category object in an (∞,1)-category, groupoid object
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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
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.
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.
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
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 $V \mapsto V^*$.
Last revised on May 5, 2022 at 16:17:28. See the history of this page for a list of all contributions to it.