One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity topos among locally Cartesian closed infinity categories. This is generally presented as a commonly accepted fact but not proven yet, for example, in the introduction of the HoTT book, one can read "in particular questions of coherence and strictness remains to be addressed"


Roughly,I would like to understand what are the difficulties for proving this (it seems to be an extremely natural result), or more precisely what does mean the sentence I quoted from the book ?


Thank you !



$\infty$-categorical interpretation of type theory
Mike Shulman has a good discussion of what exactly needs to be shown in his


* Michael Shulman, "The univalence axiom for inverse diagrams" ([arXiv:1203.3253](http://arxiv.org/abs/1203.3253))


The issue is that in the type theory the classification of objects is by strict pullbacks. So roughly the problem is that one has to show the [object classifier](http://ncatlab.org/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos) of a random $\infty$-topos is, when the latter is presented by a [type-theoretic model category](http://ncatlab.org/nlab/show/type-theoretic%20model%20category), presented by an object which classifies other objects by strict pullbacks with strict respect for composition. Establishing this in generality is (or would be) a kind of semi-strictification result for $\infty$-toposes.


See also the [other references](http://ncatlab.org/nlab/show/univalence+axiom#References) in the nLab entry.


