FROM MULTISETS TO SETS IN HOMOTOPY TYPE THEORY

Sets in homotopy type theory

Homotopy Type Theory may be seen as an internal language for the ∞category of weak ∞-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak ∞-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of)...

Multisets in Type Theory

A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin-Löf Type Theory, in the presence of Voevodsky’s Univalence Axio...

A model of type theory in simplicial sets: A brief introduction to Voevodsky's homotopy type theory

As observed in [HS] identity types in intensional type theory endow every type with the structure of a weak higher dimensional groupoid. The simplest and oldest notion of weak higher dimensional groupoid is given by Kan complexes within the topos sSet of simplicial sets. This was observed around 2006 independently by V. Voevodsky and the author. The aim of this note is to describe in an accessi...

Introduction - from type theory and homotopy theory to univalent foundations

Homotopy Type Theory

Introduction 3 1 A short guide to constructive type theory 7 1.1 A dependent type over a type . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.1 Dependent products . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.1.2 Dependent sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Defining types inductively . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2 Type theor...