smooth infinity-stack

> For more see at _[[smooth ∞-groupoid]]_.

#Contents#
* table of contents
{:toc}

## Idea

The concept of smooth $\infty$-stack is essentially that of

* [[smooth ∞-groupoid]].

* [[orbifold|∞-orbifold]].

Following the logic described at

* [[motivation for sheaves, cohomology and higher stacks]]

* [[(∞,1)-topos]]

a _smooth $\infty$-stack_ is the [[∞-categorification]] of [[smooth space]] and [[differentiable stack]]. It is an [[∞-stack]] on the ([[essentially small category|essentially small]]) [[site]] [[Diff]] of smooth [[manifolds]], or correspondingly on $Ball \subset Diff$ or [[CartSp]] $\subset Diff$ (see [[smooth space]] for more on that).

So smooth $\infty$-stacks are the objects in the [[(∞,1)-topos]] that computes _smooth_ generalized [[cohomology]]. (See [[schreiber:Differential Nonabelian Cohomology|differential nonabelian cohomology]] and the disucssion under "Models" below for more on that).

## Definition

Let $CartSp = \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff| n,m \in \mathbb{N}\} \subset Diff$ be the full subcategory [of [[Diff]] on the [[Cartesian spaces]] of the simple form $\mathbb{R}^n$, equipped with the standard structure of a [[site]] with the [[coverage]] given by [[open covers]] of manifolds.

Then

$$
  Smooth\infty Grpd := (\infty,1)Sh(CartSp)
$$

is the [[(∞,1)-topos]] given by the [[(∞,1)-category of (∞,1)-sheaves]] on $CartSp$.

This is the [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]].

## Models

There is a large number of [[model category|model structures]] [[presentable (infinity,1)-category|presenting]] $\mathbf{H}_{Diff}$: all the [[model structure on simplicial presheaves|model structures on simplicial (pre)sheaves]] on $CartSp$.

### In terms of $\infty$-groupoids internal to smooth spaces ##

Notice for instance that there is the [[model structure on simplicial sheaves]] given by the category $SSh(CartSp)$ equipped with the injective [[local model structure on simplicial presheaves]].

But sheaves on cartesian spaces

$$
  Sh(CartSp)
  =: SmoothSp
$$

is the category of [[smooth spaces]], and $SSh(CartSp)$ is just the category of [[simplicial objects]] of that

$$
  SSh(CartSp)
  \simeq
  SmoothSp^{\Delta^{op}}
  \,.
$$

So one model for smooth $\infty$-stacks is given by [[simplicial object|simplicial]] [[smooth spaces]].

Notice that the fibrant object in $SmoothSp^{\Delta^{op}}$ are the globally [[Kan complex]]-valued sheaves under the [[equivalence of categories]]

$$
  SmoothSp^{\Delta^{op}}
  \simeq
  Sh(CartSp, SSet)
  \,,
$$

that satisfy [[descent]] (see [[descent for simplicial presheaves]]).

Being [[Kan complex]]-valued just means that the fibrant objects are sheaves on $CartSp$ with values in [[∞-groupoids]].

Moreover, the [[descent]]-condition on $CartSp$ is comparatively trivial, and in many cases (...details eventually here, but see examples below...) entirely empty, as every cartesian space is (smoothly, even) contractible.

This means that the fibrant objects in $SSh(CartSp)$ are pretty much nothing but [[∞-groupoids]] [[internalization|internal to]] [[smooth spaces]]. (But notice that the requirement that she corresponding sheaf is [[Kan complex]]-valued is a bit weaker that other notions of "$\infty$-groupoid internal to smooth spaces" that one may come up with).

In particular [[∞-groupoids]] internal to [[diffeological spaces]] are therefore a model for smooth $\infty$-stacks.

Moreover, a morphism between smooth $\infty$-stacks modeled by such internal $\infty$-groupoids is modeled as an $\infty$-[[anafunctor]] (see [[simplicial localization]], [[homotopy category]] and [[category of fibrant objects]] for details).

The model of smooth $\infty$-stacks given by $\infty$-groupoids internal to [[diffeological spaces]] with [[anafunctors]] as morphism between them is the model used in the [[John Baez|Baez]]-ian school description of [[principal infinity-bundle|higher principal bundles]] and [[schreiber:Differential Nonabelian Cohomology|differential nonabelian cohomology]].

### Examples ###

Let $G$ be a [[Lie group]]. Using the embedding

$$
  Diff \hookrightarrow SmoothSp
$$

of [[manifolds]] into [[smooth spaces]] we may regard $G$ naturally as a sheaf on CartSp.

Write $\mathbf{B} G$ for the [[delooping]] of $G$, a one-object [[groupoid]] [[internalization|internal to]] [[smooth space|SmoothSp]]. Postcomposing with the [[nerve]] functor $N : $ [[Grpd]] $\to$ [[SSet]] this yields a [[Kan complex]]-valued [[simplicial presheaf|simplicial sheaf]] $N \mathbf{B} G$ which we shall by convenient and useful abuse of notation just call $\mathbf{B} G$ itself.

Notice that $\mathbf{B} G$ does not satisfy [[descent]] when regarded as a simplicial sheaf on all of [[Diff]]: there its [[∞-stackification]] is instead $G Bund(-)$, the [[stack]] of $G$-[[principal bundles]]

$$
  G Bund(-) : U \mapsto groupoid of G-bundles on U
$$

(or rather, in our context of simplicial sheaves, a [[rectified infinity-stack|rectification]] of that).

But restricted to the [[site]] $CartSp$ the [[simplicial presheaf|simplicial sheaf]] $\mathbf{B} G$ _does_ satisfy [[descent]]: there is up to isomorphism only a single $G$-bundle on $\mathbb{R}^n$, so that one finds an [[equivalence of categories]]

$$
  G Bund(\mathbb{R}^n) \simeq (\mathbf{B} G)(\mathbb{R}^n)
  := \mathbf{B}(Diff(\mathbb{R}^n,  G))
$$

for each $\mathbb{R}^n$. This means that  $\mathbf{B}G$ is a fibrant object in the injective [[model structure on simplicial sheaves]].
So in particular all the constructions and examples discussed at [[category of fibrant objects]] apply to $\mathbf{B}G$: we get the [[generalized universal bundle|universal G-bundle]] $\mathbf{E} G \to \mathbf{B}G$ regarded as a smooth $\infty$-stack as the [[pullback]]

$$
  \array{
    \mathbf{E}G &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{d_0}{\to}& \mathbf{B}G
    \\
    \downarrow^{d_1}
    \\
    \mathbf{B}G
  }
$$

in $SmoothSp^{\Delta^{op}}$, which, do to the [[nerve]] being [[right adjoint]] is the same as the image under the nerve of the corresponding [[pullback]] in sheaves of [[groupoids]] (so that still our notational suppressing of $N$ is justified).

etc.

[[!redirects smooth infinity-stacks]]
[[!redirects smooth ∞-stack]]
[[!redirects smooth ∞-stacks]]


smooth isotopy

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _smooth isotopy_ is an [[isotopy]] that varies [[smooth function|smoothly]], hence a isotopy that, as a [[left homotopy]], is a [[smooth homotopy]].

In [[knot theory]], one typically does not want to identify [[knots]] $S^1 \to S^3$ by plain [[isotopy]], as that makes all [[tame knots]] be equivalent. A common fix is to use [[ambient isotopy]] instead. But one may also use smooth isotopy. (see e.g. [Greene 13](#Greene13) or MO discussion [here](https://math.stackexchange.com/a/1312016/58526)).

## Definition

Let $\Sigma$ and $X$ be [[smooth manifolds]], and let

$$
  \gamma_0, \gamma_1 \;\colon\; \Sigma \hookrightarrow X
$$

be two [[embeddings of smooth manifolds]]. Then a _smooth isotopy_ between them is a _[[smooth homotopy]] between them via [[embeddings of smooth manifolds|embeddings]]_: a [[smooth function]]

$$
  \eta \;\colon\; [0,1] \times \Sigma  \longrightarrow X
$$

such that

$$
  \eta(0,-) \;=\; \gamma_0
  \phantom{AAA}
  \eta(1,-) \;=\; \gamma_1
$$

and such that for each $t \in [0,1]$,

$$
  \gamma(t,-)
  \;\colon\;
  \Sigma \longrightarrow X
$$

is an [[embedding of smooth manifolds]].

(e.g. [Greene 13, Def. 1.7](#Greene13))

## Related concepts

* [[smooth homotopy]]

## References

* {#Greene13} Josh Greene, _Combinatorial methods in knot theory_, 2013 ([pdf](https://www2.bc.edu/joshua-e-greene/MT885S13/Lecture%201.pdf))


smooth locus

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###Context###
#### Synthetic differential geometry
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[[!include synthetic differential geometry - contents]]
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=--
=--

#Contents#

* table of contents
{:toc}

## Idea

A _smooth locus_ is the [[duality|formal dual]] of a finitely generated [[smooth algebra]] (or $C^\infty$-ring):

a [[space]] that behaves as if its algebra of functions is a finitely generated [[smooth algebra]].

Given that a [[smooth algebra]] is a smooth refinement of an ordinary [[ring]] with a morphism from $\mathbb{R}$, a smooth locus is the analog in [[Models for Smooth Infinitesimal Analysis|well-adapted models]] for [[synthetic differential geometry]] for what in [[algebraic geometry]] is an [[affine variety]] over $\mathbb{R}$.

## Definition

A finitely generated [[smooth algebra]] is one of the form $C^\infty(\mathbb{R}^n)/J$, for $J$ an ideal of the ordinary underlying algebra.

Write $C^\infty Ring^{fin}$ for the [[category]] of finitely generated smooth algebras.

Then the [[opposite category]] $\mathbb{L} := (C^\infty Ring^{fin})^{op}$ is the [[category]] of **smooth loci**.

### Notation

For $A \in C^\infty Ring^{fin}$ one write $\ell A$ for the corresponding object in $\mathbb{L}$.

Often one also write

$$
  R := \ell C^\infty(\mathbb{R})
$$

for the real line regarded as an object of $\mathbb{L}$.

## Properties

The category $\mathbb{L}$ has the following properties:

+-- {: .num_prop #SmoothManifoldsFullSubcategoryOfSmoothLoci}
###### Proposition

The canonical inclusion functor

$$
  SmthMfd \hookrightarrow \mathbb{L}
$$

$$
  X \mapsto \ell C^\infty(X)
$$

from the category [[Diff|SmthMfd]] of [[smooth manifolds]] is a [[full subcategory]] embedding (i.e. a [[full and faithful functor]]. Moreover, it preserves [[pullbacks]] along [[transversal maps]].

=--

+-- {: .num_prop}
###### Proposition

The [[Tietze extension theorem]] holds in $\mathbb{L}$: $R$-valued functions on closed subobjects in $\mathbb{L}$ have an extension.

=--

## Applications

There are various [[Grothendieck topology|Grothendieck topologies]] on $\mathbb{L}$ and various of its subcategories, such that [[category of sheaves|categories of sheaves]] on these are [[smooth topos]]es that are well-adapted models for [[synthetic differential geometry]].

For more on this see

* *[[Models for Smooth Infinitesimal Analysis]]*

## Related concepts

* [[smooth algebra]]

## References

See the references at [[generalized smooth algebra|C-infinity-ring]].

[[!redirects smooth loci]]


smooth loop space

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### Context
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Mapping space
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[[!include mapping space - contents]]
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=--

# Contents
* table of contents
{: toc}

## Idea

The notion of the _smooth loop space_ of a (smooth) [[smooth manifold|manifold]] is a way to make the set of smooth maps from the [[circle]] to the target manifold into an object amenable to the tools of [[differential topology]].

Let $M$ be a [[smooth manifold|smooth finite dimensional manifold]].  Then the set of smooth maps $S^1 \to M$ is not usefully the underlying set of a finite dimensional manifold.  Therefore to make it into an object that can be studied in differential topology, one has to allow for smooth spaces other than finite dimensional smooth manifolds.  That is to say, one has to work in a larger category than just that of finite dimensional smooth manifolds.

If the category is [[cartesian closed]] then the smooth loop space of a smooth space, say $X$, is simply the result of applying the [[internal hom]] functor with that space as target and the circle as source:
$$
  L X \coloneqq [S^1, X]
$$
(This is the smooth [[free loop space]]; for the smooth [[based loop space]] one applies the standard restriction.)

