17 research outputs found
Modalities in homotopy type theory
Univalent homotopy type theory (HoTT) may be seen as a language for the
category of -groupoids. It is being developed as a new foundation for
mathematics and as an internal language for (elementary) higher toposes. We
develop the theory of factorization systems, reflective subuniverses, and
modalities in homotopy type theory, including their construction using a
"localization" higher inductive type. This produces in particular the
(-connected, -truncated) factorization system as well as internal
presentations of subtoposes, through lex modalities. We also develop the
semantics of these constructions
Constructive sheaf models of type theory
We generalise sheaf models of intuitionistic logic to univalent type theory
over a small category with a Grothendieck topology. We use in a crucial way
that we have constructive models of univalence, that can then be relativized to
any presheaf models, and these sheaf models are obtained by localisation for a
left exact modality. We provide first an abstract notion of descent data which
can be thought of as a higher version of the notion of prenucleus on frames,
from which can be generated a nucleus (left exact modality) by transfinite
iteration. We then provide several examples.Comment: Simplified the definition of lex operation, simplified the encoding
of the homotopy limit and remark that the homotopy descent data is a lex
modality without using higher inductive type