As explained [here](https://ncatlab.org/nlab/show/relation+between+type+theory+and+category+theory), simply typed lambda calculus can be viewed as a syntactic language for category theory. My question is, can the following modification make it equally well a formal syntactic language for 2-category theory?


It goes something like this: we have a universe $U$. For each $X : U$, we make $X$ into the universe of a simply typed lambda calculus by introducing the necessary [type formation rules](https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus). Maps $a \rightarrow b$ for $a, b : X$ are like maps of objects, under the usual correspondence. 


In this setup, I would expect maps $X \rightarrow Y$ for $X, Y : U$ to correspond to functors in some way, and I would expect some kind of correspondence between type theories of this kind and $2$-categories. 


More generally, I expect that introducing a higher universe is like passing from $n$-categories to $n+1$-categories. For a similar idea, applied instead to homotopy type theory, we might expect to get a model for $(\infty, 1)$-categories. Has anyone pursued an approach like this?



Type theory - category theory correspondence
For the 2-categorical part there is Robert Seely's paper [Modelling computations: 2-categorical framework](http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf) from LICS 1987, and of course a bunch of papers that came afterwards [that cite the paper](https://scholar.google.si/scholar?cites=14241625227633821950&as_sdt=2005&sciodt=0,5&hl=sl). As Ivan points out, the $(\infty, 1)$-aspects of computation are current research.


