| 词条 | Draft:Fundamental groupoid |
| 释义 |
In algebraic topology, the fundamental groupoid of a topological space is a generalization of the fundamental group. It is a topological invariant, and so can be used to distinguish non-homeomorphic spaces. The fundamental groupoid captures information about both the connectedness and homotopy type of the space. The homotopy hypothesis, an important conjecture in homotopy theory formulated by Alexander Grothendieck, states that a suitable generalization of the fundamental groupoid captures all information about the space up to homotopy equivalence. {{Quote|quote = [...] In certain situations (such as descent theorems for fundamental groups à la van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids [...] |author = Alexander Grothendieck |source = Esquisse d'un Programme (Section 2, English translation) }} MotivationFormal definitionProperties{{Expand section}}The fundamental groupoid of a space {{var|X}} is connected if and only if {{var|X}} is path-connected. Examples
GeneralizationsThe fundamental weak ∞-groupoid{{See also|∞-groupoid}}The homotopy hypothesis{{Main|Homotopy hypothesis}}{{Expand section}}The homotopy hypothesis is an important conjecture in homotopy theory formulated by Alexander Grothendieck. It states that a suitable generalization of the fundamental groupoid captures all information about the space up to homotopy equivalence. In homotopy type theory{{Expand section}}In intensional intuitionistic type theory (ITT), types have the structure of weak ∞-groupoids (for details and references, see Homotopy type theory#History). This observation led to the development of homotopy type theory, in which weak ∞-groupoids are a primitive or synthetic notion (meaning they are not defined within the theory). See also
References
External links
|
| 随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。