# nLab smooth functor

Contents

## Internal $n$-category

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 $FinVect$ is viewed as a category enriched in the category of smooth manifolds.

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

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

## Examples

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

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

## Extensions

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

• 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)