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- 1. Mathematical Knowledge Management in Algebraic Topology J´nathan Heras Vicente o Supervisors: Dr. Vico Pascual Mart´ ınez-Losa Dr. Julio Rubio Garc´ ıa Department of Mathematics and Computer Science University of La Rioja Spain May 31, 2011J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 2. Table of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 3. IntroductionTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 4. Introduction ContextContext Mathematical Knowledge Management (MKM) Computation Deduction Communication J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 5. Introduction ContextContext Mathematical Knowledge Management (MKM) Computation Deduction Communication Algebraic Topology Mathematical subject which studies topological spaces by algebraic means Applications in Coding theory, Robotics, Digital Image analysis J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 6. Introduction ContextContext Mathematical Knowledge Management (MKM) Computation Deduction Communication Algebraic Topology Mathematical subject which studies topological spaces by algebraic means Applications in Coding theory, Robotics, Digital Image analysis Kenzo Computer Algebra system devoted to Algebraic Topology developed by F. Sergeraert Common Lisp package Homology groups unreachable by any other means J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 7. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 8. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 9. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 10. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 11. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules Deduction J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 12. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules Deduction Certiﬁcation of Kenzo algorithms Isabelle/Hol Coq J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 13. Introduction Particularization of MKM to Algebraic TopologyParticularization of MKM to Algebraic Topology Communication Human beings Other programs Computation Kenzo GAP New modules Deduction Certiﬁcation of Kenzo algorithms Isabelle/Hol Coq Certiﬁcation of Kenzo programs ACL2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 14. CommunicationTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 15. Communication MotivationMotivation Algebraic Topology Expert ⇒ Makes the most of Kenzo Common Lisp Expert J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 16. Communication MotivationMotivation Algebraic Topology Expert ⇒ Makes the most of Kenzo Common Lisp Expert Non Common Lisp Expert ⇒ Unfriendly front-end J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 17. Communication MotivationMotivation Algebraic Topology Expert ⇒ Makes the most of Kenzo Common Lisp Expert Non Common Lisp Expert ⇒ Unfriendly front-end Non Algebraic Topology Expert ⇒ Needs guidance J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 18. Communication MotivationGoal Develop a system which increases Kenzo accessibility J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 19. Communication MotivationGoal Develop a system which increases Kenzo accessibility Friendly front-end Mediated access to Kenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 20. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Client Kenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 21. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Client Kenzo Representation of mathematical knowledge OpenMath Independent of programming language ⇒ XML ⇒ MathML Easily interchangeable New language J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 22. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access XML-Kenzo XML-Kenzo Client Kenzo XML-Kenzo construction and computation requests Includes some mathematical knowledge Small Type System: CC, SS, SG, ASG Some restrictions about arguments (S n ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 23. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access XML-Kenzo validator XML-Kenzo XML-Kenzo Client Kenzo XML-Kenzo validator Common Lisp module Receives XML-Kenzo requests → validates them against XML schema deﬁnition Type restrictions Mathematical restrictions Kenzo restrictions J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 24. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Valid XML-Kenzo Construction Modules XML-Kenzo validator XML-Kenzo XML-Kenzo Client Kenzo Construction modules Process construction requests Check restrictions not included in XML-Kenzo (functional dependencies) 4 modules: CC, SS, SG and ASG J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 25. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo XML-Kenzo Client Kenzo Computation Modules Computation modules Process computation requests Check restrictions not included in XML-Kenzo (reduction degree) 2 modules: Homology and Homotopy J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 26. Communication Providing a mediated access to KenzoProviding a mediated access to Kenzo Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo XML-Kenzo Client Processing Kenzo Modules Computation Modules Processing modules Help construction and computation modules 2 modules: Reduction degree and Homotopy assistant J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 27. Communication The whole frameworkTowards the whole framework This architecture provides Mediated access to Kenzo Friendly front-end but . . . J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 28. Communication The whole frameworkTowards the whole framework This architecture provides Mediated access to Kenzo Friendly front-end but . . . Desirable features Diﬀerent clients Eﬃcient Extensible Adaptable to diﬀerent needs J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 29. Communication The whole frameworkTowards diﬀerent clients Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo XML-Kenzo Client Processing Kenzo Modules Computation Modules XML-Kenzo is ad-hoc J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 30. Communication The whole frameworkTowards diﬀerent clients Mediated access Construction Modules Valid XML-Kenzo XML-Kenzo validator XML-Kenzo OpenMath Client Processing Kenzo Modules Computation Modules XML-Kenzo is ad-hoc OpenMath Standard Communication with the outside world J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 31. Communication The whole frameworkTowards diﬀerent clients Mediated access Construction Modules Valid XML-Kenzo Client Phrasebook XML-Kenzo validator XML-Kenzo XML-Kenzo OpenMath Client Processing Kenzo Modules Client Computation Modules XML-Kenzo is ad-hoc OpenMath Standard Communication with the outside world Phrasebook J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 32. Communication The whole frameworkTowards Eﬃciency Mediated access Construction Modules Valid XML-Kenzo Client Phrasebook XML-Kenzo validator XML-Kenzo XML-Kenzo OpenMath Client Internal Processing Kenzo Memory Modules Client Computation Modules System roughly equivalent to Kenzo Internal memory Memoization technique Store spaces and computations J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 33. Communication The whole frameworkThe Kenzo framework Framework Microkernel Construction Modules __ Internal External __ Adapter Server Server __ __ __ XML-Kenzo __ XMLValidator __ Kenzo XML-Kenzo __ XML-Kenzo OpenMath __ Internal Processing __ Valid __ __ Clients Memory __ __ __ __ Modules __ Phrasebook __ __ __ Computation Modules Based on well-known patterns and methods J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 34. Communication The whole frameworkTowards extensibility Include new functionality J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 35. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 36. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality Include new systems Computer Algebra systems Theorem Prover tools J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 37. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality Include new systems Computer Algebra systems Theorem Prover tools Interoperability J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 38. Communication The whole frameworkTowards extensibility Include new functionality Increase Kenzo functionality Include new systems Computer Algebra systems Theorem Prover tools Interoperability Kenzo framework as a client of a plug-in framework Plug-in Framework Plug-in Manager Client-1 Client-2 __ Module Module OMDoc Plug-in Client __ __ Client-3 Client-4 Plug-in repository Module Module Plug-in Registry J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 39. Communication The whole frameworkIncluding new Kenzo functionality <code id="new-constructor"> <data format="Kf/internal-server"> new-constructor-is.lisp </data> <data format="Kf/microkernel"> new-constructor-m.lisp </data> <data format="Kf/external-server"> XML-Kenzo.xsd </data> <data format="Kf/adapter"> new-constructor-a.lisp </data> </code> Framework Microkernel Construction Modules __ Internal External __ Adapter Server Server __ __ __ XML-Kenzo __ XMLValidator __ Kenzo XML-Kenzo __ XML-Kenzo OpenMath __ Internal Processing __ Valid __ __ Clients Memory __ __ __ __ Modules __ Phrasebook __ __ __ Computation Modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 40. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 41. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 42. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) Guided by heuristics J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 43. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) Guided by heuristics Design of interaction: Noesis method J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 44. Communication The whole frameworkA distinguished client of our frameworks A Graphical User Interface implemented in Common Lisp Design decisions Functionality (Common Lisp) + Structure (XUL) + Layout (stylesheet) Guided by heuristics Design of interaction: Noesis method Introduce the dimension,n, of the loop space [Yes] Object Selected in [No] Main tab? Error: Invalid dimension Is the object, X, a Simplicial Set? Select Object X from a list of simplicial sets Is the dimension a [No] [No] natural number? [Yes] Error: Invalid Object [Yes] Construction of the loop space iterated n times of the object X J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 45. Communication The whole frameworkA customizable GUI Features Extensibility Adaptability J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 46. Communication The whole frameworkA customizable GUI Features Extensibility Adaptability The Graphical User Interface is client of Kenzo framework client of plug-in framework GUI organized through modules: basic and experimental J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 47. Communication The whole frameworkA customizable GUI Features Extensibility Adaptability The Graphical User Interface is client of Kenzo framework client of plug-in framework GUI organized through modules: basic and experimental OpenMath fKenzo Kenzo framework GUI fKenzo plug-in Kenzo framework registry plug-in registry OMDoc plug-in Plug-in framework The whole system is called fKenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 48. Communication fKenzofKenzo: A tool adaptable to diﬀerent needs Beginner student of Algebraic Topology Load and remove basic modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 49. Communication fKenzofKenzo: A tool adaptable to diﬀerent needs Beginner student of Algebraic Topology Load and remove basic modules Advanced student or researcher of Algebraic Topology Load and remove basic modules Load and remove experimental modules J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 50. Communication fKenzofKenzo: A tool adaptable to diﬀerent needs Beginner student of Algebraic Topology Load and remove basic modules Advanced student or researcher of Algebraic Topology Load and remove basic modules Load and remove experimental modules Digital images GAP ACL2 ... J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 51. Communication fKenzofKenzo summary fKenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 52. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 53. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 54. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 55. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end Extensible J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 56. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end Extensible Adaptable to diﬀerent needs J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 57. Communication fKenzofKenzo summary fKenzo Integral assistant for Algebraic Topology Constrains Kenzo functionality Provides guidance in the interaction and a friendly front-end Extensible Adaptable to diﬀerent needs Our architecture can be applied in other contexts J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 58. ComputationTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 59. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 60. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets Integrated in fKenzo J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 61. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets Integrated in fKenzo Veriﬁed using ACL2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 62. ComputationNew Kenzo modules Increase Kenzo computational capabilities Simplicial Complexes Digital Images Pushout of Simplicial Sets Integrated in fKenzo Veriﬁed using ACL2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 63. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 64. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) C∗ (X ) has ﬁnite nature Diﬀerential maps can be expressed as integer matrices Homology groups: Smith Normal Form J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 65. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) C∗ (X ) has ﬁnite nature Diﬀerential maps can be expressed as integer matrices Homology groups: Smith Normal Form C∗ (X ) has non ﬁnite nature Previous methods cannot be applied Eﬀective Homology · Sergeraert’s ideas · Provides real algorithms to compute homology groups · Implemented in the Kenzo system J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 66. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries: Intuitive Idea Compute homology groups of a space X H∗ (X ) = H∗ (C∗ (X )) C∗ (X ) has ﬁnite nature (Eﬀective Chain Complex) Diﬀerential maps can be expressed as integer matrices Homology groups: Smith Normal Form C∗ (X ) has non ﬁnite nature (Locally Eﬀective Chain Complex) Previous methods cannot be applied Eﬀective Homology · Sergeraert’s ideas · Provides real algorithms to compute homology groups · Implemented in the Kenzo system J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 67. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition An eﬀective chain complex is a free chain complex of Z-modules, C∗ = (Cn , dn )n∈N , where each group Cn is ﬁnitely generated and an algorithm returns a Z-base in each grade n an algorithm provides the diﬀerentials dn Deﬁnition A locally eﬀective chain complex is a free chain complex of Z-modules C∗ = (Cn , dn )n∈N where each group Cn can have inﬁnite nature, but there exists an algorithm such that ∀x ∈ Cn , we can compute dn (x) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 68. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition An eﬀective chain complex is a free chain complex of Z-modules, C∗ = (Cn , dn )n∈N , where each group Cn is ﬁnitely generated and an algorithm returns a Z-base in each grade n an algorithm provides the diﬀerentials dn diﬀerentials dn : Cn → Cn−1 can be expressed as integer matrices possible to compute Ker dn and Im dn+1 possible to compute the homology groups Deﬁnition A locally eﬀective chain complex is a free chain complex of Z-modules C∗ = (Cn , dn )n∈N where each group Cn can have inﬁnite nature, but there exists an algorithm such that ∀x ∈ Cn , we can compute dn (x) impossible to compute Ker dn and Im dn+1 possible to perform local computations, diﬀerential of a generator J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 69. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition A reduction ρ between two chain complexes C∗ y D∗ (denoted by ρ : C∗ ⇒ D∗ ) is a triple ρ = (f , g , h) h f +D C∗ k ∗ g satisfying the following relations 1) fg = IdD∗ ; 2) dC h + hdC = IdC∗ −gf ; 3) fh = 0; hg = 0; hh = 0. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 70. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition A reduction ρ between two chain complexes C∗ y D∗ (denoted by ρ : C∗ ⇒ D∗ ) is a triple ρ = (f , g , h) h f +D C∗ k ∗ g satisfying the following relations 1) fg = IdD∗ ; 2) dC h + hdC = IdC∗ −gf ; 3) fh = 0; hg = 0; hh = 0. Theorem If C∗ ⇒ D∗ , then C∗ ∼ D∗ ⊕ A∗ , with A∗ acyclic, which implies that = Hn (C∗ ) ∼ Hn (D∗ ) for all n. = J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 71. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐ D∗ , is a triple ⇒ ε = (B∗ , ρ, ρ ) where B∗ is a chain complex, ρ : B∗ ⇒ C∗ and ρ : B∗ ⇒ D∗ . 42 B∗ 30 s{ #+ 14 s{ #+ 21 C∗ D∗ 10 15 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 72. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐ D∗ , is a triple ⇒ ε = (B∗ , ρ, ρ ) where B∗ is a chain complex, ρ : B∗ ⇒ C∗ and ρ : B∗ ⇒ D∗ . 