Applications of “Homotopical algebra” in the set up of Lie groupoids

Question

The question is as in the title.

(What are some of the) are there any applications of Homotopical algebra (in the context of Quillen’s book “Homotopical algebra”) in better understanding (or developing new theory) of Lie groupoids?

I am reading Homotopical algebra hoping that this would help me to learn some $\infty$-category theory. Other than that, I do not have any plan. As I am familiar with Lie groupoids, I wanted to see if there is a place where these two interact in a nice way.

(As of now), I do not have any better explanation than this to ask the question.

Answer 1

There are many connections between Lie groupoids and homotopical algebra. In recent years, a particularly prominent connection is to the theory of simplicial presheaves, originally developed in the context of homotopical algebra by Joyal and Jardine in 1980s, with important precursors in the work of Illusie, Artin–Mazur, Brown, Grothendieck, and many others.

First, recall that the bicategory of Lie groupoids, smooth right principal bibundles, and smooth biequivariant isomorphisms of bibundles is equivalent to the bicategory of stacks in groupoids admitting a surjective representable submersion from a manifold, their morphisms, and 2-isomorphisms.

The latter bicategory embeds homotopically fully faithfully in the category of simplicial presheaves on the site of smooth manifolds (or just cartesian spaces) equipped with the Čech-local projective model structure.

The latter model category can be seen as an enhancement of Lie groupoids (and stacks) that has much better properties: all quotients, more generally, all homotopy colimits, exist in this model category and have the expected properties. In particular, the quotient of a Lie groupoid by a nonfree action of a Lie 2-group exists and has the desired properties.

Additionally, the model category of simplicial presheaves is cartesian, which means it contains infinite-dimensional mapping spaces of manifolds, which can be manipulated in much the same manner as other objects, e.g., we can easily define and compute the tangent bundle of Map(M,N) for two smooth manifolds M and N.

The model category of simplicial presheaves contains classifying simplicial presheaves of differential forms, principal G-bundles with connection, bundle d-gerbes with connection, etc. This allows us to compute (say) the de Rham cohomology of the classifying simplicial presheaf of principal G-bundles with connection and see that it is isomorphic to the graded algebra of invariant polynomials on the Lie algebra of G. This recovers the Chern–Weil homomorphism.

Furthermore, the notion of a derived mapping space allows us to define differential forms, principal G-bundles with connections, etc., in a conceptual way for any simplicial presheaf, and prove the desired properties of these construction (e.g., descent) very easily.

Some recommended sources:

• Daniel S. Freed, Michael J. Hopkins: Chern-Weil forms and abstract homotopy theory. (An accessible introduction to the main ideas.)

• Daniel Dugger: Sheaves and homotopy theory. (The Čech-local model structure on simplicial presheaves.)

• Araminta Amabel, Arun Debray, Peter J. Haine: Diferential Cohomology. Categories, Characteristic Classes, and Connections. (More about simplicial presheaves.)

• Patrick Iglesias-Zemmour: Diffeology. (For sheaves of sets and how to work with them.)

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff: Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction. (Some applications: characteristic classes in the language of simplicial presheaves.)