This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics, as in tensor calculus. For the different notion of a tensor in enriched category theory see under copower.
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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
Generally, a tensor is an element of a tensor product.
Traditionally this is considered in differential geometry for the following case:
for $X$ a manifold, $T X$ the tangent bundle, $T^* X$ the cotangent bundle, $\Gamma(T X)$, $\Gamma(T^* X)$ their spaces of sections and $C(X)$ the associative algebra of functions on $X$, a rank-$(p,q)$ tensor or tensor field on $X$ is an element of the tensor product of modules over $C(X)$
A rank $(p,0)$-tensor is also called a covariant tensor and a rank $(0,q)$-tensor a contravariant tensor.
(…)
A vector field is a rank $(1,0)$-tensor field.
A Riemannian metric is a symmetric rank $(0,2)$-tensor.
A differential form of degree $n$ is a skew-symmetric rank $(0,n)$-tensor.
A Poisson tensor is a skew-symmetric tensor of rank $(2,0)$.
For instance section 2.4 of