There are a variety of suitable categories listed at [[generalized smooth spaces]].

The categories of smooth spaces discussed at [[generalized smooth spaces]] are all cartesian closed and thus all admit smooth loop spaces of all objects.
However, by restricting to a subcategory that might not be cartesian closed (for example, finite dimensional smooth manifolds) one can ask for another subcategory that is large enough to contain all the resulting smooth loop spaces.  For finite dimensional smooth manifolds it turns out that one can give their smooth loop spaces the structure of an infinite dimensional [[Fréchet manifold]].

Depending on taste and application, certain smooth [[quotient objects]] of this free loop space may be and are often considered.

## Properties

### Fréchet manifold structure on $[S^1,X]$ for $X$ a manifold
 {#FrechetManifoldStructure}

In this section, we will describe the structure of a
smooth loop space of a smooth manifold
as a Fréchet manifold in detail.  The construction is quite standard, and a partial list of references and further reading can be found at the end of this page.

We start with a [[smooth manifold]], $M$, of dimension $n$.  Note that here manifolds definitely do not have a boundary.  For simplicity, we assume that it is orientable.  The point of this assumption is that it allows us to identify the model space of $L M$ with the Fréchet space $L \mathbb{R}^n = C^\infty (S^1, \mathbb{R}^n)$.  In the unorientable case, some components would have this as model space whereas others would have a _twisted_ version of this space.

The key piece of structure needed on $M$ is that of a [[local addition]], $\eta \colon T M \to M$.

Let $\eta \colon T M \to M$ be a local addition on $M$.  Let $V \subseteq M \times M$ be the image of the map $\pi \times \eta \colon T M \to M \times M$.  Although (as yet) we know nothing about the topologies of $L T M$ or of $L V$, we can at least say that the looped map, $(\pi \times \eta)^L \colon L T M \to L V$, is a bijection.

+-- {: .num_lemma #lemcharts}
###### Lemma
Let $\alpha \in L M$.  Define the set $U_\alpha \subseteq L M$ by:

$$
U_\alpha \coloneqq \{ \beta \in L M : (\alpha, \beta) \in L V\}.
$$

Then the preimage of $\{\alpha\} \times U_\alpha$ under $(\pi \times \eta)^L$ is naturally identified with $\Gamma_{S^1}(\alpha^* T M)$.  In particular, the zero section of $\alpha^* T M$ maps to $(\alpha, \alpha) \in \{\alpha\} \times U_\alpha$.
=--

+-- {: .proof}
###### Proof
We claim that there is a diagram:

$$
\begin{matrix}
L T M &\overset{(\pi \times \eta)^L}{\to} & L V \\
\uparrow & & \uparrow\mathrlap{\beta \mapsto (\alpha,\beta)} \\
\Gamma_{S^1}(\alpha^* T M) & & U_\alpha
\end{matrix}
$$

such that the bijection at the top takes the image of the left-hand vertical map to the image of the right-hand one.  Both of the vertical maps are injective - the right-hand one obviously so, we shall investigate the left-hand one in a moment - and thus the bijection $(\pi \times \eta)^L$ induces a bijection from the lower-left to the lower-right.

The left-hand vertical map, $\Gamma_{S^1}(\alpha^* T M) \to L T M$, is defined as follows: the total space of $\alpha^* T M$ is:

$$
\big\{(t, v) \in S^1 \times T M : \alpha(t) = \pi(v) \big\}.
$$

It is an embedded submanifold of $S^1 \times T M$.  Therefore, a map into $\alpha^* T M$ is smooth if and only if the compositions with the projections to $S^1$ and to $T M$ are smooth.  Now a map $S^1 \to \alpha^* T M$ is a section if and only if it projects to the identity on $S^1$.  Therefore, there is a bijection (of sets):

$$
\begin{aligned}
\Gamma_{S^1}(\alpha^* T M) &\cong \big\{ \beta \in L T M : (t, \beta(t)) \in \alpha^* T M\; \text{for all}\; t \in S^1\big\} \\
 &\cong \big\{ \beta \in L T M : \alpha(t) = \pi \beta(t) \text{for all}\; t \in S^1\big\} \\
 &\cong \big\{ \beta \in L T M : \pi^L \beta = \alpha \big\} &\eqcolon L_\alpha T M
\end{aligned}
$$

In particular, the map $\Gamma_{S^1}(\alpha^* T M) \to L T M$ is injective.

We apply $(\pi \times \eta)^L$ to the defining condition for $L_\alpha T M$ and see that $L_\alpha T M$ is the preimage under this map of everything of the form $(\alpha, \gamma)$ in $L V$.  By construction, $\gamma \in L M$ is such that $(\alpha, \gamma) \in L V$ if and only if $\gamma \in U_\alpha$.  Hence $(\pi \times \eta)^L$ identifies $L_\alpha T M$ with $\{\alpha\} \times U_\alpha$.

Finally, note that the zero section of $\alpha^* T M$ maps to the image of $\alpha$ under the zero section of $T M$.  Since $\eta$ composed with the zero section of $T M$ is the identity on $M$, the image under the zero section of $\alpha^* T M$ in $V$ is $(\alpha, \alpha)$ as required.
=--

The resulting map, let us write it $\Psi_\alpha \colon \Gamma_{S^1}(\alpha^* T M) \to U_\alpha$, has the following concrete description.  Let $\beta \colon S^1 \to \alpha^* T M$ be a section and let $\tilde{\beta} \colon S^1 \to T M$ be the corresponding loop in $T M$ (so that $\beta(t) = (t, \tilde{\beta}(t))$ when viewing $\alpha^* T M$ as a submanifold of $S^1 \times T M$).  Then $(\pi \times \eta)^L (\tilde{\beta}) = (\alpha, \eta^L (\tilde{\beta}))$ so $\Psi_\alpha(\beta) = \eta^L (\tilde{\beta})$.

As we have assumed $M$ to be orientable, $\alpha^* T M$ can be trivialised.  A smooth such trivialisation defines a linear homeomorphism $\Gamma_{S^1}(\alpha^* T M) \cong L \mathbb{R}^n$.  We use this to impose a smooth structure on $\Gamma_{S^1}(\alpha^* T M)$, noting that any two such trivialisations induce the same structure.

To investigate the transition functions, we need two loops.  In fact, let's have two of everything.

+-- {: .num_proposition #transition}
###### Proposition
Let $\eta_1, \eta_2 \colon T M \to M$ be local additions with corresponding neighbourhoods $V_1$, $V_2$ of the diagonal in $M \times M$.  Let $\alpha_1$, $\alpha_2$ be smooth loops in $M$.  Let $\Psi_1 \colon \Gamma_{S^1}(\alpha_1^* T M) \to U_1$ and  $\Psi_2 \colon \Gamma_{S^1}(\alpha_2^* T M) \to U_2$ be the corresponding maps defined as above.

The transition function:

$$
\Psi_{1 2} \coloneqq \Psi_1^{-1} \Psi_2 \colon \Psi_1^{-1}(U_1 \cap U_2) \to \Psi_2^{-1}(U_1 \cap U_2)
$$

is a diffeomorphism.
=--

+-- {: .proof}
###### Proof
We start by characterising the space $\Psi_1^{-1}(U_1 \cap U_2)$ within $\Gamma_{S^1}(\alpha_1^* T M)$.
Let $W_{1 2} \subseteq \alpha_1^* T M$ be the set:
$$
\big\{ (t,v) \in \alpha_1^* T M : (\alpha_2(t), \eta_1(v)) \in V_2\big\}.
$$

Note that this is open in $\alpha_1^* T M$ as it is the preimage of the open set $V_2$ by the continuous map $\alpha_2 \times \eta_1$.

Let $\gamma \in \Gamma_{S^1}(\alpha_1^* T M)$ and let $\tilde{\gamma} \in L T M$ be the image of $\gamma$ (so that $\gamma(t) = (t, \tilde{\gamma}(t))$.  Then $\gamma(t) \in W_{1 2}$ for all $t$ if and only if $(\alpha_2(t), \eta_1(\tilde{\gamma}(t))) \in V_2$ for all $t$.  That is to say, if and only if $(\alpha_2, \eta_1^L(\tilde{\gamma})) \in L V_2$.  This is precisely the condition that $\eta_1^L(\tilde{\gamma}) \in U_2$.  Since $\eta_1^L(\tilde{\gamma}) = \Psi_1(\gamma)$, we see that $\gamma$ takes values in $W_{1 2}$ if and only if $\Psi_1(\gamma) \in U_2$.  Since $\operatorname{Im} \Psi_1 = U_1$, we conclude that $\Gamma_{S^1}(W_{1 2}) = \Psi_1^{-1}(U_1 \cap U_2)$.

Let us define $W_{2 1} \subseteq \alpha_2^* T M$ similarly.  The idea of the proof that $\Phi_{1 2}$ is a diffeomorphism is to show that it is induced by a diffeomorphism $W_{1 2} \cong W_{2 1}$.

Let $\theta_1 \colon W_{1 2} \to T M$ be the map:
$$
\theta_1(t,v) = (\pi \times \eta_2)^{-1}(\alpha_2(t), \eta_1(v)).
$$
The definition of $W_{1 2}$ ensures that $(\alpha_2(t) ,\eta_1(v)) \in V_2$ for $(t,v) \in W_{1 2}$ and this is the image of $\pi \times \eta_2$.  Hence $\theta_1$ is well-defined.  Define $\theta_2 \colon W_{2 1} \to T M$ similarly.  These are both smooth maps.

Notice that $\pi(\pi \times \eta_i)^{-1} \colon V_i \subseteq M \times M \to M$ is the projection on to the first factor and $\eta_i(\pi \times \eta_i)^{-1} \colon V_i \to M$ is the projection on to the second.  Thus $\pi \theta_1(t,v) = \alpha_2(t)$.  Hence $\theta_1 \colon W_{1 2} \to T M$ is such that $(t, \theta_1(t,v)) \in \alpha_2^* T M$ for all $(t,v) \in W_{1 2}$.  Then:

$$
(\alpha_1(t), \eta_2(\theta_1(t,v))) = (\alpha_1(t), \eta_1(v)) \in V_1
$$

so $(t, \theta_1(t,v)) \in W_{2 1}$.  Hence we have a map $\phi_{1 2} \colon W_{1 2} \to W_{2 1}$ given by:

$$
\phi_{1 2}(t, v) = (t, \theta_1(t,v)).
$$

Similarly we have a map $\phi_{2 1} \colon W_{2 1} \to W_{1 2}$.  These are both smooth since the composition with the inclusion into $S^1 \times T M$ is smooth.

Consider the composition $\phi_{2 1}\phi_{1 2}(t,v)$.  Expanding this out yields:

$$
\begin{aligned}
\phi_{2 1} \phi_{1 2}(t,v) &= \phi_{2 1}(t, \theta_1(t,v))\\
&=(t, \theta_2(t,\theta_1(t,v))) \\
&=(t, (\pi \times \eta_1)^{-1}(\alpha_1(t), \eta_2(\theta_1(t,v)))) \\
&= (t, (\pi \times \eta_1)^{-1}(\alpha_1(t), \eta_1(v))) \\
&= (t, (\pi \times \eta_1)^{-1}(\pi(v), \eta_1(v))) \\
&= (t,v)
\end{aligned}
$$

The penultimate line is because $(t,v) \in \alpha_1^* T M$ so $\pi(v) = \alpha_1(t)$.  Hence $\phi_{2 1}$ is the inverse of $\phi_{1 2}$ and so $\phi_{1 2}$ is a diffeomorphism.  Thus the map $\phi_{1 2}^L$ is a diffeomorphism from $\Psi_1^{-1}(U_1 \cap U_2)$ to $\Psi_2^{-1}(U_1 \cap U_2)$.  We just need to show that this is the transition function.  To do this, we show that $\Psi_2 \psi_{1 2}^L = \Psi_2 \Phi_{1 2}$.  The right-hand side is, by definition, $\Psi_1$ which satisfies:

$$
\Psi_1(\gamma)(t) = \eta_1(\tilde{\gamma})(t).
$$

On the other side,

$$
\begin{aligned}
\phi_{1 2}^L (\gamma)(t) &= \phi_{1 2}(\gamma(t)) \\
= (t, \theta_1(t, \tilde{\gamma}(t))) \\
= (t, (\pi \times \eta_2)^{-1}(\alpha_2(t), \eta_1(\tilde{\gamma}(t)))).
\end{aligned}
$$

Hence

$$
\Psi_2 \phi_{1 2}^L(\gamma)(t) = \eta_2(\pi \times \eta_2)^{-1}(\alpha_2(t), \eta_1(\tilde{\gamma}(t))) = \eta_1(\tilde{\gamma})(t).
$$

Thus $\phi_{1 2}^L = \Phi_{1 2}$ and so the transition functions are diffeomorphisms.
=--

+-- {: .num_remark #extension}
###### Remark
This construction easily generalises quite widely.  Very little of the  structure of $S^1$ was used at all: that mainly came in in the smooth structure of $L \mathbb{R}^n$.  The key structure of $M$ was the local addition and thus one could regard this as a construction of [[locally additive space|locally additive spaces]].  For more on the possible extensions, see the references.
=--

### Relation between Fréchet-manifold structure and diffeological structure
 {#SmoothStruc}

For $X$ a [[smooth manifold]], the [[Fréchet manifold]] structure on $L X$, discussed [above](#FrechetManifoldStructure)
agrees with the standard [[mapping space]] [[diffeological space|diffeology]] on $L X$ in the following sense.

