Why is it interesting?

Groupoids are a natural generalization of groups and are able to describe symmetries that usual group theory cannot describe. In some sense, a groupoid is a collection of groups.

Groupoids are a modern way to think about symmetriesJohn Baez

Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups . . . In fact, though groups are indeed sufficient to characterize homogeneous structures, there are plenty of objects which exhibit what we clearly recognize as symmetry, but which admit few or no nontrivial automorphisms. It turns out that the symmetry, and hence much of the structure, of such objects can be characterized algebraically if we use groupoids and not just groups.Weinstein, A. (1996) ‘Groupoids: Unifying Internal and External Symmetry

A groupoid is a generalization of a group that is particularly handy to express local symmetries of geometrical structure.http://philsci-archive.pitt.edu/2133/1/geometrie.pdf

Next to the title of this article one sees a photograph of a herd of zebra. No explicit explanation is offered for its presence, nor is one needed. The received account as to why zebras sport stripes is that when they stand in a herd, a charging lioness is presented with a strongly patterned visual array, making it very difficult for her to detect the outline of a single member of the herd. The rationale for the choice of this picture, in which one imagines Weinstein played a part, rests in his idea that groupoids are better than groups at detecting the inner symmetry of patterns of this kind. This idea Weinstein explicitly illustrates in the article itself with a discussion of the symmetries of a set of bathroom tiles. In contrast to this rather mundane concern of the mathematician contemplating the pattern of the grouting while enjoying a soak, the cover picture makes clear that such inner symmetry is a matter of life and death. As any zebra will tell you, ‘symmetry capturable by groupoids but not by groups saves lives’. The importance of mathematical conceptualisation by David Corfield

There is no benefit today in arithmetic in Roman numerals. There is also no benefit today in insisting that the group concept is more fundamental than that of groupoid.

Ronald Brown

It is interesting in this respect to note the view of Connes [58] that Heisenberg discovered quantum mechanics by considering the groupoid of quantum transitions rather than the group of symmetry.Groupoids and crossed objects in algebraic topology by Ronald Brown

The idea of making systematic use of groupoids . . . , however evident it may look today, is to be seen as a significant conceptual advance, which has spread into the most manifold areas of mathematics . . . In my own work in algebraic geometry, I have made extensive use of groupoidsGrothendieck, as quoted in Groupoids and crossed objects in algebraic topology by Ronald Brown

Layman

What is a Groupoid? When promoting a mathematical concept, it is never a bad idea to think up an illustration from everyday life. Ronald Brown (1999, p. 4), a leading researcher in groupoid theory, has provided us with a good example by considering possible car journeys between cities of the United Kingdom. Now, one approach to capturing the topology of the British road system is to list the journeys one can make beginning and ending in Bangor, the Welsh town where Brown’s university is located. This possesses the advantage that the members of the list form a group under the obvious composition of trips, where the act of remaining in Bangor constitutes the group’s identity element. ( Note that trips are being considered here only ‘up to homotopy’. In particular, taking a trip and then retracing one’s steps is to be equated with staying at home.) However, for a country so dominated by its capital city, it might appear a little strange to privilege Bangor and the act of staying put there. Each city might be thought to deserve equal treatment. Pleasant as it is to remain in Bangor, staying put in London should surely be seen as another identity element. Moreover, if you want to know about trips from London to Birmingham, it would seem perverse to have to sift through the set of round trips from Bangor which pass through London and then Birmingham, even if all you need to know is contained therein. And if ferry journeys are excluded, this method is perfectly hopeless for finding out about trips out of Belfast. More reasonable, then, to list all trips between any pair of cities, where ordered pairs of trips can be composed, if and only if the destination of the first trip matches the starting point of the second. Something group-like remains, but with only a partial composition. On this basis Brown can claim that ‘[t]his na¨ıve viewpoint gives rise to the heretical suggestion that the natural concept is that of groupoid rather than group’ (Brown, 1999, p. 4).The importance of mathematical conceptualisation by David Corfield

A groupoid is a collection of places ("objects") together with a collection of ways to get from one place to another ("morphisms") satisfying the following list of requirements:

Staying at a place A and doing nothing counts as a way (the "identity"). A way to get from A to B, together with a way to get from B to C, gives you a way to get from A to C ("composition"). You can always backtrack, and you always ignore backtracking. That is, if you go from A to B, then there's always a backtrack ("inverse") from B back to A, and if you take it then that's the same as if you just stayed at A. The backtrack of a backtrack is the same way again.

(I've ignored associativity because stating it while staying in the non-technical language I'm using above gets a bit wordy, and in any case it's a very natural requirement that you'd probably automatically assume anyway.)

A simple family of examples of groupoids is given by taking a graph and constructing the free groupoid on it: the places are the vertices of the graph, while the ways are paths in the graph, except that we can always traverse edges in either direction and we always ignore backtracking as above.

An example you might've played with at some point is the 15 puzzle, which forms a groupoid where the places are the possible configurations of the puzzle and the ways are ways to slide tiles around to pass between configurations. Qiaochu Yuan

Student

• http://math.ucr.edu/home/baez/week248.html
• http://math.ucr.edu/home/baez/week249.html
• Groupoids: Unifying Internal and External Symmetry A Tour through Some Examples by Alan Weinstein

Researcher

Examples

Example1

Example2:

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