### Abstract

For a topological functor U : E → B, the fiber U^{-1}(b), b ∈ B, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b, b) on U^{-1} (b) is cocontinuous. It is shown that every cocontinuous left action of B(b, b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b = Ω, the subobject classifier, then B(Ω, Ω) inherits both a monoidal and a cocomplete poset structure. In the case B = Sets, all cocontinuous left actions of B(Ω, Ω) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.

Original language | English (US) |
---|---|

Pages (from-to) | 495-504 |

Number of pages | 10 |

Journal | Applied Categorical Structures |

Volume | 10 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2002 |

### Fingerprint

### Keywords

- Final lift actions
- Topological category

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Final lift actions associated with topological functors.** / Mielke, Marvin.

Research output: Contribution to journal › Article

*Applied Categorical Structures*, vol. 10, no. 5, pp. 495-504. https://doi.org/10.1023/A:1020534620672

}

TY - JOUR

T1 - Final lift actions associated with topological functors

AU - Mielke, Marvin

PY - 2002/10/1

Y1 - 2002/10/1

N2 - For a topological functor U : E → B, the fiber U-1(b), b ∈ B, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b, b) on U-1 (b) is cocontinuous. It is shown that every cocontinuous left action of B(b, b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b = Ω, the subobject classifier, then B(Ω, Ω) inherits both a monoidal and a cocomplete poset structure. In the case B = Sets, all cocontinuous left actions of B(Ω, Ω) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.

AB - For a topological functor U : E → B, the fiber U-1(b), b ∈ B, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b, b) on U-1 (b) is cocontinuous. It is shown that every cocontinuous left action of B(b, b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b = Ω, the subobject classifier, then B(Ω, Ω) inherits both a monoidal and a cocomplete poset structure. In the case B = Sets, all cocontinuous left actions of B(Ω, Ω) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.

KW - Final lift actions

KW - Topological category

UR - http://www.scopus.com/inward/record.url?scp=0036807310&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036807310&partnerID=8YFLogxK

U2 - 10.1023/A:1020534620672

DO - 10.1023/A:1020534620672

M3 - Article

AN - SCOPUS:0036807310

VL - 10

SP - 495

EP - 504

JO - Applied Categorical Structures

JF - Applied Categorical Structures

SN - 0927-2852

IS - 5

ER -