
A Note on Generalized Algebraic Theories and Categories with Families
We give a new syntax independent definition of the notion of a generaliz...
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Coherence of strict equalities in dependent type theories
We study the coherence and conservativity of extensions of dependent typ...
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Induction principles for type theories, internally to presheaf categories
We present new induction principles for the syntax of dependent type the...
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Constrained Type Families
We present an approach to support partiality in typelevel computation w...
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Bicategories in Univalent Foundations
We develop bicategory theory in univalent foundations. Guided by the not...
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Type theories in category theory
We introduce basic notions in category theory to type theorists, includi...
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A Higher Structure Identity Principle
The ordinary Structure Identity Principle states that any property of se...
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Internal ∞Categorical Models of Dependent Type Theory: Towards 2LTT Eating HoTT
Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type CON of contexts, a type family TY indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a longstanding and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP. In this paper, we develop a notion of ∞categories with families (∞CwF's). The approach to higher categories used relies on the previously suggested semiSegal types, with a new construction of identity substitutions that allow for both univalent and nonunivalent variations. The typetheoretic universe as well as the internalised syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in twolevel type theory (2LTT). Apart from introducing ∞CwF's, this paper is meant to serve as a "gentle introduction" to shortcomings of 1categories in type theory without UIP, and to difficulties of and approaches to internal higherdimensional categories.
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