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As explained here, 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. 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?

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    $\begingroup$ I really like the question, but it is more a research project than a MO question. A few steps in this direction have been made here: arxiv.org/abs/1705.07442. It's hard to say how "definitive" this answer is to the problem of $(\infty,1)$-categories. Another important reference, gathering the state of art up to 1991, is T. Streicher, Semantics of Type Theory: Correctness, Completeness, and Independence Results. $\endgroup$ – Ivan Di Liberti Jan 1 at 10:11
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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.

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    $\begingroup$ Is this really what the question is asking though? It's true that Seely uses a 2-categorical framework to model lambda calculus where the 2-cells are reductions. But in most type theories that I know, these reductions are necessarily invertible. In a general 2-category I'm not sure there is a type theory which could be considered an "internal language of a 2-category". $\endgroup$ – Ali Caglayan Jan 3 at 11:10
  • $\begingroup$ Well, the author accepted my answer :-) $\endgroup$ – Andrej Bauer Jan 3 at 21:14
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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).

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