Fulton-MacPherson operad

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###Context###
#### Higher algebra
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[[!include higher algebra - contents]]
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#### Topology
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[[!include topology - contents]]
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#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[compactification]] of [[configuration spaces of points]] was introduced in [Axelrod-Singer 93, p. 5-6](#AxelrodSinger93), [Fulton-MacPherson 94](#FultonMacPherson94) and an [[operad]]-[[structure]] defined on it by [Getzler-Jones 94](#GetzlerJones94) [Kontsevich 97](#Kontsevich97), now called the _Fulton-MacPherson operad_, [[weak equivalence|weakly equivalent]] to the [[little n-disk operad]] ([Salvatore 01, Prop. 4.9](#Salvatore01)), hence a model for an [[En-operad]].

The [[de Rham cohomology]] as well as the [[de Rham complex]] itself of the Fulton-MacPherson operad it [[quasi-isomorphism|quasi-isomorphic]] to [[graph complexes]] via [[TQFT|topological]] [[Feynman amplitudes]] (see [below](#RelationToGraphComplexes)), in fact it is the [[perturbative quantum field theory|perturbative]] computation of [[Feynman amplitudes]] in [[Chern-Simons theory]] that motivated the compactified configuration space, this way, in [Axelrod-Singer 91](#AxelrodSinger91), [Axelrod-Singer 93](#AxelrodSinger93). These two [[quasi-isomorphism]] serve to exhibit [[formal dg-algebra|formality]] of the Fulton-MacPherson operad, and hence also [[formality of the little n-disk operad]].

To see the nature of the Axelrod-Singer-Fulton-MacPherson compactification of [[configuration spaces of points]], observe that the [[configuration space of points|configuration space]] of exactly 2 points in a [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^d$ is [[homotopy equivalence]] to the [[quotient]] by [[group of order two|Z/2]] of a [[n-sphere|(d-1)-sphere]]

$$
  Conf_2( \mathbb{R}^d )
  \;\simeq\;
  S^{d-1}/(\mathbb{Z}/2)
$$

this being the sphere of [[direction of a vector|directions]] of the [[vector]] pointing from one of the two points to the other. This means that the configuration space $Conf_n( \mathbb{R} )$ of $n$ points is [[homotopy equivalent]] to a [[quotient]] by the [[symmetric group]] $\Sigma(n)$ of the [[Cartesian product]] of one such [[n-sphere|(d-1)-sphere]] of [[direction of a vector|directions]] for each [[pair]] of points and various [[open intervals]] encoding the relative [[distance]] between the points.

The _Fulton-MacPherson compactification_ is the evident [[compactification]] of this [[quotient space|quotient]] of [[Cartesian products]] of [[n-sphere|(d-1)-spheres]] with [[open intervals]], replacing the latter by their corresponding [[closed intervals]].

This means that a point on the [[boundary]] of the Fulton-MacPherson compactification corresponds to a would-be [[configuration space of points|configuration]] of points where some of the points have formally vanishing [[distance]] to each other, while still remembering a relative [[direction of a vector|direction]] to each other.

More abstractly one may sum this up succinctly by saying that the ASFM-compactification is the  [[blowup]] of the [[fat diagonal]] inside the [[Cartesian products]] of the base space with itself.

In the literature these [[boundary]] configurations are often referred to in terms of "infinitesimally close points". While there may be some formal relation to actual [[infinitesimal neighbourhoods]] in (for example) [[synthetic differential geometry]], we point out the Axelrod-Singer-Fulton-MacPherson compactifications as defined below are plain [[topological spaces]] or manifolds with corners (in some suitable sense), not [[objects]] in a [[smooth topos]] in any of the usual senses.

## Definition

For $n, d \in \mathbb{N}$, consider first the [[topological space]] of configurations of _ordered_ [[n-tuples]] of points in the [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^d$, hence the [[mapping space]] of [[injections]]/[[embeddings of topological spaces]]

$$
  Emb( \{1, \cdots, n\}, \mathbb{R}^d )
  \;\subset\;
  \left(
    \mathbb{R}^d
  \right)^n
  \,.
$$

The [[group]] of [[translation group|translations]] and of [[scale transformations]] [[action|acts]] canonically on $\mathbb{R}^d$ and hence [[diagonal action|diagonally]] on $Emb( \{1, \cdots, n\}, \mathbb{R}^d )$. The resulting [[quotient topological space]]

$$
  Emb( \{1, \cdots, n\}, \mathbb{R}^d )
  \longrightarrow
  Emb( \{1, \cdots, n\}, \mathbb{R}^d )/ \mathbb{R}^d \rtimes \mathbb{R}_{\gt 0}
$$

is clearly [[homotopy equivalence|homotopy equivalent]] to the original space.

But now for $n \geq 2$ this quotient space canonically carries the structure of a [[topological manifold]] (in fact of a [[smooth manifold]]) of [[dimension of a manifold|dimension]]

$$
  dim
  \left(
    Emb\left( \{1, \cdots, n\}, \mathbb{R}^d \right)/\left( \mathbb{R}^d \rtimes \mathbb{R}_{\gt 0}\right)
  \right)
  \;=\;
  n \cdot d - d - 1
$$

while for $n \leq 1$ the quotient is the [[point space]] (of [[dimension of a manifold|dimension]] 0).

Now for a [[pair]] $i \neq j \in \{1, \cdots, n\}$ of distinct indices, consider the [[continuous function]]

\[
  \label{ProjectionsToBoundarySpheres}
  \array{
    Emb\left( \{1, \cdots, n\}, \mathbb{R}^d \right)/\left( \mathbb{R}^d \rtimes \mathbb{R}_{\gt 0}right)
  \right)
    &\overset{\theta_{i j}}{\longrightarrow}&
    S^{d-1} \subset \mathbb{R}^d
    \\
    (x_1, \cdots, x_n)
    &\mapsto&
    \frac{
      x_j - x_i
    }{
      {\vert x_j - x_i\vert}
    }
  }
\]

which takes each configuration to the [[direction]] of the [[vector]] between the $i$th and the $j$th point in the configuration.

