TITLE: Prospects for reverse mathematics in Homotopy Type Theory
QUESTION [22 upvotes]: Reverse mathematics, as I mean here, is the study of which theorems/axioms can be used to prove other theorems/axioms over a weak base theory.  Examples include

Subsystems of Second Order Arithmetic (SOSOA).  Theorems expressible in second order arithmetic are compared over $\mathsf{RCA}_0$ (which is roughly the theory of computable sets of natural numbers). 
Constructive Reverse Mathematics.  Bridges and others use Bishop-style constructive mathematics as a base theory to compare various nonn-constructive principles.

Here are my questions.

Would it be possible to do reverse mathematics over Homotopy Type Theory (HoTT)?
Would HoTT make for a good base theory (roughly corresponding to computable objects in mathematics)? My understanding is that HoTT is a constructive theory, this makes me believe the answer is yes.
Would the univalence axiom need to be removed from the base theory?  My understanding is that the univalence axiom is incompatible with the law of excluded middle.  This would be problematic.
Would the results be similar to those in Constructive Reverse Mathematics?  (Would they be exactly the same?)
Would computability theoretic ideas be of use as they are in SOSOA reverse mathematics?  Computability theorists are drawn to SOSOA because of its computability theoretic nature.
Would the computer implementations of HoTT, in say Coq, be helpful in proving reverse-mathematics-like results.  It would be nice to be able to do reverse mathematics with the "safety net" of a proof checker.
Has reverse mathematics in HoTT been done already in any sense (formally or informally)?

Note:
There may be some relationship with this question: Forcing in homotopy type theory

REPLY [8 votes]: At present, homotopy type theory is purported to be the internal language (i.e. syntax) of $(\infty,1)$-topoi. That univalence is an axiom is to say that there is a class of higher topoi that satisfy univalence and a class of higher topoi that do not. For instance, the topos of sets, together with trivial higher morphisms, do not satisfy univalence.
Suppose we wish to pursue the reverse mathematics of any theorem that has been proven in homotopy type theory. Take any theorem, say, like in my comment, that the fundamental group of the circle is the integers. In this paper by Shulman and Licata, they mention that univalence is required to prove this theorem. As I see it, the circle in the topos of sets, where univalence does not hold, is actually just the singleton set. Every path and higher path is trivial. Hence the fundamental group of the circle, in the model of the topos of sets, is trivial. 
A reverse mathematician interested to study the strength of this topogical result would want to see how much of univalence is required. Is there a higher topos in which univalence fails yet the fundamental group of the circle is still the integers? Can we characterize those higher topoi in which the fundamental group of the circle is the integers? Can we characterize these topoi syntactically?