nLab has an article on [internal logic](https://ncatlab.org/nlab/show/internal+logic). Homotopy type theory also discusses its application in logic. Any one knows references to the correspondence study of the internal logic induced from higher category theories and higher topo theories ?



internal logic from higher categories
In addition to the [reference](http://arxiv.org/abs/1507.02648) stated in the comment. nLab does provide an entry for [internal logic from $(\infty,1)-$topos.](https://ncatlab.org/nlab/show/internal+logic+of+an+(infinity,1)-topos) 


