analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
…
…
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)
Differential calculus is one of the two halves of the infinitesimal calculus, the other being integral calculus. (The two are linked by the fundamental theorem of calculus.) The differential calculus was developed in the 18th century by Isaac Newton and Gottfried Leibniz (acting independently).
In modern terms, the original differential calculus describes the behaviour of differentiation of functions on the real line. From here, we move to the study of differential equations and then to differential geometry (de Rham complex) and eventually generalize to a wide class of analogous situations (see for instance synthetic differential geometry and D-modules).
Differential calculus on non-finite dimensional spaces is also known as variational calculus.
In the presence of Lie algebra actions a variant of differential caclulus is Cartan calculus.
Since Hochschild homology and cyclic homology (see there) is closely related to differential forms and de Rham differentials, generalizations and abstractions of Hochschild (co)homology are also used as generalized differential calculi .
We list how aspects of differential calculus are captured by the axioms of cohesion and differential cohesion.
With the axioms of cohesion one may characterize the line object $\mathbb{A}^1 = \mathbb{R}^1$ (by the fact that its A1-homotopy localization is the localization modality which is exhibited by the given shape modality) and then one may essentially characterize a function
which is the “universal de Rham differential”. For any function $f \;\colon\; X \longrightarrow \mathbb{R}$, the composite
is interpreted as the de Rham differential of $f$. This is discussed in some detail at geometry of physics in the section 4. Differentiation.
A slight variant of this construction, using pullbacks of flat modal types yields variational calculus. See there the section In terms of smooth spaces.
With the stronger axioms of differential cohesion one may similarly ask for the infinitesimal interval $D \hookrightarrow \mathbb{R}$. With this one can proceeed as in synthetic differential geometry and formulate for instance differential equations synthetically, as discussed there in the section In terms of synthetic differential geometry.
Moreover, differential cohesion induces de Rham spaces and hence the geometry in the slice over them, dependent types over the infinitesimal shape modality. This is D-geometry which is a general way of talking about differential equations.
Discussion of differential calculus in terms of coinduction is in