There is a [[functor]]

$$
  F: Frech \to Diff
$$

from [[Frechet manifold|Fréchet manifold]]s to [[diffeological spaces]] defined in the same way as the well-known functor from [[smooth manifold]]s to [[diffeological space]]s: a plot is precisely a smooth map $c : U \to L X$, where $U$ is an object in the domain category, e.g. an open subset of some $\mathbb{R}^n$.

(Notice that a theorem of M. Losik says that the functor F is [[full and faithful functor|full and faithful]], just like that including manifolds into diffeological spaces!)

Now there are two diffeologies on $L X$: this is the structure of a [[sheaf]] on the category [[CartSp]] obtained as regarding $L X$ as the [[internal hom]] with respect to the [[closed monoidal structure on sheaves]]

$$
  L X = [S^1,X] : U \mapsto Hom_{Sh(CartSp)}(S^1 \times Y(U), X)
$$

(where $Y$ denotes the [[Yoneda embedding]]).

This means that a map $c : U \to L X$ is a plot if and only if the associated map $U \times S^1 \to X$ is smooth. The second diffeology is the one obtained from the functor $F$.

These two diffeologies _coincide_  -- in the sense that every plot of one is a plot of the other. In particular, they have the same sets of smooth functions.

A detailed proof of this is in ([Waldorf, lemma A.1.7](#Waldorf))

### $G$-Structures on smooth loop spaces
 {#GStructures}

The usual notions of _[[G-structures]]_ for [[manifolds]], such as [[orientation]], [[spin structure]], [[string structure]], etc. do not carry over directly to their smooth loop spaces, but they are closely related by [[transgression]]: a [[spin structure]] on $X$ is supposed to induce a kind of orientation structure on $\Omega X$, a string structure on $X$ is supposed to induce a kind of spin structure on $\Omega X$.

Formalizations of such "smooth loop space $G$-structures" have been proposed in  ([Stolz-Teichner 2005](#Stolz)), ([Waldorf 2011](#Waldorf11))  and ([Waldorf 2016](#Waldorf16)). In particular an equivalence between spin structures on smooth loop space and [[string structure]] on the underlying space is discussed in ([Waldorf 14](#Waldorf14)).

### As the loop space object of the smooth path groupoid
 {#AsLoopObj}

Given the notion of the smooth [[path groupoid]] $\mathcal{P}_1(X)$ of the smooth space $X$, we may think of the smooth loop _space_ as the corresponding [[loop space object]].

We say this in detail now.

#### Recollection of the situation in $Top$

First briefly recall the situation for [[topological space]]s and their [[loop space object]]s, which are the topological [[loop space]]s:

The ordinary procedure is that we regard a [[topological space]] as an [[object]] in the [[(∞,1)-category]] [[Top]] and produce its [[loop space]] at a chosen [[pointed object|base point]] $x : {*} \to X$ as the [[limit in a quasi-category|(∞,1)-pullback]] of the base point along itself:

$$
  \array{
    \Omega_x X &\to& {*}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{x}}
    \\
    {*} &\stackrel{x}{\to}& X
  }
  \,.
$$

**In words** this is the simple statement that $\Omega_x X$ is the space of all [[homotopy|homotopies]] in the [[(∞,1)-category]] [[Top]] from the map $x : {*} \to X$ to itself. Any one such homotopy is itself a continuous map $\gamma : I \to X$ from the [[interval object|standard interval]] $I = [0,1]$ to $X$, such that restricted to its endpoints it produces the map $x$. Clearly, these are precisely the loops in $X$, based at $x$.

#### Generalization to $\infty$-Lie groupoids

You might think: well, now let $X \in Sh_{(\infty,1)}(Diff)$ be a [[smooth manifold]] regarded as a [[representable functor|representable]] object in the [[(∞,1)-category of (∞,1)-sheaves]] of [[∞-stack]]s on [[Diff]] or maybe better in $Sh_{(\infty,1)}(CartSp)$ on [[CartSp]] -- i.e. of [[Lie ∞-groupoid]]s -- , we play the same trick and compute the [[homotopy pullback]] $Q$

$$
  \array{
    Q &\to& {*}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{x}}
    \\
    {*} &\stackrel{x}{\to}& X
  }
  \;\;\;\;\;\;\;\;\;\;\;\;
  \in Sh_{(\infty,1)}(CartSp)
$$

to obtain the smooth loop space. But it doesn't: the $Q$ here is $Q = {*}$! That's because $X$ regarded as a representable object in $Sh_{(\infty,1)}(Diff)$ is really a [[discrete category|categorically discrete]] [[Lie groupoid]]: it has a smooth space $X$ of objects, but no nontrivial morphisms. And the "homotopies" in the [[homotopy pullback]] are homotopies as seen by the _[[morphisms]]_ in $X$. There is just the identity morphism in $X$ going from ${*} \to X$ to itself, so the homotopy pullback is the point.

Why then did it work in [[Top]]? Because, if you look closely, there really we did something different! Regarded as an [[(∞,1)-category]], [[Top]] is really the collection of [[∞-stack]]s _on the point_ :

$$
  Sing : Top \stackrel{\simeq}{\to} Sh_{(\infty,1)}({*})
  \,.
$$

Under this identification, a topological space is _not_ identified with a [[representable functor|representable]] object! The only representable object in $Sh_{(\infty,1)}({*})$ is of course the [[point]] itself, the [[terminal object]]. Instead, as the notation above already suggests, under this identification a topological space is really identified with its _singular simplicial complex_ $Sing(X)$. But that's really to be thought of as the topological [[fundamental ∞-groupoid]] of $X$. We should write

$$
  \Pi(X) = Sing(X) = Hom_{Top}(\Delta^\bullet_{Top},X) \in SSet
$$

instead of $X$ when we regard the topological space $X$ as an object of the [[(∞,1)-category]] of topological spaces! For more on this, see the discussion at [[homotopy hypothesis]].

If instead we had interpreted the topological space $X$ as a representable object, hence as a [[discrete category|categorically discrete]] object in the $(\infty,1)$-category of [[topological ∞-groupoid]]s, we would have seen the same phenomenon as for the smooth $X$ above: its [[loop space object]] would have been the point.
From this perspective now it is clear how the abstract notion of [[loop space object]] corresponds to the geometrically expected one: for a geometric [[space]] $X$, its loop space is the [[loop space object]] of its [[fundamental ∞-groupoid]].

This statement can be given sense in all contexts where the underlying [[topos]] of our ambient [[(∞,1)-topos]] of [[space]]s is a [[lined topos]]: we need to know which object $R$ is the standard _line_ or [[interval object]]. This determines the _geometric paths_ in a space. Taking these geometric paths to be the [[morphism]]s of a [[fundamental ∞-groupoid]] then makes the geometric paths into "categorical paths", i.e. into morphisms. These then are what the abstract definition of [[loop space object]] can see.

And indeed, whenever the underlying [[topos]] of [[space]]s that we are looking at is a [[lined topos]] the corresponding [[(∞,1)-topos]] comes equipped with a generalization of the topological [[fundamental ∞-groupoid]] construction: we can associate to every [[space]] $X$ its
[[schreiber:path ∞-groupoid]] $\Pi(X)$: the morphisms of $\Pi(X)$ are given by paths in $X$ as seen by the given [[interval object]] $I$. All entirely analogous to the familiar situation for [[Top]], only that now we are testing our generalized [[space]]s over test objects in an arbitrary [[site]] and are using a correspondingly different notion of [[interval object]].

#### The concrete definition

So for $X$ a [[smooth manifold]], regarded as a representable object in the [[(∞,1)-topos]] $Sh_{(\infty,1)}(CartSp)$ of [[Lie ∞-groupoid]]s we have now that the homotopy pullback of any point ${*} \to X \hookrightarrow \Pi(X)$ along itself in the [[schreiber:path ∞-groupoid]] $\Pi(X)$ does indeed produce the expected [[Lie ∞-groupoid]] $\Omega_x X$  in

$$
  \array{
    \Omega_x X &\to& {*}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{x}}
    \\
    {*} &\stackrel{x}{\to}& \Pi(X)
  }
  \;\;\;\;\;\;\;\;\;\;\;\;
  \in Sh_{(\infty,1)}(CartSp)
$$

whose

* smooth space of objects is the smooth space of smooth loops in $X$ based at $x$;

* smooth space of morphisms is the smooth space of smooth $I$-homotopies between smooth loops in $X$

* etc;

Here we want not the full loop [[∞-groupoid]], but just some sort of truncation to a [[0-category|0-groupoid]] just of loops. There are several choices for how exactly to do this, depending on which higher morphisms we just discard, and which we use to identify 1-morphisms. Whatever we do, we end up with some notion of smooth [[path groupoid]] $\mathcal{P}_1(X)$ of $X$, whose 1-morphisms are certain smooth quotient space of the smooth space of 1-morphisms in $\Pi(X)$.

Then, accordingly, forming the [[loop space object]] of this [[path groupoid]] $\mathcal{P}_1(X)$ yields the smooth space $LoopSpace(X)$

$$
  \array{
    LoopSpace_x(X) &\to& {*}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{x}}
    \\
    {*} &\stackrel{x}{\to}& \mathcal{P}_1(X)
  }
$$

which is the smooth subspace of the smooth space of morphisms in $\mathcal{P}_1(X)$ of those morphisms that start and end at $x$.

When unwrapping what all this means, one sees that the object $LoopSpace_x(X) \in Sh_{(\infty,1)}(CartSp)$ that we obtain this way is nothing but the image under the embedding $Sh(CartSp) \hookrightarrow Sh_{(\infty,1)}(CartSp)$ of ordinary [[sheaf|sheaves]] into $\infty$-stacks of some [[quotient object|quotient]] of the [[internal hom]] $[I,X]$ in the [[closed monoidal structure on sheaves]]. Being an internal hom of representables, this is a [[concrete sheaf]] and as such it is precisely the smooth loop space regarded as a [[diffeological space]].

## Related concepts

* [[Dirac operator on smooth loop space]]

* [[Dirac-Ramond operator]], [[Witten genus]]

* [[free loop orbifold]], [[inertia orbifold]]

* [[loop space]], [[free loop space]]

* [[stringor bundle]]

## References

A general standard reference on generalized smooth spaces is

* {#km} Kriegl and Michor, _A Convenient Setting of Global Analysis_

The structure of loop spaces as Fréchet manifolds is covered in chapter 42 of [KM](#km) and in various other articles, many of which cover extensions of the basic construction to other mapping spaces.  In particular,

* Michor, _Manifolds of differentiable mappings_ [MR583436 (83g:58009)](http://www.ams.org/mathscinet-getitem?mr=583436)

* Michor, _A convenient setting for differential geometry and global analysis (I and II)_ [MR764972 (86g:58014a)](http://www.ams.org/mathscinet-getitem?mr=764972)

* [[Andrew Stacey]], _The Differential Topology of Loop Spaces_ [main page](http://www.math.ntnu.no/~stacey/Seminars/diffloop.html) (Note: this was designed as an "easy reader" version of [KM](#km))

* [[Andrew Stacey]], _Constructing Smooth Manifolds of Loop Spaces_ [main page](http://www.math.ntnu.no/~stacey/Research/Papers/smooth.html)

The relation of this [[Fréchet manifold]]  structure to the canonical [[diffeological space]] structure is discussed in

* {#Waldorf} [[Konrad Waldorf]], _Transgression to Loop Spaces and its Inverse I_ ([arXiv:0911.3212](http://arxiv.org/abs/0911.3212))

Discussion of [[G-structures]] on smooth loop spaces is in the following articles.

For [[orientation]] structure on loop space and its [[transgression]] from [[spin structure]] on target space (cf. *[[stringor bundle]]*):

* {#Waldorf11} [[Konrad Waldorf]], *A Loop Space Formulation for Geometric Lifting Problems*, Journal of the Australian Mathematical Society **90** 1 (2011) 129-144  [[arXiv:1007.5373](http://arxiv.org/abs/1007.5373), [doi:10.1017/S1446788711001182 ](https://doi.org/10.1017/S1446788711001182 )]

For [[spin structure]] on loop spaces and its [[transgression]] from [[string structure]] on target space:

* {#Stolz} [[Stephan Stolz]], [[Peter Teichner]], _The spinor bundle on loop space_ (2005) [[pdf](http://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/MPI.pdf), [[Stolz-Teichner-SpinorOnLoopSpace.pdf:file]]]

* {#Waldorf16} [[Konrad Waldorf]], _Spin structures on loop spaces that characterize string manifolds_, Algebr. Geom. Topol. **16** (2016) 675-709 [[arXiv:1209.1731](http://arxiv.org/abs/1209.1731), [doi:10.2140/agt.2016.16.675](https://doi.org/10.2140/agt.2016.16.675)]

* {#Waldorf14} [[Konrad Waldorf]], _String geometry vs. spin geometry on loop spaces_ ([arXiv:1403.5656](http://arxiv.org/abs/1403.5656))

See also:

* {#SperaWurzbacher07} Mauro Spera, Tilman Wurzbacher, *Twistor spaces and spinors over loop spaces*, Math. Ann. **338** (2007) 801–843 [[doi:10.1007/s00208-007-0085-3](https://doi.org/10.1007/s00208-007-0085-3)]

* Alessandra Capotosti, *[[From String structures to Spin structures on loop spaces]]*, Ph. D. thesis, Università  degli Studi Roma Tre, Rome (2016)

There are also sketchy notes in

* {#Melrose} [[Richard Melrose]], _Analysis on loop spaces_, Lecture notes, 2013 ([pdf](http://math.mit.edu/~rbm/18.158-S13/18-158-S13.pdf))

For the motivation of much of this via the [[Dirac-Ramond operator]] and the [[Witten genus]] see the references there.

This entry was created in parallel with [this MO thread](http://mathoverflow.net/questions/12652/loop-spaces-as-generalized-smooth-spaces-or-as-infinite-dimensional-manifolds) from which parts of it is taken.

[[!redirects smooth loop space]]
[[!redirects smooth loop spaces]]


smooth Lorentzian space

> **Under Construction**

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## Definition

### Lorentzian manifold

A _Lorentzian manifold_ $(X, \eta)$ of dimension $(d+1)$ is a [[smooth manifold]] equipped with a [[pseudo-Riemannian metric]] $\eta$ of signature $[+--\cdots -]$ (but note that the complementary choice $[-++\cdots +]$ is also used in the literature). This equips the [[tangent bundle|tangent space]] $T_x X$ at every point $x \in X$ canonically with the structure of a $(d+1)$-dimensional [[Minkowski space]].  Accordingly, tangent vectors $v \in T_x X$ of $X$ are called _[[timelike]]_ , _[[lightlike]]_ or _[[spacelike]]_ , if their norm-square $\mu_x(v,v)$ is positive, zero or negative, respectively.  See also at _[[causal structure]]_.

### Causal structure

A **time orientation** on a Lorentzian manifold $X$ is a smooth (or, depending on the author, continuous) [[vector field]] $\nu \in \Gamma(T X)$ such that at all points $x \in X$ the vector $\nu_x$ is timelike. In general, a Lorentzian manifold does not have globally defined timelike continuous vector fields. Sometimes only Lorentzian manifolds admitting a time orientation are also called [[spacetime]]s.

Given a time orientation $\nu$, a vector $v \in T_x X$ is **future directed** if it is timelike or light-like and its inner product with the time orientation vector at that point is positive, $\mu_x(\nu,v_x) \gt 0$.

Since $\nu$ itself is smooth, it follows that it is future directed with respect to itself at every point.

A smooth curve in $X$, i.e. a smooth map $\gamma : [0,1] \to X$ is called a **timelike curve** or a **lightlike curve** or a **spacelike curve** or a **future-directed curve** precisely if all of its tangent vectors $(\gamma_* \partial_s) \in T_{\gamma(s)} X$ are.

We say that a point $y \in X$ **lies in the future** of a point $x \in X$ if $y = x$ or if there exists a future-directed curve $\gamma : [0,1] \to X$ with $\gamma(0) = x$ and $\gamma(1) = y$. Equivalently, in this case $x$ **lies in the past** of $y$.

#### Notions of causality

We say that $(X,\mu)$ has **closed timelike curves** (closed future-directed curves) if there exists a non-constant timelike (future-dierected) curve starting and ending at some point $x$. Spacetimes which do not contain closed timelike curves are called **chronological**, spacetimes which do not contain closed future directed (i.e. non-spacelike) curves are called **causal**.

+-- {: .un_def}
###### Definition
Given a time orientated spacetime L, the **chronological future** $I^+(p)$ of a point $p \in L$ is the set of events that can be reached by a future directed timelike curve starting from p:
$$
I^+(p) := \{ q \in L | \text{there exists a future directed timelike curve} \lambda(t) \text{with} \lambda(0) = p and \lambda(1)=q \}
$$

The **causal future** $J^+(p)$ of $p$ is defined in the same way with future directed timelike replaced by future directed causal aka non-spacelike.
=--

+-- {: .un_def}
###### Definition
A subset S of a time orientated spacetime L is said to be **achronal** if no two points in S can be connected by a future directed timelike curve, i.e. for all $p, q \in L$ we have $q \notin I^+(p)$.
=--

+-- {: .un_theorem}
###### Theorem
**boundary of chronological future**
Let L be a time orientated spacetime and $S \subset L$. Then the boundary of $I^+(S)$ is either empty or an achronal, three-dimensional, embedded, $C^0$-submanifold of L.
=--
This is theorem 8.1.3 in the book of Wald.

Examples of non-chronological Lorentzian manifolds are the [[anti de Sitter space]] and the [[Kerr spacetime]].

While the former is more of a theoretical interest due to the maximality of the symmetry group, the latter is usually seen as a solution with relevance to actual _physics_, despite the fact that causality does not hold everywhere.

Note that the property of being chronological is not strong enough to enforce causality as understood in everyday life: Even if there are no _closed_ future-directed curves, there still may be e.g. nonclosed _ergodic_ future-directed curves (they come close to every point they already passed in the "past"). An often used stronger condition that models the everyday notion of causality is that the manifold has to be [[globally hyperbolic manifold|globally hyperbolic]] ([Wikipedia](http://en.wikipedia.org/wiki/Globally_hyperbolic)), which, as already mentioned, excludes certain solutions modelling e.g. black holes.

#### Being causal means being a poset

Precisely if the Lorentzian space is causal in that there are _no_ closed future-directed curves is the [[relation]]

* $(x \leq y) \Leftrightarrow$ "$y$ is in the future of $x$"

a [[poset]], hence a [[category]] with at most a single morphism between any two objects:

The [[object]]s of this category are the points of $X$. A [[morphism]] $x \to y$ is a pair of points $x \leq y$ with $y$ in the future of $x$. Composition of morphisms is transitivity of the relation. The identity morphism on $x$ is the reflexivity $x \leq x$.

The _anti-symmetry_ $(x \leq y \leq x) \Rightarrow (x = y)$ is precisely the absence of closed future-directed curves in $X$.

Conversely, from just knowing $X$ as a smooth manifold and knowing this [[poset]] structure on $X$, one can reconstruct the **light cone structure** of $(X,\mu)$, i.e. the information about which tangent vectors are timelike, lightlike, etc.

One can see

> (...reference...)

that the pseudo-Riemannian metric $\mu$ may be reconstructed from the lightcone structure and the [[volume density]] that it induces. In this sense a Lorentzian manifold without future-directed curves is equivalently a smooth [[poset]] equipped with a smooth [[measure]] on its space of objects.

### Generalized smooth Lorentzian spaces

...

> A **smooth Lorentzian space** is supposed to be like a Lorentzian manifold, but whose underlying space is not necessarily a smooth manifold, but a  [[generalized smooth space]].

> So this is "something like" a [[partial order|poset]] [[internal category|internal]] to a category of measure spaces, or a [[poset]]-valued 2-[[stack]] on something like [[CartSp]] or the like.

...

## Construction on a smooth Lorentzian space

### Causal subsets

Let $(X,\mu,\nu)$ be a time-oriented Lorentzian space regarded as a smooth [[category]] that is a [[poset]].

A **causal subset** of a $X$ is one of its  [[under-over category|under-over categories]] $x \downarrow X \downarrow y$ for a pair $x,y \in X$ of points in $X$.

Its objects form the collection of all points $z \in X$ that are both in the future of $x$ as well as in the past of $y$.

### The path $(\infty,1)$-category of a Lorentzian space {#PathnCategory}

To every [[Lie ∞-groupoid]] $X$ is associated its [[schreiber:path ∞-groupoid]] $\mathbf{\Pi}(X)$. But more generally, to a smooth [[(∞,1)-category]] is associated a **path $(\infty,1)$-category**. See [[fundamental (infinity,1)-category]].

A causal Lorentzian manifold may naturally be regarded as a smooth category (a smooth [[poset]]) and as such has a path [[(n,r)-category|(2,1)]]-category.
Its invertible [[morphism]]s are smooth spacelike curves, and its non-invertible morphisms contain future-directed paths. This $(2,1)$-category plays the role of the [[path groupoid]] of a plain manifold and is akin to the path 2-groupoid of paths in an [[orbifold]], only that where the latter has all morphisms invertible, crucially in the path 2-groupoid of a Lorentzian space, there are non-invertible morphisms, reflecting the causal structure of that space.

To put this construction into context, we therefore first recall the story for paths in an orbifold.

#### Prelude: the path 2-groupoid of an orbifold

To an ordinary [[smooth manifold]] or [[generalized smooth space]] $X$ is associated its [[fundamental groupoid]] $\Pi_1(X)$ and its smooth [[path groupoid]] $P_1(X)$: [[categories]] whose [[object]]s are the points of $X$ and whose [[morphism]]s are certain equivalence classes of smooth paths between these objects.

This construction generalizes from paths in plain spaces, to paths in spaces that are themselves smooth groupoids: notably to [[orbifold]]s $X$.

Given an orbifold $X$ with space of objects $X_0$ and space of morphisms $X_1$, paths in it form a smooth [[2-groupoid]] $P_1(X)$ which looks as follows:

* objects of $P_1(X)$ are the points of $X_0$;

* the morphisms of $P_1(X)$ are formal composites of two types of morphisms

  1. the smooth paths $X_0$, i.e. the morphisms of $P_1(X_0)$;

     $\gamma : x \to y$

  1. the original morphisms of $X$, i.e. the elements of $X_1$. Since the orbifold is locally given by a [[group]] $G$, we may think of these morphisms as being of the form $x \to g\cdot x$, connecting a point $x \in X_0$ to the point $g\cdot x$ that it is [[isomorphism|isomorphic]] to under the orbifold action.

* the [[2-morphism]]s of $P_1(X)$ are paths in $X_1$, i.e. morphisms of $P_1(X_1)$. These we may picture as

  $$
    \array{
      x &\stackrel{\gamma}{\to}& y
      \\
      \downarrow &\swArrow& \downarrow
      \\
      g\cdot x &\underset{g\cdot \gamma}{\to}& g \cdot y
    }
    \,.
  $$

  This is a path $(\gamma, g\cdot \gamma)$ of pairs of points that are related under the orbifold action.

The path 2-groups $P_1(X)$ of the orbifold encodes the correct notion of trajectories in the orbifold: such a trajectory may proceed along smooth paths, and intermittently it may jump between the "orbifold sectors". Notably an [[automorphism]] in $P_1(X)$ on a point $x$ may be given by a smooth path $x \to g x$ that does not come back to $x$ but just to one of its mirror-images, composed with the jump-morphism $g x \to x$ back to $x$. Sometimes (notably in [[string theory]]) such loops are called _twisted sectors_ of loop configurations.

A detailed description of the smooth [[2-groupoid]] of paths in a smooth 2-groupoid may be found in [section 2.1](http://arxiv.org/PS_cache/arxiv/pdf/0808/0808.1923v1.pdf#page=19) of

* U.S., [[Konrad Waldorf]], _Connections on nonabelian gerbes and their holonomy_ ([arXiv:0808.1923](http://arxiv.org/abs/0808.1923))

There the groupoid $X$ is taken to be a [[Cech nerve|Cech groupoid]], but the general mechanism of the construction does not depend on this. A fully general description of paths in (higher) smooth groupoids is also at [[schreiber:path ∞-groupoid]].

It is immediate how this construction generalizes when the smooth groupoid $X$ is replaced by a smooth category. This we turn to now.

#### The path $(2,1)$-category of a Lorentzian space

Now we discuss the same as above, where now $X = (X_1 \stackrel{\to}{\to} X_0)$ is not a smooth groupoid, but a smooth category, notably the smooth [[poset]] determined by a smooth Lorentzian space.

So let $X$ be a smooth causal Lorentzian manifold, regarded as a [[poset]]. So $X_0$ is the underlying manifold and $X_1 \subset X_0 \times X_0$ is the collection of pairs of points with one in the future of the other.

We equip this with the structure of a [[internal category|category internal]] to [[diffeological space]]s, hence with the structure of a category-valued [[presheaf]] on the [[site]] [[CartSp]], by declaring that a plot $\phi : U \to X_0$  of the space of objects is a _spacelike_ smooth map $U \to X_0$: the push-forward along $\phi$ of every tangent vector of $U \in CartSp$ yields a spacelike vector in $X_0$.

Analogously, we declare a plot $\phi  : U \to X_1$ to be a pair of plots into $X_0$ such that pointwise this assigns a point and one point in its future.

From now on, by abuse of notation, by $X$ we shall mean this category internal to diffeological spaces, regarded as a category-valued presheaf on [[CartSp]].

Then the path $(2,1)$-category $P_1(X)$ is defined as follows:

* its space of objects is again the diffeological space $X_0$;

* the elements of its space of morphisms are generated from

  * morphisms $\gamma : x \to y$
    given by reparameterization or thin-hoimotopy
    classes of smooth spacelike curves $\gamma : [0,1] \to X_0$;

  * morphisms of the form $x \to y$ for every $x \leq y$ in the causal structure of $X$.

  There is an evident diffeology on this space (a quotient of a disjoint union of product diffeologies). This defines the diffeological space $(P_1(X))$.

* the elements of its space of 2-morphisms are generated from 2-morphisms

  $$
    \array{
      & \nearrow \searrow^{\mathrlap{[\gamma_1]}}
      \\
      x_1  && y_1
      \\
      \downarrow &\swArrow& \downarrow
      \\
      x_2  && y_2
      \\
      & \searrow \nearrow_{\mathrlap{[\gamma_2]}}
    }
    \;\;\;\;
    \forall t \in [0,1] : \gamma_1(t) \leq \gamma_2(t)
  $$

  given from classes of smooth paths in $X_1$, i.e. from classes of paths of pairs of points, one in the future of the other.

  There is an evident diffeology and evident composition operations on this. Notice that the generating 2-cells are 2-isomorphisms, but that their source and target morphisms are not generally invertible.

## Related concepts

* [[pseudo-Riemannian manifold]]

  * [[Lorentzian manifold]]

    * [[Lorentzian geometry]]

    * [[spacetime]]

  * [[globally hyperbolic Lorentzian manifold]]

* [[conal manifold]]

* [[Lorentzian orbifold]]

## References

Named after [[Hendrik Lorentz]].

A classic reference for general relativity is

* [[Robert Wald]], _General Relativity_

A textbook dedicated to the classical [[differential geometry|diffential geometric]] aspects Lorentzian manifolds is

* John K. Beem; Paul E. Ehrlich, ; Kevin L. Easley, _Global Lorentzian geometry_ ([ZMATH entry](http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0846.53001&format=complete))

A classical influential text on the nature of Lorentzian space is

* [[Roger Penrose]], [[Wolfgang Rindler]], _Spinors and space time_, in 2 vols. Cambridge Univ. Press 1984/1988.

On Lorentz structure as [[G-structure]]:

* Leandro A. Lichtenfelz, [[Paolo Piccione]], Abdelghani Zeghib, _On the Isometry Group of Lorentz Manifolds_, in: Miguel Sanchez, Miguel Ortega, Alfonso Romero (eds.) _Recent Trends in Lorentzian
Geometry_, Springer 2013 ([doi:10.1007/978-1-4614-4897-6_12](https://doi.org/10.1007/978-1-4614-4897-6_12))

The relation between causality and [[poset]]-structure (see also [[causal set]]) is reviewed for instance in

* Ettore Minguzzi, _Time and Causality in General Relativity_ , talk notes, Ponta Delgada, July 2009 ([pdf](http://fqxi.org/data/documents/Minguzzi%20%20Azores%20Talk.pdf))

More details are discussed in the context of [[domain theory]], see for instance

* [[Keye Martin]] and [[Prakash Panangaden]],
Domain Theory and the Causal Structure of Space-Time

Some vaguely related blog discussion is at

* The Nature of Time

* Canonical Measures on Configuration Spaces

[[!redirects smooth Lorentzian space]]
[[!redirects smooth Lorentzian spaces]]

[[!redirects Lorentzian manifold]]
[[!redirects Lorentzian manifolds]]
[[!redirects smooth Lorentzian manifold]]
[[!redirects smooth Lorentzian manifolds]]

[[!redirects Lorentzian space]]
[[!redirects Lorentzian spaces]]

[[!redirects Lorentzian spacetime]]
[[!redirects Lorentzian spacetimes]]

[[!redirects Lorentzian structure]]
[[!redirects Lorentzian structures]]


smooth manifold

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## Idea

A [[differentiable manifold]] is a [[topological space]] which is _locally_ [[homeomorphism|homeomorphic]] to a [[Euclidean space]] (a [[topological manifold]]) and such that the [[gluing functions]] which relate these Euclidean [[local charts]] to each other are [[differentiable functions]], for a fixed degree of differentiability. If one considers arbitrary differentiablity, then one speaks of _smooth manifolds_. For a general discussion see at _[[manifold]]_.

Differential and smooth manifolds are the basis for much of [[differential geometry]]. They are the analogs in differential geometry of what [[schemes]] are in [[algebraic geometry]].

If one relaxes the condition from being locally isomorphic to a Euclidean space to admitting local smooth maps from a Euclidean space, then one obtains the concept of [[diffeological spaces]] or even [[smooth sets]], see at _[[generalized smooth space]]_ for more on this.

The generalization of differentiable manifolds to [[higher differential geometry]] are [[orbifolds]] and more generally [[differentiable stacks]]. If one combines this with the generalization to [[smooth sets]] then one obtains the concept of [[smooth stacks]] and eventually [[smooth infinity-stacks]].

Smooth manifolds form a category, [[SmoothManifolds]].

## Definition

### Traditional definition

For the traditional definition see at _[[differentiable manifold]]_.

### Patching as idempotent splitting

In an exercise of his 1973 Perugia lectures [[F. William Lawvere]] reported a somewhat surprising observation:

In the case of [[smooth manifolds]] the process of piecing together the local data can be elegantly summed up as [[Karoubi envelope|splitting of idempotents]] in a category of open subsets of Euclidean spaces. More precisely:

Let $Diff$ be the category of [[smooth manifolds]] and [[smooth maps]], where by a "smooth manifold", we mean a finite-dimensional, second-countable, Hausdorff, $C^\infty$ [[manifold]] without boundary.  Let $i: Open \hookrightarrow Diff$ be the [[full subcategory]] whose objects are the [[open subspaces]] of finite-dimensional [[Cartesian spaces]].

+-- {: .num_theorem}
###### Theorem
The subcategory $i: Open \hookrightarrow Diff$ exhibits $Diff$ as an idempotent-splitting completion of $Open$.
=--

+-- {: .proof}
###### Proof
By a general [[Karoubi envelope#def2|lemma for idempotent splittings]], it suffices to prove that

* Every smooth manifold is a smooth retract of an open set in Euclidean space;

* If $p : U \to U$ is a smooth idempotent on an open set $U \subseteq \mathbb{R}^n$, then the subset $Fix(p) \hookrightarrow U$ is an [[embedding of smooth manifolds|embedded]] submanifold.

For the first statement, we use the fact that any manifold $M$ can be realized as a closed submanifold of some $\mathbb{R}^n$, and every closed submanifold has a [[tubular neighborhood theorem|tubular neighborhood]] $U \subseteq \mathbb{R}^n$. In this case $U$ carries a structure of vector bundle over $M$ in such a way that the inclusion $M \hookrightarrow U$ is identified with the zero section, so that the bundle projection $U \to M$ provides a retraction, with right inverse given by the zero section.

For the second statement, assume that the origin $0$ is a fixed point of $p$, and let $T_0(U) \cong \mathbb{R}^n$ be its tangent space (observe the presence of a _canonical_ isomorphism to $\mathbb{R}^n$). Thus we have idempotent linear maps $d p(0), Id-d p(0): T_0(U) \to T_0(U)$ where the latter factors through the inclusion $\ker \; d p(0) \hookrightarrow T_0(U)$ via a projection map $\pi: T_0(U) \to \ker \; d p(0)$. We have a map $f: U \to \mathbb{R}^n$ that takes $x \in U$ to $x - p(x)$; let $g$ denote the composite

$$U \stackrel{f}{\longrightarrow} \mathbb{R}^n \cong T_0(U) \stackrel{\pi}{\longrightarrow} \ker\; d p(0).$$

Now we make some easy observations:

1. $Fix(p) \subseteq g^{-1}(0)$.

1. The map $p: U \to U$ restricts to a map $p: g^{-1}(0) \to g^{-1}(0)$, by idempotence of $p$.

1. The derivative $d g(0): T_0(U) \to T_0(\ker \; d p(0)) \cong \ker \; d p(0)$ is $\pi$ again since $Id - d p(0)$ is idempotent. Thus $d g(0)$ has full rank ($m$ say), and so the restriction of $g$ to some neighborhood $V$ has $0$ as a regular value, and $g^{-1}(0) \cap V$ is a manifold of dimension $m$ by the [[implicit function theorem]]. The tangent space $T_0(g^{-1}(0) \cap V)$ is canonically identified with $im(d p(0))$.

1. There are smaller neighborhoods $V'' \subseteq V' \subseteq V$ so that $p$ restricts to maps $p_1, p_2$ as in the following diagram ($i, i', i''$ are inclusion maps, all taking a domain element $x$ to itself):
$$\array{
g^{-1}(0) \cap V'' & \stackrel{i''}{\hookrightarrow} & g^{-1}(0) \\
 _\mathllap{p_2} \downarrow & & \downarrow _\mathrlap{p} \\
g^{-1}(0) \cap V' & \stackrel{i'}{\hookrightarrow} & g^{-1}(0) \\
 _\mathllap{p_1} \downarrow & & \downarrow _\mathrlap{p} \\
g^{-1}(0) \cap V & \stackrel{i}{\hookrightarrow} & g^{-1}(0)
}$$
and such that $p_1, p_2$ are diffeomorphisms by the [[implicit function theorem|inverse function theorem]] (noting here that $d p_i(0): im(d p(0)) \to im(d p(0))$ is the identity map, by idempotence of $p$).

1. Letting $q: g^{-1}(0) \cap V' \to g^{-1}(0) \cap V''$ denote the smooth inverse to $p_2$, we calculate $i' = p \circ i'' \circ q$, and
$$i p_1 = p i' = p p i''q = p i'' q = i',$$
so that $p_1(x) = x$ for every $x \in g^{-1}(0) \cap V'$. Hence $g^{-1}(0) \cap V' \subseteq Fix(p)$.

