My research focuses on higher geometric groupoids and the morphisms between them, with particular emphasis on higher Lie theory and differential geometry. Groupoids provide a natural framework for describing geometric structures with symmetries, while their higher-categorical generalizations, higher groupoids, encode richer geometric and homotopical information. Guided by the Yoneda philosophy, I am interested in understanding these objects through the morphisms between them and the notions of equivalence they induce. In geometry, strict morphisms often fail to capture the appropriate notion of equivalence between geometric structures, motivating the study of generalized morphisms represented by principal bundles and bibundles. These lead naturally to Morita equivalences and correspondences, which provide a categorical framework for comparing geometric objects. My research explores these ideas in the higher-categorical setting, particularly through higher Morita equivalences and higher correspondences of geometric infinity-groupoids.
Publications
2026
arXiv
Morphisms of double Lie groupoids: a simplicial approach
Motivated by recent developments in generalized Kähler geometry, we study morphisms of double Lie groupoids from a simplicial viewpoint. We show that the codiagonal functor extends from objects to horizontal and vertical principal bibundles and to square-shaped (1,1)-morphisms. The latter determine 2-dimensional morphisms of square and globular shape between anafunctors of Lie 2-groupoids. We illustrate these constructions through two applications: First, we revisit the equivalence between two models of nonabelian gerbes: groupoid bundle gerbes and principal 2-bundles. We show that a principal 2-bundle and its associated groupoid bundle gerbe determine equivalent anafunctors from a manifold to the corresponding structure Lie 2-group. Second, we refine the integration of Manin triples in transitive Courant algebroids. We show that integrations associated with different choices of suitably transverse Manin triples in the same Courant algebroid are related by symplectic Morita equivalences. Consequently, the resulting integration of the background Courant algebroid is well defined up to symplectic Morita equivalence.
@article{AlvarezKrishnaRonchi2026,title={Morphisms of double Lie groupoids: a simplicial approach},author={Álvarez, Daniel and Krishna, Kalin and Ronchi, Stefano},year={2026},eprint={2609.32553},archiveprefix={arXiv},primaryclass={math.DG},}