Moreover, for each [[triple]] $i,j,k$ of mutually distinct indices, consider the [[continuous function]]

$$
  \array{
    Emb\left( \{1, \cdots, n\}, \mathbb{R}^d \right)/\left( \mathbb{R}^d \rtimes \mathbb{R}_{\gt 0}\right)
    &\overset{\delta_{i j k}}{\longrightarrow}&
    (0,\infty) \subset [0,\infty]
    \\
    (x_1, \cdots, x_n)
    &\mapsto&
    \frac{ {\vert x_i - x_j\vert} }{ {\vert  x_j - x_k \vert} }
  }
$$

which takes each configuration to the [[ratio]] of the [[distance]] between the $i$th and the $j$th point over that of the $j$th to the $k$th point.

Together these functions induce one total [[continuous function]]

\[
  \label{EmbeddingOfTheOrderedConfigurationSpace}
  \iota
  \;\colon\;
  Emb\left( \{1, \cdots, n\}, \mathbb{R}^d \right)/\left( \mathbb{R}^d \rtimes \mathbb{R}_{\gt 0}\right)
  \overset{ (\theta_{i j}, \delta_{i j k}) }{\hookrightarrow}
  \left( S^{d-1} \right)^{ n(n-1) }
  \times [0,\infty]^{ \left(  n(n-1)(n-2) \right) }
\]

from the ordered configuration space into the [[Cartesian product]] of all these [[n-sphere|(d-1)-spheres]] and [[intervals]].

A moment of reflection shows that every ordered configuration modulo translation and rescaling, hence every point on the left, may be uniquely reconstructed from its direction vectors and distance ratios, hence from a point on the right, hence that this function is a [[homeomorphism]] onto its [[image]] ([Sinha 03, lemma 3.18](#Sinha03)).

+-- {: .num_defn #FultonMacPhersonCompactification}
###### Definition
**(Fulton-MacPherson compactification)**

The _Fultan-MacPherson compactification_ of the [[configuration space of points|configuration space]] of ordered [[n-tuples]] of points in the [[Cartesian space]]/[[Euclidean space]] $\mathbb{R}^d$ is the [[topological closure]] of the [[image]] of (eq:EmbeddingOfTheOrderedConfigurationSpace):

$$
  FM_n(\mathbb{R}^d)
  \;\coloneqq\;
  \overline{ \iota\left(
      Emb\left( \{1, \cdots, n\}, \mathbb{R}^d \right)/\left( \mathbb{R}^d \rtimes \mathbb{R}_{\gt 0}\right)
  \right) }
$$

=--

(e.g. [Lambrechts-Volic 14, Def. 5.1](#LambrechtsVolic14))

The [[symmetric group]] $\Sigma(n)$ still canonically [[action|acts]] on $FM_n(\mathbb{R}^d)$ and hence the FM-compactification of the actual [[configuration space of points]] $Conf_n(\mathbb{R}^d)$ is the [[quotient space]]

$$
  \overline{Conf_n(\mathbb{R}^d)}
  \;=\;
  FM_n(\mathbb{R}^d)/\Sigma(n)
  \,.
$$

## Pictorial notation
 {#Pictorial}

By the above, the points on the [[boundary]] of a Fulton-MacPherson compactification of [[configuration spaces of points]] (Def. \ref{FultonMacPhersonCompactification}) correspond to configurations that involve [[triples]] of points in space $x,y,z$, such that the [[distance]] between two of them is "vanishing in ratio to the distance of both to the third".

Hence where a configuration of two points in space looks like

a configuration of such a boundary case of three points may be visualized as

where the "funnel" might be thought of as an infinite magnifying glass held to the location of the second point and revealing that it really consists of two points of vanishing distance to each other, relative to their joint distance to the remaining point.

In this fashion, for instance a boundary point in the Fulton-MacPherson compactification depicted as follows

would consist of

* points 3, 4 and 5  being arbitrarily close to each other with respect to their distance to points 1, 2 and 6

* points 3 and 5 also being arbitrarily close to each other with respect to their joint distance to 4

* points 1, 2 and 6 being arbitrarily close to each other in comparison to their joint distances to 3, 4 and 5

* points 1 and 2 also being arbitrarily close to each other with respect to their joint distance to 6.

> graphics grabbed from [Lambrechts-Volic 14, Figures 3 and 4](#LambrechtsVolic14)

This pictorial notation was introduced in [Sinha 03](#Sinha03). It immediately suggests the correct [[operad]]-[[structure]] on the Fulton-MacPherson compactifications.

## Properties

### Relation to the little $n$-disk operad

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

The Fulton-MacPherson operad $FM_\bullet\left( \mathbb{R}^d\right)$ is [[weak equivalence|weakly equivalent]] in the [[model structure on operads]] with respect to the [[classical model structure on topological spaces]], to the [[little n-disk operad|little d-disk operad]].

=--

([Salvatore 01, Prop. 4.9](#Salvatore01), summarized as [Lambrechts-Volic 14, Prop. 5.6](#LambrechtsVolic14))

### de Rham cohomology
  {#deRhamCohomology}

We discuss the [[de Rham complex]] $\Omega^\bullet(FM_n(\mathbb{R}^d))$ and the [[de Rham cohomology]] $H^\bullet_{dR}( FM_n(\mathbb{R}^d) )$ of Fulton-MacPherson compactifications.

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

Let $d \in \mathbb{N}$ ([[dimension of a manifold|dimension]] of [[Euclidean space]]),  and $n \in \mathbb{N}$ (number of points) and consider the Fulton-MacPherson compactification $FM_n(\mathbb{R}^d)$ from Def. \ref{FultonMacPhersonCompactification}.

Let

$$
  dvol_{\left(S^{d-1}\right)}
  \;\in\;
  \Omega^\bullet\left( S^{d-1} \right)
$$

be the standard [[volume form]] on the standard [[n-sphere|d-sphere]].

For $i \neq j \in \{1, \cdots, n\} $ two distinct point labels, write

$$
  g_{i j}
  \;\coloneqq\;
  \left(\theta_{i j}\right)^\ast\left( dvol_{\left( S^{d-1}\right)}\right)
  \;\in\;
  \Omega^\bullet\left( FM_n\left(\mathbb{R}^d\right) \right)
$$

be the [[pullback of differential forms]] of this standard volume form along the [[projection]] $\theta_{i j} \;\colon\; FM_n\left( \mathbb{R}^{d}\right) \longrightarrow S^{d-1}$ from (eq:ProjectionsToBoundarySpheres).

