### Abstract

We propose a model of evolutionary systems characterized by internal measurement, or endo-physics, that is based on a particular graph morphism. If we take an internal stance, we have to focus on the influence of the outside of the domain of choice. The outside of the domain is not explicitly expressed as a particular ensemble in advance, although a formal expression generally requires an explicit form of the outside as the possible forms. In our model, a system is expressed by a directed graph, and the time transition is expressed as a particular graph morphism referring to the outside of the domain of the morphism. Referring to the outside of the domain is expressed by dynamical decomposition and synthesis of the relationship between a graph and its possible forms expressed as an underlying free category. Because the transformations from a graph to a category and vice versa are expressed as two adjunctive functors, a graph morphism is defined by a sequence of the operation of a free functor and a special operator standing for invalidation of the operation of the free functor. Time development of this system can generate a particular universal structure like a limit and co-limit as a part of a directed graph, which stands for generation of a higher order. We also argue that the process toward the generation of a universal structure can reveal intermittency, and that an emergent boundary in pattern formation can be demonstrated by the generation of this universal structure. In addition, we discuss emergent computation such as the origin of primitive recursive functions through the origination of a universal structure.

Original language | English |
---|---|

Pages (from-to) | 283-313 |

Number of pages | 31 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 156 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2001 Aug 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Category theory
- Emergent boundary
- Internal measurement
- Origin of universality

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*156*(3-4), 283-313. https://doi.org/10.1016/S0167-2789(01)00237-8

**The origin of universality : Making and invalidating a free category.** / Gunji, Yukio; Higashi, Hideki.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 156, no. 3-4, pp. 283-313. https://doi.org/10.1016/S0167-2789(01)00237-8

}

TY - JOUR

T1 - The origin of universality

T2 - Making and invalidating a free category

AU - Gunji, Yukio

AU - Higashi, Hideki

PY - 2001/8/15

Y1 - 2001/8/15

N2 - We propose a model of evolutionary systems characterized by internal measurement, or endo-physics, that is based on a particular graph morphism. If we take an internal stance, we have to focus on the influence of the outside of the domain of choice. The outside of the domain is not explicitly expressed as a particular ensemble in advance, although a formal expression generally requires an explicit form of the outside as the possible forms. In our model, a system is expressed by a directed graph, and the time transition is expressed as a particular graph morphism referring to the outside of the domain of the morphism. Referring to the outside of the domain is expressed by dynamical decomposition and synthesis of the relationship between a graph and its possible forms expressed as an underlying free category. Because the transformations from a graph to a category and vice versa are expressed as two adjunctive functors, a graph morphism is defined by a sequence of the operation of a free functor and a special operator standing for invalidation of the operation of the free functor. Time development of this system can generate a particular universal structure like a limit and co-limit as a part of a directed graph, which stands for generation of a higher order. We also argue that the process toward the generation of a universal structure can reveal intermittency, and that an emergent boundary in pattern formation can be demonstrated by the generation of this universal structure. In addition, we discuss emergent computation such as the origin of primitive recursive functions through the origination of a universal structure.

AB - We propose a model of evolutionary systems characterized by internal measurement, or endo-physics, that is based on a particular graph morphism. If we take an internal stance, we have to focus on the influence of the outside of the domain of choice. The outside of the domain is not explicitly expressed as a particular ensemble in advance, although a formal expression generally requires an explicit form of the outside as the possible forms. In our model, a system is expressed by a directed graph, and the time transition is expressed as a particular graph morphism referring to the outside of the domain of the morphism. Referring to the outside of the domain is expressed by dynamical decomposition and synthesis of the relationship between a graph and its possible forms expressed as an underlying free category. Because the transformations from a graph to a category and vice versa are expressed as two adjunctive functors, a graph morphism is defined by a sequence of the operation of a free functor and a special operator standing for invalidation of the operation of the free functor. Time development of this system can generate a particular universal structure like a limit and co-limit as a part of a directed graph, which stands for generation of a higher order. We also argue that the process toward the generation of a universal structure can reveal intermittency, and that an emergent boundary in pattern formation can be demonstrated by the generation of this universal structure. In addition, we discuss emergent computation such as the origin of primitive recursive functions through the origination of a universal structure.

KW - Category theory

KW - Emergent boundary

KW - Internal measurement

KW - Origin of universality

UR - http://www.scopus.com/inward/record.url?scp=0035882706&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035882706&partnerID=8YFLogxK

U2 - 10.1016/S0167-2789(01)00237-8

DO - 10.1016/S0167-2789(01)00237-8

M3 - Article

AN - SCOPUS:0035882706

VL - 156

SP - 283

EP - 313

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -