smooth set

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

# Smooth spaces
* table of contents
{: toc}

## Idea

The concept of a _smooth set_ or _smooth space_, in the sense discussed here, is a generalization of that of [[smooth manifolds]] beyond that of [[diffeological spaces]]: A smooth set is a [[generalized smooth space]] that may be _probed_ by smooth [[Cartesian spaces]].

For expository details see at _[[geometry of physics -- smooth sets]]_.

Alternatively, the smooth test spaces may be taken to be more generally all [[smooth manifolds]]. But since [[manifolds]] themselves are built from gluing together smooth [[open balls]] $D^n_{int} \subset \mathbb{R}^n$ or equivalently [[Cartesian spaces]] $\mathbb{R}^n$, one may just as well consider Cartesian spaces test spaces. Finally, since $D^n$ is diffeomorphic to $\mathbb{R}^n$, one can just as well take just the cartesian smooth spaces $\mathbb{R}^n$ as test objects.

## Definition

The category of **smooth spaces** is the [[sheaf topos]]

$$
  SmoothSp := Sh(Diff)
$$

of [[sheaves]] on the [[site]] [[Diff]] of smooth manifolds equipped with its standard [[coverage]] ([[Grothendieck topology]]) given by open covers of manifolds.

Since $Diff$ is [[equivalence of categories|equivalent]] to the category of [[manifolds]] _embedded_ into $\mathbb{R}^\infty$, $Diff$ is an [[essentially small category]], so there are no size issues involved in this definition.

But since manifolds themselves are defined in terms of gluing conditons, the [[Grothendieck topos]] $SmoothSp$ depends on much less than all of $Diff$.

Let

$$
  Ball := \{ (D^n_{int} \to D^m_{int}) \in Diff | n,m \in \mathbb{N}\}
$$

and

$$
  CartSp := \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff  | n,m \in \mathbb{N}\}
$$

be the full [[subcategory|subcategories]] $Ball$ and [[CartSp]] of $Diff$ on open balls and on cartesian spaces, respectively. Then the corresponding [[sheaf]] [[topos|toposes]] are still those of smooth spaces:

$$
  \begin{aligned}
    SmoothSp &\simeq Sh(Ball)
    \\
     & \simeq Sh(CartSp)
  \end{aligned}
  \,.
$$

## Examples

* The category of ordinary [[manifolds]] is a
  full subcategory of smooth spaces:
  $$
    Diff \hookrightarrow SmoothSp
    \,.
  $$
  When one regards smooth spaces concretely as sheaves
  on $Diff$, then this inclusion is of course just the
  [[Yoneda embedding]].

* The full [[subcategory]]
  $$
    DiffSp \subset SmoothSp
  $$
  on [[concrete sheaf|concrete sheaves]] is
  called the
  category of [[diffeological spaces]].

  * The standard class of examples of smooth spaces that motivate their use even in cases where one starts out being intersted just in [[smooth manifolds]] are **mapping spaces**: for $X$ and $\Sigma$ two smooth spaces (possibly just ordinary smooth manifolds), by the [[closed monoidal structure on presheaves]] the **mapping space** $[\Sigma,X]$, i.e. the space of smooth maps $\Sigma \to X$ exists again naturally as a smooth. By the general formula it is given as a [[sheaf]] by the assignment

    $$
      [\Sigma,X] : U \mapsto SmoothSp(\Sigma \times U, X)
     \,.
    $$

    If $X$ and $\Sigma$ are ordinary manifolds, then the [[hom-set]] on the right sits inside that of the underlying sets $SmoothSp(\Sigma \times U , X) \subset Set(|\Sigma| \times |U|, |X| )$ so that $[\Sigma,X]$ is a [[diffeological space]].

    The above formula says that a $U$-parameterized family of maps $\Sigma \to X$ is smooth as a map into the smooth space $[\Sigma,X ]$ precisely if the corresponding map of sets $U \times \Sigma \to X$ is an ordinary morphism of smooth manifolds.

* The canonical examples of smooth spaces that
  are not diffeological spaces are the sheaves of
  (closed) differential forms:
  $$
    K^n : U \mapsto \Omega^n_{closed}(U)
    \,.
  $$

* The category
  $$
    SimpSmoothSp :=
    SmoothSp^{\Delta^{op}}
  $$
  equivalently that of sheaves on $Diff$ with
  values in [[simplicial sets]]
  $$
    \cdots \simeq Sh(Diff, SSet)
  $$
  of [[simplicial objects]] in smooth spaces
  naturally carries the structure of a
  [[homotopical category]]
  (for instance the
    [[model structure on simplicial sheaves]]
  or that of a Brown [[category of fibrant objects]]
  (if one restricts to locally Kan simplicial sheaves))
  and as such is a
  [[presentable (∞,1)-category|presentation]]
  for the [[(∞,1)-topos]] of
  [[smooth ∞-stacks]].

## Properties
 {#Properties}

### Cohesion
 {#Cohesion}

+-- {: .num_prop #SmoothSetsFormACohesiveTopos}
###### Proposition
**([[smooth sets]] form a [[cohesive topos]])**

The [[category]] $SmoothSet$ of [[smooth sets]] is a [[cohesive topos]]

\[
  \label{SheafToposAdjointQuadruple}
  SmoothSet
    \array{
      \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow}
      \\
      \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow}
      \\
      \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow}
      \\
      \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow}
    }
  Set
\]

=--

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

First of all (by [this Prop](geometry+of+physics+--+smooth+sets#CategoryOfSmoothSets)) smooth sets indeed form a [[sheaf topos]], over the [[site]] [[CartSp]] of [[Cartesian spaces]] $\mathbb{R}^n$ with [[smooth functions]] between them, and equipped with the [[coverage]] of differentiably-[[good open covers]] ([this def.](geometry+of+physics+--+smooth+sets#DifferentiallyGoodOpenCover))

$$
  SmoothSet \simeq Sh(CartSp)
  \,.
$$

Hence, by Prop. \ref{CategoriesOfSheavesOnCohesiveSiteIsCohesive}, it is now sufficient to see that [[CartSp]] is a [[cohesive site]] (Def. \ref{OneCohesiveSite}).

It clearly has [[finite products]]: The [[terminal object]] is the [[point]], given by the 0-[[dimension|dimensional]] [[Cartesian space]]

$$
  \ast = \mathbb{R}^0
$$

and the [[Cartesian product]] of two [[Cartesian spaces]] is the Cartesian space whose [[dimension]] is the [[sum]] of the two separate dimensions:

$$
  \mathbb{R}^{n_1} \times \mathbb{R}^{n_2}
  \;\simeq\;
  \mathbb{R}^{ n_1 + n_2 }
  \,.
$$

This establishes the first clause in Def. \ref{OneCohesiveSite}.

For the second clause, consider a differentiably-[[good open cover]] $\{U_i \overset{}{\to} \mathbb{R}^n\}$ ([this def.](geometry+of+physics+--+smooth+sets#DifferentiallyGoodOpenCover)). This being a [[good cover]] implies that its [[Cech groupoid]] is, as an [[internal groupoid]] (via [this remark](geometry+of+physics+--+categories+and+toposes#PresheavesOfGroupoidsAsInternalGroupoidsInPresheaves)), of the form

\[
  \label{CechGroupoidForCartSp}
  C(\{U_i\}_i)
  \;\simeq\;
  \left(
    \array{
       \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j)
       \\
       \big\downarrow \big\uparrow \big\downarrow
       \\
       \underset{i}{\coprod} y(U_i)
    }
  \right)
  \,.
\]

where we used the defining property of [[good open covers]] to identify $y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )$.

The [[colimit]] of (eq:CechGroupoidForCartSp), regarded just as a [[presheaf]] of [[reflexive graph|reflexive]] [[directed graph|directed]] [[graphs]] (hence ignoring [[composition]] for the moment), is readily seen to be the [[graph]] of the [[colimit]] of the components (the [[universal property]] follows immediately from that of the component colimits):

\[
  \label{ColimitOfCechGroupoidOverCartSp}
  \begin{aligned}
    \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim}
    C(\{U_i\}_i)
    & \simeq
    \left(
      \array{
        \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim}
         \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j)
         \\
         \big\downarrow \big\uparrow \big\downarrow
         \\
         \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim}
         \underset{i}{\coprod} y(U_i)
      }
    \right)
    \\
    & \simeq
    \left(
      \array{
         \underset{i,j}{\coprod}
         \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim}
         y(U_i \underset{\mathbb{R}^n}{\cap} U_j)
         \\
         \big\downarrow \big\uparrow \big\downarrow
         \\
         \underset{i}{\coprod}
         \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim}
         y(U_i)
      }
    \right)
    \\
    & \simeq
    \left(
      \array{
         \underset{i,j}{\coprod} \ast
          \\
         \big\downarrow \big\uparrow \big\downarrow
         \\
         \underset{i}{\coprod} \ast
      }
    \right)
  \end{aligned}
  \,.
\]

Here we first used that [[colimits commute with colimits]], hence in particular with [[coproducts]] ([this prop.](geometry+of+physics+--+categories+and+toposes#LimitsCommuteWithLimits)) and then that the colimit of a [[representable presheaf]] is [[generalized the|the]] [[singleton]] set ([this Lemma](geometry+of+physics+--+categories+and+toposes#ColimitOfRepresentableIsSingleton)).

This colimiting [[graph]] carries a unique [[composition]] structure making it a [[groupoid]], since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point.

Precisely this operation on [[Cech groupoids]] of [[good open covers]] of [[topological spaces]] is what _[[Borsuk's nerve theorem]]_ is about, a classical result in [[topology]]/[[homotopy theory]]. This theorem implies directly that the set of [[connected components]] of the groupoid (eq:ColimitOfCechGroupoidOverCartSp) is in [[bijection]] with the set of [[connected components]] of the [[Cartesian space]] $\mathbb{R}^n$, regarded as a [[topological space]]. But this is evidently a [[connected topological space]], which finally shows that, indeed

$$
  \pi_0
  \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim}
    C(\{U_i\}_i)
  \;\simeq\;
  \ast
  \,.
$$

The second item of the second clause in Def. \ref{OneCohesiveSite} follows similarly, but more easily: The [[limit]] of the [[Cech groupoid]] is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since $CartSp$ has a [[terminal object]] $\ast = \mathbb{R}^0$, which is hence an [[initial object]] in the [[opposite category]] $CartSp^{op}$, limits over $CartSp^{op}$ yield simply the evaluation on that object:

\[
  \label{ColimitOfCechGroupoidOverCartSp}
  \begin{aligned}
    \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim}
    C(\{U_i\}_i)
    & \simeq
    \left(
      \array{
        \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim}
         \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j)
         \\
         \big\downarrow \big\uparrow \big\downarrow
         \\
         \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim}
         \underset{i}{\coprod} y(U_i)
      }
      \phantom{A}
    \right)
    \\
    & \simeq
    \left(
      \array{
         \underset{i,j}{\coprod}
         Hom_{CartSp}\left( \ast,  U_i \underset{\mathbb{R}^n}{\cap} U_j \right)
         \\
         \big\downarrow \big\uparrow \big\downarrow
         \\
         \underset{i}{\coprod}
         Hom_{CartSp}( \ast, U_i )
      }
    \right)
  \end{aligned}
  \,.
\]

Here we used that [[colimits]] (here [[coproducts]]) of [[presheaves]] are computed objectwise, and then the definition of the [[Yoneda embedding]] $y$.

But the [[equivalence relation]] induced by this graph on its set of objects $\underset{i}{\coprod}  Hom_{CartSp}( \ast, U_i )$ precisely identifies pairs of points, one in $U_i$ the other in $U_j$, that are actually the same point of the $\mathbb{R}^n$ being covered. Hence the set of [[equivalence classes]] is the set of points of $\mathbb{R}^n$, which is just what remained to be shown:

$$
  \pi_0
  \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim}
    C(\{U_i\}_i)
  \;\simeq\;
  Hom_{CartSp}(\ast, \mathbb{R}^n)
  \,.
$$

=--

### Topos points and stalks

+-- {: .num_lemma }
###### Lemma

For every $n \in N$ there is a
[[point of a topos|topos point]]

$$
  D^n : Set \stackrel{\stackrel{(D^n)^*}{\leftarrow}}
    {\stackrel{D^n_*}{\to}}
    SmoothSp
$$

where the [[inverse image]] morphism -- the [[stalk]] --
is given on $A \in SmoothSp$ by

$$
  (D^n)^* A :=
  \colim_{\mathbb{R}^n \supset U \ni 0} A(U)
  \,,
$$

where the colimit is over all open neighbourhoods of the
origin in $\mathbb{R}^n$.

=--

+-- {: .num_lemma }
###### Lemma

SmoothSp has [[point of a topos|enough points]]:
they are given by the
$D^n$ for $n \in \mathbb{N}$.

=--

### Distribution theory

Since a space of [[smooth functions]] on a [[smooth manifold]] is canonically a smooth set, it is natural to consider the _smooth_ [[linear functionals]] on such [[mapping spaces]]. These turn out to be equivalent to the [[continuous linear functionals]], hence to [[distributional densities]]. See at _[[distributions are the smooth linear functionals]]_ for details.

## Variants and generalizations
 {#VariantsAndGeneralizations}

### Synthetic differential geometry

The [[site]] [[CartSp]]${}_{smooth}$ may be replaced by the site [[CartSp]]${}_{th}$ (see there) whose objects are products of smooth Cartesian spaces with [[infinitesimally thickened points]]. The corresponding [[sheaf topos]] $Sh(CartSp_{th})$ is called the _[[Cahiers topos]]_. It contains smooth spaces with possibly infinitesimal extension and is a model for [[synthetic differential geometry]] (a "[[smooth topos]]"), which $Sh(CartSp)$ is not.

The two toposes are related by an [[adjoint quadruple]] of functors that witness the fact that the objects of $Sh(CartSp_{th})$ are possiby infinitesimal extensions of objects in $Sh(CartSp)$.
For more discussion of this see [[synthetic differential ∞-groupoid]].

### Higher smooth geometry

The topos of smooth spaces has an evident generalization from [[geometry]] to [[higher geometry]], hence from [[differential geometry]] to [[higher differential geometry]]: to an [[(∞,1)-topos]] of _[[smooth ∞-groupoids]]_. See there for more details.

$\,$

## Related concepts

[[!include geometries of physics -- table]]

$\,$

## References

The notion appears as a model for [[homotopy types]] in

* [[Denis-Charles Cisinski]], Ch. 6 in: *Faisceaux localement asphériques* (2003) [[pdf](https://cisinski.app.uni-regensburg.de/mtest2.pdf), [[Cisinski-FaisceauxLocAsph.pdf:file]]]

and in the context of [[smooth ∞-groupoids]] in

* [[Daniel Dugger]], section 3.4, from page 29 on in:  _Sheaves and Homotopy Theory_ [[web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf)]

  > (the [[point of a topos|topos points]] of $Sh(Diff)$ are discussed there in example 4.1.2 on p. 36, mentioned before on p. 31)

and then more explicitly, as a [[cohesive topos]] using the small site [[CartSp]], in

* [[Urs Schreiber]], Def. 1.2.197 of: _[[schreiber:differential cohomology in a cohesive topos]]_

Lecture notes:

* [[Urs Schreiber]]: _[[geometry of physics -- smooth sets]]_

Discussion of smooth sets as a [[convenient category of spaces|convenient category]] for [[variational calculus]] of [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:

* [[Grigorios Giotopoulos]],  *Classical field theory in the topos of smooth sets*, [talk at](Center+for+Quantum+and+Topological+Systems#GiotopoulosOct2023) [[CQTS]] (Oct 2023) [[[Giotopoulos-FieldTheoryInSmoothSets.pdf:file]], video:[YT](https://youtu.be/7Bw9CJct8QY)]

* {#GiotopoulosSati23} [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

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

[[!redirects smooth sets]]

[[!redirects SmoothSet]]


smooth spectrum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[sheaf of spectra]] on the [[site]] of all [[smooth manifolds]] may be thought of as a [[spectrum]] equipped with generalized [[smooth structure]], in just the same way as an [[(∞,1)-sheaf]] on this site may be thought of as a [[smooth ∞-groupoid]]. Therefore one might speak of the [[stable (∞,1)-category]]

$$
  Sh_\infty(SmoothMfd, Spectra)
  \simeq
  Stab(Sh_\infty(SmoothMfd))
  =
  Stab(Smooth \infty Grpd)
$$

which is the [[stabilization]] of that of [[smooth ∞-groupoids]] as being the $\infty$-category of _smooth spectra_, just as the [[stable (∞,1)-category of spectra]] itself is the [[stabilization]] of that of bare [[∞-groupoids]].

Together with [[smooth ∞-groupoids]] smooth spectra sit inside the [[tangent cohesive (∞,1)-topos]] over smooth manifolds. By the discussion [there](tangent+cohesive+∞-topos#CohesiveAndDifferentialRefinement), every smooth spectrum sits in a hexagonal [[differential cohomology diagram]] which exhibits it ([Bunke-Nikolaus-Völkl 13](#BunkeNikolausVoelkl13)) as the moduli of a generalized [[differential cohomology]] theory  (in generalization of how every ordinary spectrum, via the [[Brown representability theorem]], corresponds to a bare [[generalized (Eilenberg-Steenrod) cohomology theory]]).

## Properties

### From chain complexes of smooth modules
 {#FromChainComplexesOfSmoothModules}

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

Write

* $Smooth0Type \coloneqq Sh(SmthMfd)$ for the [[topos]] of [[smooth spaces]];

* $\mathbf{R} \in Smooth0Type$ for the sheaf of [[real number]]-valued [[smooth functions]] (the canonical [[line object]] in $Smooth0Type$);

* $\mathbf{R} Mod$ for the category of [[abelian sheaves]] over smooth manifolds which are $\mathbf{R}$-[[modules]].

=--

+-- {: .num_defn #RModulesAsInfinityPresheaves}
###### Definition (Notation)

Let $C_\bullet \in Ch_\bullet(\mathbf{R}Mod)$ be a [[chain complex]] (unbounded) of [[abelian sheaves]] of $\mathbf{R}$-modules. Via the projective [[model structure on functors]] this defines an [[(∞,1)-presheaf]] of [[chain complexes]]

$$
  Ch_\bullet(\mathbf{R}Mod)
  \longrightarrow
  Sh(SmthMfd, Ch_{\bullet})
  \longrightarrow
  L_{qi} PSh(SmthMfd, Ch_\bullet)
  \simeq
  PSh_\infty(SmthMfs, Ch_\bullet)
  \,.
$$

We still write $C_\bullet\in PSh_\infty(SmthMfd, Ch_\bullet)$ for this [[(∞,1)-presheaf]] of chain complexes.

=--

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

Under the [[stable Dold-Kan correspondence]]

$$
  DK \;\colon\; Ch_\bullet \longrightarrow Spectra
$$

a chain complex of $\mathbf{R}$-modules $C_\bullet \in Ch_\bullet(\mathbf{R}Mod)$, regarded as an [[(∞,1)-presheaf]] of spectra on $SmthMfd$ as in def. \ref{RModulesAsInfinityPresheaves}, is already an [[(∞,1)-sheaf]], hence a smooth spectrum (i.e. without further [[∞-stackification]]).

=--

This appears as ([Bunke-Nikolaus-Völkl 13, lemma 7.12](#BunkeNikolausVoelkl13)).

## Examples

### De Rham spectra
 {#ExamplesDeRhamSpectra}

Write $Ch_\bullet$ for the [[(∞,1)-category of chain complexes]] (of [[abelian groups]], hence over the [[ring]] $\mathbb{Z}$ of [[integers]]). It is convenient to choose for $A_\bullet \in Ch_\bullet$ the grading convention

$$
  \array{
    \vdots
    \\
    \downarrow
    \\
    A_{-1}
    \\
    \downarrow
    \\
    A_0
    \\
    \downarrow
    \\
    A_1
    \\
    \downarrow
    \\
    \vdots
   }
$$

such that under the [[stable Dold-Kan correspondence]]

$$
  DK \;\colon\; Ch_\bullet \stackrel{}{\longrightarrow} Spectra
$$

the [[homotopy groups]] of spectra relate to the [[homology groups]] by

$$
  \pi_n(DK(A_\bullet)) \simeq  H_{-n}(A_\bullet)
  \,.
$$

In particular for $A \in $ [[Ab]] an [[abelian group]] then $A[n]$ denotes the chain complex concentrated on $A$ in degree $-n$ in this counting.

The grading is such as to harmonize well with the central example of a sheaf of chain complexes over the site of [[smooth manifolds]], which is the [[de Rham complex]], regarded as a [[smooth spectrum]] via the discussion at _[smooth spectrum -- from chain complexes of smooth modules](smooth+spectrum#FromChainComplexesOfSmoothModules)_

$$
  \Omega^\bullet \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
$$

$$
  \Omega^{\bullet} \;\colon\;  X\mapsto (\cdots \to 0 \to 0 \to \Omega^0(X) \stackrel{\mathbf{d}}{\to} \Omega^1(X)\stackrel{\mathbf{d}}{\to} \cdots)
$$

with $\Omega^0(X) = C^\infty(X, \mathbb{R})$ in degree 0.

We also need for $n \in \mathbb{N}$ the truncated sheaf of complexes

$$
  \Omega^{\bullet \geq n} \in Sh_\infty(SmthMfd, Ch_\bullet) \longrightarrow Sh_\infty(SmthMfd, Spectra) \hookrightarrow T \mathbf{H}
$$

$$
  \Omega^{\bullet \geq n} \;\colon\;  X\mapsto (\cdots \to 0 \to 0 \to \Omega^n(X) \stackrel{\mathbf{d}}{\to} \Omega^{n+1}(X)\stackrel{\mathbf{d}}{\to} \cdots)
$$

with $\Omega^n(X)$ in degree $n$.

More genereally, for $C \in Ch_\bullet$ any [[chain complex]], there is $(\Omega \otimes C)^{\bullet \geq n}$ given over each manifold $X$ by the [[tensor product of chain complexes]] followed by truncation.

Hence

$$
  (\Omega \otimes C)^{\bullet \geq n}
  =
  (\cdots \to 0 \to 0 \to \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k} \stackrel{\mathbf{d} \pm d_{C}}{\to} \oplus_{k \in \mathbb{N}} \Omega^{k}(X) \otimes C_{n-k+1}\stackrel{\mathbf{d}\pm d_{C}}{\to} \cdots)
 \,.
$$

### Algebraic K-theory of smooth manifolds

see at _[[algebraic K-theory of smooth manifolds]]_

## References

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

[[!redirects smooth spectra]]


smooth structure

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

#Contents#
* table of contents
{:toc}

## Idea

A _differential structure_ on a [[topological space]] $X$ is the extra structure of a [[differential manifold]] on $X$. A _smooth structure_ on $X$ is the extra structure of a [[smooth manifold]].

## Definition

+-- {: .num_defn #SmoothStructure}
###### Definition
**(smooth structure)**

Let $X$ be a [[topological manifold]] and let

$$
  \left(
    \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X
  \right)_{i \in I}
  \phantom{AAA}
   \text{and}
  \phantom{AAA}
  \left(
    \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X
  \right)_{j \in J}
$$

be two [[atlases]], both making $X$ into a [[smooth manifold]] ([this def.](differentiable+manifold#DifferentiableManifold)).

Then there is a [[diffeomorphism]] of the form

$$
  f
  \;\colon\;
  \left(
    X
      \;,\;
    \left(
      \mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X
    \right)_{i \in I}
  \right)
    \overset{\simeq}{\longrightarrow}
  \left(
    X\;,\;
    \left(
      \mathbb{R}^{n} \underoverset{\simeq}{\psi_j}{\longrightarrow} V_j \subset X
    \right)_{j \in J}
  \right)
$$

precisely if the [[identity function]] on the underlying set of $X$ constitutes such a diffeomorphism.
(Because if $f$ is a diffeomorphism, then also $f^{-1}\circ f = id_X$ is a diffeomorphism.)

That the identity function is a diffeomorphism between $X$ equipped with these two atlases means (by [definition](differentiable+manifold#DifferentiableManifold)) that

$$
  \underset{{i \in I} \atop {j \in J}}{\forall}
  \left(
    \phi_i^{-1}(V_j) \overset{\phi_i}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^n
    \phantom{AA}
    \text{is smooth}
  \right)
  \,.
$$

Hence diffeomorphsm induces an [[equivalence relation]] on the set of
smooth atlases that exist on a given [[topological manifold]] $X$. An [[equivalence class]]
with respect to this equivalence relation is called a _smooth structure_ on $X$.

=--

## Properties

+-- {: .num_theorem #UniquenessOfSmoothStructures}
###### Theorem
**(uniqueness of smooth structure on [[Euclidean space]] in $d \neq 4$)

For $n \in \mathbb{N}$ a [[natural number]] with $n \neq 4$, there is a unique (up to [[isomorphism]]) smooth structure on the [[Cartesian space]] $\mathbb{R}^n$.

=--

This was shown in ([Stallings 62](#Stallings62)).

+-- {: .num_theorem}
###### Theorem

In $d = 4$ the analog of this statement is false. One says that on $\mathbb{R}^4$ there exist [[exotic smooth structure]]s.

=--

### Exotic smooth structures

Many topological spaces have canonical or "obvious" smooth structures. For instance a [[Cartesian space]] $\mathbb{R}^n$ has the evident smooth structure induced from the fact that it can be covered by a single chart -- itself.

From this example, various topological spaces inherit a canonical smooth structure by embedding. For instance the $n$-[[sphere]] may naturally be thought of as the collection of points

$$
  S^n \hookrightarrow \mathbb{R}^n
$$

given by $S^n = \{\vec x \in \mathbb{R}^n | \sum_i (x^i)^2 = 1\}$ and this induces a smooth structure of $\mathbb{S}^n$.

But there may be other, non-equivalent smooth structures than these canonical ones. These are called [[exotic smooth structure]]s. See there for more details.

## Related concepts

* [[smooth structure on a topos]]

## References

* {#Stallings62} [[John Stallings]], _The piecewise linear structure of Euclidean space_ , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. ([pdf](http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=2118140))

See also

* Wikipedia, _[Smooth structure](https://en.wikipedia.org/wiki/Smooth_structure)_

[[!redirects differential structure]]
[[!redirects differential structures]]
[[!redirects smooth structures]]


smooth structure from submanifolds

{:myproof: .proof style="margin-left:2em;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}
{:goal: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

+-- {: goal}
###### Goal
The purpose of this page is to investigate the question:

> How much of a [[smooth manifold]]'s structure can be determined by knowing its [[submanifolds]]?
=--

This can be interpreted in several ways mathematically.  We start with two smooth manifolds, $M$ and $N$, and a bijection $\Phi \colon M \to N$.  We want to learn something about what $\Phi$ must or can be by testing it on submanifolds.  Ideally, we would like a condition equivalent to $\Phi$ being a diffeomorphism.  Some conditions to consider are the following.

1. $S \subseteq M$ is a submanifold if and only if $\Phi(S) \subseteq N$ is a submanifold.
1. $S \subseteq M$ is a submanifold of dimension $m$ if and only if $\Phi(S) \subseteq N$ is a submanifold of dimension $m$.
1. For some fixed $m$ and for all $k \le m$, $S \subseteq M$ is a submanifold of dimension $k$ if and only if $\Phi(S) \subseteq N$ is a submanifold of dimension $k$.
1. For some fixed $m$, $S \subseteq M$ is a submanifold of dimension $m$ if and only if $\Phi(S) \subseteq N$ is a submanifold of dimension $m$.

## Ideology ##

One reason that [[Andrew Stacey|I]] am interested in this is that it is a variation of the idea of [[generalised smooth spaces]].  In a generalised smooth space, one defines "smoothness" to be "that which is detectable when tested on known spaces".  The point is that the _known spaces_ are fixed and independent of the object being tested.  Here, the known spaces can (and, indeed, do) vary with the object being tested.  So each object has its own family of test objects.  Closely related to this is the question of automorphisms of a category.  If we have a category with test objects inside it then applying an automorphism will change the test objects so to study this situation we have to find some other way of characterising the test objects.  In [[comparative smootheology]], I showed that there were no (essentially) non-trivial automorphisms of the category of [[Frölicher spaces]] by showing that one could identify the real line by its endomorphism monoid.

## Preliminary Thoughts ##

For $\mathbb{R}$, it is the case that preserving submanifolds is equivalent to being a homeomorphism so long as we are considering submanifolds of dimension $1$.  By cardinality, we can separate submanifolds of dimension $1$ and $0$ and so from any of the statements we see that $\Phi$ takes submanifolds of dimension $1$ to submanifolds of dimension $1$.  But submanifolds of $\mathbb{R}$ of dimension $1$ are precisely the open sets, whence $\Phi$ is a homeomorphism.

This extends more generally.  If $\Phi \colon \mathbb{R}^n \to \mathbb{R}^m$ is a bijection with the property that $\Phi(S)$ is a submanifold of $\mathbb{R}^m$ if and only if $S$ is a submanifold of $\mathbb{R}^n$ then $\Phi$ is a homeomorphism.  The proof is slightly different depending on whether we allow the dimension of a submanifold to be [[locally constant]] or insist that it be [[constant]].

We start with an open set $U \subseteq \mathbb{R}^n$.  As this is a submanifold, $\Phi(U)$ is a submanifold of $\mathbb{R}^m$.  For cardinality reasons, $\Phi(U)$ cannot be zero dimensional everywhere.  Indeed, it can be zero dimensional at at most a countable number of points.  Choose a point $p \in U$ such that $\dim_{\Phi(p)} \Phi(U) \ne 0$.

We choose an open set $V$ with $p \in V \subseteq \overline{V} \subseteq U$.  By assumption, $\Phi(V)$ is a submanifold of $\mathbb{R}^m$ and $\Phi(p) \in \Phi(V)$.  Suppose that $\dim_{\Phi(p)} \Phi(U) \ne m$.  Then there is a connected submanifold $T$ of $\mathbb{R}^m$ of dimension at least $1$ which meets $\Phi(U)$ transversally at $\Phi(p)$.  In particular $T \cap \Phi(U) = \{\Phi(p)\}$.  Applying $\Phi^{-1}$, we find that $\Phi^{-1}(T)$ is a submanifold of $\mathbb{R}^n$ which meets $U$ only at $p$.  Thus $p$ must be an isolated point of $\Phi^{-1}(T)$.  Since the closure of $V$ is contained within $U$, the union $V \cup \Phi^{-1}(T)$ is therefore a submanifold of $\mathbb{R}^n$.  Applying $\Phi$, we see that $\Phi(V) \cup T$ is a submanifold of $\mathbb{R}^m$.  But $\Phi(V)$ and $T$ both have non-zero dimension at $p$ and meet transversally there, so their union cannot be a submanifold of $\mathbb{R}^m$.

Thus at those points where $\dim \Phi(U) \ne 0$, it must be of dimension $m$.  To conclude the proof, we just need to show that there are no points where $\dim \Phi(U) = 0$.  Suppose that there is such a point, say $p$.  Then $\Phi(p)$ is an isolated point from $\Phi(U)$.  Choose an open set $W \subseteq \mathbb{R}^m$ such that $\Phi(p) \in \partial W$ and $W \cap \Phi(U) = \emptyset$.  Then $W \cup \{\Phi(p)\}$ is not a submanifold of $\mathbb{R}^m$, but $\Phi(W) \cap U = \emptyset$ so $\Phi(W) \cap \{p\}$ is a submanifold of $\mathbb{R}^n$.  Hence there cannot be any such points.

In conclusion, $\Phi(U)$ is a submanifold of $\mathbb{R}^m$ of dimension $m$, whence an open subset.  As the same holds for $\Phi^{-1}$, we conclude that $\Phi$ is a homeomorphism.

## Initial Motivation ##

The initial motivation for this page came from [this question](http://math.stackexchange.com/q/26954/2907) on <http://maths.stackexchange.com>.


smooth structure of the path groupoid

### Introduction ###

The [[path groupoid]] is a groupoid internal to some category of [[generalized smooth spaces]].  Its space of morphisms is defined by taking a quotient of a variant of a [[path space]] by the notion of [[thin homotopy]].  The path space is, generally, the same in all proposed categories of [[generalized smooth space]] but as the path groupoid is formed by taking a quotient, its structure will vary depending on the choice.

This page was started in response to a question on MathOverflow by Theo Johnson-Freyd.  The question was "[What is the infinite dimensional manifold structure on the space of smooth paths  mod thin homotopy?](http://mathoverflow.net/questions/17803/what-is-the-infinite-dimensional-manifold-structure-on-the-space-of-smooth-paths)".

### Linear Structure ###

To find a linear structure, we consider the path groupoid in a vector space, specifically $\mathbb{R}^n$.  Thus we consider the space of all smooth paths in $\mathbb{R}^n$ and quotient by the equivalence relation of thin homotopy.

The most important point to realise about the linear structure on this quotient is that it doesn't exist.

The quotient is not by cosets of a subspace and so the linear structure on $P\mathbb{R}^n$ does not descend to $P^1\mathbb{R}^n$.  For it to descend we would need to know that the sum of two paths which are both thin homotopic to the zero path is again thin homotopic to the zero path.  Here's an example where that fails:

$$
\begin{aligned}
\alpha(t) &= (\cos(2\pi t), 0) \\
H_\alpha(s,t) &= (s\cos(2\pi t),0) \\
\beta(t) &= (0,\sin(2\pi t)) \\
H_\beta(s,t) &= (0,s\sin(2\pi t)) \\
(\alpha + \beta)(t) &= (\cos(2\pi t), \sin(2\pi t))
\end{aligned}
$$

Note that, for simplicity, we are defining the path groupoid as the quotient of **all** smooth paths by the corresponding thin homotopical relation.  The resulting space is the same as that formed by taking paths with sitting instants, or paths which are flat at the end-points.

The reason for this failure is seen when we examine how the sum of two paths is formed.  The sum of two arbitrary paths $\alpha$ and $\beta$ is defined as the composition

$$
I \to I \times I \to \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n.
$$

Here, and hereafter, $I = [0,1]$.

If $\alpha$ and $\beta$ are thinly homotopic to the zero path, then the obvious way to construct a thin homotopy from the sum to the zero path is to take the sum of the thin homotopies.

$$
I^2 \to I^2 \times I^2 \to \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n.
$$

One cannot compute the exact rank of this without knowing the maps involved, but one can get bounds.  In particular, the upper bound on the rank is the minimum of the upper bounds of each piece.  The crucial point here is that (unless $n = 1$), the map with the lowest rank is the map $I^2 \times I^2 \to \mathbb{R}^n \times \mathbb{R}^n$.  This is constructed from two thin homotopies, and so has maximum rank $2$!

Of course, this does not **prove** that it does not always work (the counterexample given above does that) since it might be the case that the maps are always such that this upper bound is not achieved.  What this does is explain why the existence of the counterexample is not surprising: it would take a considerable number of coincidences for no counterexample to exist.

Extending this further, we observe that the process of taking the path groupoid is not a product-preserving functor.  This is obvious from considering the case of $\mathbb{R}^2$: in $\mathbb{R}$, every path is thinly homotopic to every other (with the same end-points) and so $P^1\mathbb{R}$ simply has objects $\mathbb{R}$ with a single morphism between each two objects.  However, $\mathbb{R}^2$ is much, much larger than the corresponding product as it has plenty of paths not thinly homotopic to each other (even with the same end-points).

Finally, the lack of a linear structure on $P^1 \mathbb{R}^n$ rules out any interesting manifold structure on $P^1 M$.

### Diffeological Structure ###

The diffeological structure is simple to explain.  A map $\phi \colon U \to P^1M$ is a plot if and only if it locally lifts to $P M$.  That is, if there is a cover $\mathcal{V}$ of $U$ and for each $V \in \mathcal{V}$ a lift of $\phi|_V \colon V \to P^1 M$ to a smooth map $V \to P M$.

The fact that plots have to lift locally does mean that it is not always possible to patch together smooth curves, even with suitable reparametrisation.  That is, if two smooth curves $\alpha, \beta \colon I \to P^1 M$ are such that $\alpha(1) = \beta(0)$ and each is flat at that point, then it may not be possible to concatenate them to a smooth path $\alpha # \beta$.  Here is an example.  Note that a smooth path in $P^1 M$ locally lifts to a smooth path in $P M$, which adjoints to a smooth surface in $M$.  So we define $\alpha$ and $\beta$ as maps $I^2 \to M$ with the first parameter being the parameter along the path in $P^1 M$.

$$
\begin{aligned}
\alpha(s,t) &= \big((1 - s) \sin(2 \pi t),1 - \cos(2 \pi t)\big) \\
\beta(s,t) &= \big(s \sin(2 \pi t), -1 + \cos(2 \pi t)\big)
\end{aligned}
$$

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We then reparametrise these in $s$ so that $\alpha$ and $\beta$ are flat (but without sitting instants) at the appropriate end-points ($s = 1$ for $\alpha$, $s = 0$ for $\beta$).  At the end-points, we have $\alpha(1,t) = (0,\sin(2 \pi t))$ which is thinly homotopic to the zero map. Similarly, $\beta(0,t) = (0,-\sin(2\pi t))$ which is, again, thinly homotopic to the zero map.  So $\alpha(1,t) = \beta(0,t)$ but even with the reparametrisation, there is no way to lift the concatenation, $\alpha # \beta$, to $P^1 M$.

If we include a sitting instant at the concatentation point then it is possible to concatentate these paths.  This is because the quotient sets are themselves path-connected, so there is a smooth path between $\alpha(1,t)$ and $\beta(0,t)$ which lies wholly within the space of paths thinly homotopic to the zero map.  While we "sit" at the concatenation point in $P^1 M$, the lifted map is frantically dashing from $\alpha(1,t)$ to $\beta(0,t)$.

### Frölicher Structure ###

The Frölicher structure on $P^1 M$ is determined by the fact that smooth functions $P^1 M \to \mathbb{R}$ are those functions that compose to smooth functions $P M \to \mathbb{R}$.  Then the smooth curves are those that compose to smooth curves under all smooth functions.  This means that concatenations such as the one above are allowed.


smooth structure on a topos

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

#Contents#
* table of contents
{:toc}

## Idea

A [[topos]] can be viewed as a generalization of a [[topological space]].  A _smooth structure on a topos_ is a corresponding generalization of the notion of [[smooth manifold]], in that to put a smooth structure on $Sh(X)$, for a topological space $X$, is (at least closely related to) putting the structure of a smooth manifold on $X$.  However, since it is phrased internally with reference to the (Cauchy and Dedekind) [[real numbers objects]], it is applicable to any topos.

## Definition

Given a [[topos]] $\mathbf{H}$, write

* $\mathbb{N} \in \mathbf{H}$ for its [[natural numbers object]];

* $\mathbb{R}_{C} \in \mathbf{H}$ for its [[Cauchy real numbers object]];

* $\mathbb{R}_{D} \in \mathbf{H}$ for its [[Dedekind real numbers object]];

* $\mathbb{R} \in $ [[Set]] for the external [[real numbers]].

There are canononical [[subobject]] inclusions

$$
  \mathbb{N} \hookrightarrow \mathbb{R}_C \hookrightarrow \mathbb{R}_D
  \,.
$$

+-- {: .num_defn #StandardFunction}
###### Definition
([Fourman 75, def. 3.6](#Fourman75))
A morphism

$$
  g \colon \mathbb{R}_D^{n_1} \longrightarrow \mathbb{R}_D^{n_2}
$$

in the topos $\mathbf{H}$ is called a __standard function__, precisely if

1. it is a [[continuous function]] in the internal sense that

   $$
     \underset{\epsilon \gt 0}{\forall}
     \underset{\delta \gt 0}{\exists}
     \underset{\vec x, \vec y \in \mathbb{R}_D^{n_1}}{\forall}
     \left(
       \left(
         \underset{i}{max} {\vert x_i - y_i \vert}
         \lt \delta
       \right)
       \Rightarrow
       \left(
         \underset{i}{max} {\vert g(\vec x)_i - g(\vec y)_i \vert}
         \lt \delta
       \right)
     \right)
   $$

1. it respects the sub-object of Cauchy reals, in that

   $$
     \underset{\vec x \in \mathbb{R}_C^{n_1}}{\forall}
     \left(
        g(\vec x) \in \mathbb{R}_{C}^{n_2}
     \right)
     \,.
   $$

=--

We will furthermore consider __smooth standard functions__, meaning standard functions that satisfy the internalized ordinary definition of [[smooth function]] (i.e. $C^\infty$).

We define an [[equivalence relation]] on $\mathbb{R}_D^n$ by taking two elements to be equivalent if there are smooth standard functions taking them into each other:

$$
  \vdash
  (r \simeq s)
  \coloneqq
  \underset{{f,g } \atop {smooth \; standard\;functions}}{\exists}
  \left(
     \left(
       f(r) = s
     \right)
        \wedge
     \left(
       g(s) = r
     \right)
  \right)
  \,.
$$

+-- {: .num_defn #SmoothStructureDimN}
###### Definition
([Fourman 75, def. 4.1](#Fourman75))
A __smooth structure of dimension $n$__ on a topos $\mathbf{H}$ is an equivalence class for the above equivalence relation on $\mathbb{R}_D^n$.  In other words, the object of smooth structures of dimension $n$ is the quotient of $\mathbb{R}_D^n$ by this equivalence relation.
=--

+-- {: .num_defn}
###### Definition
([Fourman 75, def. 4.4](#Fourman75))
Given a smooth structure $S$ of dimension $n$ on the topos $\mathbf{H}$ according to def. \ref{SmoothStructureDimN}, then the __smooth real number object__ is the [[subobject]]

$$
  \mathbb{R}_S \hookrightarrow \mathbb{R}_D
$$

defined internally as the set of all images of points in $S\subseteq \mathbb{R}_D^n$ under smooth standard functions $\mathbb{R}_D^n \to \mathbb{R}_D$.  In symbols, it is given by

$$
  \mathbb{R}_S
  \coloneqq
  \left\{
    x \in \mathbb{R}_D
    \;|\;
    \underset{{smooth \; standard\;funct.}\atop{\mathbb{R}_D^n \stackrel{f}{\to} \mathbb{R}_D}}{\exists}
    \underset{s \in S}{\exists}
    (x = f(s))
  \right\}
  \,.
$$

=--

Note that any function constant at a Cauchy real is standard.  Therefore, every Cauchy real is a smooth real.

## Examples

+-- {: .num_example}
###### Example
([Fourman 75, example 4.3 1](#Fourman75))
Let $X$ be a [[smooth manifold]] of [[dimension]] $n$, with [[sheaf topos]] $\mathbf{H} \coloneqq Sh(X)$.  As shown at [[real numbers object]], $\mathbb{R}_D$ is then the sheaf of *continuous* real-valued functions on $X$.

Let $S\subseteq \mathbb{R}_D^n$ be the sheaf of local [[coordinate systems]], i.e. $S(U)$ is the set of real-valued functions $U\to \mathbb{R}^n$ that are smooth and are locally diffeomorphisms onto their images.  Then $S$ is a smooth structure on $Sh(X)$ of dimension $n$, according to def. \ref{SmoothStructureDimN}.

The corresponding object $\mathbb{R}_S$ of smooth reals is the sheaf of *smooth* real-valued functions $X\to \mathbb{R}$.  Note that since $X$ is locally connected, the Cauchy real numbers object $\mathbb{R}_C$ in $Sh(X)$ is the sheaf of locally constant real-valued functions, so $\mathbb{R}_S$ sits strictly in between $\mathbb{R}_C$ and $\mathbb{R}_D$.

=--

+-- {: .num_example}
###### Example
Let $h:1\to \mathbb{R}_D^n$ be any global section of $\mathbb{R}_D^n$.  Then the equivalence class of $h$, under the above equivalence relation, is a smooth structure.  In particular, if $h$ is a tuple of Cauchy real numbers (such as $0$), then so is every point in $S$, and thus $\mathbb{R}_S = \mathbb{R}_C$.  This gives the "discrete" smooth structure on $\mathbf{H}$.

For $\mathbf{H}=Sh(X)$, such a global section is a continuous map $h:X\to \mathbb{R}^n$, and this gives the smooth structure "cogenerated by" $h$, in that it makes $h$ smooth and only those other functions that must be smooth if $h$ is.  The case when $h$ is a Cauchy real corresponds to $h:X\to \mathbb{R}^n$ being locally constant, in which case all smooth functions are locally constant; this is the "minimal" smooth structure on a topological space $X$.
=--

## References

* {#Fourman75} [[Michael Fourman]], _Comparaison des Réels d'un Topos - Structures Lisses sur un Topos Elémentaire_ , Cah. Top. Géom. Diff. Cat. **16** (1975) pp.233-239. ( _Colloque Amiens 1975 proceedings_ ) (p. 18-24 in [NUMDAM](http://www.numdam.org/item?id=CTGDC_1975__16_3_217_0)))


smooth super infinity-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

> this entry is under construction

#Contents#
* table of contents
{:toc}

## Idea

The notion of _smooth super $\infty$-groupoid_ or _smooth super [[geometric homotopy type]]_ is the combination of of _[[super ∞-groupoid]]_ and _[[smooth ∞-groupoid]]_. The [[cohesive (∞,1)-topos]] of smooth super-$\infty$-groupoids is a context that realizes [[higher geometry|higher]] [[supergeometry]].

Smooth super $\infty$-groupoids include
[[supermanifolds]], [[super Lie groups]] and their [[deloopings]] etc. Under [[Lie differentiation]] these map to [[super L-∞ algebras]].

## Definition

We consider one of at least two possible definitions, that differ (only) slightly in some fine technical detail. The other is at _[[super smooth infinity-groupoid]]_.

We take a _smooth super $\infty$-groupoid_ to be a [[smooth ∞-groupoid]] but not over the [[base topos]] [[∞Grpd]] of bare [[∞-groupoid]]s, but over the [[base topos]] [[Super∞Grpd]] of [[super ∞-groupoids]].

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

Write $sCartSp$ for the [[full subcategory]] of that of [[supermanifolds]] on the [[super Cartesian spaces]] $\{\mathbb{R}^{p|q}\}_{p,q \in \mathbb{N}}$. Regard this as a [[site]] by taking the [[coverage]] the product coverage of the [[good open cover]] coverage of [[CartSp]] and the trivial coverage on [[superpoints]].

=--

+-- {: .num_remark}
###### Remark

So a [[covering]] family of $\mathbb{R}^{p|q}$ is of the form

$$
  \{
    U_i \times \mathbb{R}^{0|q} \longrightarrow \mathbb{R}^{p|q}
  \}_{i}
$$

for

$$
  \{
    U_i  \longrightarrow \mathbb{R}^{p}
  \}_{i}
$$

a differentiably [[good open cover]] of $\mathbb{R}^{p}$.

=--

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

Let

$$
  SmoothSuper\infty Grpd := Sh_{(\infty,1)}(sCartSp, Super \infty Grpd)
$$

be the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] over the site $sCartSp$, def. \ref{SuperCartesianSpaces}.

=--

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

The [[(∞,1)-topos]] $Smooth \infty Grpd$ of def. \ref{SmoothSuperInfinityGroupoids} is  a [[cohesive (∞,1)-topos]] over [[∞Grpd]].

=--

This and the stronger statement that it is in fact it is actually cohesive over [[Super∞Grpd]] is discussed [below](CohesionOverBaseToposes), see cor. \ref{SystemOfCohesion}.

## Properties

### Cohesion over smooth $\infty$-groupoids and over super $\infty$-groupoids
 {#CohesionOverBaseToposes}

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

$Smooth Super \infty Grpd$ is a [[cohesive (∞,1)-topos]]
over [[Super∞Grpd]].

$$
  Smooth Super \infty Grpd
   \stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}}
  Super \infty Grpd
 \,.
$$

=--

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

By definition of the [[coverage]] on $sCartSp$ in def. \ref{SuperCartesianSpaces}, the proof of the cohesion of [[Smooth∞Grpd]] = $Sh_\infty(CartSp)$ goes through verbatim _for each fixed [[superpoint]]_ and that gives precisely the claim.

=--

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

[[Super∞Grpd]] is [[infinitesimal cohesion|infinitesimally cohesive]] over [[∞Grpd]].

=--

By the discussion at [[Super∞Grpd]].

+-- {: .num_cor #SystemOfCohesion}
###### Corollary

$SmoothSuper\infty Grpd$ is cohesive and in

In fact we have a [[commutative diagram]] of [[cohesive (∞,1)-topos]]

$$
  \array{
     Smooth Super \infty Grpd
   &\stackrel{\overset{\Pi_{Super}}{\longrightarrow}}{\stackrel{\overset{Disc_{Super}}{\leftarrow}}{\stackrel{\overset{\Gamma_{Super}}{\longrightarrow}}{\underset{coDisc_{super}}{\leftarrow}}}}
   &
   Super \infty Grpd
     \\
     \downarrow \uparrow && \downarrow \uparrow
     \\
     Smooth \infty Grpd
    &
   \stackrel{\overset{\Pi}{\longrightarrow}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\longrightarrow}}{\underset{coDisc}{\leftarrow}}}}
    & \infty Grpd
  }
$$

where the right vertical adjoints exhibit [[infinitesimal cohesion]].

=--

## Structures

We discuss realizations of the general abstract
structures in a cohesive (∞,1)-topos realized in $Smooth Super \infty Grpd$.

### Exponentiated super $L_\infty$-algebras

A [[super L-∞ algebra]] $\mathfrak{g}$ is an [[L-∞ algebra]] internal to
$Sh(SuperPoint)$.

The [[Lie integration]] of $\mathfrak{g}$ is ...

## Applications

* [[D'Auria-Fre formulation of supergravity]]

* [[schreiber:The brane bouquet]] of [[Green-Schwarz action functionals]] for super $p$-[[brane]] [[sigma-models]].

## Related concepts

* [[cohesive (∞,1)-topos]]

  * [[discrete ∞-groupoid]]

  * [[Euclidean-topological ∞-groupoid]]

  * [[smooth ∞-groupoid]]

  * [[synthetic differential ∞-groupoid]]

  * [[super ∞-groupoid]]

  * **smooth super ∞-groupoid**

    * [[Super Gerbes]]

  * [[super formal smooth infinity-groupoid]]

  * [[synthetic differential super ∞-groupoid]]

## References

For general references see the references at _[[super ∞-groupoid]]_ .

Discussion of smooth super $\infty$-groupoids:

* [[Urs Schreiber]], section 4.5 of: _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Hisham Sati]], [[Urs Schreiber]], Section 3.1.3 of: _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

* [[Urs Schreiber]], _[[schreiber:Introduction to Higher Supergeometry]]_, lecture at _[[Higher Structures in M-Theory 2018]]_, [Durham Symposium](http://www.maths.dur.ac.uk/lms/)

  published in parts as: _Higher Structures in M-Theory_ (with [[nLab:Branislav Jurčo]], [[nLab:Christian Saemann]], [[nLab:Martin Wolf]]), Fortschritte der Physik (2019) ([arXiv:1903.02807](https://arxiv.org/abs/1903.02807), [doi:10.1002/prop.201910001]( https://doi.org/10.1002/prop.201910001))

[[!redirects super smooth ∞-groupoid]]
[[!redirects super smooth infintiy-groupoid]]

[[!redirects smooth super ∞-groupoid]]

[[!redirects smooth super infinity-groupoids]]
[[!redirects smooth super ∞-groupoids]]

[[!redirects SuperSmooth∞Grpd]]
[[!redirects SmoothSuper∞Grpd]]

[[!redirects smooth super homotopy type]]
[[!redirects smooth super homotopy types]]

[[!redirects smooth super ∞-group]]
[[!redirects smooth super ∞-groups]]

[[!redirects smooth super-n-group]]
[[!redirects smooth super-n-groups]]

[[!redirects smooth super infinity-group]]
[[!redirects smooth super infinity-groups]]

[[!redirects smooth super space]]
[[!redirects smooth super spaces]]


smooth superalgebra

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _smooth superalgebra_ (or _$C^\infty$-superalgebra_) is a [[superalgebra]]-version of the notion of _[[smooth algebra]]_.

## References

* [[David Yetter]], _Models for synthetic supergeometry_, Cahiers, 29, 2 (1988) ([NUMDAM](http://www.numdam.org/item?id=CTGDC_1988__29_2_87_0))

* [[David Carchedi]], [[Dmitry Roytenberg]], *On Theories of Superalgebras of Differentiable Functions* [[arXiv:1211.6134](http://arxiv.org/abs/1211.6134)]

[[!redirects smooth superalgebras]]

[[!redirects C-infinity superalgebra]]
[[!redirects C-infinity superalgebras]]


smooth topos

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

#Contents#

* automatic table of contents goes here
{:toc}

#Idea#

A __smooth topos__ or __smooth [[lined topos]]__ is
the kind of [[topos]] studied in [[synthetic differential geometry]], a [[category]] of [[generalized smooth spaces]] for which a notion of [[infinitesimal space]] exists.

It is defined to be a [[category]] of objects that behave like [[spaces]], one of which --- the _line_ object $R$ --- is equipped with the structure of a commutative [[algebra]], such that for [[infinitesimal object | infinitesimal objects]] $S \subset R^n$ all morphisms $S \to R$ are _linear_  --- i.e. such that the [[Kock-Lawvere axiom]] holds.

# Definition #

There is a standard definition and various straightforward variations.

## Standard definition ##

For $(\mathcal{T},R)$ a [[lined topos]] there is the notion of an $R$-algebra object in $\mathcal{T}$. For $A$ and $B$ any two $R$-algebra objects, let $R Alg_\mathcal{T}(A,B)$ denote the [[subobject]] of $B^A$ consisting of morphisms $A \to B$ which are algebra homomorphisms. Let $Spec(A)$ denote the algebra spectrum $R Alg_\mathcal{T}(A,R)$ of $A$ in $\mathcal{T}$.

+-- {: .num_defn }
###### Definition
An **$R$-[[infinitesimal object|Weil algebra]]** $W$ is an $R$-algebra of the form $W = R \oplus J$,
where $J$ is an $R$-finite-dimensional nilpotent ideal.

=--

+-- {: .num_defn }
###### Definition
A [[lined topos]] $(\mathcal{T},R)$ is a **smooth topos** if, for each $R$-[[infinitesimal object|Weil algebras]] $W$, the functor $(-)^{Spec W} : \mathcal{T} \to \mathcal{T}$ has a [[right adjoint]] and the canonical morphism

  $$
    W \to R^{Spec(W)}
  $$

  is an [[isomorphism]] in $\mathcal{T}$.
=--

That is to say, each $R$-Weil algebra in a lined topos is [[infinitesimal object|infinitesimal]] and satisfies the [[Kock-Lawvere axiom]]. The right adjoint of $(-)^{Spec W}$ is known as the "[[amazing right adjoint]]".

# Examples #

## Well-adapted models ##

A smooth topos $(\mathcal{T},R)$ is called a **well adapted model** if there is a [[full and faithful functor]]

$$
  Diff \hookrightarrow \mathcal{T}
$$

from the [[category]] [[Diff]] of [[smooth manifold | smooth manifolds]] into it, that takes the real line $\mathbb{R}$ to the [[lined topos|line object]] $R$.

In these _well adapted models_ ordinary [[differential geometry]] is therefore faithfully embedded.

For a list of examples of well adapted models see

* [[Models for Smooth Infinitesimal Analysis]].

Notice that by far not all models are of this form, as the following examples show. On the contrary, the axioms of [[synthetic differential geometry]] may be regarded as providing a unified framework in particular for [[differential geometry]] of [[manifold | manifolds]] and [[algebraic geometry]] of [[algebraic space | algebraic spaces]], [[scheme | schemes]] and other objects.

## Models for supergeometry ##

It is straightforward to slightly enhance the axioms of smooth toposes such as to incorporate the step from [[differential geometry]] to [[supergeometry]], one just requires that algebra structure on the [[lined topos|line object]] $R$ is further refined to that of a [[superalgebra]]. The result is called a [[super smooth topos]]. See there for a list of models of these.

## Models for algebraic geometry ##

A simple model of a [[smooth topos]] that may be regarded as a context inside which much of [[algebraic geometry]] takes place is the following:

Let $k$ be a [[field]] and let $ (k-Alg^{finp})^{op}$ be the [[opposite category]] of the [[category]] of finitely presented $k$-[[algebra | algebras]]. Then the [[presheaf]] category $\mathcal{T} = PSh((k-Alg^{finp})^{op})$ equipped with the [[lined topos|line object]] $R = k[T]$ (the algebra of polynomials over $k$ in one variable $T$)

$$
  (
    \mathcal{T} = PSh((k-Alg^{finp})^{op}),
    R = k[T]
  )
$$

is a smooth topos. This is described in section 9.3 of

* [[Anders Kock]], _Synthetic geometry of manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

Notice that despite the name of that book, this model is _not_ a well adapted model in that ordinary [[smooth manifold | smooth manifolds]] do not embed full and faithfully into this topos.

Instead, interpreting the internal notion of manifold described in that book -- called _formal manifolds_ in the model $\mathcal{T} = PSh((k-Alg^{finp})^{op})$ produces something like [[formal scheme | formal schemes]] over $k$.

Indeed, much of [[algebraic geometry]] over $k$ may be thought of as being concerned with this model for a smooth topos. A main difference is that in [[algebraic geometry]] attention is usually focused on particularly well behaved objects inside $\mathcal{T} = PSh((k-Alg^{finp})^{op})$: those that satisfy a [[sheaf]] condition with respect to a [[Grothendieck topology]] on $(k-Alg^{finp})^{op}$ and among those moreover those that are _locally isomorphic_ to a [[representable functor|representable]]: these are the [[scheme | schemes]] or [[algebraic space | algebraic spaces]] over $k$.

Probably the [[category of sheaves]] [[localization]] of $PSh((k-Alg^{finp})^{op})$ with respect to one of the standard topologies (Zariski or etale) is still a [[smooth topos]], so that this condition can be enforced by passing to a more restrictive model.

But since [[scheme | schemes]] alone will never form a [[topos]], and smooth topos axiomatization of [[algebraic geometry]] will always contain more general -- also more "pathological" -- objects than [[scheme | schemes]]. It's an example of an old dictum by [[Grothendieck]] that it is more useful to have a nicely behaved category that contains pathological objects than a badly-behaved category of only nice objects: the [[topos]]-[[category theory|general nonsense]] allows useful general constructions in $\mathcal{T}$ which may in each individual case be checked for whether they land in the sub-category of [[schemes]] or [[algebraic space | algebraic spaces]] or not.

A similar comment of course applies to the "well-adapted models" mentioned above: into these ordinary [[manifold | manifolds]] only embed, they contain "smooth space"s much more general than manifolds (such as [[diffeological space | diffeological spaces]]) but also possibly "pathological" ones, from some perspective or other.

See also

* [[synthetic differential geometry applied to algebraic geometry]].

## Warning on preservation of (co)limits ##

Most of the examples above provided [[topos | toposes]] into which a [[category]] $NiceSpaces$ of nicely behaved [[space | spaces]], such as [[manifold | manifolds]] or [[scheme | schemes]] embeds [[full and faithful functor|full and faithfully]].

$$
  NiceSpaces \hookrightarrow \mathcal{T}
  \,.
$$

But the inclusion functor will in general not preserve all [[limit | limits]] and [[colimit | colimits]] that exist in $NiceSpaces$.

For instance for the well-adopted models the full and faithful inclusion of [[Diff]] typically respects only [[pullback | pullbacks]] of [[manifold | manifolds]] along [[transversal map]]. This is because this is the case for the inclusion $Diff \hookrightarrow \mathbb{L}$ of [[Diff]] into the category of [[smooth locus|smooth loci]] and the [[Grothendieck topology]] for these models is typically [[subcanonical coverage|subcanonical]]. See the discussion at [[smooth locus]] for more on this.

This means that universal constructions in a smooth topos may yield different results than in a smaller category of more nicely behaved spaces. However, it is noteworthy that in the above example of manifolds, one may argue that the ordinary [[pullback]] of manifolds along non-[[transversal map | transversal maps]] is the _wrong_ pullback in any case: it doesn't behave well with the [[cohomology]] of manifolds. One motivation for derived geometry, in the case of manifolds specifically the motivation for considering [[derived smooth manifold | derived smooth manifolds]], is to pass from objects in a [[topos]] instead more generally to objects in an [[(∞,1)-topos]] of [[derived stack|stack]] [[∞-stack | ∞-stacks]].

[[!redirects smooth topoi]]
[[!redirects smooth toposes]]


smoothing kernels

Compact operators


SmoothManifolds

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

#Contents#
* table of contents
{:toc}

## Definition

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

$SmoothManifolds$ is the [[category]] whose

* [[objects]] are [[smooth manifolds]]

* [[morphisms]] are [[smooth functions]] between these.

Similarly, for $n \in \mathbb{N}$

$DifferentiableManifolds_n$ is the [[category]] whose

* [[objects]] are [[differentiable manifolds]]

* [[morphisms]] are [[differentiable functions]] between these

for $n$-fold differentiabiliy.

Each of these categories is also commonly denoted $Man$ or $Mfd$ or [[Diff]] etc.

=--

## Properties

### As a site

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

The category $SmoothManifolds$ becomes a [[large site]] by equipping it with the [[coverage]] consisting of [[open cover]]s.

This is an [[essentially small site]]: a [[dense sub-site]] for $SmoothManifolds$ is given by [[CartSp]]${}_{smooth}$.

=--

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

The first statement follows trivially as for [[Top]]: the preimage of an [[open subset]] under a [[continuous function]] is again open (by definition of continuouss function).

For the second statement one needs that every [[paracompact manifold]] admits a _differentially [[good open cover]]_ : an [[open cover]] by [[open ball]]s that are [[diffeomorphic]] to a [[Cartesian space]]s. The proof for this is spelled out at [[good open cover]].

=--

+-- {: .un_corollary}
###### Corollary

The [[sheaf topos]] over $SmoothManifolds$ is a [[cohesive topos]].

The [[hypercompletion]] of the [[(∞,1)-sheaf (∞,1)-topos]] over $SmoothManifolds$ is a [[cohesive (∞,1)-topos]].

=--

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

For the first statement, use that by the _comparison lemma_ discussed at [[dense sub-site]] we have an [[equivalence of categories]]

$$
  Sh(SmoothManifolds) \simeq Sh(CartSp_{smooth})
  \,.
$$

By the discussion at [[CartSp]] we have that $CartSp_{smooth}$ is a [[cohesive site]]. By the discussion there the claim follows.

For the second statement observe that the Joyal-Jardine [[model structure on simplicial sheaves]] $Sh(SmoothManifolds)^{\Delta^{op}}_{loc}$ is a [[presentable (∞,1)-category|presentation]] for the [[hypercompletion]] of the [[(∞,1)-category of (∞,1)-sheaves]] $\hat Sh_{(\infty,1)}(SmoothManifolds)$ (see [[presentations of (∞,1)-sheaf (∞,1)-toposes]]). By the above result it follows that there is an [[equivalence of (∞,1)-categories]] between the [[hypercompletion]]s

$$
  \hat Sh_{(\infty,1)}(SmoothManifolds) \simeq \hat Sh_{(\infty,1)}(CartSp_{smooth})
  \,.
$$

Now [[CartSp]]${}_{smooth}$ is even an [[∞-cohesive site]]. By the discussion there it follows that $Sh_{(\infty,1)}(CartSp_{smooth})$ (before [[hypercompletion]]) is a [[cohesive (∞,1)-topos]]. This means that it is in particular a [[local (∞,1)-topos]]. But this implies (as discussed there), that the [[(∞,1)-category of (∞,1)-sheaves]] already is the [[hypercomplete (∞,1)-topos]]. Therefore finally

$$
  \cdots \simeq Sh_{(\infty,1)}(CartSp_{smooth})
  \,.
$$

=--

+-- {: .un_remark}
###### Remark

The [[cohesive topos]] $Sh(SmoothManifolds) \simeq Sh(CartSp_{smooth})$ is in particular the home of [[diffeological space]]s. See there for more details.

The [[cohesive (∞,1)-topos]]

$$
  Smooth \infty Grp := Sh_{(\infty,1)}(SmoothManifolds) \simeq Sh_{(\infty,1)}(CartSp_{smooth})
$$

is that of [[smooth ∞-groupoid]]s. Discussed at [[Smooth∞Grpd]].

The theory of [[differentiable stacks]] is that of [[geometric stack]]s in the [[(2,1)-sheaf]] [[(2,1)-topos]]

$$
  Sh_{(2,1)}(SmoothManifolds) \simeq Sh_{(2,1)}(CartSp_{smooth})
   \simeq
  \tau_{\leq 1} Sh_{(\infty,1)}(CartSp_{smooth})
$$

=--

## Related concepts

* [[CartSp]]${}_{top}$ ,  [[TopMfd]]

* [[CartSp]]${}_{smth}$

* [[Diff]]

category: category

[[!redirects Man]]
[[!redirects Mfd]]

[[!redirects SmoothMfd]]
[[!redirects SmthMfd]]
[[!redirects SmthManifolds]]


smothering 2-functor

# Smothering 2-functors

* table of contents
{:toc}

## Idea

A *smothering 2-functor* is the [[strict 2-category|2-categorical analogue]] of a [[smothering functor]].

## Definition

A 2-functor $F:C\to D$ is **smothering** if

1. It is [[surjective on objects]], and
1. It is *locally smothering*, i.e. each hom-functor $C(x,y) \to D(F x, F y)$ is a [[smothering functor]].  Thus, $F$ is full on 1-cells, full on 2-cells, and conservative on 2-cells.

## Properties

* Smothering 2-functors reflect equivalences: if $f:x\to y$ is in $C$ and $F(f)$ is an equivalence, then $f$ is an equivalence.  For by fullness on 1-cells, any inverse of $F(f)$ lifts to $C$, by fullness on 2-cells, the comparison 2-cells lift to $C$, and then by conservativity those lifts are again isomorphisms.

* Applying fullness on 1-cells again, if $x,y\in C$ and $F x \simeq F y$ in $D$, then $x\simeq y$ in $C$.

* Smothering 2-functors lift and reflect adjoints: any [[adjunction]] in $D$ is the image of some adjunction in $C$, and the lift can make use of any specified lifts of the objects, 1-cells, and either the unit or the counit 2-cell.  This is [RV, 4.5.2](#RV); the idea is to lift the missing 2-cell arbitrarily and then "fix" it by composing with the inverse of one [[triangle identity]] composite (which is an isomorphism by 2-cell conservativity).

## Examples

* For any [[∞-cosmos]] $K$, there is a smothering 2-functor $(K/A)_2 \to K_2/A$, where $K_2$ denotes the [[homotopy 2-category]] of an [[∞-cosmos]].  In the special case when $K=QCat$ consists of [[quasi-categories]], this is [RV, 3.4.7](#RV).

## Related pages

* [[smothering functor]]

## References

* [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, [arXiv](http://arxiv.org/abs/1306.5144)
 {#RV}


smothering functor

# Smothering functor

* table of contents
{:toc}

## Idea

A *smothering functor* is a [[functor]] that is "almost an [[equivalence of categories]]" except that it may not be [[faithful functor|faithful]].  Smothering functors tend to arise when comparing "different [[homotopy categories]]" of the same category that impose more or less refined notions of [[homotopy equivalence]].  Frequently they can be treated more or less like equivalences.

The notion is due to [Riehl and Verity](#RV).

## Definition

A functor $F:C\to D$ is **smothering** if it is

1. [[surjective on objects functor|surjective on objects]],
1. [[full functor|full]], and
1. [[conservative functor|conservative]].

If instead of being surjective on objects $F$ is [[essentially surjective]] on objects, we may say that $F$ is **weakly smothering**.

## Properties

* Each (strict) [[fiber]] of a smothering functor is an [[inhabited]] [[connected]] [[groupoid]]. ([RV, 3.3.2](#RV))
* If $F:C\to D$ is smothering and $x,y\in C$ satisfy $F x \cong F y$ in $D$, then $x \cong y$ in $C$, by fullness combined with conservativity.  In other words, $F$ is "full on isomorphisms" (but since it is not faithful, it is not [[pseudomonic]]).
* Further combining this with surjectivity on objects, every smothering functor is an [[isofibration]].

## Examples

* For any [[model category]] $C$, the functor $Ho(C^{\mathbf{2}}) \to Ho(C)^{\mathbf{2}}$ is weakly smothering, where $\mathbf{2}$ denotes the [[interval category]].  This property appears in the axiom (Der5) for [[derivators]].

## Related pages

* [[smothering 2-functor]]
* [[derivator]]

## References

* [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics 280 (2015) 549 - 642, [arXiv](http://arxiv.org/abs/1306.5144)
 {#RV}

[[!redirects smothering functors]]
[[!redirects weakly smothering functor]]
[[!redirects weakly smothering functors]]


Snaith theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The original theorem by Snaith ([Snaith 79](#Snaith79)) identifies the [[complex K-theory spectrum]] [[KU]] simply as the localization of the [[∞-group ∞-ring]] $\mathbb{S}[B U(1)]$ of the [[circle 2-group]] away from the [[Bott element]] $\beta$:

$$
  KU \simeq \mathbb{S}[B U(1)][\beta^{-1}]
  \,.
$$

Later, further instances of such characterizations of familiar [[E-∞ rings]] have been given:

* in ([Gepner-Snaith 08](#GepnerSnaith08)) the analogous statement for [[algebraic K-theory]] is formulated in terms of [[motivic spectra]];

* in ([Westerland 12](#Westerland12)) the tower of [[Morava E-theories]] (or some [[homotopy fixed point]] [[spectra]] of these) is shown to be localizations of the [[∞-group ∞-rings]] of $B^{n+1} \mathbb{Z}_p$, for all $n \in \mathbb{N}$.

## Statement

### Preliminaries

For $A$ an [[abelian ∞-group]] write $E \coloneqq \mathbb{S}[A] = \Sigma^\infty_+ A$ for its [[∞-group E-∞ ring]].

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

For $\beta \in \pi_n(E)$ an element of the $n$th stable [[homotopy group]], then _multiplication by $\beta$_ is a homomorphism

$$
  \beta_\ast
  \;\colon\;
  E
  \to
  \Sigma^{-n} E
  \,.
$$

The _localization_ of $E$ at $\beta$ is the [[homotopy colimit]] over the iterated multiplication with $\beta$

$$
  E[\beta^{-1}]
  \coloneqq
  \underset{\to}{\lim}
  \left[
    E \stackrel{\beta_\ast}{\to}
    \Sigma^{-n}E
    \stackrel{\Sigma^{-n} \beta_\ast}{\to}
   \Sigma^{-2n} E
   \to
   \cdots
  \right]
$$

which has the [[universal property]] that $\mu_\beta$ becomes an [[equivalence]] on $E[\beta^{-1}]$.

=--

### For complex topological K-theory
 {#ForComplexTopologicalKTheory}

The original formulation of Snaith's theorem ([Snaith 79, theorem 2.12](#Snaith70), spring) for complex [[topological K-theory]].

#### Preliminaries

Write $(Ho(Top), \times, \ast)$ for the [[classical homotopy category]], regarded as a [[symmetric monoidal category]] under forming [[product spaces]], with [[tensor unit]] the [[point space]].

Write $(Ho(Top^{\ast/}), \wedge , S^0)$ for the homotopy category of [[pointed topological spaces]] with [[tensor product]] the [[smash product]] of pointed spaces and [[tensor unit]] the [[0-sphere]]

Write $(Ho(Spectra), \wedge, \mathbb{S})$ be the [[stable homotopy category]] with its [[symmetric monoidal smash product of spectra]] $\wedge$ whose [[tensor unit]] is the [[sphere spectrum]] $\mathbb{S}$.

For $A,B \in Ho(Spectra)$ two [[spectra]], we write

$$
  [A,B] \coloneqq Hom_{Ho(Spectra)}(A,B) \in Ab
$$

for [[hom-set]] in the [[stable homotopy category]] and write

$$
  [A,B]_\bullet \coloneqq [\Sigma^\bullet A, B]
$$

for the corresponding $\mathbb{Z}$-graded group ([this def.](Introduction+to+Stable+homotopy+theory+--+1-1#GradedAbelianGroupStructureOnHomsInTheHomotopyCategory)).

There are two pairs of ([[derived functor|derived]]) [[adjoint functors]]

$$
  Ho(Spectra)
    \underoverset
      {\underset{\Omega^\infty}{\longrightarrow}}
      {\overset{\Sigma^\infty}{\longleftarrow}}
      {}
  Ho(Top^{\ast/})
    \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{(-)_+}{\longleftarrow}}
      {\bot}
  Ho(Top)
$$

which are [[strong monoidal functors]] (by [this example](pointed+object#WedgeAndSmashOfBasePointAdjoinedTopologicalSpaces) and [this prop](Introduction+to+Stable+homotopy+theory+--+1-1#StableHomotopyCategoryIsIndeedStabilizationOfClassicalHomotopyCategory)). The left adjoint composite

$$
  \mathbb{S}[-]
    \;\colon
  \Sigma^\infty((-)_+)
$$

hence takes [[H-spaces]] and in particular [[H-groups]] $G$ to [[ring spectra]].

More in detail, an [[H-space]] structure is a $G \in Ho(Top)$ equipped with morphisms

$\mu \;\colon\; G \times G \longrightarrow G$

$e \;\colon\; \ast \longrightarrow G$

satisfying [[associativity]] and [[unitality]] in $Ho(Spectra)$, and the corresponding ring spectrum has product

$$

  \left(\Sigma^\infty(G_+)\right)
    \wedge
  \left( \Sigma^\infty G_+\right)
   \;\simeq\;
  \Sigma^\infty( (G \times G)_+ )
    \overset{\Sigma^\infty (e_+)}{\longrightarrow}
  \Sigma^\infty (G_+)
$$

and unit

$$
  \mathbb{S}
    \simeq
  \Sigma^\infty( \ast_+)
    \overset{ \Sigma^\infty(e_+) }{\longrightarrow}
  \Sigma^\infty(G_+)
$$

We call

$$
  \mathbb{S}[G] \coloneqq \Sigma^\infty(G_+)
$$

equipped with this monoid structure
the _[[H-group ring spectrum]]_ of $G$. See [there](∞-group+∞-ring#HGroupRingSpectra) for more.

#### The ring spectrum of the circle 2-group

One such [[H-group]] is the _[[circle 2-group]]_, hence (the [[homotopy type]] of) the [[classifying space]] $B U(1)$ for [[complex line bundles]], equivalently the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$, canonically presented by the [[complex projective space]] $\mathbb{C}P^\infty$

$$
  K(\mathbb{Z},2) \simeq B U(1) \simeq \mathbb{C}P^\infty
  \;\in\;
  Ho(Spaces)
  \,.
$$

This being the [[classifying space]] for [[complex line bundles]], it becomes an [[H-group]] via the map

$$
  B U(1) \times B U(1) \longrightarrow B U(1)
$$

which classifies the [[tensor product of vector bundles|tensor product of line bundles]], with inverses given by the map

$$
  B U(1) \longrightarrow B U(1)
$$

which form [[dual vector bundle|dual line bundles]].

Hence its [[H-group ring spectrum]] is

$$
  \mathbb{S}[B U(1)]
   \;=\;
  \Sigma^\infty( B U(1)_+ )
  \,.
$$

Therefore for $X \in Ho(Top^{\ast/})$ a [[pointed topological space]], then

$$
  [\Sigma^\infty X,\Sigma^\infty(G_+)]_\bullet
$$

is a graded ring, with the product of elements

$$
  \left(
    \Sigma^{n_i} \Sigma^\infty X
     \overset{\alpha_i}{\longrightarrow}
   \Sigma^\infty(G_+)
  \right)
  \;\in\;
  [\Sigma^\infty X, \Sigma^\infty(G_+)]_{n_i}
$$

for $i \in \{1,2\}$ given by

$$
  \Sigma^{n_1 + n_2}
  \Sigma^\infty X
    \overset{\Sigma^{n_1+ n_2} \Sigma^\infty \Delta_X}{\longrightarrow}
  \Sigma^{n_1 + n_2}
  \Sigma^\infty (X \wedge X)
   \simeq
  \left(
    \Sigma^{n_1} \Sigma^\infty X
  \right)
    \wedge
  \left(
     \Sigma^{n_2} \Sigma^\infty X
  \right)
    \overset{ \alpha_1 \wedge \alpha_2 }{\longrightarrow}
  \left(
    \Sigma^\infty (G_+)
  \right)
    \wedge
  \left(
    \Sigma^\infty (G_+)
  \right)
    \overset{\Sigma^\infty (\mu_+)}{\longrightarrow}
  \Sigma^\infty (G_+)
  \,.
$$

Here the isomorphism on the left is the combination of the strong monoidalness of $\Sigma^\infty$ with the respect of suspension $\Sigma$ for the smash product of spectra (the [[tensor triangulated category]] structure on $Ho(Spectra)$, [this prop.](Introduction+to+Stable+homotopy+theory+--+1-2#TensorTriangulatedStructureOnStableHomotopyCategory)).

Observe that we have a splitting

$$
  \Sigma^\infty(B U(1)_+)
   \;\simeq\;
  \left(
    \Sigma^\infty (B U(1))
  \right)
    \;\oplus\;
  \mathbb{S}
$$

(by [this remark](∞-group+∞-ring#HGroupRingSpectrumSplitsAsDirectSumWithSphereSpectrum)) and hence a canonical morphism

$$
  \Sigma^\infty (B U(1))
    \longrightarrow
  \Sigma^\infty (B U(1)_+)
  \,.
$$

Via this splitting, the morphism in $Ho(Top^{\ast/})$

$$
  h \colon S^2 \longrightarrow B U(1)
$$

in $[S^2, B U(1)] \simeq \pi_2(B U(1)) \simeq \mathbb{Z}$ which classifies the [[basic complex line bundle on the 2-sphere]] and represents $1 \in \mathbb{Z}$, induces a morphism

$$
  \beta
    \;\colon\;
  \Sigma^2 \mathbb{S}
    \simeq
  \Sigma^\infty S^2
    \overset{ \Sigma^\infty( h ) }{\longrightarrow}
  \Sigma^\infty B U(1)
    \overset{}{\longrightarrow}
  \Sigma^\infty (B U(1)_+)
$$

hence an element in $[\mathbb{S}, \Sigma^\infty(B U(1)_+)]_2$.

#### The map to the K-theory spectrum
 {#MapToKTheorySpacetum}

The [[spectrum]] [[KU]] which [[Brown representability theorem|represents]]  complex [[topological K-theory]]
has in degree 0 the the [[product space]]

$$
  \Omega^\infty KU \simeq B U \times \mathbb{Z} \in Ho(Top^{\ast/})
$$

of the stable [[classifying space]] $B U$ for [[complex vector bundles]] and the [[integers]]. The base point is $(\ast, 0)$.

The [[projection]]

$$
  B U \times \mathbb{Z} \longrightarrow \mathbb{Z}
$$

classifies the virtual [[rank of a vector bundle|rank]] of [[virtual vector bundle]].

Hence the inclusion of classifying spaces

$$
  B U \simeq B U \times \{0\} \hookrightarrow  B U \times \mathbb{Z}
$$

classifies the inclusion $\tilde K_{\mathbb{C}}(-) \hookrightarrow K_{\mathbb{C}}(-)$ of [[reduced K-theory]].

There is a canonical morphism

$$
  \Sigma^\infty(B U(1)_+)
    \overset{\Phi}{\longrightarrow}
  K U
$$

in $Ho(Spectra)$, being the $(\Sigma^\infty \dashv \Omega^\infty)$-[[adjunct]] of

$$
  B U(1)_+ \longrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U
$$

in $Ho(Top^{\ast/})$, which in turn is the $((-)_+ \dashv U)$-[[adjunct]] of the canonical

$$
  \mathcal{O}(1) \;\colon\; B U(1) \longrightarrow B U \simeq B U \times \{1\} \hookrightarrow B U \times \mathbb{Z}
$$

in $Ho(Top)$.

Since the [[formal group law]] for K-theory says that $\mu^\ast ( \mathcal{O}(1) ) \simeq pr_1^\ast (\mathcal{O}(1)) \otimes pr_2^\ast(\mathcal{O}(2))$ $\Phi$ is  a homomoprhism of ring spectra.

Under the above splitting, the morphism $\Phi$ decomposes as

$$
  \Phi
    \;\colon\;
  \Sigma^\infty (B U(1)_+)
    \simeq
  \Sigma^\infty B U(1)  \oplus \mathbb{S}
    \overset{ ( \tilde i , e_{KU})  }{\longrightarrow}
  KU
$$

Since, by the above, morphisms $\Sigma^\infty B U(1) \longrightarrow K U$ in $Ho(Spectra)$,
hence equivalently morphisms $B U(1) \longrightarrow B U \times \mathbb{Z}$ in $Ho(Top^{\ast/})$,
hence equivalently morphisms $B U(1) \to B U$ in $Ho(Top)$  correspond to the reduced K-theory of
$B U(1)$, and since morphisms $\mathbb{S} \to KU$ in $Ho(Spectra)$, hence equivalently morphism $\ast \to B U \times \mathbb{Z}$
in $Ho(Top)$ correspond to the K-theory of the point, and since over (colimits of) [[compact topological spaces]]
K-theory splits as $K(X) \simeq \tilde K(X) \oplus K(\ast)$ via $[E]- n \mapsto ([E] - rk(E)) + ( rk(E) - n )$
([this prop](topological+K-theory#KGrupDirectSummandReducedKGroup)) it follows that

1. $\tilde i$ takes the canonical line bundle $\mathcal{O}(1)$ on $B U(1)$ to its image $\mathcal{O}(1)-1$ in reduced K-theory

   $$
     B U(1) \overset{\mathcal{O}(1) \mapsto (\mathcal{O}(1)-1)}{\longrightarrow} B U \simeq B U \times \{0\} \hookrightarrow B U \times \mathbb{Z} \simeq \Omega^\infty K U
   $$

1. $e_{K U}$ is adjunct to $\ast \simeq (\ast,1) \hookrightarrow B U \times \mathbb{Z}$ ( is the ring spectrum unit of $K U$).

Now observe that $\Phi$ takes multiplication by $\beta$ to multiplication with the [[Bott element]] $(h-1)$:

This is because multiplication by $\beta$ is the outer right boundary of the following diagram

$$
  \array{
     \Sigma^\infty S^2 \wedge \Sigma^n \Sigma^\infty X
     \\
     {}^{\mathllap{ (h \oplus 0) \wedge id }}\downarrow
     \\
     \left(\Sigma^\infty( B U(1) ) \oplus \mathbb{S}\right) \wedge \Sigma^n \Sigma^\infty X
     \\
     {}^{\mathllap{id \wedge \alpha}}\downarrow
     \\
     \Sigma^\infty(B U(1)_+) \wedge \Sigma^n \Sigma^\infty( B U(1)_+ )
       &\overset{\Sigma^n\Sigma^\infty (\mu_+)}{\longrightarrow}&
     \Sigma^n \Sigma^\infty (B U(1)_+)
     \\
     \downarrow && \downarrow
     \\
     K U \wedge \Sigma^n K U &\longrightarrow & \Sigma^n K U
  }
$$

and since the bottom square commutes (since tensor product of line bundles corresponds to their product in K-theory)
this is equivalent to the left and bottom boundary, which, by the above discussion, is multiplication with the
[[Bott element]] $(h-1)$.

Since the Bott element is invertible in $K U$, this means for all $X \in Ho(Top^\ast/)$ that the morphism

$$
  [\Sigma^\infty X , \Sigma^\infty ( B (1)_+ ) ]_\bullet
    \overset{\Phi_X \coloneqq \Phi \circ (-)}{\longrightarrow}
  [\Sigma^\infty X, K U]_\bullet
    \simeq
  \tilde K_{\mathbb{C}}^\bullet(X)
$$

extends to the quotient ring

$$
  [\Sigma^\infty X, \Sigma^\infty ( B U(1)_+ )]_{\bullet}[\beta^{-1}]
$$

in which two elements are identified if they differ by multiplication by $\beta$, as above:

$$
  \array{
    [\Sigma^\infty X, \Sigma^\infty( B U(1)_+ )]  &\overset{\Phi}{\longrightarrow}& \tilde K_{\mathbb{C}}^\bullet(X)
    \\
    \downarrow & \nearrow_{\mathrlap{\exists \Phi}}
    \\
    [\Sigma^\infty X, \Sigma^\infty( B U(1)_+ )]_\bullet [\beta^{-1}]
  }
  \,.
$$

#### Isomorphy

+-- {: .num_theorem}
###### Theorem

If $X \in Ho(Top^{\ast/})$ has the homotopy type of a [[finite CW-complex]], then the [[natural transformation]]

$$
  \Phi_X
    \;\colon\;
  [X, \Sigma^\infty ( B U(1)_+ )]_\bullet [\beta^{-1}]
    \longrightarrow
  \tilde K^\bullet(X)
$$

is a [[natural isomorphism]].