From all this it follows that $Fix(p) \cap V' = g^{-1}(0) \cap V'$, meaning $Fix(p)$ is locally diffeomorphic to $\mathbb{R}^m$, and so $Fix(p)$ is an embedded submanifold of $\mathbb{R}^n$.
=--

+-- {: .num_remark}
###### Remark
Another proof of this result may be found [here](http://www.mat.univie.ac.at/~michor/dgbook.pdf).
=--

[[F. William Lawvere|Lawvere]] comments on this fact as follows:

> {#LawvereComment} "This powerful theorem justifies bypassing the complicated considerations of charts, coordinate transformations, and atlases commonly offered as a "basic" definition of the concept of manifold. For example the 2-sphere, a manifold but not an open set of any Euclidean space, may be fully specified with its smooth structure by considering any open set $A$ in 3-space $E$ which contains it but not its center (taken to be $0$) and the smooth idempotent endomap of $A$ given by $e(x) = x/{|x|}$. All general constructions (i.e., functors into categories which are Cauchy complete) on manifolds now follow easily (without any need to check whether they are compatible with coverings, etc.) provided they are known on the opens of Euclidean spaces: for example, the tangent bundle on the sphere is obtained by splitting the idempotent $e'$ on the tangent bundle $A \times V$ of $A$ ($V$ being the vector space of translations of $E$) which is obtained by differentiating $e$. The same for cohomology groups, etc."  ([Lawvere 1989](#Law89), p.267)

### General abstract geometric definition
 {#GeneralAbstractCharacterization}

There is a fundamental and [[category theory|general abstract]] way to think of smooth manifolds, which realizes their theory as a special case of general constructions in [[higher geometry]].

In this context one specifies for instance $\mathcal{G}$ a [[geometry (for structured (∞,1)-toposes)]] and then plenty of geometric notions are defined canonically in terms of $\mathcal{G}$. The theory of smooth manifolds appears if one takes $\mathcal{G} = $ [[CartSp]].

Alternatively one can specify [[differential cohesion]] and proceed as discussed at _[differential cohesion -- structures - Cohesive manifolds (separated)](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#CohesivemanifoldsSeparated)_.

This is discussed in [The geometry CartSp](#TheGeometryCartSp) below.

#### The geometry $CartSp$ {#TheGeometryCartSp}

Let [[CartSp]] be the [[category]] of [[Cartesian space | Cartesian spaces]] and [[smooth function | smooth functions]] between them. This has finite [[product | products]] and is in fact (the [[syntactic category]]) of a [[Lawvere theory]]: the theory of [[smooth algebras]].

Moreover, $CartSp$ is naturally equipped with the [[good open cover]] [[coverage]] that makes it a [[site]].

Both properties together make it a [[pregeometry (for structured (∞,1)-toposes)]] (if the notion of [[Grothendieck topology]] is relaxed to that of [[coverage]] in [StrSp](#StrSp)).

For $\mathcal{X}$ a [[topos]], a [[product]]-preserving [[functor]]

$$
  \mathcal{O} : \mathcal{G} \to \mathcal{X}
$$

is a $\mathcal{G}$-[[algebra over an algebraic theory|algebra]] in $\mathcal{X}$. This makes $\mathcal{X}$ is $\mathcal{G}$-[[ringed topos]]. For $\mathcal{G} = $ [[CartSp]] this algebra is a [[smooth algebra]] in $\mathcal{X}$. If $\mathcal{X}$ has a [[site]] of definition $X$, then this is a [[sheaf]] of [[smooth algebra | smooth algebras]] on $X$.

If $\mathcal{O}$ sends [[covering]] families $\{U_i \to U\}$ in $\mathcal{G}$ to [[effective epimorphism]] $\coprod_i \mathcal{O}(U_i) \to \mathcal{O}(U)$ we say that it is a _local $\mathcal{G}$-algebra_ in $\mathcal{X}$, making $\mathcal{X}$ a $\mathcal{G}$-[[locally ringed topos]].

The [[big topos]] $Sh(\mathcal{G})$ itself is canonically equipped with such a local $\mathcal{G}$-algebra, given by the [[Yoneda embedding]] $j$ followed by [[sheafification]] $L$

$$
  \mathcal{O} : \mathcal{G} \stackrel{j}{\to} PSh(\mathcal{G}) \stackrel{L}{\to}
  Sh(\mathcal{G})
  \,.
$$

It is important in the context of [[locally representable structured (infinity,1)-topos|locally representable locally ringed toposes]] that we regard $Sh(\mathcal{G})$ as equipped with this local $\mathcal{G}$-algebra. This is what remembers the [[site]] and gives a notion of local representability in the first place.

The [[big topos]] $Sh(CartSp)$ is a [[cohesive topos]] of [[generalized smooth space | generalized smooth spaces]]. Its [[concrete sheaves]] are precisely the [[diffeological space | diffeological spaces]]. See there for more details. We now discuss how with $Sh(CartSp)$ regarded as a $CartSp$-structured topos, smooth manifolds are precisely its [[locally representable structured (infinity,1)-toposes|locally representable objects]].

#### Cartesian spaces as representable objects of $Sh(CartSp)$

The representables themselves should evidently be locally representable and canonically have the structure of $CartSp$-structured toposes.

Indeed, every object $U \in \mathrm{CartSp}$ is canonically a [[CartSp]]-[[ringed space]], meaning a [[topological space]] equipped with a local sheaf of [[smooth algebra | smooth algebras]]. More generally: every object $U \in CartSp$ is canonically incarnated as the $CartSp$-[[structured (∞,1)-topos]]

$$
  (\mathcal{X}, \mathcal{O}_{\mathcal{X}})
  :=
  (Sh_{(\infty,1)}(CartSp)/U ,
    \;\;\; \mathcal{O}_U : CartSp \stackrel{j}{\to} Sh_{(\infty,1)}(CartSp) \stackrel{U^*}{\to} Sh_{(\infty,1)}(CartSp)/U)
$$

given by the [[over-(∞,1)-topos]] of the [[big topos|big]] [[(∞,1)-sheaf (∞,1)-topos]] over $CartSp$ and the [[structure sheaf]] given by the composite of the [[(∞,1)-Yoneda embedding]] and the [[inverse image]] of the [[etale geometric morphism]] induced by $U$.

#### Smooth manifolds as locally representable objects of $Sh(CartSp)$ {#smoothManifoldsAsLocallyRepresentableObjects}

+-- {: .num_defn}
###### Definition

Say a [[concrete sheaf|concrete object]] $X$ in the [[sheaf topos]] $Sh(CartSp)$ -- a [[diffeological space]] -- is _locally representable_ if there exists a family of open embeddings $\{U_i \hookrightarrow X\}_{i \in X}$ with $U_i \in CartSp \stackrel{j}{\hookrightarrow} Sh(CartSp)$ such that the canonical morphism out of the [[coproduct]]

$$
  \coprod_i U_i \to X
$$

is an [[effective epimorphism]] in $Sh(CartSp)$.

Let $LocRep(CartSp) \hookrightarrow Sh(CartSp)$ be the full [[subcategory]] on locally representable sheaves.

=--

+-- {: .num_prop}
###### Proposition

There is an [[equivalence of categories]]

$$
  Diff \simeq LocRep(CartSp)
$$

of the category [[Diff]] of smooth manifolds with that of locally representable sheaves for the [[geometry (for structured (infinity,1)-toposes)|pre-geometry]] $CartSp$.

=--

+-- {: .proof}
###### Proof

Define a functor $Diff \to LocRep(CartSp)$ by sending each smooth manifold to the sheaf over $CartSp$ that it naturally represents. By definition of [[manifold]] there is an [[open cover]] $\{U_i \hookrightarrow X\}$. We claim that $\coprod_i U_i \to X$ is an [[effective epimorphism]], so that this functor indeed lands in $LocRep(CartSp)$. (This is a standard argument of sheaf theory in [[Diff]], we really only need to observe that it goes through over [[CartSp]], too.)

For that we need to show that

$$
  \coprod_{i, j} U_i \times_X U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_i U_i \to X
$$

is a [[coequalizer]] diagram in $Sh(CartSp)$ (that the [[Cech groupoid]] of the cover is equivalent to $X$.). Notice that the [[fiber product]] here is just the intersection in $X$ $U_i \times_X U_j \simeq U_i \cap U_j$. By the fact that the [[sheaf topos]] $Sh(CartSp)$ is by definition a [[reflective subcategory]] of the [[presheaf topos]] $PSh(CartSp)$ we have that [[colimit | colimits]] in $Sh(CartSp)$ are computed as the [[sheafification]] of the corresponding colimit in $PSh(CartSp)$. The colimit in $PSh(CartSp)$ in turn is computed objectwise. Using this, we see that that we have a coequalizer diagram

$$
  \coprod_{i, j} U_i \times_X U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_i U_i \to S(\{U_i\})
$$

in $PSh(CartSp)$, where $S(\{U_i\})$ is the [[sieve]] corresponding to the cover: the [[subfunctor]] $S(\{U_i\}) \hookrightarrow X$ of the functor $X : CartSp^{op} \to Set $ which assigns to $V \in CartSp$ the set of [[smooth function | smooth functions]] $V \to X$ that have the property that they factor through any one of the $U_i$.

Essentially by the definition of the [[coverage]] on $CartSp$, it follows that [[sheafification]] takes this subfunctor inclusion to an [[isomorphism]]. This shows that $X$ is indeed the tip of the coequalizer in $Sh(CartSp)$ as above, and hence that it is a locally representable sheaf.

Conversely, suppose that for $X \in Conc(Sh(CartSp)) \hookrightarrow Sh(CartSp)$ there is a family of open embeddings $\{U_i \hookrightarrow X\}$ such that we have a coequalizer diagram

$$
  \coprod_{i, j} U_i \times_X U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} U_i \to X
$$

in $Sh(CartSp)$, which is the sheafification of the corresponding coequalizer in $PSh(CartSp)$. By evaluating this on the point, we find that the underlying set of $X$ is the coequalizer of the underlying set of the $U_i$ in $Set$. Since every plot of $X$ factors locally through one of the $U_i$ it follows that $X$ is a [[diffeological space]].

It follows that in the pullback diagrams

$$
  \array{
    U_i \times_X U_j &\to& U_j
   \\
    \downarrow && \downarrow
   \\
   U_i &\to& X
  }
$$

the object $U_i \cap U_j$ is the diffeological space whose underlying topological space is the intersection of $U_i$ and $U_j$ in the topological space underlying $X$. In particular the inclusions $U_i \times_X U_j \hookrightarrow U_i$ are open embeddings.

=--

#### As locally representable $CartSp$-structured $(\infty,1)$-toposes

We may switch from regarding smooth manifolds as objects in the [[big topos]] $X \in Sh(CartSp)$ to regrading them as [[topos | toposes]] themselves, by passing to the [[over-topos]] $Sh(CartSp)/X$. This remembers the extra (smooth) structure on the [[topological space]] $X$ by being canonically a [[locally ringed topos]] with the [[structure sheaf]] of [[smooth function | smooth functions]] on $X$: a [[CartSp]]-[[structured (∞,1)-toposes]]

For every choice of [[geometry (for structured (∞,1)-toposes)]]   there is a notion of $\mathcal{G}$-[[locally representable structured (∞,1)-topos]] ([StrSp](#StrSp)).

+-- {: .num_prop}
###### Claim

Smooth manifolds are equivalently the [[n-localic (infinity,1)-topos|0-localic]] [[CartSp]]-[[generalized scheme | generalized schemes]] of locally finite presentation.

=--

+-- {: .proof}
###### Sketch of proof

The statement says that a smooth manifold $X$ may be identified with an [[∞-stack]] on [[CartSp]] (an [[∞-Lie groupoid]]) which is represented by a [[CartSp]]-[[structured (∞,1)-topos]] $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ such that

1. $\mathcal{X}$ is a [[n-localic (∞,1)-topos|0-localic (∞,1)-topos]];

1. There exists a family of objects $\{U_i \in \mathcal{X}\}$ such that the canonical morphism $\coprod_i U_i \to *_{\mathcal{X}}$ to the [[terminal object in an (∞,1)-category|terminal object]] in $\mathcal{X}$ is a [[regular epimorphism]];

1. For every $i \in I$ there is an equivalence

  $$
    (\mathcal{X}/U_i, \mathcal{O}_{\mathcal{X}|U_i})
    \underoverset{\simeq}{t_i}{\to} (Sh_{(\infty,1)}(\mathbb{R}^n), \mathcal{O}(\mathbb{R}^n))
     \,.
  $$

The second and third condition say in words that $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is locally equivalent to the ordinary cannonically [[CartSp]]-[[locally ringed space]] $\mathbb{R}^n$ (for $n \in \mathbb{N}$ the [[dimension]]. The first condition then says that these local identifications cover $\mathcal{X}$.

(...)

(...)

