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 |
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Keywords
- Final lift actions
- Topological category
ASJC Scopus subject areas
- Applied Mathematics
Cite this
Final lift actions associated with topological functors. / Mielke, Marvin.
In: Applied Categorical Structures, Vol. 10, No. 5, 01.10.2002, p. 495-504.Research output: Contribution to journal › Article
}
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 -