Hyperorder

Hyperorder is a concept in mathematics and theoretical computer science that refers to the study of the structure and properties of infinite-dimensional spaces. It was introduced by Vladimir Voevodsky in the early 2000s as a way to understand the homotopy theory of higher-dimensional spaces, which are spaces that have a rich structure of higher-dimensional shapes and holes.

In traditional topology, spaces are studied using continuous maps and homotopies, which are ways of deforming one map into another. However, these tools are not sufficient to capture the full complexity of higher-dimensional spaces. Hyperorder provides a framework for studying these spaces by introducing the notion of a "higher homotopy groupoid," which is a generalization of the fundamental groupoid of a space.

A higher homotopy groupoid is a category whose objects are points in the space and whose morphisms are paths between these points, up to homotopy. This allows for the study of higher-dimensional shapes and holes in a space, such as 2-dimensional surfaces and 3-dimensional volumes. Hyperorder also provides tools for studying the relationships between these higher-dimensional shapes and holes, such as the notion of a "higher homotopy limit."

One of the key applications of hyperorder is in the study of higher-dimensional algebraic geometry, where it is used to understand the structure of moduli spaces of higher-dimensional algebraic varieties. Hyperorder has also been used in the study of higher-dimensional homotopy theory, where it provides a way to understand the relationships between different higher-dimensional spaces.