nLab groupoidification

Idea

Groupoidification is a program based on the observation that the operation of pull-pushing bundles of groupoids

$$\array{ \Psi \\ \downarrow \\ X }$$

through spans

$$\array{ && S \\ & \swarrow && \searrow \\ X &&&& Y }$$

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.

Remarks

• Groupoidification is closely related to – or maybe even synonymous to – geometric function theory.

De-groupoidification is similar to passing to motivic functions.

References

John Baez keeps a web page with relevant links and background material

• John Baez Groupoidification (web)

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)

Relation to representation theory

Groupoidification seems to be a central underlying governing principle in representation theory in its incarnation in geometric function theory.

Relation to quantization

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)

• pdf

Some related remarks are in

• Schreiber: Nonabelian cocycles and their sigma model QFTs

• Cafe: new_paper_on_the_hecke_bicateg