfa-0378
  124. Suggested Readingcfa-0379
24 Predictive State Representations in a Topos44 sections
  1. 24.1 Predictive State Representationscfa-0380
  2. 24.1.1 Action–Observation Testscfa-0381
  3. 24.1.2 Predictive Statecfa-0382
  4. 24.2 Workflows as Intervention Sequencescfa-0383
  5. 24.2.1 Judo Calculus Interpretationcfa-0384
  6. 24.3 Predictive State Representations as Sheaf Sectionscfa-0385
  7. 24.3.1 Action–Observation Testscfa-0386
  8. 24.3.2 Local Predictive State Spacescfa-0387
  9. 24.3.3 Restriction Mapscfa-0388
  10. 24.3.4 Sheaf Conditioncfa-0389
  11. 24.3.5 Interpretationcfa-0390
  12. 24.3.6 Failure of Gluing as a Diagnosticcfa-0391
  13. 24.3.7 Relation to Workflow Extractioncfa-0392
  14. 24.4 Local Predictive Statescfa-0393
  15. 24.5 Sheaf of Predictive Statescfa-0394
  16. 24.5.1 Restriction Mapscfa-0395
  17. 24.5.2 Gluingcfa-0396
  18. 24.5.3 Failure of Gluingcfa-0397
  19. 24.5.4 A Judo Calculus for Predictive Testscfa-0398
  20. 24.6 A Toy Examplecfa-0399
  21. 24.7 A Worked Nike Examplecfa-0400
  22. 24.7.1 Actions as Interventionscfa-0401
  23. 24.7.2 Predictive Testscfa-0402
  24. 24.7.3 Predictive Updatecfa-0403
  25. 24.7.4 Local Contextscfa-0404
  26. 24.7.5 Monoidal Compositioncfa-0405
  27. 24.7.6 Gluing and Consistencycfa-0406
  28. 24.7.7 Interpretationcfa-0407
  29. 24.7.8 Role of ROCKETcfa-0408
  30. 24.7.9 Summarycfa-0409
  31. 24.7.10 A Failure-to-Glue Scenariocfa-0410
  32. 24.7.11 Subobject Classifiers and Intuitionistic Predictive Semanticscfa-0411
  33. 24.7.12 Descent Defects and a Cohomological Caveatcfa-0412
  34. 24.8 Relation to BASKET and ROCKETcfa-0413
  35. 24.9 Connection to Topos Causal Modelscfa-0414
  36. 24.10 Experimental Resultscfa-0415
  37. 24.10.1 What the Implementation Measurescfa-0416
  38. 24.10.2 PSR Variant Comparisoncfa-0417
  39. 24.10.3 ROCKET Variant Comparisoncfa-0418
  40. 24.10.4 Representative Predictive Testscfa-0419
  41. 24.10.5 Gluing and Predictive Obstructioncfa-0420
  42. 24.11 Temporal Structure of Predictive Obstructioncfa-0421
  43. 24.11.1 Summarycfa-0422
  44. Suggested Readingcfa-0423
25 Causal Density Functions12 sections
  1. 25.1 Introductioncfa-0424
  2. 25.2 Causal Density Functionssec-cdf
  3. 25.3 Analytic Distinctions and Statistical Scopesec-cdf-scope
  4. 25.4 Estimating Causal Density Functionsalg-cdf-revised
  5. 25.5 Experimental Resultscfa-0425
  6. 25.5.1 PISA 2022 Socio–Economic Panelapp-pisa2022
  7. 25.5.2 Sachs Protein Signalingsec-sachs11
  8. 25.5.3 Multi-Regime Chain Experimentsec-sheaf
  9. 25.5.4 Discussion: Causal Density and Sheaf Coherencecfa-0426
  10. 25.5.5 Synthetic Benchmark Experimentsapp-synthetic
  11. 25.6 Summarycfa-0427
  12. Suggested Readingcfa-0428