=--

## Properties

### Faithful embedding into $cAlg^{op}$

+-- {: .num_prop}
###### Proposition
**(Milnor's exercise)**

The [[functor]]

$$
  C^\infty(-) \colon SmthMfd \hookrightarrow cAlg_{\mathbb{R}}^{op}
$$

(from the category of smooth manifolds to the [[opposite category]] of [[commutative algebras]] over the [[real numbers]]) that sends a smooth manifold $X$ to its commutative $\mathbb{R}$-[[associative algebra|algebra]] of [[smooth functions]] $X \to \mathbb{R}$ is a [[fully faithful functor]], hence exhibits $SmthMfd$ as a [[full subcategory]] of $cAlg^{op}$.

=--

([Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10](#KolarSlovakMichor93))

For more see at _[[embedding of smooth manifolds into formal duals of R-algebras]]_.

### Further properties

* [[Borel's theorem]]

* [[Tietze extension theorem]]

* [[Whitney extension theorem]]

* [[Steenrod-Wockel approximation theorem]]

* [[derivations of smooth functions are vector fields]]

## Classes of examples

* [[CR manifold]]

* [[symplectic manifold]]

* etc.

## Related concepts

* [[embedding of smooth manifolds]]

* [[infinite dimensional smooth manifold]]

* [[pro-manifold]]

* [[diffeological space]], [[smooth set]]

* [[orbifold]]

* [[differentiable stack]], [[smooth stack]], [[smooth infinity-stack]]

* [differential cohesion -- structures - Cohesive manifolds (separated)](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#CohesivemanifoldsSeparated)

## References

The vision of the modern notion of [[manifolds]] is attributed to:

* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

Further historical contributions:

* {#VeblenWhitehead1931} Oswald Veblen and [[J.H.C. Whitehead]], _A set of axioms for differential geometry_, Proc. Natl. Acad. Sci. USA 17, 551-561 (1931)

* {#VeblenWhitehead1932} Oswald Veblen and [[J.H.C. Whitehead]], _The foundations of differential geometry_, Camb. Tracts Math. Math. Phys. 29, Cambridge University Press, Cambridge (1932)

* {#Whitney1936} Hassler [[Whitney]], _Differentiable manifolds_, Ann. Math. (2) 37, 645-680 (1936)

Early account with an eye towards [[cobordism theory]] and the [[Pontrjagin-Thom isomorphism]]:

* {#Pontrjagin55} [[Lev Pontrjagin]], _Smooth manifolds and their applications in Homotopy theory_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/pont4.pdf), [doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001))

Textbook accounts:

* [[Georges de Rham]], Chapter I of: *Differentiable Manifolds -- Forms, Currents, Harmonic Forms*, Grundlehren **266**, Springer (1984) [[doi:10.1007/978-3-642-61752-2](https://doi.org/10.1007/978-3-642-61752-2)]

* {#MilnorStasheff74} [[John Milnor]], [[James Stasheff]], Section 1 of: _Characteristic Classes_, Annals of Mathematics Studies, Princeton University Press 1974 ([ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76))

* [[Yvonne Choquet-Bruhat]], [[Cécile DeWitt-Morette]], *Analysis, manifolds and physics*, North Holland (1982, 2001) $[$[ISBN:9780444860170](https://www.elsevier.com/books/analysis-manifolds-and-physics-revised-edition/choquet-bruhat/978-0-444-86017-0)$]$

  > (with an eye towards [[mathematical physics]])

* {#KolarSlovakMichor93} [[Ivan Kolar]], [[Jan Slovak]] and [[Peter Michor]], _Natural operations in differential geometry_, 1993, 1999 ([web](http://www.emis.de/monographs/KSM/))

* {#Kosinski93} [[Antoni Kosinski]], _Differential manifolds_, Academic Press 1993 ([pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/kosinski.pdf))

* {#Lee09}  [[Jeffrey Lee]], _Manifolds and differential geometry_, Graduate Studies in Mathematics **107**  (2009) ([ISBN: 978-0-8218-4815-9](https://bookstore.ams.org/gsm-107), [doi:10.1090/gsm/107](https://doi.org/10.1090/gsm/107))

* {#Lee12} [[John Lee]], *Introduction to Smooth Manifolds*, Springer (2012) [[doi:10.1007/978-1-4419-9982-5](https://doi.org/10.1007/978-1-4419-9982-5), [book webpage](https://sites.math.washington.edu/~lee/Books/ISM/), [pdf](https://math.berkeley.edu/~jchaidez/materials/reu/lee_smooth_manifolds.pdf)]

* [[Theodore Frankel]], chapter 1 of _[[The Geometry of Physics - An Introduction]]_, Cambridge University Press (1997, 2004, 2012)

* Jet Nestruev, _Smooth Manifolds and Observables_ , Springer LNM **220** Heidelberg 2003 ([doi:10.1007/b98871](https://link.springer.com/book/10.1007/b98871))

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 3 of _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), <a href="http://alpha.sinp.msu.ru/~panov/LibBooks/GRAV/(Graduate_Student_Series_in_Physics)Mikio_Nakahara-Geometry,_Topology_and_Physics,_Second_Edition_(Graduate_Student_Series_in_Physics)-Institute_of_Physics_Publishing(2003).pdf">pdf</a>)

Smooth manifolds are defined as locally ringed spaces in

* Juan A. Navarro González, and Juan B. Sancho de Salas, _$C^\infty$-Differentiable Spaces_, Springer LNM **1824** 2003.

Discussion of smooth manifolds as colimits of the Cech nerves of their good open covers is also at

* MathOverflow, _[Gluing of manifolds and the Hausdorff condition](http://mathoverflow.net/questions/65684/gluing-of-manifolds-and-the-hausdorff-condition)_

The general abstract framework of [[higher geometry]] referred to above is discussed in

* {#StrSp} [[Jacob Lurie]], _[[Structured Spaces]]_

The proof that idempotents split in the category of smooth manifolds was adapted from this MO answer:

* Zack (http://mathoverflow.net/users/300/zack), Idempotents split in category of smooth manifolds?, URL (version: 2014-04-06): http://mathoverflow.net/q/162556 ([web](http://mathoverflow.net/a/162556/2926))

Which provides a solution to exercise 3.21 in

* [[William Lawvere]], _Perugia Notes - Theory of Categories over a Base Topos_ , Ms. Università di Perugia 1973.

The [above comment](#LawvereComment) by Lawvere is taken from

* {#Law89}  [[F. William Lawvere]], _Qualitative distinctions between some toposes of generalized graphs_, Contemporary Mathematics 92 (1989), 261-299. ([pdf](http://conceptualmathematics.files.wordpress.com/2013/01/toposesofgeneralizedgraphs.pdf))

[[!redirects smooth manifolds]]


smooth map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[function]] which is [[differentiable function]] to arbitrary order is called a _smooth function_.

### In the real numbers

#### Epsilon-delta definition

Let $\mathbb{R}$ be the [[real numbers]]. A function $f:\mathbb{R} \to \mathbb{R}$ is **smooth** if it comes with a [[sequence]] of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that

* for every real number $x \in \mathbb{R}$, $(D^{0}f)(x) = f(x)$

* for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every real number $x \in \mathbb{R}$,
$$|(D^{n}f)(x + h) - (D^{n}f)(x) - h (D^{n + 1}f)(x)| \lt \epsilon |h|$$

Unwrapping the recursive definition above, a function $f:\mathbb{R} \to \mathbb{R}$ is **smooth** if it comes with a [[sequence]] of functions $D^{(-)}f:\mathbb{N} \to (\mathbb{R} \to \mathbb{R})$ and a sequence of functions $M^{(-)}f:\mathbb{N} \to (\mathbb{Q}_+ \to \mathbb{Q}_+)$ in the positive rational numbers, such that

* for every real number $x \in \mathbb{R}$, $(D^{0}f)(x) = f(x)$

* for every natural number $n \in \mathbb{N}$, for every positive rational number $\epsilon \in \mathbb{Q}_+$, for every real number $h \in \mathbb{R}$ such that $0 \lt | h | \lt M^{n}f(\epsilon)$, and for every real number $x \in \mathbb{R}$,
$$\left|f(x + h) - \sum_{i=0}^n \frac{h^i (D^{i}f)(x)}{i!}\right| \lt \epsilon |h^n|$$

#### Infinitesimal definition

Given a predicate $P$ on the real numbers $\mathbb{R}$, let $I$ denote the set of all elements in $\mathbb{R}$ for which $P$ holds. A [[partial function]] $f:\mathbb{R} \to \mathbb{R}$ is equivalently a function $f:I \to \mathbb{R}$ for any such predicate $P$ and set $I$.

A function $f:I \to \mathbb{R}$ is **smooth at a subset** $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in S$, such that for all Archimedean ordered Artinian local $\mathbb{R}$-algebras $A$ with ring homomorphism $h_A:\mathbb{R} \to A$ and [[nilradical]] $D$, natural numbers $n \in \mathbb{N}$, and purely infinitesimal elements $\epsilon \in D$ such that $\epsilon^{n + 1} = 0$
$$f_A(h_A(j(a)) + \epsilon) = \sum_{i = 0}^{n} \frac{1}{i!} h_A\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

Equivalently, let $\mathbb{R}[[\epsilon]]$ denote the ring of univariate [[formal power series]] on $\mathbb{R}$. $\mathbb{R}[[\epsilon]]$ is a [[Artinian local ring|Artinian local]] $\mathbb{R}$-algebra with homomorphism $h:\mathbb{R} \to \mathbb{R}[[\epsilon]]$. A function $f:I \to \mathbb{R}$ is **smooth at a subset** $S \subseteq I$ with injection $j:S \hookrightarrow \mathbb{R}$ if it has a function $\frac{d^{-} f}{d x^{-}}:\mathbb{N} \times S \to \mathbb{R}$ with $\frac{d^0 f}{d x^0}\left(a\right) = a$ for all $a \in S$, such that for all natural numbers $n \in \mathbb{N}$
$$f_A(h(j(a)) + \epsilon) = \sum_{i = 0}^{\infty} \frac{1}{i!} h\left(\frac{d^i f}{d x^i}\left(a\right)\right) \epsilon^i$$

A function $f:I \to \mathbb{R}$ is **smooth at an element** $a \in I$ if it is smooth at the [[singleton subset]] $\{a\}$, and a function $f:I \to \mathbb{R}$ is **smooth** if it is smooth at the [[improper subset]] of $I$.

### Between Cartesian spaces

A [[function]] on (some open subset of) a [[cartesian space]] $\mathbb{R}^n$ with values in the [[real line]] $\mathbb{R}$ is **smooth**, or __infinitely differentiable__, if all its [[derivative]]s exist at all points. More generally, if $A \subseteq \mathbb{R}^n$ is any subset, a function $f: A \to \mathbb{R}$ is defined to be _smooth_ if it has a smooth extension to an open subset containing $A$.

By [[coinduction]]: A [[function]] $f : \mathbb{R} \to \mathbb{R}$ is _smooth_ if (1) its [[derivative]] exists and (2) the derivative is itself a smooth function.

For $A \subseteq \mathbb{R}^n$, a **smooth map** $\phi: A \to \mathbb{R}^m$ is a function such that $\pi \circ \phi$ is a smooth function for every linear functional $\pi: \mathbb{R}^m \to \mathbb{R}$. (In the case of finite-dimensional codomains as here, it suffices to take the $\pi$ to range over the $m$ coordinate projections.)

The concept can be generalised from cartesian spaces to [[Banach spaces]] and some other infinite-dimensional spaces.  There is a [[locale]]-based analogue suitable for [[constructive mathematics]] which is not described as a function of points but as a special case of a [[continuous map]] (in the localic sense).

### Between smooth manifolds

A [[topological manifold]] whose transition functions are smooth maps is a [[smooth manifold]]. A smooth function between smooth manifolds is a function that (co-)restricts to a smooth function between subsets of Cartesian spaces, as above, with respect to any choice of [[atlases]], hence which is a $k$-fold [[differentiable function]] (see there for more details), for all $k$ The [[category]] [[Diff]] is the category whose objects are smooth manifolds and whose morphisms are smooth maps between them.

### Between generalized smooth spaces

There are various [[categories]] of [[generalised smooth spaces]] whose [[morphisms]] are generalized [[smooth functions]].

For details see for example at _[[smooth set]]_.