=--

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

Let $d \in \mathbb{N}$ ([[dimension of a manifold|dimension]] of [[Euclidean space]]),  and $n \in \mathbb{N}$ (number of points) and consider the Fulton-MacPherson compactification $FM_n(\mathbb{R}^d)$ from Def. \ref{FultonMacPhersonCompactification}.

The [[de Rham cohomology]] of $FM_n\left( \mathbb{R}^d\right)$ is, as a [[graded-commutative algebras]], [[isomorphism|isomorphic]] to the [[quotient algebra]] of the [[free construction|free]] [[graded commutative algebra]] [[generators and relations|generated]] by the classes of the forms $g_{i j}$ from Def. \ref{SphereClassGeneratorsOfDeRhamCohomology}

$$
  H^\bullet_{dR}\left(
    FM_n\left( \mathbb{R}^d \right)
  \right)
  \;\simeq\;
  \wedge_{\mathbb{R}}^\bullet
  \left\langle
    \left[g_{i j}\right] \;\vert\; i \neq j \in \{1, \cdots, n\}
  \right\rangle / \sim
$$

by the following three relations (for all distinct $i,j,k \in \{1,\cdots, n\}$):

1. $\left[g_{i j}\right] \;\sim\; (-1)^d \left[g_{j i}\right]$;

1. $\left[g_{i j}\right] \wedge \left[g_{i j}\right] \;\sim\; 0$;

1.  $
     \left[g_{i j}\right] \wedge \left[ g_{j k} \right]
     +
     \left[g_{j k}\right] \wedge \left[ g_{k i} \right]
     +
     \left[g_{k i}\right] \wedge \left[ g_{i j} \right]
     \;\sim\;
     0
   $ $\;\;\;\;$("3-term relation")

=--

This is due to ([Arnold 69](#Arnold69), [Cohen 73](#Cohen73)).

See also at _[[configuration space of points]]_ the section _[Cohomology](configuration+space+of+points#Cohomology)_.

+-- {: .num_remark #RealCohomologyOfFMCompactificationInTermsOfGraphCohomology}
###### Remark
**([[real cohomology]] of the FM-compactification in terms of [[graph cohomology]])**

In the [[graph complex]]-model for the [[rational homotopy type]] of the ordered [[configuration space of points]] $Conf_n\big( \mathbb{R}^D\big)$ the three relations in Prop. \ref{DeRhamCohomologyOfFMCompactification} are incarnated as follows:

1. a graph changes sign when one of its edges is reversed ([this Def.](graph+complex#SignRulesForGraphs))

1. a graph with [[parallel edges]] is a vanishing graph ([this Def.](graph+complex#VanishingGraphs))

1. the graph coboundary of a single trivalent internal vertex ([this Example](graph+complex#ThreeTermRelation)).

=--

### Relation to Graph complexes and Formality theorem
 {#RelationToGraphComplexes}

We have that [[the Fulton-MacPherson operad is formal]] in the sense that for each of its component [[manifolds]] $FM_n\left( \mathbb{R}^d\right)$ (Def. \ref{FultonMacPhersonCompactification}) there is a [[zig-zag]] of [[quasi-isomorphisms]] between their [[de Rham cohomology]] and their [[de Rham complex]], and such that these morphisms are compatible with the induced [[cooperad]]-[[structure]] on both sides.

Concretely, the [[zig-zags]] may be taken to consist of one [[span]] of [[quasi-isomorphisms]] out of a suitable [[graph complex]] to the [[de Rham cohomology]]/[[de Rham complex]] of the [[Fulton-MacPherson operad]].

Here the morphism from the [[graph complex]] to the [[de Rham complex]] of the [[Fulton-MacPherson operad]] regards the latter as the [[compactification]] of a [[configuration space of points]], regards [[functions]]/[[differential forms]] on [[configuration spaces of points]] as [[n-point functions]] of a [[topological quantum field theory|topological]] [[quantum field theory]], regards suitable [[graphs]] as [[Feynman diagrams]] and proceeds by sending each such graph/Feynman diagram to a corresponding [[Feynman amplitude]].

This idea of a proof was sketched in [Kontsevich 99](#Kontsevich99), a full account is due to [Lambrechts-Volic 14](#LambrechtsVolic14).

+-- {: .num_defn #TheGraphComplexdgcAlgebra}
###### Definition
**(the [[graph complex]])**

For $n\in \mathbb{N}$, write

$$
  Graphs_n\left( \mathbb{R}^d \right)
  \;\in\;
  dgcAlg_{\mathbb{R}}
$$

for the [[graph complex]] of "[[Feynman graphs]]" with $n$ external [[vertices]], regarded with its [[structure]] of a [[differential graded-commutative algebra]] over the [[real numbers]].

=--

([Lambrechts-Volic 14, Def. 6.19](#LambrechtsVolic14))

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

The [[linear function]]

$$
  \overline{I}
  \;\colon\;
  Graphs_n\left( \mathbb{R}^d\right)
  \overset{ \phantom{AA}\simeq \phantom{AA} }{\longrightarrow}
  H^\bullet_{dR}\left(  FM_n\left( \mathbb{R}^d \right) \right)
$$

from the [[graph complex]] [[dgc-algebra]] (Def. \ref{TheGraphComplexdgcAlgebra}) to the [[de Rham cohomology]] (Prop. \ref{DeRhamCohomologyOfFMCompactification}) of the Fulton-MacPherson compactification (Def. \ref{FultonMacPhersonCompactification}) given on generators by

$$
  \overline{I}\left( \Gamma \right)
  \;=\;
  \left\{
    \array{
      \left[g_{i j}\right] &\vert& \Gamma \, {\text{is diagram without internal vertices} \atop \text{and single edge between external vertices}\, i \, \text{and}\, j}
      \\
      0 &\vert& \Gamma \, \text{has internal vertices}
    }
  \right.
$$

(where $g_{i j} \in \Omega^\bullet\left(  FM_n\left( \mathbb{R}^d\right) \right)$ are the [[differential forms]] from Def. \ref{SphereClassGeneratorsOfDeRhamCohomology})

extends to a [[homomorphism]] of [[dgc-algebras]] which is a [[quasi-isomorphism]].

