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groupoids
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Several classes of groupoids and large groupoids shall be considered in this topic with pertinent examples that illustrate the construction of groupoids through several extensions of the much simpler (and global) group symmetry to both higher order symmetries and dimensions, as well as internal (or local, partial) plus external symmetry. Considered as powerful tools for investigating both Abelian and non-Abelian structures, groupoids are now essential for understanding topology, and are one of the important-if not the most important- concepts in algebraic topology ([1])
Groupoids are generalizations or extensions of the concept of group, supergroup, `virtual group', and paragroup, in several ways; one may simply extend the notion of a group viewed as an one-object category to a many-object category with group-like elements and all invertible morphisms. Another enrichment of the notion of a group-as in the case of topological groups- is the concept of topological groupoid
 . One can also think of a groupoid as a class of linked groups, and further extend the latter groupoid definition to higher dimensions through `geometric'-algebraic constructions, for example, to double groupoids, cubic groupoids, ..., groupoid categories, groupoid supercategories, and so on. Crossed modules of groups and crossed complexes also correspond to such extended groupoids.
For precise definitions of specific classes of groupoids, see also groupoid and topological groupoid definitions, as well as those entries listed next as examples.
of major classes of groupoids defining the several extensions and enrichment possibilities of the notions of group and group symmetry introduced in the above definition are the subject of several other entries:
- 2-groupoids (please see groupoid categories)
- Double groupoids; homotopy double groupoid of a Hausdorff space
- Higher homotopy groupoids and the higher dimensional, generalized van Kampen theorems
- Groupoid category
- Crossed complexes
- Higher dimensional algebra (HDA)
- Groupoid super-categories ($n$ -categories, etc.)
- Groupoid supercategories
- 1
- R. Brown. 2006. Topology and Groupoids. Booksurge PLC.
- 2
- R. Brown. 2008. Nonabelian Algebraic Topology. preprint, (two volumes).
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"groupoids" is owned by bci1. [ full author list (2) ]
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See Also: groupoid, groupoid categories, groupoid homomorphism, homotopy double groupoid of a Hausdorff space, topological groupoid, quantum groups, higher dimensional generalized Van Kampen theorems (HD-VKT), equivalent representations of groupoids, groupoid and group representations related to quantum symmetries, locally compact groupoids, quantum operator algebras in quantum field theories, symmetry and groupoid representations in functional biology, groupoid representations induced by measure, algebroid structures and extended symmetries, groupoid C*-dynamical system, Borel groupoid, Borel morphism
| Other names: |
groupoid categories, topological groupoids, supergroups |
| Keywords: |
groupoid categories, groupoid categories, topological groupoid, Lie groupoid, examples of groupoids, higher homotopy groupoids, HDA, higher dimensional algebraic topology (HDAT), groupoid and group representations related to quantum symmetries |
This object's parent.
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Cross-references: super-categories, higher dimensional algebra, theorems, higher homotopy groupoids, homotopy double groupoid of a Hausdorff space, definitions, complexes, modules, supercategories, double groupoids, morphisms, invertible, category, virtual group, algebraic, topology, non-Abelian structures, abelian, plus, dimensions, order, symmetry, group, extensions, classes
There are 45 references to this entry.
This is version 40 of groupoids, born on 2008-07-23, modified 2009-02-01.
Object id is 10856, canonical name is Groupoids.
Accessed 2550 times total.
Classification:
| AMS MSC: | 18B40 (Category theory; homological algebra :: Special categories :: Groupoids, semigroupoids, semigroups, groups ) | | | 18G55 (Category theory; homological algebra :: Homological algebra :: Homotopical algebra) | | | 55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory) | | | 55U35 (Algebraic topology :: Applied homological algebra and category theory :: Abstract and axiomatic homotopy theory) | | | 55U05 (Algebraic topology :: Applied homological algebra and category theory :: Abstract complexes) |
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Pending Errata and Addenda
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