nLab geometry of physics -- differentiation

This entry contains one chapter of the material at geometry of physics.

Differentiation

So far we have dealt with cohesive structures for which there is a notion of smooth variation, say of the position of a particle along a trajectory in spacetime. The idea of differentiation is to measure the difference between the position of two points on a cohesive trajectory in space as the difference between their worldline coordinates “tends to 0” without actually becoming 0. One also says that differentiation is forming “infinitesimal differences” of a cohesive process – and we will make precise here what this means.

There are two stages to the theory of differentiation:

1. We may think of differentiation as just a means to analyze more in detail the cohesive structure already given, without adding new structure, hence without a priori refining our notion of what a “cohesive trajectory” is. Indeed, given any line object $\mathbb{R}$ in a cohesive ∞-topos, there is canonically a homomorphism of cohesive spaces

   $$\mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl}(-,\mathbb{R})$$

   from the line to the cohesive moduli space of closed differential 1-forms, which is such that it sends a cohesive curve on the line to the differential form on this curve whose value at each point is the differential of the curve, its rate of infinitesimal change at that point.

   Below in Differentiation of smooth functions and differetial forms we discuss this construction in the standard model of smooth cohesion for smooth spaces, where it reproduces what traditionally is called the de Rham differential $\mathbf{d}$.

   Further below in Maurer-Cartan forms – Cohesive differentiation we show how $\mathbf{d}$ comes out from just the abstract axioms of cohesionn.

2. We may think of differentiation as reflecting a refinement of smooth cohesion such that infinitesimal cohesive trajectories actually exist. Here, on top of having a measure for how a cohesive trajectory changes infinitesimally at a given point, it makes sense to ask concretely if two points on a trajectory are infinitesimally close to each other. In this approach the very notion of cohesion is refined to include explicitly a way to speak not just about a “cohesive blob of points”, but to decide whether it is in fact just an “infinitesimal cohesive blob of points” – an infinitesimally thickened point.

   Differential geometry with such an explicit notion of infinitesimals is known as synthetic differential geometry: the axioms here allow one to synthesize an infinitesimally thickened point and not just to reason about it as if it existed.

   In such a synthetic differential context then the differential $\mathbf{d}$ from above not just exists as a whole, but we can “take it apart and re-synthesize it” by realizing its value at each point literally as the ordinary difference between two infinitesimally close points. Similarly, various other fundamental constructions in differential geometry, such as that of tangent bundles and jet bundles have a usefully transparent axiomatic characterization in the presence of synthetic infinitesimals. (Sophus Lie, one of the founding fathers of differential geometry famously said that he indeed found his theorems using such synthetic reasoning intuitively, and just did not publish them this way due to a lack of formalization of this language – at his time. ) This we discuss in the Mod Layer in D-geometry below.

In the Differentiation semantic layer below we formalize differentiation, and these two aspects of it, by adding to the notion of cohesive topos that of an infinitesimal cohesive neighbourhood.

Recalling that a cohesive topos is an abstract cohesive blob, an infinitesimal cohesive neighbourhood is accordingly an infinitesimally thicked cohesive blob (which is itself again a cohesive blob):

$$\array{ CohesiveBlob &&\hookrightarrow&& InfinitesimallyThickenedCohesiveBlob \\ & \searrow && \swarrow \\ && Point } \;\;\;\;\;\;\;\; \array{ \mathbf{H} &&\stackrel{}{\hookrightarrow}&& \mathbf{H}_{th} \\ & \searrow && \swarrow \\ && DiscSpaces } \,.$$

Model Layer

We discuss

• Differentiation of smooth functions and differential forms

  • first just on coordinate patches

  • and then on general smooth spaces.

By considering fiber products of smooth mapping spaces with discrete spaces of boundary configurations, we obtain from this the differentiation theory called

• Variational calculus

  • with the notion of Smooth functional

  • and Variational derivative of smooth functionals.

  • The central class of examples of this of interest in physics is the variation of action functionals that yields the Euler-Lagrange equations of motion in classical field theory. This we discuss in Euler-Lagrange equations.

