گروهواره

در ریاضیات، به‌خصوص در نظریه رسته‌ها و نظریه هموتوپی، **گروهواره** (Groupoid) (اصطلاحات با رواج کمتر: **گروهواره براندت** یا **گروه مجازی**)، مفهوم گروه را به طرق معادل مختلفی تعمیم می‌دهد. گروهواره را می‌توان حداقل به دو صورت زیر دید:
* گروهی که عملگر دوتایی آن با تابعی جزئی جایگزین شده باشد؛
* رسته‌ای که در آن هر ریخت معکوس‌پذیر باشد. رسته‌ای از این نوع را می‌توان به صورت رسته مجهز به عمل یگانی دید که به این عمل *معکوس* گفته می‌شود (در قیاس با نظریه گروه‌ها). گروهواره‌ای که تنها یک شیء داشته باشد همان گروه رایج است.

## ارجاعات

## منابع

* 
* Brown, Ronald, 1987, "From groups to groupoids: a brief survey," *Bull. London Math. Soc.* **19**: 113-34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
* —, 2006. *[http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html Topology and groupoids.]* Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
* —, Higher dimensional group theory Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology. Many references.
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* 
* F. Borceux, G. Janelidze, 2001, *[https://archive.today/20121223050454/http://www.cup.cam.ac.uk/catalogue/catalogue.asp?isbn=9780521803090 Galois theories.]* Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids.
* Cannas da Silva, A., and A. Weinstein, *[http://www.math.ist.utl.pt/~acannas/Books/models_final.pdf Geometric Models for Noncommutative Algebras.]* Especially Part VI.
* Golubitsky, M., Ian Stewart, 2006, "Nonlinear dynamics of networks: the groupoid formalism", *Bull. Amer. Math. Soc.* **43**: 305-64
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* Higgins, P. J. , "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145—149.
* Higgins, P. J. and Taylor, J. , "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math. , Volume 962. Springer, Berlin (1982), 115—122.
* Higgins, P. J. , 1971. *Categories and groupoids.* Van Nostrand Notes in Mathematics. Republished in *Reprints in Theory and Applications of Categories*, No. 7 (2005) pp. 1–195; freely downloadable. Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g. fundamental groupoid.
* Mackenzie, K. C. H. , 2005. *[https://web.archive.org/web/20050310034123/http://www.shef.ac.uk/~pm1kchm/gt.html General theory of Lie groupoids and Lie algebroids.]* Cambridge Univ. Press.
* Weinstein, Alan, "Groupoids: unifying internal and external symmetry — A tour through some examples." Also available in Postscript., Notices of the AMS, July 1996, pp. 744–752.
* Weinstein, Alan, "The Geometry of Momentum" (2002)
* R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In *Algebraic and geometric combinatorics*, volume 423 of *Contemp. Math*. , 305–324. Amer. Math. Soc. , Providence, RI (2006)
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رده:ساختارهای جبری
رده:نظریه رسته‌ها
رده:نظریه هموتوپی