Non-abelian Hodge theory for positive characteristic and crystalline structures

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The purpose of this project is to extend non-abelian Hodge theory to algebraic varieties for positive characteristic or arithmetic varieties. For this purpose I have been studying general theories for some construction of crystalline topos using higher category theory. It seems that higher topos theory is a suitable language to describe real crystalline topoi. Various higher category theories (even higher topos theories) have been constructed in these ten twenty years by many researchers (C. Simson, B. Then, J. Lurie). Until last year I used mainly Simpson's methods, but after reading recent work of J. Lurie I realized that it is suitable to use Lurie's higher topos theory for our constructions. This project will be important for arithmetic geometry and mathematical physics in the near future. On the other hand, to describe higher crystalline structure, it seems that formulations of the structure over the ring of ratioinal integers are not adequate. A recent work of N. Durov construes the field with one element F_1 which has been expected to exist. If higher crystalline topos theory could be constructed over the field F_1, it should be useful. A construction of such theory is the next theme for this project.