McCartan Spaces: A Generalization of Manifolds for Non-Commutative Geometry
McCartan is a mathematical structure that generalizes the notion of a manifold. It was introduced by John McCartan in the 1990s as a way to study non-commutative geometry and the geometry of spaces with a non-trivial fundamental group.
A McCartan space is a topological space that is equipped with a sheaf of rings, called the McCartan sheaf, which encodes the geometry of the space. The McCartan sheaf is a generalization of the sheaf of functions on a manifold, and it includes additional structure such as a notion of "differential" that is not necessarily commutative.
One of the key features of McCartan spaces is that they can have a non-trivial fundamental group, which means that the space is not necessarily path-connected. This is in contrast to manifolds, which are always path-connected. The non-trivial fundamental group of a McCartan space allows for the study of more exotic geometric structures, such as those found in non-commutative geometry and the geometry of spaces with a non-trivial fundamental group.
McCartan spaces have found applications in a variety of fields, including algebraic geometry, number theory, and mathematical physics. They provide a way to study geometric objects that are not necessarily commutative, and they have been used to study a wide range of problems, from the geometry of algebraic varieties to the study of quantum field theories.
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