Fundamental pro-groupoids and covering projections
We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs(X) and an induced category pro(π crs(X), Sets) such that for any to...
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 1998 |
| País: | España |
| Recursos: | Universidad de La Rioja (UR) |
| Repositorio: | RIUR. Repositorio Institucional de la Universidad de La Rioja |
| OAI Identifier: | oai:portal.dialnet.es:doc/5bbc68a3b750603269e80ca7 |
| Acesso em linha: | https://investigacion.unirioja.es/documentos/5bbc68a3b750603269e80ca7 |
| Access Level: | acceso abierto |
| Palavra-chave: | ÄŒech fundamental group ÄŒech fundamental pro-groupoid Category of fractions Covering projection Covering reduced sieve Covering transformation Fundamental groupoid G-sets Locally constant presheaf Pro-groupoid Subdivision |
| Resumo: | We introduce a new notion of covering projection E → X of a topological space X which reduces to the usual notion if X is locally connected. We use locally constant presheaves and covering reduced sieves to find a pro-groupoid π crs(X) and an induced category pro(π crs(X), Sets) such that for any topological space X the category of covering projections and transformations of X is equivalent to the category pro(π crs(X), Sets). We also prove that the latter category is equivalent to pro(πCX, Sets), where πCX is the Čech fundamental pro-groupoid of X. If X is locally path-connected and semilocally 1-connected, we show that π crs(X) is weakly equivalent to πX, the standard fundamental groupoid of X, and in this case pro(π crs(X), Sets) is equivalent to the functor category SetsπX. If (X, *) is a pointed connected compact metrisable space and if (X, *) is 1-movable, then the category of covering projections of X is equivalent to the category of continuous π̌(X, *)-sets, where π̌1(X, *) is the Čech fundamental group provided with the inverse limit topology. |
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