Differentiation of smooth functions and differential forms

Differentiation on coordinate patches

By definition to smooth function $f \colon \mathbb{R} \to \mathbb{R}$ is associated its derivative, a smooth function $f' \colon \mathbb{R} \to \mathbb{R}$. And more generally to a smooth function $f \colon \mathbb{R}^n \to \mathbb{R}$ are associated its partial derivatives, smooth functions

$$\frac{\partial f}{\partial x^i} \colon \mathbb{R}^n \to \mathbb{R}$$

for $1 \leq i \leq n$.

The de Rham differential collects all partial derivatives of a function into a single differential 1-form

Definition

For $n \in \mathbb{N}$, The de Rham differential on smooth functions in $C^\infty(\mathbb{R}^n)$ is the function

$$\mathbf{d} \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$$

which takes $f \in C^\infty(\mathbb{R}^n)$ to

$$\mathbf{d}f \coloneqq \sum_{i = 1}^n \frac{\partial f}{\partial x^i} \mathbf{d} x^i \,.$$

Example

For $x^i \colon \mathbb{R}^n \to \mathbb{R}$ one of the coordinate functions, the de Rham differential $\mathbf{d} x^i$ indeed coincides with the basis element of the same name according to def. , using that

$$\frac{\partial x^i}{\partial x^{j}} = \left\{ \array{ 1 & | i = j \\ 0 & | otherwise } \right. \,.$$

Proposition

The de Rham differentials $\mathbf{d} \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$ for all $n \in \mathbb{N}$ are compatible with pullback of differential 1-forms, def. , in that for each coordinate transformation $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^{k}$ the diagram

$$\array{ C^\infty(\mathbb{R}^k) &\stackrel{\mathbf{d}}{\to}& \Omega^1(\mathbb{R}^k) \\ \downarrow^{\mathrlap{\phi^\ast}} && \downarrow^{\mathrlap{\phi^\ast}} \\ C^\infty(\mathbb{R}^{\tilde k}) &\stackrel{\mathbf{d}}{\to}& \Omega^1(\mathbb{R}^{\tilde k}) }$$

commutes.

Proof

This is equivalently the statement of the chain rule of differentiation: For any $f \in C^\infty(\mathbb{R}^k)$ we have on the one hand, by def. and def.

$$\begin{aligned} \mathbf{d} \phi^\ast f & = \mathbf{d} (f \circ \phi) \\ & = \sum_{j = 1}^{\tilde k} \frac{\partial (f \circ \phi)}{\partial x^j}\mathbf{d} x^j \end{aligned}$$

and on the other hand, applying the definition in the other order,

$$\begin{aligned} \phi^* \mathbf{d}f & = \phi^* \sum_{i = 1}^k \frac{\partial f}{\partial x^i} \mathbf{d}x^i \\ & = \sum_{i = 1}^k \sum_{j = 1}^{\tilde k} \left(\frac{\partial f}{\partial x^i}\circ \phi \right) \cdot \left(\frac{\partial \phi^i}{\partial x^j}\right) \mathbf{d}x^j \end{aligned} \,.$$

Both expressions agree precisely if for all $j$ we have

$$\frac{\partial (f \circ \phi)}{\partial x^j} = \sum_{i = 1}^k \left(\frac{\partial f}{\partial x^i}\circ \phi\right) \cdot \left(\frac{\partial \phi^i}{\partial x^j}\right) \;\;\; \in C^\infty(\mathbb{R}^{\tilde k}) \,.$$

This is precisely the statement of the chain rule for differentiation.

Notice that as smooth spaces $\mathbb{R} = \Omega^0 = C^\infty(-)$, by prop. . Therefore the above says that

Proposition

The de Rham differential, def. , constitutes a homomorphism of smooth spaces, def.

$$\mathbf{d} \colon \mathbb{R} \to \Omega^1$$

from the real line to the universal smooth moduli space of differential 1-forms, def. .

Remark

Below in Maurer-Cartan form on a Lie group we discuss a more general abstract origin of $\mathbf{d} \colon \mathbb{R} \to \Omega^1_{cl}$.

