ora-0026
1.14 Repair spaces
Let \(p:E\to X\) be a simplicial family of decorated decisions. A partial decorated horn \(e_\Lambda :\Lambda _i^n\to E\) over a proposed simplex \(\sigma :\Delta ^n\to X\) determines a derived lifting space
\[ \operatorname {hofib}_{(e_\Lambda ,\sigma )} \left[ \operatorname {Map}(\Delta ^n,E)\to \operatorname {Map}(\Lambda _i^n,E) \mathop{\times }_{\operatorname {Map}(\Lambda _i^n,X)} \operatorname {Map}(\Delta ^n,X) \right]. \]
This is the space of compatible fillers.
|
Repair-space shape |
Structural reading |
What a scalar can miss |
|---|---|---|
|
\(\varnothing \) |
no admissible completion |
source of incompatibility |
|
contractible |
canonical up to coherent homotopy |
internal witnesses |
|
several components |
qualitatively different repair classes |
which alternatives remain possible |
|
nontrivial higher homotopy |
coherent families of transformations |
higher relations among repairs |