=--

([Lambrechts-Volic 14, Theorem 8.1](#LambrechtsVolic14))

+-- {: .num_example #ThreeTermRelation}
###### Example
**(the "3-term relation")**

In the [[graph complex]] the [[differential]] of the [[graph]] as shown on the left below (the vertices on the horizontal line are the external vertices, that above the line is internal) is a linear combination as shown on the right:

$\phantom{AAA}\array{ \partial \\ \phantom{A} \\ \phantom{a}}$

$\phantom{A}\array{ = \\ \phantom{A} \\ \phantom{a} }\phantom{A}$

> graphics grabbed from [Lambrechts-Volic 14, Figure 1 & Figure 2](#LambrechtsVolic14)

Under the above [[quasi-isomorphism]] (Prop. \ref{QuasiIsoFromGraphComplexToDeRhamCohomology}) from the [[graph complex]] to the [[de Rham complex]] on the [[Fulton-MacPherson compactification]] of a [[configuration space of points]] given by sending each [[graph]] to its [[Chern-Simons propagator|Chern-Simons]] [[Feynman amplitude on compactified configuration spaces of points]] (Remark \ref{AsPerturbativeChernSimonsTheory}) this relation becomes the "3-term relation" from Prop. \ref{DeRhamCohomologyOfFMCompactification}

$$
     \left[g_{i j}\right] \wedge \left[ g_{j k} \right]
     +
     \left[g_{j k}\right] \wedge \left[ g_{k i} \right]
     +
     \left[g_{k i}\right] \wedge \left[ g_{i j} \right]
     \;\sim\;
     0
$$

satisfied by the [[Chern-Simons propagator]] form

$$
  \left(
    g_{i j}
    \;\in\;
    \Omega^2\left( FM_n(\mathbb{R}^d)  \right)
  \right)
  \,.
$$

=--

Now we consider the analogous but richer quasi-isomorphism from the [[graph complex]] not just to the [[de Rham cohomology]] of the compactified configuration space, but its full [[de Rham complex]].

+-- {: .num_defn #FiberIntegrationOverInternalVertices}
###### Definition
**([[fiber integration]] over internal [[vertices]])**

For $\Gamma \in Graphs_n\left( \mathbb{R}^d \right)$ an element in the [[graph complex]] (Def. \ref{TheGraphComplexdgcAlgebra}) with $k$ internal vertices, write

$$
  \array{
    FM_{n + k}\left( \mathbb{R}^d \right)
    \\
    \Big\downarrow {}^{\mathrlap{\pi_n^{n + k}}}
    \\
    FM_{n}\left( \mathbb{R}^d\right)
  }
$$

for the evident [[continuous function]] between the Fulton-MacPherson compactifications of ordered [[configuration spaces of points]] (Def. \ref{FultonMacPhersonCompactification}) obtained by forgetting the $k$ "internal" of the total of $n + k$ points in a configuration.

This canonically carries the [[structure]] of a [[fiber bundle]] of [[semi-algebraic manifolds]] and hence induces a [[fiber integration|fiber-]] [[integration of differential forms]] from minimal to partially algebraic differential forms:

$$
  \underset{
    \pi_n^{n + k}
  }{\int}
  \;\colon\;
  \Omega^\bullet_{min}
  \left(
    FM_{n + k}\left( \mathbb{R}^d \right)
  \right)
  \longrightarrow
  \Omega^{\bullet - d \cdot k}_{PA}
  \left(
    FM_n\left( \mathbb{R}^d \right)
  \right)
$$

=--

([Lambrechts-Volic 14, Theorem 5.8 and equation (111)](#LambrechtsVolic14))

+-- {: .num_prop #QuasiIsomorphismBetweenGraphComplexAndDeRhamComplexOfFM}
###### Proposition
**([[quasi-isomorphism]] between [[graph complex]] and [[de Rham complex]] of Fulton-MacPherson compactification)**

Consider the [[linear function]]

$$
  \overline{I}
  \;\colon\;
  Graphs_n\left( \mathbb{R}^d\right)
  \overset{ \phantom{AA}\simeq \phantom{AA} }{\longrightarrow}
  \Omega^\bullet_{PA}\left(  FM_n\left( \mathbb{R}^d \right) \right)
$$

from the [[graph complex]] [[dgc-algebra]] (Def. \ref{TheGraphComplexdgcAlgebra}) to the [[semialgebraic de Rham complex]] (Def. \ref{FiberIntegrationOverInternalVertices}) of the Fulton-MacPherson compactification (Def. \ref{FultonMacPhersonCompactification}) given on [[graphs]]

$$
  \Gamma \;\in\; Graphs_{n}\left( \mathbb{R}^d\right)
$$

with

$$
  k \coloneqq int(\Gamma)
$$

internal vertices by the [[integral transform]] of the canonical [[volume form]] through the [[span]]

$$
  \array{
    &&
    FM_{n + k}\left( \mathbb{R}^d\right)
    \\
    &
    {}^{ \mathllap{ \left( \theta_{a b} \vert a \neq b\in \{1, \cdots, n + k\} \right) } }\swarrow
    &&
    \searrow^{ \mathrlap{ \pi_n^{n + k}  } }
    \\
    \left(
      S^{d-1}
    \right)^{n + k}
    && &&
    FM_n\left( \mathbb{R}^d \right)
  }
$$

where

1. the [[function]] on the left, to the [[Cartesian product]] of one [[n-sphere|(d-1)-spheres]] for each pair of distinct vertices, is given by the functions $\theta_{a b}$ from (eq:ProjectionsToBoundarySpheres)

1. the function on the right is the canonical forgetful projection from Def. \ref{FiberIntegrationOverInternalVertices},

hence which sends

$$
  \Gamma
  \;\mapsto\;
  \underset{ \pi_{n}^{n + int(\Gamma)} }{\int}
  \underset{
    (a \to b) \in \Gamma
  }{\wedge}
  g_{a b}
$$

to the [[fiber integration]] over internal vertices (Def. \ref{FiberIntegrationOverInternalVertices}) of the [[wedge product]] of the [[differential forms]]

$$
  g_{a b} \coloneqq \theta_{a b}^\ast dvol_{\left(S^{d-1}\right)}
$$

(Def. \ref{SphereClassGeneratorsOfDeRhamCohomology}), one for each [[edge]] in the graph from vertex $a$ to vertex $b$.

This function constitutes a [[homomorphism]] of [[dgc-algebras]] which is a [[quasi-isomorphism]].

