Organisms, viruses, and computers

Contents

1. Essay ISeparationAn (∞)-Topos-Theoretic Separation of Organisms, Viruses, and ComputersAutonomous closure, host-relative closure, and externally maintained machinery are distinct, and the functors that identify a virus with a program do not preserve the distinction.
   28 sections · 29 numbered results

   1. 1Ambient (∞)-topos
   2. 2Typed physical systems
   3. 3The substrate invariant
   4. 4Organizational signatures
   5. 5Organizational resource data
   6. 6Homotopy invariance
   7. 7Closure is preserved by equivalence
   8. 8External supplementation
   9. 9Image of combined production
   10. 10Base change and descent
   11. 11Closure satisfies descent
   12. 12Axiomatized cells
   13. 13Axiomatized viruses
   14. 14Axiomatized conventional computers
   15. 15Cell/computer separation
   16. 16Virus/cell separation
   17. 17The virus/computer analogy
   18. 18Non-conservativity of the analogy
   19. 19The original argument, categorically
   20. 20The combined invariant
   21. 21Substrate and organization are independent
   22. 22Simulation is weaker than equivalence
   23. 23Open-system category theory
   24. 24Relation to Rosen
   25. 25Relation to autopoiesis
   26. 26Main separation theorem
   27. 27Logical form of the conclusion
   28. 28Final categorical formulation
   29. References
2. Essay IIOperadsOperadic Autopoiesis in an (∞)-ToposA cell is a requirement operad inside a least fixed point of endogenous production. A virus closes only after a homotopy pushout with a host. A computer, on the physical signature, does not close at all.
   60 sections · 36 numbered results

   1. 1Ambient (∞)-topos
   2. 2Internal (∞)-categories
   3. 3Internal symmetric monoidal (∞)-categories
   4. 4Open physical processes
   5. 5The process (∞)-operad
   6. 6Internal suboperads
   7. 7Systems as bounded operadic realizers
   8. 8Endogenous processes
   9. 9Organizational requirements
   10. 10Why one-step generation is not enough
   11. 11Production doctrine
   12. 12Recursive endogenous closure
   13. 13Operadic organizational closure
   14. 14Interpretation
   15. 15Equivalence of structured systems
   16. 16Fixed-point invariance
   17. 17Autopoiesis is invariant
   18. 18Interaction interfaces
   19. 19Operadic coupling
   20. 20Why the pushout is larger than the union
   21. 21Mixed trees
   22. 22Emergent causal capacity
   23. 23Coupled production
   24. 24Relative closure
   25. 25Strict host dependence
   26. 26The virus theorem
   27. 27The cell
   28. 28Virus and cell
   29. 29The computer
   30. 30Cell and computer
   31. 31Stronger than a list of parts
   32. 32Material typing
   33. 33The full physical invariant
   34. 34Why a virus and a program can look alike
   35. 35Non-conservativity
   36. 36The virus/program analogy
   37. 37Localization
   38. 38Simulation
   39. 39Locality
   40. 40Descent for closure
   41. 41Host coupling is not a change of context
   42. 42Three loci
   43. 43An important non-theorem
   44. 44A stratified invariant
   45. 45One invariant proves little
   46. 46A hierarchy of forgetful semantics
   47. 47Forgetfulness in general
   48. 48Main separation theorem
   49. 49What the equations say about a bacterium
   50. 50What the equations say about a virus
   51. 51What the equations say about a computer
   52. 52A sharper analogy, still not an equivalence
   53. 53Higher homotopies
   54. 54Robustness as homotopy
   55. 55Evolution as deformation
   56. 56Closure is scale-relative
   57. 57Enlarging the boundary
   58. 58Do not move the boundary in silence
   59. 59Relation to the autopoiesis problem
   60. 60Final formulation
   61. References
3. Essay IIIThermodynamicsThermodynamically Sustained Autopoiesis in an (∞)-ToposLife-like autonomy is a least fixed point of feasible production together with a greatest fixed point of viable dynamics. Organizational closure is sustained by thermodynamic openness, not by isolation.
   60 sections · 24 numbered results

   1. 1The ambient physical (∞)-topos
   2. 2Open physical processes
   3. 3The cost object
   4. 4Thermodynamic valuation
   5. 5Sequential and parallel composition
   6. 6Nonequilibrium free energy
   7. 7Entropy production
   8. 8Exergy
   9. 9Thermodynamically feasible operations
   10. 10Budget-indexed operadic composition
   11. 11The thermodynamic production doctrine
   12. 12Thermodynamic organizational closure
   13. 13State objects
   14. 14The viability subobject
   15. 15Admissible controls
   16. 16The viability predecessor
   17. 17Monotonicity
   18. 18The viability kernel
   19. 19Two fixed points
   20. 20Thermodynamically sustained autopoiesis
   21. 21Why both fixed points are required
   22. 22Living systems are not thermodynamically closed
   23. 23Maintenance cost
   24. 24An energetic obstruction
   25. 25The cumulative form
   26. 26Isolation cannot pay the maintenance
   27. 27The operadic maintenance burden
   28. 28Flux and force
   29. 29Host coupling
   30. 30Cost of a mixed tree
   31. 31Two ways a host can help
   32. 32Viral thermodynamic dependence
   33. 33Coupling is not a free lunch
   34. 34Subsystem cost is not total cost
   35. 35Industrial coupling of computers
   36. 36The dependency analogy, and its limit
   37. 37The cell, thermodynamically
   38. 38The thermodynamic invariant
   39. 39Thermodynamic equivalence
   40. 40Invariance
   41. 41Thermodynamic separation
   42. 42The virus is a different object from the infected cell
   43. 43Changing the boundary changes the object
   44. 44No self-sufficiency claim without a boundary
   45. 45Feedstock is not machinery
   46. 46Two operads, not one pile of matter
   47. 47What a bacterium imports
   48. 48The ordinary computer has the opposite architecture
   49. 49The theory does not ban artificial life
   50. 50Autopoiesis does not classify up to equivalence
   51. 51Viability as a modality
   52. 52Localization and the kernel together
   53. 53A viability localization is not a physical equivalence
   54. 54A scale of equivalence
   55. 55The full invariant
   56. 56Full invariance
   57. 57Main separation theorem
   58. 58The diagram
   59. 59The defining equations
   60. 60Conclusion
   61. References

Reading map

Notation

$\mathcal{E}$

Ambient (∞)-topos of physical contexts.

$K,{\tau }_X,\mathrm{Supp}(X)$

Component types, the typing map, and material support.

$\Sigma ,R_a,d_a,p_a$

Organizational signature, resource, demand, and endogenous production.

${\mathrm{Req}}_a,{\mathrm{End}}_a$

Images of demand and of endogenous production.

${\mathrm{Cl}}_a(X),{\mathrm{Cl}}_a(X\mid H)$

Closure in mode a, and closure after supplementation by H.

${\mathbb{O}}^{\otimes }$

Internal colored (∞)-operad of open physical processes.

${\mathbb{E}}_X,{\mathbb{R}}_X$

Endogenous process operad and organizational requirement operad.

${\Phi }_X,{\mathbb{G}}_X=\mu {\Phi }_X$

Production doctrine and its least fixed point.

$X\star H$

Coupling: homotopy pushout of endogenous operads along a common interface.

$\mathbb{Q},\Theta ,c(f)$

Thermodynamic cost quantale, valuation, and minimum cost of an operation.

${\Phi }_X^{\mathrm{th}},{\mathbb{G}}_X^{\mathrm{th}}$

Production restricted to thermodynamically feasible realizations.

$S_X,V_X,{\Psi }_X$

State object, viability subobject, and viability operator.

${\mathbb{V}}_X=\nu {\Psi }_X$

Viability kernel: greatest fixed point of admissible dynamics.

$\mathrm{TA}(X),m_X$

Thermodynamically sustained autopoiesis, and minimal maintenance power.

$U,Q,L$

Forgetful, dependency, and localization functors. None is assumed conservative.