nLab
2-category of 2-dimensional cobordisms

This is a special case of the $(\mathrm{\infty },n)$-category of cobordisms.

References

A complete generators and relations? presentation of the 2-dimensional extended unoriented oriented bordism bicategory as a symmetric monoidal bicategory? is given in

• Chris Schommer-Pries, The classification of two-dimensional extended topological field theories PhD thesis, UC Berkeley (2009) (pdf)

The 2-category theory in Schommer-Pries’ thesis relies on a thesis by a student of Ross Street:

• Paddy McCrudden, Balanced coalgebroids , Theory and Applications of Categories 7 (2000), 71-147. (tac)