nLab smooth functor

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 FinDimVect, when $FinDimVect$ is viewed as a category enriched in the category of smooth manifolds.

That is, a smooth functor is a functor $F \colon FinDimVect \to FinDimVect$ such that the map $FinDimVect(X,Y) \to FinDimVect(F(X),F(Y))$ 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 $FinDimVect^n \to FinDimVect$, and also to contravariant functors $FinDimVect^{op} \to FinDimVect$, such as the dual $V \mapsto V^*$.

Related concepts

• linear functor

References

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