Groupoidification is a program based on the observation that the operation of pull-pushing bundles of groupoids
through spans
of groupoids becomes a linear map acting on vector spaces after taking groupoid cardinality – after “degroupoidification”.
From another perspective, these over-groupoids are an example for geometric function objects as considered in the context of geometric function theory.
De-groupoidification is similar to passing to motivic functions.
John Baez keeps a web page with relevant links and background material
In particular there are the articles in preparation
John Baez, Alexander Hoffnung, Christopher Walker, Higher-dimensional algebra VII: Groupoidification, arxiv/0908.4305
John Baez, Alexander Hoffnung, Higher-dimensional algebra VIII: The Hecke Bicategory, (pdf)
Towards topological groupoidification (pdf)
Groupoidification seems to be a central underlying governing principle in representation theory in its incarnation in geometric function theory.
Groupoidification in particular seems to illuminate structures encountered in the context of quantum field theory. Discussions of groupoidification in the context of QFT are
Jeffrey Morton, Categorified algebra and quantum mechanics, Theory and Application of Categories 16 (2006), 785-854 (arXiv, tac)
Jeffrey C. Morton, 2-Vector Spaces and Groupoids (arXiv)
Some related remarks are in