In mathematics, especially in higher-dimensional algebra and homotopy theory, a double groupoid generalises the notion of groupoid and of category to a higher dimension.
a doublegroupoid generalises the notion of groupoid and of category to a higher dimension. A doublegroupoid D is a higher-dimensional groupoid involving...
homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen...
Brown, R.; Hardie, K.; Kamps, H.; Porter, T. (2002), "The homotopy doublegroupoid of a Hausdorff space", Theory Appl. Categories, 10 (2): 71–93. Brown...
properties and applications of homotopy groupoids and filtered spaces. Noncommutative doublegroupoids and double algebroids are only the first examples...
fundamental groupoid of the orbit space X/G is isomorphic to the orbit groupoid of the fundamental groupoid of X, i.e. the quotient of that groupoid by the...
Kampen's Theorem: A discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set Animations to introduce fundamental...
_{G}^{\bullet }(X)} which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex are Ω G n ( X ) = ⨁ 2 k + i...
to the Morita equivalence class of the unit groupoid. If N is a compact manifold with boundary, its double M can be formed by gluing together a copy of...
examples of bicomplexes that come up in nature. In particular, for a Lie groupoid, there is a bicomplex associated to itpg 7-8 which can be used to construct...
differentiated in order to obtain a Lie bialgebroid. A Poisson groupoid is a Lie groupoid (G⇉M) together with a Poisson structure π on G such that the multiplication...
by the action groupoid G′ = G ⋉ X associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits...
automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories...
characteristic of a finite groupoid is the sum of 1/ |Gi |, where we picked one representative group Gi for each connected component of the groupoid. Euler calculus...
ISBN 978-1-4020-9383-8 Jean Pradines, In Ehresmann's footsteps: from group geometries to groupoid geometries (English summary) Geometry and topology of manifolds, 87–157...
V^{**}} from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps...
iterate the procedure, and G {\displaystyle G} is homotopy equivalent to the double loop space Ω 2 B ( B G ) {\displaystyle \Omega ^{2}B(BG)} . In case G {\displaystyle...
Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.) Lang, Serge (1994). Algebraic Number Theory. Berlin, New York: Springer-Verlag...
model featured in Giraud's theory of gerbes, which are roughly sheaves of groupoids over M. In 1994 Murray introduced bundle gerbes, which are geometric realizations...
e = ett* for some projection e. In a *-semigroup, PI(S) is an ordered groupoid with the partial product given by s⋅t = st if s*s = tt*. In terms of examples...
replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop...
Neumann algebras of a measurable equivalence relation and a measurable groupoid can be defined. These examples generalise von Neumann group algebras and...
This leads to the idea of modding out the gauge group to obtain the gauge groupoid as the closest description of the gauge connection in quantum field theory...
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