Christinel,
Thanks for the positive assessment of my paper. I gave your paper a pretty high score a few weeks ago. I did this while I was on travel and I don't think I had time to write a post on your blog page. I will try to write a comment, which will probably require rereading your paper.
There is a paper by Schreiber on directly applying HOTT to physics. This is a difficult and in some ways foreign way of doing physics. I am less sure about the role of HOTT directly in physics, but rather that a simplified form of mathematics that connects to HOTT will become more important. It is in much the same way that physicists do not employ set theory a whole lot in theoretical physics. However, behind the analysis used by physicist there is point-set topology. We generally reduce the complexity of this mathematics. If I were to actually engage in this I would study the HOTT, and an introduction to HOTT with physics and related web pages on this site, are worth going through.
To be honest it has been a while since I have studied this. I have been working on a homotopy approach to quantum gravity. I mention some of that in my essay. This concerns Bott periodicity with respect to holography. The connection though is rather apparent. There are also some similarities to C* algebra. This work of mine connects with what is called magma, which constructs spacetimes as the product on RвЉ•V, for V a vector space,
(a, x)в--¦( b, y) = (au + bv, [x|y] - ab)
where the square bracket is an inner product. This is a Jordan product and the right component is a Lorentz metric distance. This is also the basis for magma, which leads to groupoids and ultimately topos. A more convenient "working man's" approach to HOTT is needed.
There is my sense that mathematics has a body and a soul. The body concerns things that are computed, such as what can run on a computer. The soul concerns matters with infinity, infinitesimals, abstract sets such as all the integers or reals and so forth. If you crack open a book on differential geometry or related mathematics you read in the introduction something like, "The set of all possible manifolds that are C^в€ћ with an atlas of charts with a G(n,C) group action ... ." The thing is that you are faced with ideas here that seem compelling, but from a practical calculation perspective this is infinite and in its entirety unknowable. This along with infinitesimals, or even the Peano theory result for an infinite number of natural numbers, all appears "true," but much of it is completely uncomputable.
Cheers LC