We now extend the de Rham differential to differential forms of higher degree.

Definition

For all $n \in \mathbb{N}$ let

$$\mathbf{d} \colon \Omega^\bullet(\mathbb{R}^n) \to \Omega^\bullet(\mathbb{R}^n)$$

be the unique extension of $\mathbf{d} \colon C^\infty(-) \to \Omega^1(-)$ to a degree-1 derivation with

$$\mathbf{d}\mathbf{d}x^i = 0 \,.$$

(…)

Proposition

For each $n \in \mathbb{N}$ the de Rham differential of def. constitutes a homomorphism of smooth spaces

$$\mathbf{d} \colon \Omega^n \to \Omega^{n+1}$$

form the universal smooth moduli space of differental $n$-forms to that of differential $n+1$-forms.

Differentiation on smooth spaces

We now extend the notion of derivatives and de Rham differentials from smooth functions on Cartesian spaces to smooth functions on general smooth spaces.

Recall from def. that the set of differential $n$-forms on a smooth space $X$ is $\Omega^n(X) \coloneqq Hom(X, \Omega^n)$.

Definition

For $X \in Smooth0Type$ a smooth space and $n \in \mathbb{N}$, the de Rham differential on $n$-forms over $X$ is the function

$$\mathbf{d} \colon \Omega^n(X) \to \Omega^{n+1}(X)$$

which is the postcomposition with the homomorphism of smooth spaces of prop. :

$$Hom(X,\mathbf{d}) \colon Hom(X,\Omega^n) \to Hom(X,\Omega^{n+1}) \,.$$

In particular the derivative of a smooth function $f \colon X \to \mathbb{R}$ is the composite

$$\mathbf{d}f \colon X \stackrel{f}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1_{cl} \hookrightarrow \Omega^1 \,.$$

Below in Variation is differentiation on smooth spaces we find that this notion of differentiation of smooth functions on smooth spaces subsumes what traditionally is called variational calculus of functionals on mapping spaces.

Example: The electromagnetic field strength

for instance electromagnetic potential

$$A = \phi \mathbf{d}t + A_1 \mathbf{d}x^1 + A_2 \mathbf{d}x^2 + A_3 \mathbf{d}x^3$$

then the electromagnetic field strength is

$$F \coloneqq \mathbf{d}A = E_1 \mathbf{d}x^1 \wedge \mathbf{d}t + E_2 \mathbf{d}x^2 \wedge \mathbf{d}t + E_3 \mathbf{d}x^3 \wedge \mathbf{d}t + B_1 \mathbf{d}x^2 \wedge \mathbf{d}x^3 + B_2 \mathbf{d}x^3 \wedge \mathbf{d}x^1 + B_3 \mathbf{d}x^1 \wedge \mathbf{d}x^2$$

with

$$E_i = \frac{\partial \phi}{\partial x^i} - \frac{\partial A_i}{\partial t}$$

and

$$B_1 = \frac{\partial A_2}{\partial x^3} - \frac{\partial A_3}{\partial x^2}$$

etc

This are the first 2 of 4 Maxwell equations: $\mathbf{d} F = 0$

(the other 2 are discussed below in Riemannian geometry)

for

$$A,A' : X \to \Omega^1(-)$$

a gauge transformation $A \to A'$ is $\lambda : X \to \mathbb{R}$ with

$$A' = A + \mathbf{d} \lambda$$

Variational calculus

Traditionally a functional is a function which is sufficiently like a smooth function, but defined not on a manifold, but on a mapping space between manifolds. Also traditionally, a variational derivative of such a functional is something aking to a derivative, generalized to this context, and subject to the condition that all variations preserve some boundary conditions.

We formulate this classical theory in the context of smooth spaces. Here a functional is simply a homomorphism of smooth spaces out of a smooth mapping space, as in def. . We may impose respect for boundary conditions by forming the fiber product of this mapping space with a discrete smooth space inclusion, given in def. below. Then the variational derivative is simply the ordinary derivative of def. .