=--

That this is a dgc-algebra-homomorphism compatible with the [[cooperad]]-[[structure]] is proven in [Lambrechts-Volic 14](#LambrechtsVolic14), that it is a [[quasi-isomorphism]] is proven in [Lambrechts-Volic 14, section 10](#LambrechtsVolic14).

+-- {: .num_remark #AsPerturbativeChernSimonsTheory}
###### Remark
**([[perturbative quantum field theory|perturbative]] [[Chern-Simons theory]] )**

The [[differential forms]] $g_{i j}$ from Def. \ref{SphereClassGeneratorsOfDeRhamCohomology} may be understood as being the [[Feynman propagator]] of [[perturbative quantum field theory|perturbative]] [[higher Chern-Simons theory]] regarded equivalently in its incarnation as a  [[Feynman amplitude on compactified configuration spaces of points]]. See at _[[Chern-Simons propagator]]_ for more on this.

From this perspective, the [[quasi-isomorphism]] in Prop. \ref{QuasiIsomorphismBetweenGraphComplexAndDeRhamComplexOfFM} sends each [[graph]], regarded as a [[Feynman diagram]] for [[higher Chern-Simons theory]], to its correspomnding [[Feynman amplitude]], regarded in its incarnation as a  [[Feynman amplitude on compactified configuration spaces of points]].

That this should be the case was originally suggested in [Kontsevich 94](#Kontsevich94).

=--

## Related concepts

* [[little n-disk operad]]

* [[Feynman amplitude on compactified configuration space of points]]

* [[Hilbert scheme]]

## References

The [[compactification]] of [[configuration spaces of points]] was first considered for two points in

* {#AxelrodSinger91} [[Scott Axelrod]], [[Isadore Singer]],  _Chern-Simons Perturbation Theory_, in S. Catto, A. Rocha (eds.) Proc. XXthe DGM Conf. World Scientific Singapore, 1992, 3-45; ([arXiv:hep-th/9110056](http://arxiv.org/abs/hep-th/9110056))

* {#AxelrodSinger93} [[Scott Axelrod]], [[Isadore Singer]],  _Chern--Simons Perturbation Theory II_, J. Diff. Geom. 39 (1994) 173-213 ([arXiv:hep-th/9304087](http://arxiv.org/abs/hep-th/9304087))

and then more generally in

* {#FultonMacPherson94} [[William Fulton]], [[Robert MacPherson]], _A compactification of configuration spaces_, Ann. of Math. (2), 139(1):183–225, 1994 ([jstor:2946631](https://www.jstor.org/stable/2946631))

based on the general method of [[resolution of singularities]] due to

* {#Hironaka64} [[Heisuke Hironaka]], _Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I_, Annals of Mathematics Second Series, Vol. 79, No. 1 (Jan., 1964), pp. 109-203 (95 pages) ([jstor:1970486](https://www.jstor.org/stable/1970486))

More general compactifications of spaces of _subspace arrangements_ are due to

* {#deConciniProcesi95} [[Corrado de Concini]], [[Claudio Procesi]], _Wonderful models of subspace arrangements_, Selecta Mathematica, New Series Vol. 1, No. 3, 1995 ([pdf](https://link.springer.com/content/pdf/10.1007/BF01589496.pdf))

see also

* {#Feichtner05} Eva Maria Feichner, _De Concini–Procesi Wonderful Arrangement Models:  A Discrete Geometer’s Point of View_, Math. Sci. Res. Inst. Publ 52, 2005 ([pdf](http://library.msri.org/books/Book52/files/17feich.pdf))

The [[operad]]-[[structure]] on Axelrod-Singer Fulton-MacPherson compactifications of [[configuration spaces of points]] was considered in

* {#GetzlerJones94} [[Ezra Getzler]], [[John Jones]], _Operads, homotopy algebra and iterated integrals for double loop spaces_ ([arXiv:hep-th/9403055](https://arxiv.org/abs/hep-th/9403055))

and alternatively in

* {#Kontsevich99} [[Maxim Kontsevich]], around Def. 12 of _Operads and Motives in Deformation Quantization_, Lett.Math.Phys.48:35-72,1999 ([arXiv:math/9904055](https://arxiv.org/abs/math/9904055))

* {#Kontsevich97} [[Maxim Kontsevich]], section 5.1 of _Deformation quantization of Poisson manifolds, I_, Lett.Math.Phys.66:157-216,2003 ([arXiv:q-alg/9709040](https://arxiv.org/abs/q-alg/9709040))

which was corrected in

* Giovanni Gaiffi, _Models for real subspace arrangements and stratified manifolds_, Int. Math. Res. Not. 12:627-656, 2003 ([pdf](http://www.dm.unipi.it/~gaiffi/papers/models.pdf))

and developed in detail in

* {#Sinha03} [[Dev Sinha]], _Manifold theoretic compactifications of configuration spaces_, Selecta Math. (N.S.) 10(3):391-428, 2004 ([arXiv:math/0306385](https://arxiv.org/abs/math/0306385))

which also shows the equivalence to [Fulton-MacPherson 94](#FultonMacPherson94).

Review includes

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], section 5 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

The [[de Rham cohomology]] was determined in

* {#Arnold69} [[Vladimir Arnold]], _The cohomology ring of the colored braid group_, Mat. Zametki, 1969, Volume 5, Issue 2, Pages 227–231 ([mathnet:mz6827](http://mi.mathnet.ru/eng/mz6827))

* {#Cohen73} [[Fred Cohen]], _Cohomology of braid spaces_, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 763-766 ([euclid:1183534761](https://projecteuclid.org/euclid.bams/1183534761))

The equivalence to the [[little n-disk operad]] was established in

* {#Salvatore01} [[Paolo Salvatore]], _Configuration spaces with summable labels_, Cohomological methods in homotopy theory. Birkhäuser, Basel, 2001. 375-395.

