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.

منابع

• Brandt, H (1927), "Über eine Verallgemeinerung des Gruppenbegriffes", Mathematische Annalen, 96 (1): 360–366, doi:10.1007/BF01209171, S2CID 119597988
• 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. 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.
• Dicks, Warren; Ventura, Enric (1996), The group fixed by a family of injective endomorphisms of a free group, Mathematical Surveys and Monographs, vol. 195, AMS Bookstore, ISBN 978-0-8218-0564-0
• Dokuchaev, M.; Exel, R.; Piccione, P. (2000). "Partial Representations and Partial Group Algebras". Journal of Algebra. Elsevier. 226: 505–532. arXiv:math/9903129. doi:10.1006/jabr.1999.8204. ISSN 0021-8693. S2CID 14622598.
• F. Borceux, G. Janelidze, 2001, Galois theories. Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids.
• Cannas da Silva, A., and A. Weinstein, 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
• "Groupoid", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• 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. 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)
• fundamental groupoid in nLab
• core in nLab