infinity1category

Infinity1Category is a concept in higher category theory, a branch of mathematics that studies categories and higher-dimensional structures. It refers to a category that is enriched over itself in a way that allows for the construction of higher-dimensional categories. This means that the morphisms in an Infinity1Category can themselves be equipped with additional structure, such as higher morphisms, making it a higher category.

The term "Infinity1Category" is often used to describe a category that is (∞,1)-category, where (∞,1) indicates that it is an (∞,n)-category with n=1. In this context, an (∞,n)-category is a higher-dimensional generalization of a category, where n indicates the dimension of the morphisms. An (∞,1)-category allows for the existence of higher morphisms, which are morphisms between morphisms, and so on, ad infinitum.

Infinity1Category is a fundamental concept in the study of higher categories and has applications in various areas of mathematics, including homotopy theory and algebraic topology. It provides a framework for understanding the higher-dimensional structures that arise in these fields, such as the homotopy types of topological spaces and the higher-dimensional algebraic structures that arise in algebraic geometry.

The study of Infinity1Category involves techniques from higher category theory, such as the use of simplicial sets and quasicategories to model higher-dimensional structures. These techniques allow mathematicians to study the properties of higher categories and to develop new tools for understanding the higher-dimensional structures that arise in mathematics.