Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called types.
Univalent foundations are an approach to the foundations of mathematics in which mathematical structures are built out of objects called types. Types in univalent foundations do not correspond exactly to anything in set-theoretic foundations, but they may be thought of as spaces, with equal types corresponding to homotopy equivalent spaces and with equal elements of a type corresponding to points of a space connected by a path. Univalent foundations are inspired both by the old Platonic ideas of Hermann Grassmann and Georg Cantor and by "categorical" mathematics in the style of Alexander Grothendieck. Univalent foundations depart from the use of classical predicate logic as the underlying formal deduction system, replacing it, at the moment, with a version of Martin-Löf type theory. The development of univalent foundations is closely related to the development of homotopy type theory.
Univalent foundations are compatible with structuralism, if an appropriate (i.e., categorical) notion of mathematical structure is adopted.
A fundamental characteristic of univalent foundations is that they — when combined with the Martin-Löf type theory — provide a practical system for formalization of modern mathematics. A considerable amount of mathematics has been formalized using this system and modern proof assistants such as Coq and Agda.
In the formalization system for univalent foundations that is based on Martin-Löf type theory and its descendants such as Calculus of Inductive Constructions, the higher dimensional analogs of sets are represented by types. The collection of types is stratified by the concept of h-level (or homotopy level).
Types of h-level $0$ are those equal to the one point type. They are also called contractible types.
Types of h-level $1$ are those in which any two elements are equal. Such types are called "propositions" in univalent foundations. The definition of propositions in terms of the h-level agrees with the definition suggested earlier by Awodey and Bauer. So, while all propositions are types, not all types are propositions. Being a proposition is a property of a type that requires proof. For example, the first fundamental construction in univalent foundations is called $\operatorname {iscontr}$. It is a function from types to types. If $X$ is a type then $\operatorname {iscontr} X$ is a type that has an object if and only if $X$ is contractible. It is a theorem that for any $X$ the type $\operatorname {iscontr} X$ has h-level $1$ and therefore being a contractible type is a property. This distinction between properties that are witnessed by objects of types of h-level $1$ and structures that are witnessed by objects of types of higher h-levels is very important in the univalent foundations.
Types of h-level $2$ are called sets. It is a theorem that the type of natural numbers has h-level $2$. It is claimed by the creators of univalent foundations that the univalent formalization of sets in Martin-Löf type theory is the best currently-available environment for formal reasoning about all aspects of set-theoretical mathematics, both constructive and classical.
Categories are defined as types of h-level $3$ with an additional structure that is very similar to the structure on types of h-level $2$ that defines partially ordered sets. The theory of categories in univalent foundations is somewhat different and richer than the theory of categories in the set-theoretic world with the key new distinction being that between pre-categories and categories.
Source: Wikipedia