Universal Decision Models

26 Universal Decisions with Kan Extensions38 sections
  1. 26.1 Introductioncfa-0430
  2. 26.2 Contexts and Partial Decision Modelscfa-0431
  3. 26.3 Left Kan Extensions: Generalization by Aggregationcfa-0432
  4. 26.3.1 Interpretationcfa-0433
  5. 26.3.2 Example: Interpolationcfa-0434
  6. 26.4 Right Kan Extensions: Consistency and Constraintscfa-0435
  7. 26.4.1 Interpretationcfa-0436
  8. 26.5 Incorporating Rewardscfa-0437
  9. 26.5.1 Max-Plus Semiringcfa-0438
  10. 26.5.2 Left Kan with Rewardscfa-0439
  11. 26.5.3 Right Kan with Rewardscfa-0440
  12. 26.6 Universal Decision Learnerscfa-0441
  13. 26.6.1 Interpretationcfa-0442
  14. 26.7 Connection to Classical Planningcfa-0443
  15. 26.7.1 Forward Evaluation (Left Kan)cfa-0444
  16. 26.7.2 Backward Consistency (Right Kan)cfa-0445
  17. 26.7.3 Interpretationcfa-0446
  18. 26.8 Universalitycfa-0447
  19. 26.8.1 Interpretationcfa-0448
  20. 26.9 Conceptual Summarycfa-0449
  21. 26.10 Decision Making as Kan Extensionsec-kan-decision
  22. 26.10.1 Local Behavior and Extensioncfa-0450
  23. 26.10.2 Planning as Left Kan Extensioncfa-0451
  24. 26.10.3 Reinforcement Learning as Right Kan Extensioncfa-0452
  25. 26.10.4 Coinduction and Right Kan Extensionscfa-0453
  26. 26.10.5 Unifying Perspectivecfa-0454
  27. 26.10.6 Bisimulation as Kan Invariancesec-kan-bisim
  28. 26.10.7 Quotient Morphisms and Abstractioncfa-0455
  29. 26.10.8 Behavioral Semantics via Kan Extensionscfa-0456
  30. 26.10.9 Kan Invariancecfa-0457
  31. 26.10.10 Interpretationcfa-0458
  32. 26.10.11 Semantic Kernels and Quotientscfa-0459
  33. 26.10.12 Homotopy and Observational Equivalencesec-homotopy-kan
  34. 26.10.13 Bellman Optimality and a Conditional Sheaf Analogysec-sheaf-bellman
  35. 26.10.14 Conceptual Summarycfa-0460
  36. 26.10.15 Online Learning as a Comparator Extensioncfa-0461
  37. 26.10.16 Bridge to Reinforcement Learningcfa-0462
  38. Suggested Readingcfa-0463
27 Universal Reinforcement Learning8 sections
  1. 27.1 Coalgebraic Dynamicscfa-0464
  2. 27.2 The Bellman Backup as a Pointwise Right Kan Extensionsec-url-bellman-ran-book
  3. 27.3 Exact and Sample-Based Computationcfa-0465
  4. 27.4 Behavioral Equivalence and Abstractioncfa-0466
  5. 27.5 Final Coalgebras and Coinductioncfa-0467
  6. 27.6 Information Fields and URLcfa-0468
  7. 27.7 Summarycfa-0469
  8. Suggested Readingcfa-0470
28 Deep URL with Geometric Transformers12 sections
  1. 28.1 From Coalgebra Morphisms to Deep URLsec-deep-url-overview
  2. 28.2 Algorithm: Deep URL with GT+DBsec-deep-url-algorithm
  3. 28.3 Loss Decomposition as Coalgebraic Constraintsec-coalgebraic-loss
  4. 28.4 Setupcfa-0471
  5. 28.5 Diagrammatic Backpropagation as Coalgebraic Regularizationsec-db-coalgebra
  6. 28.6 Connections to PVFs and successor representationscfa-0472
  7. 28.7 Resultscfa-0473
  8. 28.8 Empirical Evidence for Structural Inductive Biascfa-0474
  9. 28.9 Interpretationcfa-0475
  10. 28.10 Deep URL as Relational Geometry in Controlcfa-0476
  11. 28.11 Outlook: Toward Topological Planningsec-topological-planning
  12. Suggested Readingcfa-0477

Frontiers of AGI

29 Consciousness7 sections
  1. 29.1 Status of the proposalcfa-0479
  2. 29.2 Process layer: coalgebrascfa-0480
  3. 29.3 Logical layer: an ambient toposcfa-0481
  4. 29.4 Workspace interfacescfa-0482
  5. 29.5 Competition for limited capacitycfa-0483
  6. 29.6 What the framework does and does not establishcfa-0484
  7. Suggested Readingcfa-0485
30 Universal Imitation Games7 sections
  1. 30.1 A typed imitation experimentcfa-0486
  2. 30.2 What Yoneda licensescfa-0487
  3. 30.3 Static, adaptive, and population testscfa-0488
  4. 30.4 Preference learning as an extension problemcfa-0489
  5. 30.5 Auditable claims for imitation gamescfa-0490
  6. 30.6 Conclusioncfa-0491
  7. Suggested Readingcfa-0492
31 Formal Verification Map6 sections
  1. 31.1 How to Use This Mapcfa-0493
  2. 31.2 Numbered Theorem Mapcfa-0494
  3. 31.3 Chapter-to-Module Mapcfa-0495
  4. 31.4 Declaration-Level Crosswalk for the Newer Chapterscfa-0496
  5. 31.5 Maintenance Protocolcfa-0497
  6. Suggested Readingcfa-0498
32 CLIFF: An AGI Chatbot for This Textbook15 sections
  1. 32.1 Why CLIFF Belongs in This Bookcfa-0499
  2. 32.2 High-Level Architecturecfa-0500
  3. 32.3 Democritus as the Local Causal Modeling Layercfa-0501
  4. 32.4 Homotopy Localization and Causal Equivalencecfa-0502
  5. 32.5 Csql Bundles, Regime Gluing, and Topos-Theoretic Intuitioncfa-0503
  6. 32.6 Topic Covers, Partitions, and Retrieval Disciplinecfa-0504
  7. 32.7 Visualization as Categorical Inspectioncfa-0505
  8. 32.7.1 Relational manifold viewcfa-0506
  9. 32.7.2 Local causal model galleriescfa-0507
  10. 32.7.3 Corpus synthesis dashboardscfa-0508
  11. 32.8 Conceptual Map from Book to Systemcfa-0509
  12. 32.9 How Readers Should Use CLIFFcfa-0510
  13. 32.10 Repository and First Runcfa-0511
  14. 32.11 Outlookcfa-0512
  15. Suggested Readingcfa-0513
33 Code Companion and Sample Repository13 sections
  1. 33.1 How This Appendix Differs from the Lean Mapcfa-0514
  2. 33.2 Repository Layoutcfa-0515
  3. 33.3 Representative Chapter-to-Notebook Mapcfa-0516
  4. 33.4 How Readers Can Use the Repositorycfa-0517
  5. Suggested Readingcfa-0518
  6. Referencesreferences
  7. Glossary of Notationcfa-0519
  8. Categories, maps, and diagramscfa-0520
  9. Universal constructions and representationcfa-0521
  10. Geometry, topology, and local-to-global structurecfa-0522
  11. Probability and causalitycfa-0523
  12. Learning, decisions, and dynamicscfa-0524
  13. SymbolsSymbols