42 B∗ 30 s{ #+ 14 s{ #+ 21 C∗ D∗ 10 15 Deﬁnition An object with eﬀective homology is a quadruple (X , C∗ (X ), HC∗ , ε) where X is a locally eﬀective object C∗ (X ) is a (locally eﬀective) chain complex canonically associated with X , which allows the study of the homological nature of X HC∗ is an eﬀective chain complex ε is a equivalence ε : C∗ (X ) ⇐ HC∗ ⇒ J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 73. Computation Eﬀective Homology preliminariesEﬀective Homology preliminaries Deﬁnition A (strong chain) equivalence ε between C∗ and D∗ , ε : C∗ ⇐ D∗ , is a triple ⇒ ε = (B∗ , ρ, ρ ) where B∗ is a chain complex, ρ : B∗ ⇒ C∗ and ρ : B∗ ⇒ D∗ . 42 B∗ 30 s{ #+ 14 s{ #+ 21 C∗ D∗ 10 15 Deﬁnition An object with eﬀective homology is a quadruple (X , C∗ (X ), HC∗ , ε) where X is a locally eﬀective object C∗ (X ) is a (locally eﬀective) chain complex canonically associated with X , which allows the study of the homological nature of X HC∗ is an eﬀective chain complex ε is a equivalence ε : C∗ (X ) ⇐ HC∗ ⇒ Theorem Let an object with eﬀective homology (X , C∗ (X ), HC∗ , ε) then Hn (X ) ∼ Hn (HC∗ ) for = all n. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 74. Computation Pushout preliminariesPushout preliminaries Deﬁnition Let f , g morphisms, X f /Y g f Z g / P(f ,g ) f ∃!p g !, Q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 75. Computation Pushout preliminariesPushout preliminaries Deﬁnition Let f , g morphisms, the pushout of f , g is an object P(f ,g ) for which the diagram X f /Y g commutes f Z g / P(f ,g ) f ∃!p g !, Q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 76. Computation Pushout preliminariesPushout preliminaries Deﬁnition Let f , g morphisms, the pushout of f , g is an object P(f ,g ) for which the diagram X f /Y g commutes f respects the universal Z g / P(f ,g ) f property ∃!p g !, Q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 77. Computation Pushout preliminariesPushout preliminaries Deﬁnition Let f , g morphisms, the pushout of f , g is an object P(f ,g ) for which the diagram X f /Y g commutes f respects the universal Z g / P(f ,g ) f property ∃!p g !, Q Standard Construction P(f ,g ) ∼ (Y (X × I ) = Z )/ ∼ where I is the unit interval for every x ∈ X , ∼ (x, 0) ∼ f (x) ∈ Y (x, 1) ∼ g (x) ∈ Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 78. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout I Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets X f /Y g f Z g / P(f ,g ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 79. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout I Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets X f /Y g f Z g / P(f ,g ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 80. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout I Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets X f /Y g f Z g / P(f ,g ) Algorithm (Standard Construction) Input: two simplicial morphisms f : X → Y and g : X → Z where X , Y and Z are simplicial sets Output: the pushout P(f ,g ) , a simplicial set J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 81. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with eﬀective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 82. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with eﬀective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 83. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with eﬀective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 84. Computation Eﬀective Homology of the PushoutConstruction of the Eﬀective Homology of the Pushout II Given f : X → Y and g : X → Z simplicial morphisms where X , Y and Z are simplicial sets with eﬀective homology (X , C∗ (X ), HX∗ , εX ) f / (Y , C∗ (Y ), HY∗ , εY ) g (Z , C∗ (Z ), HZ∗ , εZ ) / (P(f ,g ) , C∗ (P(f ,g ) ), HP∗ , εP ) Algorithm (joint work with F. Sergeraert) Input: two simplicial morphisms f : X → Y and g : X → Z where X , Y and Z are simplicial sets with eﬀective homology Output: the eﬀective homology version of P(f ,g ) , that is, an equivalence C∗ (P(f ,g ) ) ⇐ HP∗ , where HP∗ is an eﬀective chain complex ⇒ J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 85. Computation Eﬀective Homology of the PushoutSketch of the algorithm Theorem (SES Theorems) Let 0 o 0 A∗ o σ / B∗ o ρ / C∗ o 0 j i be an eﬀective short exact sequence of chain complexes. Then three general algorithms are available SES1 : (B∗,EH , C∗,EH ) → A∗,EH SES2 : (A∗,EH , C∗,EH ) → B∗,EH SES3 : (A∗,EH , B∗,EH ) → C∗,EH producing the eﬀective homology of one chain complex when the eﬀective homology of both others is given. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 86. Computation Eﬀective Homology of the PushoutSketch of the algorithm Theorem (SES Theorems) Let 0 o 0 A∗ o σ / B∗ o ρ / C∗ o 0 j i be an eﬀective short exact sequence of chain complexes. Then three general algorithms are available SES1 : (B∗,EH , C∗,EH ) → A∗,EH SES2 : (A∗,EH , C∗,EH ) → B∗,EH SES3 : (A∗,EH , B∗,EH ) → C∗,EH producing the eﬀective homology of one chain complex when the eﬀective homology of both others is given. 0 o M∗ o σ / C∗ P o ρ / C∗ Y ⊕ C∗ Z o 0 j i M∗ is the chain complex associated with X × ∆1 but with the simplexes of X × (0) and X × (1) cancelled J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 87. Computation Eﬀective Homology of the PushoutSketch of the algorithm continued Step 1. From f : X → Y and g : X → Z simplicial morphisms, P(f ,g ) and its associated chain complex C∗ P are constructed Step 2. The eﬀective homology of M∗ is constructed · Deﬁne 0 o M∗ o σ2 / C∗ (X × ∆1 ) o / ρ2 C∗ (X × (0)) ⊕ C∗ (X × (1)) o 0 j2 i2 · Construct the chain complex M∗ · Build the eﬀective homology of C∗ (X × ∆1 ) · Construct the eﬀective homology of C∗ (X × (0)) ⊕ C∗ (X × (1)) · Construct the eﬀective homology of Cone(i2) · Construct the reduction M∗⇐ Cone(i2) ⇐ · Obtain the eﬀective homology of M∗ applying case SES1 Step 3. The eﬀective homology of C∗ X ⊕ C∗ Y is constructed Step 4. The eﬀective homology of the pushout P(f ,g ) is constructed · Deﬁne 0 o M∗ o σ / C∗ P o ρ / C∗ Y ⊕ C∗ Z o 0 j i · Construct the eﬀective homology of (C∗ Y ⊕ C∗ Z )[1] · Deﬁne the morphism shift : C∗ Y ⊕ C∗ Z → (C∗ Y ⊕ C∗ Z )[1] · Deﬁne the chain complex morphism χ : M∗ → (C∗ Y ⊕ C∗ Z )[1] · Construct the eﬀective homology of Cone(χ) · Obtain the eﬀective homology of P(f ,g ) applying case SES2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 88. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) SL2 (Z) Group of 2 × 2 matrices with determinant 1 over Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 89. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) SL2 (Z) Group of 2 × 2 matrices with determinant 1 over Z Isomorphic to Z4 ∗Z2 Z6 J. P. Serre. Trees. Springer-Verlag, 1980 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 90. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) SL2 (Z) Group of 2 × 2 matrices with determinant 1 over Z Isomorphic to Z4 ∗Z2 Z6 J. P. Serre. Trees. Springer-Verlag, 1980 i1 K (Z2 , 1) / K (Z4 , 1) i2 K (Z6 , 1) / K (SL2 (Z), 1) K. S. Brown. Cohomology of Groups. Springer-Verlag, 1982 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 91. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 92. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) Firstly, we deﬁne K (Z2 , 1), K (Z4 , 1) and K (Z6 , 1) (setf kz2 (k-zp-1 2)) [K2 Abelian-Simplicial-Group] (setf kz4 (k-zp-1 4)) [K15 Abelian-Simplicial-Group] (setf kz6 (k-zp-1 6)) [K28 Abelian-Simplicial-Group] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 93. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) Firstly, we deﬁne K (Z2 , 1), K (Z4 , 1) and K (Z6 , 1) (setf kz2 (k-zp-1 2)) [K2 Abelian-Simplicial-Group] (setf kz4 (k-zp-1 4)) [K15 Abelian-Simplicial-Group] (setf kz6 (k-zp-1 6)) [K28 Abelian-Simplicial-Group] Subsequently, we deﬁne the two simplicial morphisms i1 and i2 (setf i1 (kzps-incl 2 4)) [K40 Simplicial-Morphism K2 - K15] (setf i2 (kzps-incl 2 6)) [K41 Simplicial-Morphism K2 - K28] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 94. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Construction of the object K (SL2 (Z, 1)) Firstly, we deﬁne K (Z2 , 1), K (Z4 , 1) and K (Z6 , 1) (setf kz2 (k-zp-1 2)) [K2 Abelian-Simplicial-Group] (setf kz4 (k-zp-1 4)) [K15 Abelian-Simplicial-Group] (setf kz6 (k-zp-1 6)) [K28 Abelian-Simplicial-Group] Subsequently, we deﬁne the two simplicial morphisms i1 and i2 (setf i1 (kzps-incl 2 4)) [K40 Simplicial-Morphism K2 - K15] (setf i2 (kzps-incl 2 6)) [K41 Simplicial-Morphism K2 - K28] Finally, we construct the pushout of i1 and i2 (K (SL2 (Z), 1)) (setf ksl2z (pushout i1 i2)) [K52 Simplicial-Set] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 95. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 96. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) Firstly, we deﬁne Σ(K (SL2 (Z), 1)) (setf sksl2z (suspension ksl2z)) [K62 Simplicial-Set] J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 97. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) Firstly, we deﬁne Σ(K (SL2 (Z), 1)) (setf sksl2z (suspension ksl2z)) [K62 Simplicial-Set] Computing Homotopy groups (Hurewicz theorem) (homology sksl2z 1 3) Homology in dimension 1: Homology in dimension 2: Component Z/12Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 98. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computation of π∗ (Σ(SL2 (Z))) Firstly, we deﬁne Σ(K (SL2 (Z), 1)) (setf sksl2z (suspension ksl2z)) [K62 Simplicial-Set] Computing Homotopy groups (Hurewicz theorem) (homology sksl2z 1 3) Homology in dimension 1: Homology in dimension 2: Component Z/12Z π1 (Σ(SL2 (Z))) = 0, π2 (Σ(SL2 (Z))) = Z/12Z J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 99. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computing Homotopy groups continued (Whitehead tower) (setf tau (zp-whitehead 12 sksl2z (chml-clss sksl2z 2))) [K91 Fibration K62 - K79] (setf x (fibration-total tau)) [K97 Simplicial-Set] (homology x 3) Homology in dimension 3: Component Z/12Z (setf tau2 (zp-whitehead 12 x (chml-clss x 3))) [K228 Fibration K97 - K214] (setf x2 (fibration-total tau2)) [K234 Simplicial-Set] (homology x2 4) Homology in dimension 4: Component Z/12Z Component Z/2Z ... J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 100. Computation Eﬀective Homology of the PushoutExample: π∗ (Σ(SL2 (Z))) Computing Homotopy groups continued (Whitehead tower) (setf tau (zp-whitehead 12 sksl2z (chml-clss sksl2z 2))) [K91 Fibration K62 - K79] (setf x (fibration-total tau)) [K97 Simplicial-Set] (homology x 3) Homology in dimension 3: Component Z/12Z (setf tau2 (zp-whitehead 12 x (chml-clss x 3))) [K228 Fibration K97 - K214] (setf x2 (fibration-total tau2)) [K234 Simplicial-Set] (homology x2 4) Homology in dimension 4: Component Z/12Z Component Z/2Z ... π3 (Σ(SL2 (Z))) = Z/12Z, π4 (Σ(SL2 (Z))) = Z/12Z ⊕ Z/2Z, . . . J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 101. DeductionTable of Contents 1 Introduction 2 Communication 3 Computation 4 Deduction 5 Conclusions and Further work J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 102. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 103. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) Programming Language First-Order Logic Theorem Prover J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 104. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) Programming Language First-Order Logic Theorem Prover Proof techniques Simpliﬁcation Induction “The Method” J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 105. Deduction ACL2ACL2 ACL2 (A Computational Logic for an Applicative Common Lisp) Programming Language First-Order Logic Theorem Prover Proof techniques Simpliﬁcation Induction “The Method” Encapsulate principle Simulation of Higher-Order Logic J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 106. Deduction GoalGoal Simpliﬁed view of Kenzo way of working 1 Construction of initial spaces 2 Construction of new spaces from other ones 3 Computation of homology groups J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 107. Deduction GoalGoal Simpliﬁed view of Kenzo way of working 1 Construction of initial spaces Certiﬁcation of the correctness of Kenzo constructors of simplicial sets 2 Construction of new spaces from other ones Easy Perturbation Lemma Composition of reductions Cone Equivalence Theorem ... 3 Computation of homology groups .............................................................................................................. (sphere 3) [K1 Simplicial-Set] .............................................................................................................. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 108. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Deﬁnition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 109. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Deﬁnition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, The elements of K q are called q-simplexes J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 110. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Deﬁnition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, The elements of K q are called q-simplexes q−1 A q-simplex x is degenerate if x = ηi y for some simplex y ∈ K q−1 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 111. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Simplicial Sets Deﬁnition A simplicial set K , is a union K = K q , where the K q are disjoints sets, together with functions q≥0 q ∂i : K q → K q−1 , q 0, i = 0, . . . , q, q ηi : K q → K q+1 , q ≥ 0, i = 0, . . . , q, subject to the relations q−1 q q−1 q (1) ∂i ∂j = ∂j−1 ∂i if i j, q+1 q q+1 q (2) ηi ηj = ηj+1 ηi if i ≤ j, q+1 q q−1 q (3) ∂i ηj = ηj−1 ∂i if i j, q+1 q q+1 q (4) ∂i ηi = identity = ∂i+1 ηi , q+1 q q−1 q (5) ∂i ηj = ηj ∂i−1 if i j + 1, The elements of K q are called q-simplexes q−1 A q-simplex x is degenerate if x = ηi y for some simplex y ∈ K q−1 Otherwise x is called non-degenerate (or geometric) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 112. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Example c 0-simplexes: vertices: (a), (b), (c), (d) non-degenerate 1-simplexes: edges: (a b),(a c),(a d),(b c),(b d),(c d) non-degenerate 2-simplexes: d (ﬁlled) triangles: (a b c),(a b d),(a c d),(b c d) non-degenerate 3-simplexes: a b (ﬁlled) tetrahedron: (a b c d) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 113. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Example c 0-simplexes: vertices: (a), (b), (c), (d) non-degenerate 1-simplexes: edges: (a b),(a c),(a d),(b c),(b d),(c d) non-degenerate 2-simplexes: d (ﬁlled) triangles: (a b c),(a b d),(a c d),(b c d) non-degenerate 3-simplexes: a b (ﬁlled) tetrahedron: (a b c d) (b c) if i = 0 face: ∂i (a b c) = (a c) if i = 1 (a b) if i = 2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 114. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: Example c 0-simplexes: vertices: (a), (b), (c), (d) non-degenerate 1-simplexes: edges: (a b),(a c),(a d),(b c),(b d),(c d) non-degenerate 2-simplexes: d (ﬁlled) triangles: (a b c),(a b d),(a c d),(b c d) non-degenerate 3-simplexes: a b (ﬁlled) tetrahedron: (a b c d) (b c) if i = 0 face: ∂i (a b c) = (a c) if i = 1 (a b) if i = 2 (a a b c) if i = 0 degeneracy: ηi (a b c) = (a b b c) if i = 1 non-geometrical meaning (a b c c) if i = 2 J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 115. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes Proposition Let K be a simplicial set. Any n-simplex x ∈ K n can be expressed in a unique way as a (possibly) iterated degeneracy of a non-degenerate simplex y in the following way x = ηj . . . ηj1 y k r with y ∈ K , k = n − r ≥ 0, and 0 ≤ j1 · · · jk n. J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 116. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes Proposition Let K be a simplicial set. Any n-simplex x ∈ K n can be expressed in a unique way as a (possibly) iterated degeneracy of a non-degenerate simplex y in the following way x = ηj . . . ηj1 y k r with y ∈ K , k = n − r ≥ 0, and 0 ≤ j1 · · · jk n. abstract simplex dgop is a strictly decreasing sequence of degeneracy maps (dgop gmsm) := gmsm is a geometric simplex J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 117. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context: abstract simplexes Proposition Let K be a simplicial set. Any n-simplex x ∈ K n can be expressed in a unique way as a (possibly) iterated degeneracy of a non-degenerate simplex y in the following way x = ηj . . . ηj1 y k r with y ∈ K , k = n − r ≥ 0, and 0 ≤ j1 · · · jk n. abstract simplex dgop is a strictly decreasing sequence of degeneracy maps (dgop gmsm) := gmsm is a geometric simplex Examples simplex abstract simplex non-degenerate (a b) (∅ (a b)) degenerate (a a b c) (η0 (a b c)) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 118. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context degeneracy operator ηiq (dgop gmsm) := (ηiq ◦ dgop gmsm) J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 119. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context degeneracy operator ηiq (dgop gmsm) := (ηiq ◦ dgop gmsm) face operator (∂iq ◦ dgop gmsm) if ηi ∈ dgop ∨ ηi−1 ∈ dgop ∂iq (dgop gmsm) := (∂iq ◦ dgop ∂k gmsm) r otherwise; where r = q − {number of degeneracies in dgop} and k = i − {number of degeneracies in dgop with index lower than i} J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology
- 120. Deduction Certifying the correctness of Kenzo simplicial setsMathematical context degeneracy operator ηiq (dgop gmsm) := (ηiq ◦ dgop gmsm) face operator (∂iq ◦ dgop gmsm) if ηi ∈ dgop ∨ ηi−1 ∈ dgop ∂iq (dgop gmsm) := (∂iq ◦ dgop ∂k gmsm) r otherwise; where r = q − {number of degeneracies in dgop} and k = i − {number of degeneracies in dgop with index lower than i} invariant operator (dgop gmsm) ∈ K q (length dgop) q gmsm ∈ K r where r = q − (length dgop) index of the ﬁrst degeneracy in dgop is lower than q J´nathan Heras Vicente o Mathematical Knowledge Management in Algebraic Topology

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