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Groupoid


There are at least three definitions of "groupoid" currently in use.

The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set S returns a value which is itself a member of S). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88-103). A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having n=1, 2, ... elements are 1, 10, 3330, 178981952, ... (OEIS A001329), and the corresponding numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ... (OEIS A001424). An associative groupoid is called a semigroup.

The second type of groupoid is, roughly, a category which is "group-like" in the sense that every morphism (or arrow) is invertible. To make this notion more precise, one says that a groupoid is a category G consisting of a set G_0 of objects and a set G_1 of arrows whereby each arrow g:s(g)|->t(g) in G_1 has an inverse arrow g^(-1):t(g)|->s(g) (also in G_1) subject to the identities gg^(-1)=I_y and g^(-1)g=I_x. Here, x=s(g) denotes the source of the arrow g, y=t(g) denotes the target of g, and I_a equals the identity arrow I:a|->a of an object a in G_0. This notion of groupoid has become widely applied throughout modern mathematics and is often seen to generalize many group-theoretic notions in a number of fields; in particular, one can define the fundamental groupoid of a manifold M, as well as more general objects such as Lie groupoids, holonomy groupoids, Étale groupoids, etc. (Moerdijk and Mrčun 2003).

The third type of groupoid is an algebraic structure first defined by Brandt (1926) and also known as a virtual group. A groupoid with base B (or "over B") is a set G with mappings alpha and beta from G onto B and a partially defined binary operation (g,h)|->gh, satisfying the following four conditions:

1. gh is defined whenever beta(g)=alpha(h), and in this case alpha(gh)=alpha(g) and beta(gh)=beta(h).

2. Associativity: if either of (gh)k and g(hk) are defined so is the other and they are equal.

3. For each g in G, there are left- and right-identity elements lambda_g and rho_g respectively, satisfying lambda_gg=g=grho_g.

4. Each g in G has an inverse g^(-1) satisfying g^(-1)g=rho_g and gg^(-1)=lambda_g.

Any group is a groupoid with base a single point.

The most basic example of groupoids with base B is the pair groupoid, where G=B×B, and alpha(x,y)=x, beta(x,y)=y, and with multiplication (x,y)(y,z)=(x,z). Any equivalence relation on B defines a subgroupoid of the pair groupoid.

A useful way to think of a groupoid of the third type is as a parametrized equivalence relation on B, as follows. Given a groupoid over B, define an equivalence relation on B by alpha(g)∼beta(g) for each g in G. This equivalence relation is "parameterized" because there may be more than one element in G which give rise to the same equivalence,that is, g and g^' such that alpha(g)=alpha(g^') and beta(g)=beta(g^').

Though it is not obvious, one can show with a bit of work that the second and third definitions are actually equivalent.


See also

Binary Operator, Etale Space, Fundamental Group, Fundamental Groupoid, Holonomy Group, Inverse Semigroup, Lie Algebra, Lie Algebroid, Lie Group, Lie Groupoid, Monoid, Quasigroup, Semigroup, Stack of Groupoids, Topological Groupoid

Portions of this entry contributed by Christopher Stover

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References

Brandt, W. "Über eine Verallgemeinerung des Gruppengriffes." Math. Ann. 96, 360-366, 1926.Brown, R. "From Groups to Groupoids: A Brief Survey." Bull. London Math. Soc. 19, 113-134, 1987.Brown, R. Topology: A Geometric Account of General Topology, Homotopy Types, and the Fundamental Groupoid. New York: Halsted Press, 1988.Higgins, P. J. Notes on Categories and Groupoids. London: Van Nostrand Reinhold, 1971.Moerdijk, I. and Mrčun, J. Introduction to Foliations and Lie Groupoids. New York: Cambridge University Press, 2003.Renault, J. and Ramazan, B. (Eds.). "Groupoids Home Page." http://unr.edu/homepage/ramazan/groupoid/.Rosenfeld, A. An Introduction to Algebraic Structures. New York: Holden-Day, 1968.Sloane, N. J. A. Sequences A001329/M4760 and A001424 in "The On-Line Encyclopedia of Integer Sequences."Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.

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Groupoid

Cite this as:

Stover, Christopher and Weisstein, Eric W. "Groupoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Groupoid.html

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