Discrete points of a smooth space

Definition

For $X \in Smooth0Type$ a smooth space, write

$$\Gamma X \coloneqq Hom(*,X) \in Set$$

for its set of points, the set of homomorphisms, def. , from the point to $X$.

Write

$$\flat X \coloneqq Disc (\Gamma(X))$$

for the discrete smooth space, def. , on this set of points.

Definition

For every smooth space $X$ there is a canonical homomorphism of smooth spaces

$$\flat X \to X \,.$$

This sends a plot $U \to \flat X$, which by definition of $Disc(-)$ is a point in $\Gamma X$, hence a homomorphism $x \colon * \to X$, to the plot $U \to * \stackrel{x}{\to} X$ of $X$.

Smooth functionals

Let $X$ be a smooth manifold. Let $\Sigma$ be a smooth manifold with boundary $\partial \Sigma \hookrightarrow \Sigma$.

Write

$$[\Sigma, X] \in Smooth0Type$$

for the smooth space (a diffeological space) which is the mapping space from $\Sigma$ to $X$, hence such that for each $U \in$ CartSp we have

$$[\Sigma, X](U) = C^\infty(U \times \Sigma, X) \,.$$

Definition

Write

$$[\Sigma, X]_{\partial \Sigma} \coloneqq [\Sigma, X] \times_{[\partial \Sigma,X]} \flat [\partial \Sigma,X]$$

for the pullback in smooth spaces

$$\array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,,$$

where

• the bottom morphism is the restriction $[\partial \Sigma \hookrightarrow \Sigma, X]$ of configurations to the boundary;

• the right vertical morphism is the homomorphism from def. .

Proposition

The smooth space $[\Sigma, X]_{\partial \Sigma}$ is a diffeological space whose underlying set is $C^\infty(\Sigma,X)$ and whose $U$-plots for $U \in$ CartSp are smooth functions $\phi \colon U \times \Sigma \to X$ such that $\phi(-,s) \colon U \to X$ is the constant function for all $s \in \partial \Sigma \hookrightarrow \Sigma$.

Definition

A functional on the mapping space $[\Sigma, X]$ is a homomorphism of smooth spaces

$$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R} \,.$$

Functional derivative / variational derivative

Write

$$\mathbf{d} \colon \mathbb{R} \to \Omega^1$$

for the de Rham differential incarnated as a homomorphism of smooth spaces from the real line to the smooth moduli space of differential 1-forms.

Definition

The functional derivative

$$\mathbf{d}S \in \Omega^1([\Sigma,X]_{\partial \Sigma})$$

of a functional $S$, def. , is simply its de Rham differential as a homomorphism of smooth spaces, hence the composite

$$\mathbf{d} S \colon [ \Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.$$

Definition

This means that for each smooth parameter space $U \in$ CartSp and for each smooth $U$-parameterized collection

$$\phi \colon U \times \Sigma \to X$$

of smooth functions $\Sigma \to X$, constant in the parameter $U$ in some neighbourhood of the boundary of $\Sigma$,

$$S_\phi \colon [\Sigma,X]_{\partial \Sigma}(U) \to C^\infty(U,\mathbb{R})$$

is the function that sends each such $U$-collection of configurations to the $U$-collection of their values under $S$. Then

$$(\mathbf{d}S)_\phi \in \Omega^1(U)$$

is the smooth differential 1-form

$$(\mathbf{d}S)_\phi = \mathbf{d}(S(\phi)) \,.$$

Example

Let $\Sigma = [0,1] \hookrightarrow \mathbb{R}$ be the standard interval. Let

$$L(-,-) \mathbf{d}t \in \Omega^1([0,1], C^\infty(\mathbb{R}^2))$$

and consider the functional

$$S \colon ([0,1] \stackrel{\gamma}{\to} X) \mapsto \int_{0}^1 L(\gamma(t), \dot \gamma(t)) d t \,.$$

