nLab cohesive homotopy theory

Idea

By cohesive homotopy theory we mean homotopy theory equipped with structure/axioms of cohesion:

• While bare homotopy theory is concerned (just) with geometrically discrete homotopy types/$\infty$-groupoids,

• and while geometric homotopy theory ($\infty$-topos theory) generalizes this to “geometric homotopy types” ($\infty$-stacks) for extremely general notions of “geometry”,

• cohesive homotopy theory is a form of geometric homotopy theory but constrained such that the notion of geometry it encodes satisfies basic properties familiar from differential topology and differential cohomology-theory.

Semantically, cohesive homotopy theory is modeled by cohesive $\infty$-toposes, while syntactically it is described by cohesive homotopy type theory.

Or rather, all this pertains to the default notion of “cohesion”. There are further enhancements such as to differential cohesion modeled by differential cohesive $\infty$-toposes, which reflects a richer notion of higher synthetic differential geometry.

References

See the references at

• cohesive $\infty$-topos

• cohesive homotopy type theory