=--

([Snaith 81, theorem 2.12](#Snaith81), [Hopkins-Mathew](#HopkinsMathew))

That (before localization) the map is an epimorphism is due to ([Segal 73, prop. 1](#Segal73)), see [this prop.](complex+projective+space#HGroupRingSpectrumSurjectsOntoTopologicalKTheory). The analog of this statement for [[real projective space]] $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$ instead of [[complex projective space]] $\mathbb{C}P^\infty \simeq B U(1)$ is the [[Kahn-Priddy theorem]].

### For periodic complex cobordism

+-- {: .num_theorem}
###### Theorem

The [[periodic complex cobordism spectrum]] is the [[∞-group ∞-ring]] of the [[classifying space]] for stable [[complex vector bundles]] (the classifying space for [[topological K-theory]]) localized at the [[Bott element]] $\beta$:

   $$
     PMU \simeq (\mathbb{S}[B U])[\beta^{-1}]
     \,.
   $$

=--

([Snaith 81, theorem 2.7](#Snaith81))

### For algebraic K-theory

+-- {: .num_remark}
###### Remark

The analog of this result for the periodic [[algebraic cobordism spectrum]] and [[algebraic K-theory]] as [[motivic spectra]] is discussed in ([GepnerSnaith 08](#GepnerSnaith08)).

=--

### For smooth spectra and differential K-theory

Refinement of the Snaith theorem for [[KU]] to [[smooth spectra]] and to [[differential K-theory]] is in ([Bunke-Nikolaus-Völkl 13, section 6.3](#BunkeNikolausVoelkl13)). See at _[differential cohomology diagram -- Smooth Snaith K-theory](differential%20cohomology%20diagram#ViaSmoothSnaithTheorem)_.

### Snaith-like theorem for Morava $E$-theories
 {#ForMoravaETheory}

Write $\Gamma_n$ for the [[Honda formal group]]. The [[automorphism group]] $Aut(\Gamma_n)$ induces for each prime $p$ a canonical [[determinant]] morphism

$$
  det \;\colon\; \mathbb{G}_n \longrightarrow \mathbb{Z}_p^\times
$$

from the [[Morava stabilizer group]] $\mathbb{G}_n \coloneqq Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) \ltimes Aut(\Gamma_n)$.

Write

$$
  S \mathbb{G}_n \coloneqq ker(det)
$$

for the [[kernel]]. This naturally [[infinity-action|acts]] on the [[Morava E-theory]] spectrum $E_n$. Write $E^{S\mathbb{G}_n}$ for the corresponding [[homotopy fixed point]] [[spectrum]]. ([Westerland 12, 1.1](#Westerland12)).

Write $L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p]$ for the $K(n)$-localization of the [[∞-group ∞-ring]] of the [[n-group|(n+1)-group]] $B^n \mathbb{Z}_p$.

+-- {: .num_theorem}
###### Theorem

There is a [[generalized element]] $\rho_n$ of the [[E-∞ ring]] $L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p]$ such that localization at that element yields the [[Morava E-theory]] spectrum $S\mathbb{G}_n$[[homotopy fixed points]]:

$$
  L_{K(n)} \mathbb{S}[B^{n+1} \mathbb{Z}_p] [\rho_n^{-1}]

  \stackrel{\simeq}{\longrightarrow}
  E_n^{S\mathbb{G}_n}
  \,.
$$

=--

([Westerland 12, theorem 1.2](#Westerland12))

## Related concepts

* [[splitting principle]]

* [[algebraic cobordism]]

## References

The theorem is due to

* {#Snaith79} [[Victor Snaith]], _Algebraic Cobordism and K-theory_, Mem. Amer. Math. Soc. no 221 (1979)

with a simpler proof given in

* {#Snaith81} [[Victor Snaith]], _Localized stable homotopy of some classifying spaces_, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 2, 325-330. MR 600247 (82g:55006) ([pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100058205), [doi:10.1017/S0305004100058205](https://doi.org/10.1017/S0305004100058205))

using results from

* {#Segal73} [[Graeme Segal]], _The stable homotopy of complex of projective space_, The quarterly journal of mathematics (1973) 24 (1): 1-5. ([[Segal72.pdf:file]], [doi:10.1093/qmath/24.1.1]( https://doi.org/10.1093/qmath/24.1.1))

Another proof due to [[Mike Hopkins]] is in

* {#HopkinsMathew} [[Akhil Mathew]] (following [[Mike Hopkins]]), _Snaith's construction of complex K-theory_ ([pdf](http://math.uchicago.edu/~amathew/snaith.pdf), [[MathewSnaithTheorem.pdf:file]])

Refinement to [[smooth spectra]] and [[differential K-theory]] is in

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 6.3 of _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

Discussion of the [[E-∞ ring]] structure involved is around theorem 3.1 of

* [[Jacob Lurie]],  _[[A Survey of Elliptic Cohomology]]_

A version for [[motivic spectra]] [[algebraic K-theory]] is discussed in

* [[David Gepner]], [[Victor Snaith]], _On the motivic spectra representing algebraic cobordism and algebraic K-theory_, Documenta Math.  2008 ([arXiv:0712.2817](http://arxiv.org/abs/0712.2817))
 {#GepnerSnaith08}

and for [[motivic cohomology]] in

* [[Markus Spitzweck]], [[Paul Arne Østvær]], _The Bott inverted infinite projective space is homotopy algebraic K-theory_ ([pdf](http://folk.uio.no/paularne/bott.pdf))

Higher [[chromatic homotopy theory|chromatic]] analogs for [[Morava E-theory]] are discussed in

* {#Westerland12} [[Craig Westerland]], _A higher chromatic analogue of the image of J_ ([arXiv:1210.2472](http://arxiv.org/abs/1210.2472))

* {#LindSatiWesterland16} [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], _A higher categorical analogue of topological T-duality for sphere bundles_ ([arXiv:1601.06285](http://arxiv.org/abs/1601.06285))

A unifying [[general abstract]] perspective is discussed in

* {#Arndt17} [[Peter Arndt]], section 3.2 of _Abstract motivic homotopy theory_, thesis 2017 ([web](https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full),  [pdf](https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf), [[ArndtAbstractMotivic.pdf:file]])

   {#Arndth19} exposition: lecture at _[Geometry in Modal HoTT](https://felix-cherubini.de/abstracts.html)_, 2019 ([recording I](https://www.youtube.com/watch?v=f0wpcNs8hQo), [recording II](https://www.youtube.com/watch?v=sTl8637a2Zo))

See also at _[[spherical T-duality]]_.

[[!redirects Snaith's theorem]]


snake lemma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Diagram chasing lemmas
+-- {: .hide}
[[!include diagram chasing lemmas - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}

## Idea

A basic [[lemma]] in [[homological algebra]]: it constructs [[connecting homomorphisms]].

## Statement

+-- {: .num_lemma}
###### Lemma

Let

$$
  \array{
    && A' &\to & B' &\stackrel{p}{\to}& C' &\to & 0
    \\
    && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{h}}
    \\
    0 &\to& A &\stackrel{i}{\to} & B &\to& C
  }
$$

be a [[commuting diagram]] in an [[abelian category]] $\mathcal{A}$ such that the two rows are [[exact sequences]].

Then there is a [[long exact sequence]] of [[kernels]] and [[cokernels]] of the form

$$
  ker(f)  \to ker(g) \to ker(h) \stackrel{\partial}{\to}
  coker(f) \to coker(g) \to coker(h)
  \,.
$$

Moreover

* if $A' \to B'$ is a [[monomorphism]] then so is $ker(f) \to ker(g)$

* if $B \to C$ is an [[epimorphism]], then so is $coker(g) \to coker(h)$.

If $\mathcal{A}$ is [[Freyd-Mitchell embedding theorem|realized]] as a ([[full subcategory]] of) a category of $R$-[[modules]], then the _[[connecting homomorphism]]_ $\partial$ here can be defined on elements $c' \in ker(h) \subset C'$ by

$$
  \partial (c') := i^{-1} \,g\, p^{-1}(c')
  \,,
$$

where $i^{-1}(-)$ and $p^{-1}(-)$ denote any choice of [[pre-image]] (the total formula is independent of that choice).

=--

+-- {: .num_remark}
###### Remark

The snake lemma derives its name from the fact that one may draw the connecting homomorphism $\partial$ that it constructs diagrammatically as follows:

[[SnakeLemma.png:pic]]

=--

## Related concepts

* [[connecting homomorphism]]

* [[salamander lemma]]

* [[3x3 lemma]], [[5-lemma]],

* [[horseshoe lemma]]

## References

An early occurence of the snake lemma is as lemma (5.8) of

* D. A. Buchsbaum, _Exact categories and duality_, Transactions of the American Mathematical Society Vol. 80, No. 1 (1955), pp. 1-34 ([JSTOR](http://www.jstor.org/stable/1993003))

In

* [[Charles Weibel]], _[[An introduction to homological algebra]]_

it appears as lemma 1.3.2.

A purely [[category theory|category-theoretic]] proof is given in

* Temple Fay, [[Keith Hardie]], [[Peter Hilton]], _The two-square lemma_, Publicacions Matemàtiques, Vol 33 (1989) ([pdf](http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1989_33_01_10.pdf))

and in

* {#Wise} [[Jonathan Wise]], *A non-elementary proof of the snake lemma* (2011, 2023) [[pdf](https://math.colorado.edu/%7Ejonathan.wise/papers/snake.pdf), [[Wise-SnakeLemma.pdf:file]], [MO:a/7531](https://mathoverflow.net/a/7531/381)]

See also

* Wikipedia, _[Snake lemma](http://secure.wikimedia.org/wikipedia/en/wiki/Snake_lemma)_


SnapPy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_SnapPy_ or _SnapPea_ is a [[computer program]] for [[low dimensional topology]], notably computing properties of [[3-manifolds]], and in particular of [[hyperbolic 3-manifold]].

## Applications

* [[complex volume]]

* [[volume conjecture]]

## References

* [SnapPy webpage](https://snappy.math.uic.edu/)

* Wikipedia, _[SnapPea](https://en.wikipedia.org/wiki/SnapPea)_

* {#Zickert07} [[Christian Zickert]], _The volume and Chern-Simons invariant of a representation_, Duke Math. J., 150 (3):489-532, 2009 ([arXiv:0710.2049](http://arxiv.org/abs/0710.2049))

[[!redirects SnapPea]]


Snigdhayan Mahanta

* [webpage](http://www.mathematik.uni-regensburg.de/mahanta/)

###Papers/preprints include

* _Noncommutative Geometry in the Framework of Differential Graded Categories_, [ArXiv 0805.1628](http://arxiv.org/abs/0805.16280

* _Noncommutative correspondence categories, simplicial sets and $pro C^\ast$-algebras_, [ArXiv 0906.5400](http://arxiv.org/abs/0906.5400)

* [[I. Barnea]], [[M. Joachim]], [[Snigdhayan Mahanta|S. Mahanta]], (2015). _Model structure on projective systems of $C^\ast$-algebras and bivariant homology theories_, [arXiv:1508.04283](http://arxiv.org/abs/1508.04283)

category: people

[[!redirects Snigdahayan Mahanta]]
[[!redirects S. Mahanta]]


Snyder space

__Snyder space__ is historically one of the earliest nontrivial examples of noncommutative space time considered in physics.

* H. S. Snyder, _Quantized Space-Time_, Phys. Rev. __71__ (1947) [doi](http://dx.doi.org/38.10.1103/PhysRev.71.38) [pdf](http://prola.aps.org/pdf/PR/v71/i1/p38_1)
* Marco Valerio Battisti, Stjepan Meljanac, _Scalar field theory on non-commutative Snyder space-time_, [arxiv/1003.2108](http://arxiv.org/abs/1003.2108)
* [[Florian Girelli]], _Snyder space-time: K-loop and [[Lie triple system]]_, SIGMA Symmetry Integrability Geom. Methods Appl. __6__ (2010), Paper 074, 19 pp. [journal](http://www.emis.de/journals/SIGMA/2010/074) [pdf](http://www.emis.de/journals/SIGMA/2010/074/sigma10-074.pdf) [arxiv/1009.4762](http://arxiv.org/abs/1009.4762)
* F. Girelli, E. R. Livine, _Scalar field theory in Snyder space-time: alternatives_, [arXiv:1004.0621](http://arxiv.org/abs/1004.0621)
* Lei Lu, A. Stern, _Snyder space revisited_, [arxiv/1108.1832](http://arxiv.org/abs/1108.1832)

[[!redirects Snyder space-time]]