{#ReferencesPropagator} The interpretation of the quasi-isomorphism establishing [[formality of the little n-disk operad]] as sending edges to [[Feynman propagators]] of some [[topological quantum field theory]] such as [[Chern-Simons theory]] is due to

* {#AxelrodSinger93} [[Scott Axelrod]], [[Isadore Singer]], Section 2 of _Chern--Simons Perturbation Theory II_, J. Diff. Geom. 39 (1994) 173-213 ([arXiv:hep-th/9304087](http://arxiv.org/abs/hep-th/9304087))

which is recalled and recast in somewhat more generality in

* {#BottCattaneo97} [[Raoul Bott]], [[Alberto Cattaneo]], Remark 3.6 in _Integral invariants of 3-manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))

* {#CattaneoMnev10} [[Alberto Cattaneo]], [[Pavel Mnev]], Remark 11 in _Remarks on Chern-Simons invariants_, Commun.Math.Phys.293:803-836,2010 ([arXiv:0811.2045](https://arxiv.org/abs/0811.2045))

That this should exhibit a graph complex model for the cohomology of compactified configuration spaces was suggested in

* {#Kontsevich94} [[Maxim Kontsevich]], pages 11-12 of _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

and spelled out in more detail in

* [Kontsevich 99, section 3](#Kontsevich99)

with a careful proof being layed out in

* [Lambrechts-Volic 14](#LambrechtsVolic14)

[[!redirects Fulton-MacPherson operads]]

[[!redirects Fulton-MacPherson compactification]]
[[!redirects Fulton-MacPherson compactifications]]


Fumikazu Nagasato

* [personal page](https://ccmath.meijo-u.ac.jp/~fukky/index.html)

## Selected writings

On [[Lie algebra weight systems]] for the [[special linear Lie algebra]]:

* [[Fumikazu Nagasato]], *A Diagrammatic Construction of the $(sl(N,\mathbb{C}, \rho))$-Weight System*, Interdisciplinary Information Sciences, **9** 1 (2003) 43-51 ([doi:10.4036/iis.2003.43](https://doi.org/10.4036/iis.2003.43))

* [[Fumikazu Nagasato]], *An Exposition of the $\mathfrak{sl}(N,\mathbb{N})$-Weight System*, Journal of Lie Theory 17 (2007), No. 2, 263--281, Heldermann Verlag 2007 ([jlt:17014](https://www.heldermann.de/JLT/JLT17/JLT172/jlt17014.htm))

category: people


funcoid

Funcoids were first named and studied by [[Victor Porton]], an independent researcher, as an approach to [[general topology]], in particular as a massive generalisation of [[proximity spaces]], [[pretopological spaces]], etc. -- somewhat along the same lines described at [[syntopogenous space]] where nearness notions are taken to be primitive.

Let $\mathfrak{F}(A)$ and $\mathfrak{F}(B)$ be the [[reverse lattices of filters]] for the [[sets]] $A$ and $B$.

Then a __funcoid__ from $A$ to $B$ consists of [[functions]] $\alpha\colon \mathfrak{F}(A) \to \mathfrak{F}(B)$ and $\beta\colon \mathfrak{F}(B) \to \mathfrak{F}(A)$ such that
$$\forall\mathcal{X}\in\mathfrak{F}(A),\mathcal{Y}\in\mathfrak{F}(B): (\mathcal{Y}\sqcap\alpha \mathcal{X} \ne 0^{\mathfrak{F}(A)} \Leftrightarrow \mathcal{X}\sqcap\beta \mathcal{Y} \ne 0^{\mathfrak{F}(B)}).$$
That is, for every [[filter]] $\mathcal{X}$ on $A$ and every filter $\mathcal{Y}$ on $B$, the filter (on $B$) generated by the [[union]] of $\mathcal{Y}$ and $\alpha(\mathcal{X})$ is not the entire [[power set]] $\mathcal{P}(B)$ iff the filter (on $A$) generated by the union of $\mathcal{X}$ and $\beta(\mathcal{Y})$ is not the entire power set $\mathcal{P}(A)$.

The above formulation is due to [[Victor Porton]]. A slight reformulation is that funcoids can be seen as morphisms between certain types of [[Chu spaces]]. [[Todd Trimble]] has written some notes on funcoids from this point of view [here](http://ncatlab.org/toddtrimble/published/topogeny), where they are also shown to be equivalent to [[topogenous relations]] provided that the reflexivity condition of the latter is dropped (which is reasonable from an abstract perspective). The category of endofuncoids and continuous functions, and the full subcategory of reflexive endofuncoids, are each [[topological functor|topological]] over $Set$.


function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Set theory
+-- {: .hide}
[[!include set theory - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In a strict sense of the term, a **function** is a [[homomorphism]] $f \colon S \to T$ of [[sets]]: a prescription which takes any [[element]] $s \in S$ to a unique element $f(s) \in T$, this also called the *[[value]]* of the function at $s$.

Instead of "function" one also speaks of _[[map]]_ or _mapping_, but those terms are used in other ways in other contexts.

A function from a [[set]] $A$ to a set $B$ is determined by giving, for each element of $A$, a specified element of $B$.  The process of passing from elements of $A$ to elements of $B$ is called [[function application]].  The set $A$ is called the [[domain]] of $f$, and $B$ is called its [[codomain]].

A function is sometimes called a __total function__ to distinguish it from a [[partial function]].

More generally, every [[morphism]] between [[objects]] in a [[category]] may be thought of as a function in a generalized sense. This generalized use of the word is wide spread (and justified) in [[type theory]], where for $S$ and $T$ two [[types]], there is a [[function type]] denoted $S \to T$ and then the expression $f : S \to T$ means that $f$ is a [[term]] of [[function type]], hence is a function.

In this generalized sense, functions between sets are the [[morphisms]] in the [[category]] [[Set]].  This is [[cartesian closed category|cartesian closed]], and the [[function type]] $S \to T$ is then the [[function set]].

For more on this more general use of "function" see at _[[function type]]_.

## Foundations

The formal definition of a function depends on the [[foundations]] chosen.

### In set theory

* In [[material set theory]], a function $f$ is often defined to be a set of [[ordered pairs]] such that for every $x$, there is at most one $y$ such that $(x,y)\in f$.  The [[domain]] of $f$ is then the set of all $x$ for which there exists some such $y$.  This definition is not entirely satisfactory since it does not determine the codomain (since not every element of the codomain may be in the [[image]]); thus to be completely precise it is better to define a function to be an ordered triple $(f,A,B)$ where $A$ is the domain and $B$ the codomain.

* In [[structural set theory]], the role of functions depends on the particular axiomatization chosen.  In [[ETCS]], functions are among the undefined things, whereas in [[SEAR]], functions are defined to be particular relations (which in turn are undefined things).