Then for $U = \mathbb{R}$ and any smooth $U$-parameterized collection $\{\gamma_{u} \colon \Sigma \to X\}_{u \in U}$ the functional derivative takes the value

$$\begin{aligned} \mathbf{d}S_{\gamma_{(-)}} & = \left( \frac{d}{d u} \int_0^1 L(\gamma_u(t), \dot \gamma_u(t)) dt \right) \mathbf{d}u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial \dot \gamma_u(t)}{\partial u} \right) dt \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{d \gamma_u(t)}{d u} + \frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \frac{\partial }{\partial t}\frac{\partial \gamma_u(t)}{\partial u} \right) dt \mathbf{d} u \\ & = \int_{0}^1 \left( \frac{\partial L}{\partial \gamma}(\gamma_u(t), \dot \gamma_u(t)) - \frac{\partial}{\partial t}\frac{\partial L}{\partial \dot \gamma}(\gamma_u(t), \dot \gamma_u(t)) \right) \frac{\partial \gamma_u(t)}{\partial u} dt \mathbf{d}u \end{aligned} \,.$$

Here we used integration by parts and used that the boundary term vanishes

$$\begin{aligned} \int_{0}^1 \frac{\partial}{\partial t} \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(s), \dot \gamma_u(s)) \frac{\partial \gamma_u(s)}{\partial u} \right) d s & = \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(1), \dot \gamma_u(1)) \frac{\partial \gamma_u(1)}{\partial u} \right) - \left( \frac{\partial}{\partial \dot\gamma} L(\gamma_u(0), \dot \gamma_u(0)) \frac{\partial \gamma_u(0)}{\partial u} \right) \\ & = 0 \end{aligned}$$

since by prop. $\gamma_{(-)}(1)$ and $\gamma_{(-)}(0)$ are constant.

Euler-Lagrange equations

• critical locus

• Euler-Lagrange equation

• equations of motion

$\mathcal{D}$-geometry

• D-geometry

Infinitesimal smooth loci

Definition

Write

$$SmoothLoci \coloneqq SmoothAlgebras^{op} \coloneqq Func^\times(CartSp,Set)$$

for the category of spaces which are formally dual to smooth algebras: the opposite category of that of smooth algebras. This is called the category of smooth loci.

Definition

A smooth Artin algebra (also called a “Weil algebra” in the synthetic differential geometry-literature) is a smooth algebra $A$ whose underlying $\mathbb{R}$-vector space is a direct sum of the form

$$A = \mathbb{R} \oplus V \,,$$

where $V$ is of finite dimension and such that every element $v \in V \subset A$ is nilpotent, in that there is $n \in \mathbb{N}$ such that the $n$-fold product of $v$ with itself in $A$ vanishes: $v^n = 0$.

Example

The smallest smooth Artin algebra is the ring of dual numbers, def. , for which $V = \mathbb{R}$.

Definition

Write

$$InfPoint \hookrightarrow SmoothLoci$$

for the full subcategory of $SmoothLoci$, def. , of those that are duals of Artin algebras, def. .

We call this the category of infinitesimally thickened points.

de Rham space

• de Rham space

Jet bundles

• jet bundle

Semantic Layer

Synthetic differential geometry

• synthetic differential geometry

We have two full and faithful functors

$$CartSp \hookrightarrow SmoothLoci$$

$$InfPoint \hookrightarrow SmoothLoci \,.$$

Definition

Write CartSp${}_{th} \hookrightarrow SmoothLocus$ for the full subcategory of that of smooth loci, def. , on those of the form

$$\mathbf{U} = U \times D$$

with $U \in CartSp \hookrightarrow SmoothLoci$ and $D \in InfPoint \hookrightarrow SmoothLoci$.

We may call this the category of infinitesimally thickened Cartesian spaces or or formal smooth Cartesian spaces.

The category $CartSp_{th}$ carries several coverages of interest. One is this:

Definition

For $\mathbf{U} = U \times D \in CartSp_{th}$ say that a covering family is a set of morphisms in $CartSp_{th}$ of the form $\{ U_i \times D \stackrel{(\phi_i, id_D)}{\to} U \times D\}_i$ such that $\{ U_i \stackrel{\phi_i}{\to} U\}_i$ is a covering family in CartSp.

The corresponding sheaf topos $Sh(CartSp_{th})$ is known as the Cahiers topos.

Proposition

The Cahiers topos $Sh(CartSp_{th})$, def. ,
is a cohesive topos.

Definition

We write

$$SynthDiff0Type \coloneqq Sh(CartSp_{th}) \,.$$

• synthetic differential infinity-groupoid

Tangent bundle

Write $D \in Sh(CartSp_{th})$ for the smooth locus formally dual to the ring of dual numbers, def. . Write

$$i \colon * \to D$$

for the unique point inclusion.

Definition

For $X \in SynthDiff0Type$, the internal hom

$$T X \coloneqq [D,X] \in SynthDiff0Type$$

equipped with the morphism

$$[i,X] \colon [D,X] \to [*,X] \simeq X$$

is the tangent bundle of $X$.

Definition

For $X \in SynthDiff0Type$, a vector field $v$ on $X$ is a section

$$\array{ X &&\stackrel{v}{\to}&& T X \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{[i,D]}} \\ && X } \,.$$

The smooth space of vector fields is the internal hom in the slice topos over $X$

$$\mathbf{\Gamma}_X(T X ) \coloneqq [X,T X]_X \,.$$

Differential equations

• equation

• differential equation

Differential cohesion of the topos of smooth spaces

Recall the sites

• CartSp, def. ;

• InfPoint, def. ;

• CartSp${}_{th}$, def. .

Definition

Define functors

$$CartSp \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}} CartSp_{th} \stackrel{\iota}{\leftarrow} InfPoint$$

by

• $i \colon U \mapsto U \times *$;

• $p \colon U \times D \mapsto U$

• $\iota \colon D \mapsto * \times D$

Proposition

All three functors in def. are morphisms of sites. The induced geometric morphism of sheaf toposes is of the form

$$Sh(CartSp) \stackrel{}{\stackrel{\overset{Lan_i}{\hookrightarrow}}{\stackrel{\overset{(-)\circ i}{\leftarrow}}{\stackrel{\overset{(-)\circ p}{\hookrightarrow}}{\underset{Ran_p}{\leftarrow}}}}} Sh(CartSp_{th}) \stackrel{\overset{Lan_\iota}{\leftarrow}}{\underset{(-)\circ \iota}{\to}} Sh(InfPoint)$$

where hence the morphism on the left is in particular an essential geometric embedding.

Proposition

The sequence of geometric morphisms

$$Sh(CartSp) \stackrel{p^*}{\to} Sh(CartSp_{th}) \stackrel{\iota^*}{\to} Sh(InfPoint)$$

exhibit a homotopy cofiber sequence in the (2,1)-category Topos.

Proof

By the discussion at (∞,1)Topos – Existence of limits and colimits the statement is equivalently that the inverse image functors

$$Sh(CartSp) \stackrel{Lan_p}{\leftarrow} Sh(CartSp_{th}) \stackrel{Lan_\iota}{\leftarrow} Sh(InfPoint)$$

form a homotopy fiber sequence in (∞,1)Cat. Computing this in the model structure for quasi-categories after passing to nerves, the morphism $N(Lan_p)$ is clearly an inner Kan fibration because of the subcategory inclusion $Sh(CartSp) \hookrightarrow Sh(CartSp_{th})$. So by the general discussion at homotopy pullback the homotopy fiber is given by the 1-categorical fiber of $N(Lan_p)$ in sSet. By the discussion at Left Kan extension - on representables $Lan_p$ acts as $p$ on representables. The 1-categorical fiber of $N(p) \colon N(CartSp_{th}) \to N(CartSp)$ is evidently $N(InfPoint)$. Since $Lan_\iota$ is a left adjoint it preserves colimits and since ever sheaf is a colimit of representables, this is sufficient to imply the claim.

Differential cohesion

We axiomatize the existence of infinitesimals by further modalities on a cohesive topos.

