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Groupoid
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=== Čech groupoid === {{See also|Simplicial manifold|Nerve of a covering}} A Čech groupoid<ref name=":0">{{cite arXiv|last1=Block|first1=Jonathan|last2=Daenzer|first2=Calder|date=2009-01-09|title=Mukai duality for gerbes with connection|class=math.QA|eprint=0803.1529}}</ref><sup>p. 5</sup> is a special kind of groupoid associated to an equivalence relation given by an open cover <math>\mathcal{U} = \{U_i\}_{i\in I}</math> of some manifold {{tmath|1= X }}. Its objects are given by the disjoint union <math display="block">\mathcal{G}_0 = \coprod U_i ,</math> and its arrows are the intersections <math display=block>\mathcal{G}_1 = \coprod U_{ij} .</math> The source and target maps are then given by the induced maps<blockquote><math>\begin{align} s = \phi_j: U_{ij} \to U_j\\ t = \phi_i: U_{ij} \to U_i \end{align}</math></blockquote>and the inclusion map<blockquote><math>\varepsilon: U_i \to U_{ii}</math></blockquote>giving the structure of a groupoid. In fact, this can be further extended by setting<blockquote><math>\mathcal{G}_n = \mathcal{G}_1\times_{\mathcal{G}_0} \cdots \times_{\mathcal{G}_0}\mathcal{G}_1</math></blockquote>as the <math>n</math>-iterated fiber product where the <math>\mathcal{G}_n</math> represents <math>n</math>-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since<blockquote><math>\begin{matrix} U_{ijk} & \to & U_{ij} \\ \downarrow & & \downarrow \\ U_{ik} & \to & U_{i} \end{matrix}</math></blockquote>is a cartesian diagram where the maps to <math>U_i</math> are the target maps. This construction can be seen as a model for some [[∞-groupoid]]s. Also, another artifact of this construction is [[Čech cohomology|k-cocycles]]<blockquote><math>[\sigma] \in \check{H}^k(\mathcal{U},\underline{A})</math></blockquote>for some constant [[sheaf of abelian groups]] can be represented as a function<blockquote><math>\sigma:\coprod U_{i_1\cdots i_k} \to A</math></blockquote>giving an explicit representation of cohomology classes.
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