It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming speci€c instances of non-parametricity. We also address the interaction between classical axioms and the existence of automorphisms of a type universe. We work over intensional Martin-Löf dependent type theory, and ...

In this paper, we determined conditions on β, αi and fi(z) so that the integral operator { β ∫ z 0 tβ−1 n ∏ i=1 ( fi(t) t ) 1 αi dt } 1 β is univalent in the open unit disk for the two subclasses analytic functions. 2000 Mathematical Subject Classification: Primary 30C80, Secondary 30C45.

Let U = {z ∈ C : |z| < 1} be the unit disk in the complex plane, and let A be the class of functions which are analytic in the unit disk normalized with f (0) = f ′ (0)− 1 = 0. We denote by P the class of the functions p which are analytic in U , p (0) = 1 and Re p (z) > 0, for all z ∈ U . Let S be the subclass of A, consisting of all univalent functions f in U , and we consider S∗ the subclass...

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened “homotopy type theory”. In this direction, Vladimir Voevodsky observed that it is possible to model type theory using simplicial sets and that this model satisfies an additional property, cal...