### In dependent type theory

In [[dependent type theory]], a [[type]] $A$ is a [[contractible type]] if it comes with an element $a:A$ and a family of identities $x:A \vdash c(x):a =_A x$ indicating that $a$ is a [[center of contraction]]. A **function** between types $A$ and $B$ could be defined as

* a family of elements $x:A \vdash f(x):B$
* a [[span]] $(C; z:C \vdash g(z):A; z:C \vdash h(z):B)$ for which the dependent type $\sum_{z:C} g(z) =_A x$ is a [[contractible type]] for all $x:A$
* a [[multivalued partial function]] $(x:A \vdash P(x); x:A, p:P(x) \vdash f(x, p):B)$ for which the dependent type $P(x)$ is a [[contractible type]] for all $x:A$
* a [[correspondence]] $x:A, y:B \vdash R(x, y)$ for which the dependent type $\sum_{y:B} R(x, y)$ is a [[contractible type]] for all $x:A$.
* a family of types $y:B \vdash A'(y)$
* an element of the [[function type]] $f:A \to B$, in dependent type theories with function types.

The first four definitions are interdefinable with each other in [[dependent type theories]] with [[identity types]] and [[dependent sum types]]:

* From every family of elements $x:A \vdash f(x):B$, one could define a [[span]] by defining the type $C$ to be a [[copy]] of $A$, $C \coloneqq \mathrm{Copy}(A)$, the family of elements $z:C \vdash g(z):A$ to be $g(z) \coloneqq \pi_1(\alpha(x))$ since for all elements $z:C$, the dependent type $\sum_{x:A} \mathrm{copy}(x) =_C z$ is contractible and with [[center of contraction]] $z:C \vdash \alpha(z):\sum_{x:A} \mathrm{copy}(x) =_C z$, and the family of elements $x:C \vdash h(x):B$ to be $h(x) \coloneqq f(g(x))$. For every element $x:A$, the dependent type $\sum_{z:C} g(z) =_A x$ is always a [[contractible type]].

* From every [[span]] $(C; z:C \vdash g(z):A; z:C \vdash h(z):B)$ for which the dependent type $\sum_{z:C} g(z) =_A x$ is a [[contractible type]] for all $x:A$, one can define a family of elements $x:A \vdash f(x):B$ as $f(x) \coloneqq h(\pi_1(\alpha(x)))$ for all [[centers of contraction]] $x:A \vdash \alpha(x):\sum_{z:C} g(z) =_A x$.

* From every family of elements $x:A \vdash f(x):B$, one could define a [[multivalued partial function]] by defining $x:A \vdash P(x)$ to be $P(x) \coloneqq \sum_{y:A} x =_A y$ and $x:A, p:P(x) \vdash f'(x, p):B$ to be $f'(x, p) \coloneqq f(x)$. For every element $x:A$, the dependent type $\sum_{y:A} x =_A y$ is always a [[contractible type]].

* From every [[multivalued partial function]] $(x:A \vdash P(x); x:A, p:P(x) \vdash f(x, p):B)$ for which the dependent type $P(x)$ is a [[contractible type]] for all $x:A$, one could define a family of elements $x:A \vdash f'(x):B$ as $f'(x) \coloneqq f(x, \alpha(x))$ for all [[centers of contraction]] $x:A \vdash \alpha(x):P(x)$.

* From every family of elements $x:A \vdash f(x):B$, one could define a [[correspondence]] by defining $x:A, y:B \vdash R(x, y)$ to be $R(x, y) \coloneqq f(x) =_B y$. For every element $x:A$, the dependent type $\sum_{y:B} f(x) =_B y$ is always a [[contractible type]].

* From every [[correspondence]] $x:A, y:B \vdash R(x, y)$ for which the dependent type $\sum_{y:B} R(x, y)$ is a [[contractible type]] for all $x:A$, one could define a family of elements $x:A \vdash f(x):B$ as $f(x) \coloneqq \pi_1(\alpha(x))$ for all [[centers of contraction]] $x:A \vdash \alpha(x):\sum_{y:B} R(x, y)$.

* The second, third, and fourth definitions of a function are interdefinable via the following relations between [[spans]], [[multivalued partial functions]], and [[correspondences]]:

  * From every span one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:C} g(y) =_A x$ and the family of elements $x:A, p:P(x) \vdash f(x, p):B$ as $f(x, p) \coloneqq h(\pi_1(x))$.

  * From every multivalued partial function one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} P(x)$ and the family of elements $x:C \vdash g(x):A$ as $g(x) \coloneqq \pi_1(x)$.

  * From every multivalued partial function one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{p:P(x)} f(x, p) =_B y$.

  * From every correspondence one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:B} R(x, y)$, and the family of elements $x:A, p:P(x) \vdash h(x, p):B$ as $h(x, p) \coloneqq \pi_1(p)$

  * From every span one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{z:C} (g(z) =_A x) \times (h(z) =_B y)$.

  * From every correspondence one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} \sum_{y:B} R(x, y)$, the family of elements $z:C \vdash g(z):A$ as $g(z) \coloneqq \pi_1(z)$, and the function $z:C \vdash h(z):B$ as $h(z) \coloneqq \pi_1(\pi_2(z))$

The first and fifth definitions are interdefinable by the following

* From every family of elements $x:A \vdash f(x):B$ one could get a family of types by $y:B \vdash \sum_{x:A} f(x) =_B y$ where $A'(y) \coloneqq \sum_{x:A} f(x) =_B y$.

* From every family of types $y:B \vdash A'(y)$ one could get a family of elements $x:\sum_{y:B} A'(y) \vdash \pi_1(x):B$ where $A \coloneqq \sum_{y:B} A'(y)$.

If the dependent type theory has [[function types]], then

* From every element of the function type $f:A \to B$, one could define a family of elements $x:A \vdash f'(x):B$ as $f'(x) \coloneqq \mathrm{ev}(f, x)$, where $f:A \to B, x:A \vdash \mathrm{ev}(f, x):B$ appears in the [[elimination rules]] for function types.

* From every family of elements $x:A \vdash f(x):B$, one could define an element of the function type $f':A \to B$ as $f' \coloneqq x \mapsto f(x)$, where $x:A, f(x):B \vdash x \mapsto f(x):A \to B$ appears in the [[introduction rules]] for function types.