## Properties

Basic facts about smooth functions are

* the [[Hadamard lemma]]

* [[Borel's theorem]]

* the [[Tietze extension theorem]]

* the [[Steenrod-Wockel approximation theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[derivations of smooth functions are vector fields]]

## Examples

Every [[analytic functions]] (for instance a [[holomorphic function]]) is also a smooth function.

A crucial property of smooth functions, however, is that they contain also [[bump functions]].

## Related concepts

* [[differentiable function]]

* [[smooth function with rapidly decreasing derivatives]]

* [[smooth homotopy]], [[smooth isotopy]]

* [[nonlinear functional]], [[linear functional]]

[[!include infinitesimal and local - table]]

## References

An early account, in the context of [[Cohomotopy]], [[cobordism theory]] and the [[Pontryagin-Thom construction]]:

* {#Pontrjagin55} [[Lev Pontrjagin]], Chapter I of: _Smooth manifolds and their applications in Homotopy theory_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

[[!redirects smooth function]]
[[!redirects smooth functions]]
[[!redirects smooth map]]
[[!redirects smooth maps]]
[[!redirects infinitely differentiable function]]
[[!redirects infinitely differentiable functions]]
[[!redirects infinitely differentiable map]]
[[!redirects infinitely differentiable maps]]


smooth morphism

* [[smooth morphism of schemes]]

[[!redirects smooth morphisms]]


smooth morphism of schemes

#Contents#
* table of contents
{:toc}

This entry is about smooth morphisms of schemes. There are many notions of smoothness in algebra and algebraic geometry, many under the name of (such and such) regularity (nonsingularity); even in [[EGA]] there are something like 11 notions, one of which is called the smooth morphism of schemes. In the literature there are sometimes even small variations of the latter (e.g. whether we allow globally varying dimension of the smooth morphism or not). In $n$Lab a prominent role is played by the [[formally smooth morphism|formal smoothness]], which is weaker than smoothness.

## Idea

Smooth morphism is a relativization of the notion of a [[smooth scheme]].

## Definition

A [[morphism]] $f:X\to Y$ of [[schemes]] is **smooth** if

* it is [[flat morphism|flat]]

* and finitely presented (cf. [[relativization in algebraic geometry]])

* and if all the [[fiber]]s $f^{-1}(y)$ where $y\in Y$ are [[smooth schemes]] over the corresponding [[residue field]]s $k_y$.

## Properties

Smoothness of a morphism is a higher dimensional analogue of the notion of a morphism being [[etale morphism|étale]] (which is a smooth morphism of relative dimension $0$), but stronger than the notion of [[formally smooth morphism|formal smoothness]].

### Smoothness versus formal smoothness

For a morphism $f:X\to Y$ of schemes, and $x$ a point of $X$, the following are equivalent

(i) $f$ is a smooth morphism at $x$

(ii) $f$ is locally of finite presentation at $x$ and there is an open neighborhood $U\subset X$ of $x$ such that $f|_U: U\to Y$ is formally smooth

(iii) $f$ is flat at $x$, locally of finite presentation at $x$ and the sheaf of [[Kähler differential]]s $\Omega_{X/Y}$ is locally free in a neighborhood of $x$

The relative dimension of $f$ at $x$ will equal the rank of the module of Kähler differentials.

This is ([[EGAIV]]${}_4$ 17.5.2 and 17.15.15)

## Related concepts

A smooth morphism of [[relative dimension]] 0 is an [[étale morphism]].

See also [[formally smooth morphism]].

* The [[lisse-étale site]] of a [[scheme]] $X$ consists of smooth morphisms $U \to X$.

## References

* Arthur Ogus, _Smooth morphisms_ ([pdf](http://math.berkeley.edu/~ogus/Math%20_256A--08/smooth.pdf))

* Peter Bruin, _Smooth morphisms_ ([ps](http://www.google.com/url?sa=t&source=web&cd=8&ved=0CEkQFjAH&url=http%3A%2F%2Fwww.math.leidenuniv.nl%2F~pbruin%2Fsmooth.ps&rct=j&q=%22relative%20dimension%22%20smooth&ei=1zn9TOesBIrrOZbr1NQK&usg=AFQjCNFRuNB5BNwI3hXEA9jsLajt0tZldQ&cad=rja))

[[!redirects smooth morphisms of schemes]]


smooth natural numbers

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#### Synthetic differential geometry
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#Contents#
* automatic table of contents goes here
{:toc}

## Idea

In every [[smooth topos]] there is a notion of [[infinitesimal object]] and of [[infinitesimal number]].  The most common such infinitesimal numbers are [[nilpotent]], but in some special smooth toposes, there is in addition a notion of _invertible infinitesimal_.  In these toposes there is likewise an object of _smooth natural numbers_, which contains _infinite_ or _[[nonstandard number|nonstandard natural numbers]]_ (and whose inverses are invertible infinitesimals).

Some examples of such smooth toposes are discussed at [[Models for Smooth Infinitesimal Analysis]].

## The central mechanism

The phenomenon of "smooth" nonstandard natural numbers in a [[Grothendieck topos]] arises from the following simple general principle:

Consider any [[Grothendieck topos|sheaf topos]] $\mathcal{T} = Sh(C)$ such that

1. the category [[Set]] embeds [[full and faithful functor|full and faithfully]] into $\mathcal{T}$.

   (For [[smooth topos]]es we have, if they are "well adapted", even a full and faithful inclusion of [[Diff]] and then the one of [[Set]] is the one induced by the inclusion $Set \hookrightarrow Diff$.)

1. the [[Grothendieck topology]] on $C$ is on each object given by _finite_ covering families.

In such a case there are two objects in $\mathcal{T}$ that both look like they should qualify as the [[internalization|internal object]] of [[natural number]], but that are different:

1. The image $N$ of the set $\mathbb{N}$ under the given full and faithful embedding.

   This yields, trivially, a sheaf such that morphisms from any other set into it are given by arbitrary $\mathbb{N}$-valued functions on this set.

1. The abstractly defined [[natural numbers object]] $\Delta(\mathbb{N})$:

   this is the [[sheafification]] of the [[presheaf]] that is constant on the set $\mathbb{N}$. A morphism into this presheaf is a _constant_ $\mathbb{N}$-valued function. And since we are sheafifying, by assumption, with respect to _finite_ covers, a morphism from a set into its sheafification is a function into $\mathbb{N}$ that is constant on each patch of a _finite_ cover of that set and hence is a _bounded_ $\mathbb{N}$-valued function.

The unbounded functions thus represent [[infinite number|infinite]] or [[nonstandard number|non-standard]] "smooth natural numbers."  In particular, a [[generalized element]] $n \in \Delta(\mathbb{N})$ with domain of definition $\mathbb{N}$ (regarded as an object of $\mathcal{T}$) is a _bounded_ sequence of integers, whereas a similarly defined [[generalized element]] $\nu \in N$ is a possibly _unbounded_ sequence of integers.  This is intuitively similar to the unbounded sequences of numbers that represent infinitely large numbers in the ultrafilter approach to [[nonstandard analysis]] (a different way of making [[infinitesimal numbers]] precise).

## The generic nonstandard number

The _generic non-standard natural number_ is the [[generalized element]] of $N$ on the domain of definition $\ell C^\infty(\mathbb{N})/{NullTail}$ given by the canonical injection $\ell C^\infty(\mathbb{N})/NullTail \to N$ that is dual to the canonical projection of the ring onto its quotient.  Here $NullTail$ is the ideal of sequences of real numbers that vanish above some integer.

The ring $C^\infty(\mathbb{N})/NullTail$ here is a quotient ring of sequences as above, where two sequences are identified if they agree above some integer. So $\ell C^\infty(\mathbb{N})/NullTail$ is the [[smooth locus]] whose function algebra is similar to a nonstandard extension of $\mathbb{R}$.

The generic non-standard natural number is discussed on page 252 of the Moerdijk-Reyes book below.

## References

See in

* [[Ieke Moerdijk]], [[Gonzalo Reyes]], _[[Models for Smooth Infinitesimal Analysis]]_

chapter VI -- there section 1.6 section 2 -- and chapter VII.

[[!redirects smooth natural number]]


smooth scheme

#Contents#
* table of contents
{:toc}

## Idea

[[Grothendieck]] developed in [[EGA]] a number of notions of smoothness for a [[scheme]] and, more generally, for a [[smooth morphism of schemes|morphism]] of schemes. For algebraic varieties over a field, one already had a classical notion of a nonsingular variety.

## Definition

### For schemes

A [[scheme]] of finite type over a field $k$ is __smooth__ if after extension of scalars from $k$ to the [[algebraic closure]] $\bar{k}$ it becomes a __regular scheme__, i.e. the stalks of its structure sheaf are [[regular local ring]]s in the sense of commutative algebra.

A relative version of a smooth scheme is the notion of [[smooth morphism of schemes]].

### For affine schemes

Specifically, a finitely presented commutative [[associative algebra]] $A$ over a [[field]] $k$ is smooth if either of the following equivalent conditions holds

* the $A$-[[module]] $\Omega^1_k(A)$ of [[Kähler differential form]]s is a [[projective object]] in $A Mod$;

* with $A$ regarded as an $A$-[[bimodule]] in the evident way, it has a [[projective object|projective]] [[resolution]].

## Related concepts

* [[Bondal-Orlov reconstruction theorem]]

## References

For commutative $k$-algebras a discussion is for instance around theorem 9.1.2 in

* [[Victor Ginzburg]], _Lectures on noncommutative geometry_ ([arXiv:math/0506603](http://arxiv.org/abs/math.AG/0506603))

[[!redirects smooth schemes]]

[[!redirects smooth variety]]
[[!redirects smooth varieties]]


smooth Serre-Swan theorem

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#Contents#
* table of contents
{:toc}

## Idea

The _smooth Serre-Swan theorem_ ([Nestruev 03, 11.33](#Nestruev03)) states that over a [[smooth manifold]] $X$,

1. the [[section]] [[functor]]

   $$
     \Gamma_X(-)
      \;\colon\;
     SmoothVectorBundles_{/X}
       \hookrightarrow
     C^\infty(X) Mod
   $$

   that sends [[smooth vector bundles]] (with fibers of uniformly bounded dimension) over $X$ of [[finite number|finite]] [[rank]] to their [[spaces of sections|spaces of]] smooth [[sections]], regarded as [[modules]] over the [[algebra of functions|algebra of]] [[smooth functions]] on $X$, is a [[fully faithful functor]]

   ([Nestruev 03, theorem 11.29](#Nestruev03));

1. its [[essential image]] consists precisely of the [[finitely generated object|finitely generated]] [[projective modules]]

   ([Nestruev 03, 11.32](#Nestruev03)).

This is the variant for [[differential geometry]] of what the [[Serre-Swan theorem]] asserts in [[algebraic geometry]] and in [[topology]].

+-- {: .num_remark #BaseManifoldNotRequiredToBeCompact}
###### Remark
**(base [[smooth manifold]] not required to be [[compact space|compact]])

Contrary to the original theorem of [Swan 62](Serre-Swan+theorem#Swan) for [[topological vector bundles]], here in the smooth case the base [[smooth manifold]] $X$ of the [[smooth vector bundle]] is _not required_ to be [[compact space|compact]]. While for [[topological spaces]] [[compact space|compactness]] is needed to deduce that every [[topological vector bundle]] is a [[direct sum|direct summand]] of a [[trivial vector bundle]] ([this prop.](topological+vector+bundle#TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle)) for [[smooth vector bundles]] this conclusion follows without assuming compactness, by [[embedding of smooth manifolds]] into [[Cartesian spaces]] ([this prop.](embedding+of+differentiable+manifolds#ManifoldEmbedsIntoLargeDimensionalEuclideanSpace)).

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###### Remark
**(other algebraic apects of differential geometry)**

Together with the [[embedding of smooth manifolds into formal duals of R-algebras]], the smooth Serre-Swan theorem states that that differential geometry is "more algebraic" than it may superficially seem.
A third fact in this vein is that _[[derivations of smooth functions are vector fields]]_.

=--

## Related theorems

* the [[Hadamard lemma]]

* the [[Tietze extension theorem]]

* the [[Whitney extension theorem]]

* the [[Steenrod-Wockel approximation theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[derivations of smooth functions are vector fields]]

## References

* {#Nestruev03} [[Jet Nestruev]], _Smooth manifolds and observables_, Graduate texts in mathematics, 220, Springer-Verlag, ISBN 0-387-95543-7 (2003)

[[!redirects smooth Serre–Swan theorem]]