### Integrability of Lie Brackets

#### Posted by Urs Schreiber

I would like to advertise the beautiful review

Marius Crainic, Rui Loja Fernandes
*Lectures on Integrability of Lie Brackets*

arXiv:math/0611259

on the integration of Lie algebroids $(g,A)$ to Lie groupoids $C(g,A)$.

Section 3.2 has a nice review of the method of integrating Lie algebras to Lie groups using equivalence classes of paths in the Lie algebra. Then in 3.3 it is discussed how this generalizes to Lie algebroids.

In section 5.3 of *On action Lie $\infty$-groups and action Lie $\infty$-algebras* (pdf) I describe how this integration method is secretly (well, it’s pretty obvious, but still deserves to be made explicit) nothing but forming the fundamental path groupoid $\Pi_1(-)$ of the smooth classifying space $S(\mathrm{CE}(g,A))$ of $(g,A)$-valued differential forms:
$C(g,A) = \Pi_1(S(CE(g,A)))
\,.$

## Re: Integrability of Lie Brackets

Do these folks get a smooth groupoid from any Lie algebroid, or just a stacky Lie groupoid?

The latter approach, I claim, is just a consequence of working in an insufficiently nice category of smooth spaces.