nLab smooth homotopy type

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Cohesive \infty-Toposes

Contents

Idea

In view of the congruence of the notions of homotopy type and type in homotopy type theory it makes sense to refer to an object in a cohesive (∞,1)-topos H\mathbf{H} such as as Smooth∞Grpd as a smooth homotopy type or smooth infinity-groupoid. Accordingly then an n-truncated object in H\mathbf{H} is a smooth nn-type.

For instance a smooth 0-type is then an object in the sheaf topos Sh(CartSp)Sh (CartSp)HSh(CartSp) \hookrightarrow Sh_\infty(CartSp) \simeq \mathbf{H} of smooth sets.

References

Last revised on February 3, 2021 at 15:57:29. See the history of this page for a list of all contributions to it.