Type theory - category theory correspondence

For the 2-categorical part there is Robert Seely's paper Modelling computations: 2-categorical framework from LICS 1987, and of course a bunch of papers that came afterwards that cite the paper. As Ivan points out, the $(\infty, 1)$-aspects of computation are current research.

Some references to some approachesthat exist in the literature have been given in Andrej's answer and Ivan's comment. A couple others are Licata-Harper's work on two-dimensional directed type theory and Licata-Riley-Shulman's work on fibrational frameworks (specifically the "mode theory" is a 2-dimensional type theory).