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Groupoids

Intuitive 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

Concrete

By a groupoid we mean simply a small category in which every morphism is an isomorphism. Thus a group may be considered as a groupoid with one object. Groupoids and crossed objects in algebraic topology by Ronald Brown

While the categorical definition of groupoid is the most concise, you can also think of a groupoid as being like a group, except where multiplication is only partially defined, rather than being defined for any pair of elements. https://mathoverflow.net/a/1456

Let us now consider the definition of a groupoid and its motivation.6 A groupoid is composed of two sets, A and B, two functions, a and b, from B to A, and an associative partial composition, s·t, of pairs of elements of B with a(s)=b(t), such that a(s·t)=a(t) and b(s·t)=b(s). Furthermore, there is a function, c, from A to B such that a(c(x))=x=b(c(x)) and such that c(x)·s=s for all s with b(s)=x and t·c(x)=t for all t with a(t)=x. Finally, there is a function, i, from B to B such that, for all s, i(s)·s=c(a(s)) and s·i(s)=c(b(s)). This may seem like a highly convoluted definition, but it can be illustrated in Brown’s picture. We simply take A to be the set of cities, while B is the set of trips. The start and finish of a trip are given by applying a and b respectively. Applying c to a city results in the staying-put trip. Finally, i sends a trip to the same trip in reverse.7 This illustration should prompt anyone acquainted with category theory to realise that a groupoid may be defined concisely in its terms. Indeed, a groupoid is just a small category in which every arrow is invertible. This much curter definition points to an important association of groupoid theory with category theory, as we shall see later. From this perspective, groups can be seen to be special cases of groupoids; that is, they are groupoids with only one object. Alternatively, in terms of the definition above, a group may be represented as a groupoid in which the set A is a singleton, and where B corresponds to the set of group elements seen as permutation maps on the group.

The importance of mathematical conceptualisation by David Corfield

Groupoids possess many of the features which give groups their power and importance, but apply in situations which lack the symmetry which is characteristic of group theory and its applications. Though only developed since the mid 20th century, the modern concept of Lie groupoid is as much entitled as is the familiar concept of Lie group to be regarded as the rigorous formulation of the 19th century notion which went under the then vague term ‘continuous group of local transformations’; a case could be made that the modern concept of Lie group has been a transitional stage in the evolution of the notion of Lie groupoid.

Groups arise primarily, though not exclusively, in connection with symmetry; that is, as sets of automorphisms of geometric or other mathematical structures. From this viewpoint, groupoids are the natural formulation of a symmetry system for objects which have a bundle structure. The most immediate illustration from geometry is to think of a tangent bundle: with each tangent space there is at first associated a general linear group, and the presence of a geometric structure on the manifold — such as a metric, or a complex structure — is reflected in the replacement of this group by a subgroup — such as the orthogonal or complex linear group. But a tangent space is a linear approximation to the manifold only near a single point and any geometrical study will involve moving from point to point within the manifold.General Theory of Lie Groupoids and Lie Algebroids by K. Mackenzie

"it's common but arguably very misleading to think of a groupoid as being just a collection of groups. The practical problem is that there are many kinds of extra structure you can place on groupoids, and structured groupoids are usually much richer than structured groups. For example, groupoids with an action of a group G, topological groupoids, and Lie groupoids are all much richer objects than unions of groups with an action of a group G, topological groups, or Lie groups. The categorical problem, which foreshadows the practical problem, is that in order to identify a groupoid with a collection of groups you need to pick a bunch of basepoints."Qiaochu Yuan

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

The concept of groupoid is one of the means by which the twentieth century reclaims the original domain of application of the group concept. The modern, rigorous concept of group is far too restrictive for the range of geometrical application envisaged in the work of Lie. Mackenzie, K. (1987) Lie Groupoids and Lie Algebroids in Differential Geometry

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  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

From […] to the objects of field values for gauge field theory, which need to keep track of gauge equivalences, we find we are treating geometric forms of groupoid. If in the latter case instead we take the simple quotient, the plain set of equivalences classes, which amount to taking gauge symmetries to be redundant, the physics goes wrong in some sense, in that we cannot retain a local quantum field theory. Furthermore, in the case of gauge equivalence, we do not just have one level of arrows between points or objects, but arrows between arrows and so on, representing equivalences between gauge equivalences. And this process continues indefinetely to "higher" groupoids, needed to capture the higher symmetries of string theory (Schreiber 2013).

. Now, copies of a non-rigid structure cannot be canonically identified, even if they are identical (this point has been discussed at length in Ref. (Catren, 2008a)). This statement presupposes a detrivialization of the notion of identity, in the sense that an entity can be identical (i.e. identified) to itself in many non-trivial ways. More generally, two entities can be identical in many different ways. The notion of groupoid is the mathematical notion that formalizes the situations in which one deals with multiple possible identifications between identical structures. In turn, the notion of group is a particular case of the notion of groupoid, in the sense that a group (understood as a category) is a groupoid with a unique object. Whereas groups encode the multiple identities (or, in more usual terms, the symmetries) of non-rigid structures, groupoids encode the (self- and hetero-)identifications in a family of non-rigid structures. The formalism of groupoids permits us to understand the transition between Klein's Erlangen program and Ehresmann's theory of fiber bundlesd made possible by Cartan's generalization of the former - as a transition from the group-theoretical selfidentifications of a single structure (e.g. Klein geometries) to the groupoid-theoretical (self- and hetero-)identifications in a family of identical structures. Klein-Weyl's program and the ontology of gauge and quantum systems by Gabriel Catren 