Possible final project topics

By the phrase “the homotopy category of widgets” we mean¹ we have some notion of homotopy between maps of widgets and we’re considering the category whose objects are widgets and whose morphisms are homotopy classes of maps. For the case of groupoids, the homotopies are simply natural isomorphisms, while for 1-types they are just the usual homotopies.

The equivalence is given, in one direction, by the fundamental groupoid functor and in the other by the classifying space functor. Showing the equivalence boils down to showing that any groupoid is equivalent to the fundamental groupoid of its classifying space and that any 1-type is weakly homotopy equivalent to the classifying space of its fundamental groupoid.

To read about the classifying space of a group see subsection \(K(G,1)\) Spaces in section 1.4 of Hatcher’s Algebraic Topology, or section 16.5 in May’s Concise Course in Algebraic Topology.

There is no reasonable analogue of van Kampen’s theorem for the higher homotopy groups, because the groups “don’t remember enough of the structure of the space”. But it is poosible to give a notion of \(n\)-groupoid, so that a space² has a fundamental \(n\)-groupoid and there is a van Kampen type theorem that let’s you compute these. These results are due to Ronald Brown and his collaborators. The most comprehensive reference is the recent book Nonabelian Algebraic Topology. To narrow this down to a reasonable chunk for a final tutorial project, it is reasonable to talk just about the two dimensional case. There are actually two relevant notions of 2-groupoid: crossed modules (which are perhaps more suited to computation) and double groupoids (which are better suited for proving van Kampen!). Chapter 2 of the book describes crossed modules and the two dimensional van Kampen theorem; chapter 6 introduces dobule groupoids and proves the theorem.

Galois theory is analogous and closely related to the theory of connected covering spaces on a connected base. To get an algebraic analogue of the theory of all covering spaces, one must turn to commutative rings. Then, as in the case of covering spaces over a non-connected base, one needs a Galois groupoid in place of the usual Galois group. I recommend reading the following paper:

O. E. Villamayor and D. Zelinsky, Galois theory for rings with finitely many idempotents, Nagoya Math. J. Volume 27, Part 2 (1966), 721-731.

Villamayor and Zelinsky show that given commutative rings \(R \subset S\) satisfying some conditions (among which is that the image of \(R\) in \(S\) has finitely many idempotents as per the paper’s title), there is a bijection between projective, separable subalgebras of \(S\) and so-called fat subgroups of the automorphism group of \(S\) over \(R\). They establish the bijection using certain subgroupoids of the groupoid of isomorphisms between indecomposable summands of \(S\).

For more general extensions of rings there is A. R. Magid’s Galois Groupoids (Journal of Algebra 1971), an exposition of which can be found in F. Borceux and G. Janelidze’s book Galois Theories (Cambridge University Press 2001). Both of these references are more advanced and technical in nature, but of course, offer more general results. For a gentler introduction to the sort of abstract categorical generalization of Galois theory treated in Borceux and Janelidze’s book, look at E. Dubuc and C. Sánchez de la Vega’s paper On the Galois Theory of Grothendieck –but they only treat the case of “covers of a connected base” and so use only groups, not groupoids.

I haven’t checked out these references yet, my guess is that in classical Galois theory (that is, the theory dealing with fields), the groupoid that turns up is the groupoid of subfields of the algebraic closure of \(F\) which are \(F\)-isomorphic to some fixed extension \(E\) of \(F\). This might be a way to do a project relating to Galois theory which is less technical than Galois theory for rings.

R. Baer, Beiträge zur Galoisschen Theorie, S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl. 1928, Abh. 14.

L. Michler, Uber eine Verallgemeinerung des Hauptsatzes der Galoisschen Theorie, Wiss. Z. Hochsch. Schwermaschinenbau Magdeburg 11 (1956-7).

Joyal’s elegant theory of species is a neat way to organize the use of generating functions in enumerative combinatorics. The key idea is that it is possible to give a general but precise definitions of “a kind of combinatorial object one might want to count”, or, as Joyal calls them, a “combinatorial species”. There is a species of graphs, a species of trees, a species of partial orders, a species of permutations, etc. A species is a groupoid whose objects are the structures of that particular species (e.g., the objects of the species of graphs are just graphs), and whose morphisms are just the isomorphisms of such structures; this groupoid must come equipped with a covering morphism to the groupoid of finite sets and bijections that represents the operation of taking “the set of vertices” of a structure of the species. Equivalently, by (the inverse of) the Grothendieck construction, a species is a functor from the groupoid of finite sets and bijections to the category of sets. One can associate several types of generating functions to a species and nice combinatorial operations on species will correspond to simple algebraic operations on the associated generating functions.

There is a good textbook on species which unfortunately vastly underplays the role of groupoids in the theory: Combinatorial Species and Tree-like Structures by F. Bergeron, G. Labelle and P. Leroux. Joyal’s original paper on species, Une théorie combinatoire des séries formelles is a great read and does take advantage of the theory of groupoids. I suggest looking at Joyal’s paper unless you really can’t read French.

I imagine this might be an attractive topic, and I think with we might be able to swing more than project on something related to species.

Stuff types are a generalization of combinatorial species due to John Baez and James Dolan. A species is a covering morphism of groupoids from some groupoid \(G\) to the groupoid \(FB\) of finite sets and bijections; a stuff type is an arbitrary morphism of groupoids \(G \to FB\). You can read about them in the very fun paper From Finite Sets to Feynman Diagrams where they were originally introduced, or in Simon Bryne’s Macquarie University Honors thesis, On groupoids and stuff.

This is a way to do linear algebra with groupoids instead of vector spaces. Given a groupoid \(G\) we can form the vector space \(V_G\) of formal linear combinations of isomorphism classes of objects; and given a span of groupoids, that is, a pair of morphisms of groupoids \(G \leftarrow S \to H\), we can define a linear transformation \(V_G \to V_H\) for which \(S\) is analogous to the matrix of the linear transformation. All of this is very clearly explained by John Baez in The Tale of Groupoidification. (I heartily recommend all of his writing.) One thing that might be interesting is writing about the generating functions associated to a species from the point of view of groupoidification and groupoid cardinality.

I’m trying to track down references for more applications, but certainly Higgins’ book Categories and Groupoids deserves a look. In chapter 14, Applications to group theory, he uses groupoids to give much more conceptual proofs of the Nielsen-Schreir, Kurosh and Grushko theorems in combinatorial group theory.

I’ll just quote the abstract of M. Golubitsky and I. Stewart’s interesting paper Nonlinear dynamics of networks: the groupoid formalism:

A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos.

Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the “input sets”. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of `robust’ patterns of synchrony in terms of the combinatorial structure of the network.

Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables–the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology–which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.

• Lie groupoids
• Convolution algebras of groupoids