A good place to start is : L. Breen. Notes on 1-and 2-gerbes. In J. Baez and J. May, editors, Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, pages 193–235. Springer, 2009.

There are lots of other sources and various different approaches to the subject, but I have found Breen's notes very useful. Which approach suits you, of course, depends on your interests and background.

There are many other more recent treatments.

Remember that principal G bundles are also called G-torsors so search for 2-torsors. There are various papers with 2-torsors in their title, but note there are various levels of generalisation. For further information try the n-Lab page: principal 2-bundle and there you will find a reference to Toby Bartels, Higher gauge theory I: 2-Bundles, amongst lots of other useful sources. Which ones fit best with you is difficult to guess!!!

A good place to start is :

• Larry Breen. Notes on 1-and 2-gerbes. In J. Baez and J. May, editors, Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, pages 193–235. Springer, 2009. arXiv:math/0611317

There are lots of other sources and various different approaches to the subject, but I have found Breen's notes very useful. Which approach suits you, of course, depends on your interests and background.

There are many other more recent treatments.

Remember that principal G bundles are also called G-torsors so search for 2-torsors. There are various papers with 2-torsors in their title, but note there are various levels of generalisation. For further information try the n-Lab page: principal 2-bundle and there you will find a reference to Toby Bartels, Higher gauge theory I: 2-Bundles, amongst lots of other useful sources. Which ones fit best with you is difficult to guess!!!

A good place to start is : L. Breen. Notes on 1-and 2-gerbes. In J. Baez and J. May, editors, Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, pages 193–235. Springer, 2009.

There are lots of other sources and various different approaches to the subject, but I have found Breen's notes very useful. Which approach suits you, of course, depends on your interests and background.

There are many other more recent treatments.

Remember that principal G bundles are also called G-torsors so search for 2-torsors. There are various papers with 2-torsors in their title, but note there are various levels of generalisation. For further information try the n-Lab page: https://ncatlab.org/nlab/show/principal+2-bundle and there you will find a reference to Toby Bartels, https://arxiv.org/abs/math/0410328, amongst lots of other useful sources. Which ones fit best with you is difficult to guess!!!

A good place to start is : L. Breen. Notes on 1-and 2-gerbes. In J. Baez and J. May, editors, Towards Higher Categories, volume 152 of The IMA Volumes in Mathematics and its Applications, pages 193–235. Springer, 2009.

There are lots of other sources and various different approaches to the subject, but I have found Breen's notes very useful. Which approach suits you, of course, depends on your interests and background.

There are many other more recent treatments.

Remember that principal G bundles are also called G-torsors so search for 2-torsors. There are various papers with 2-torsors in their title, but note there are various levels of generalisation. For further information try the n-Lab page: principal 2-bundle and there you will find a reference to Toby Bartels, Higher gauge theory I: 2-Bundles, amongst lots of other useful sources. Which ones fit best with you is difficult to guess!!!