Definition

Given a cohesive topos $\mathbf{H} = Cohesive0Type$ over a base topos Discrete0Type, a structure of differential cohesion on $\mathbf{H}$ is an geometric embedding into a cohesive topos $\mathbf{H}_{th} = InfThickenedCohesive0Type$ with an extra left eadjoint? that preserves the terminal object:

$$\array{ Cohesive0Type &&\stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}}}&& InfThickenedCohesive0Type \\ & {}_{\mathllap{\Gamma}}\searrow \nwarrow^{\mathrlap{Disc}} && {}^{\mathllap{\Gamma_{th}}}\swarrow \nearrow_{\mathrlap{Disc_{th}}} \\ && Discrete0Type }$$

Definition

Given differential cohesion, def. , define the monad/comonad adjunction

$$(Red \dashv \Pi_{inf}) \colon \mathbf{H}_{th} \stackrel{\overset{i_*}{\leftarrow}}{\underset{i^*}{\to}} \mathbf{H} \stackrel{\overset{i_!}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H} \,.$$

We call $Red(X)$ the reduced type of $X$ and $\Pi_{inf}(X)$ the infinitesimal path ∞-groupoid of $X$.

For the $(i_* \dashv i^*)$-unit we write

$$InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)$$

and call it the constant infinitesimal path inclusion on $X$.

The $(i_* \dashv i^*)$-counit

$$Red X \to X$$

we call the inclusion of the reduced part of $X$.

Definition

Given a geometric embedding of ∞-toposes

$$CohesiveType \stackrel{i}{\hookrightarrow} InfThickenedCohesiveType$$

exhibiting differential cohesion, write

$$CohesiveType \stackrel{i}{\hookrightarrow} InfThickenedCohesiveType \to InfinitesimalType$$

for the corresponding homotopy cofiber sequence in (∞,1)-topos. The full sub-(∞,1)-category that is the kernel of the global section geometric morphism of $InfininitesimalType$ we call the (∞,1)-category of synthetic ∞-Lie algebras

$$L_\infty Algebra \coloneqq ker(\Gamma) \hookrightarrow InfinitesimalType \stackrerl{\Gamma}{\to} DiscreteType \,.$$

Example

For the moment see at Synthetic differential infinity-groupoid – Lie differentiation.

Proposition

Setting

• $\mathbf{H}_{th} \coloneqq Sh(CartSp_{th})$

• $\mathbf{H} \coloneqq Sh(CartSp)$

• $\mathbf{H}_{inf} \coloneqq Sh(InfPoint)$

makes prop. exhibit differential cohesive structure.

de Rham space

Definition

For $X \in \mathbf{H}_{th}$ we call $\Pi_{inf}(X) \in \mathbf{H}$ the de Rham space object of $X$.

Jet bundle

Definition

For $X \in \mathbf{H}$

$$Jet_X \coloneqq (InfinitesimalPathInclusion_X)_* \colon \mathbf{H}_{th}/X \to \mathbf{H}_{th}/\Pi_{inf}(X) \,.$$

For $(E \to X) \in \mathbf{H}_{th}/X$, $Jet_X(E)$ is the jet bundle of $E$.

Formally étale / formally unramified / formally smooth

Definition

A morphism $f \;\colon\; X \to Y$ in $\mathbf{H}$ is called a formally étale morphism if the naturality square of the $\mathbf{\Pi}_{inf}$-unit

$$\array{ X &\stackrel{}{\to}& \mathbf{\Pi}_{inf}(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\mathbf{\Pi}_{inf}(f)}} \\ Y &\to& \mathbf{\Pi}_{inf}(Y) }$$

is an (∞,1)-pullback.

Proposition

If $X, Y \in$ SmoothMfd $\hookrightarrow$ $\mathbf{H} \stackrel{i_!}{\to} \mathbf{H}_{th}$ then for $f \colon X \to Y$ any morphism

1. $f$ is formally étale morphism precisely if $f$ is a submersion of smooth manifolds;

2. $f$ is a formally unramified morphism precisely if it is an immersion of smooth manifolds;

3. $f$ is a formally smooth morphism precisely if it is a diffeomorphism.

Syntactic Layer

Differential homotopy type theory

• differential homotopy type theory

(…)