nLab smooth functor

Contents

Context

Internal categories

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

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)

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

General

By a smooth functor one may generally understand an internal functor in the category of SmoothManifolds, hence a homomorphism between internal categories or internal groupoids (Lie groupoids) in SmoothManifolds.

Slightly alternatively, a smooth functor may be understood as an enriched functor between enriched categories over SmoothManifolds.

In global analysis

Much more specifically, “smooth functor” is used by Kriegl & Michor 97, Sec 29.5 to refer to an endofunctor of FinVect, when FinVectFinVect is viewed as a category enriched in the category of smooth manifolds.

That is, a smooth functor is a functor F:FinVectFinVectF \colon FinVect \to FinVect such that the map FinVect(X,Y)FinVect(F(X),F(Y))FinVect(X,Y) \to FinVect(F(X),F(Y)) is smooth for every XX, YY.

Currently, the remainder of this entry focuses on this specific notion.

Examples

The iterated tensor product XX nX \mapsto X^{\otimes n} is a smooth functor.

The iterated wedge product X i=1 nXX \mapsto \bigwedge_{i=1}^n X is a smooth functor

Extensions

The concept of smooth functor can be extended to multivariate functors FinVect nFinVectFinVect^n \to FinVect, and also to contravariant functors FinVect opFinVectFinVect^{op} \to FinVect, such as the dual VV *V \mapsto V^*.

References

  • Andreas Kriegl and Peter Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, 1997 (ISBN: 978-0-8218-0780-4)

Last revised on May 5, 2022 at 16:17:28. See the history of this page for a list of all contributions to it.