### In formal category theory

In [[formal category theory]], sets are [[discrete categories]], and a **function** is a [[functor]] between [[sets]]. The functoriality structure becomes the property that a function preserves [[equality]]:

\[ x = y \Rightarrow f(x) = f(y) .\]

### For classes

One can also speak of functions between proper [[classes]], although the precise details may vary depending on the status of classes with respect to the formal theory.

In [[ZFC]] for example, proper classes are by design not formal objects in the theory; rather they are proxied by a formula in the language (for instance, the class $V$ of all sets is proxied by the formula $x = x$; intuitively we may think of $V$ as consisting of all $x$ satisfying that formula). Then functions $f: A \to B$ between classes are again classes given by suitable formulas; see for example the MathOverflow discussion [what-are-maps-between-proper-classes](http://mathoverflow.net/questions/63265/what-are-maps-between-proper-classes). If (as described above for material set theory) one wants to describe a function as an ordered triple $(f, A, B)$, then this too can be accommodated if one defines ordered triples/pairs of classes appropriately; see [here](/nlab/show/ordered+pair#class) for one possibility. Thus functors between categories whose objects and morphisms form proper classes can similarly be described in the language.

Such technical hacks can be avoided by choosing a different foundations. For example, Mac Lane in his [[Categories for the Working Mathematician]] assumes ZFC with a [[universe]] in which some sets are considered large, such as the set of small sets, so that a category like $Set$ (the category of small sets) is again a formal object of the theory.

## Alternative terms

Useful terms, more or less synonymous with *function*, are *assignment*, _assignation_ or more specifically *assignation on objects*. These do not have standard meanings but are useful to signal to readers that the domain of the 'function' under consideration is large, or that one is more interested in [[functorial]] extensions of this partial assignation (cf. e.g. Richard Garner, Homomorphisms of higher categories, Adv. Math. 224 (2010) 2269-2311 for many examples). In mathematical writing "assignment" is usually synonymous with _[[function]]_ or _[[map]]_ or "mapping". For example one might speak of "assigning to each positive number its square root" to refer to the function $\sqrt{(-)} \colon \mathbb{R}_{\geq 0} \to \mathbb{R}$.

Authors may resort to verb forms such as "assigns" or "associates" or "sends" in informal writing, perhaps to avoid the bother of specifying an axiomatic framework in which a formal notion like "function" is ensconced. For example, according to Wikipedia, Jacobson defines a functor $F: C \to D$ between categories as a "mapping" that "associates" to each object $X$ in $C$ an object $F(X)$ in $D$, etc. No clarity would be gained by making this any more formal (which as we saw in the case of functions between classes, such as classes of objects of categories, may involve annoyingly technical hacks).

Sometimes the word "assignment" is understood more generally as _[[relation]]_, often when authors define a function to be something that "assigns unique values" (for instance  [here](https://books.google.de/books?id=jYniBQAAQBAJ&pg=PA27&lpg=PA27&dq=function+map+assignment+mathematics&source=bl&ots=2aHFjcZ3c9&sig=1HkugZIeSQdnV70U5gjSx_H21ug&hl=en&sa=X&ved=0ahUKEwjpxuPey6nUAhWIfhoKHe9BDy0Q6AEIVTAJ#v=onepage&q=function%20map%20assignment%20mathematics&f=false)).

## Examples

* [[empty function]]

* [[constant function]]

Many functions that carry special names map some [[ring]] or [[field]] like the [[real numbers]] or [[complex numbers]] to itself:

* [[polynomial]]

* [[exponential]]

* [[step function]]

* [[trigonometric function]]

* [[hypergeometric function]]

* [[special function]]

* ...

Special properties these may have:

* [[continuous function]], [[analytic function]], [[smooth function]]

* [[periodic function]]

## Related concepts

* [[map]], [[operation]]

* [[partial function]], [[partial recursive function]]

* [[anafunction]]

* [[tuple]], [[sequence]], **function**

* [[dependent tuple]], [[dependent sequence]], [[dependent function]]

* [[function set]]

* [[function algebra]], [[pullback of functions]]

* [[continuous function]], [[differentiable function]], [[smooth function]]

* [[periodic function]]

* [[linear function]], [[quadratic function]], [[polynomial function]]

* [[bounded function]]

* [[fixed point]]

* [[motivic function]]

* [[functor]]

* [[2-functor]] / [[pseudofunctor]]

  * [[monoidal functor]]

* [[n-functor]]

* [[(∞,1)-functor]]

* [[(∞,n)-functor]]

[[!redirects function]]
[[!redirects functions]]

[[!redirects total function]]
[[!redirects total functions]]
[[!redirects assignation]]
[[!redirects assignment]]


function algebras on infinity-stacks

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Abstract

For $T$ any abelian [[Lawvere theory]], here we discuss -- in a variation of the theme of [[Isbell conjugation]], in generalization of ([Toën](#Toen)) and following ([Stel](#Stel)) --  a [[simplicial Quillen adjunction]] between [[model category]] structures on cosimplicial $T$-algebras and on [[simplicial presheaves]] over duals of $T$-algebras. We find mild general conditions under which this descends to the local model structure that models [[∞-stacks]] over duals of $T$-algebras. In these cases the left adjoint of the Quillen adjunction is given by sending $\infty$-stacks to their cosimplicial $T$-algebras of functions with values in the canonical $T$-[[line object]], and the adjunction models small objects relative to a choice of a small full subcategory $T\subset C \subset T Alg^{op}$  of the localization

$$
  \mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C )
$$

of the $(\infty,1)$-topos of $(\infty,1)$-sheaves over duals of $T$-algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical $T$-[[line object]].

For the special case where $T$ is the theory of ordinary commutative algebras this reproduces the situation of ([Toën](#Toen)) and many statements are straightforward generalizations from that situation. For the case that $T$ is the theory of _[[smooth algebra]]s_ ($C^\infty$-rings) we obtain a refinement of this to the context of synthetic differential geometry. In these cases, in as far as objects in $\mathbf{H}$ may be understood as [[∞-Lie groupoids]], the objects in $\mathbf{L}$ may be understood as [[∞-Lie algebroids]].

As an application, we show how [[Anders Kock]]'s simplicial model for synthetic combinatorial [[differential forms]] finds a natural interpretation as the differentiable $\infty$-stack of infinitesimal paths of a manifold. This construction is an $\infty$-categorical and synthetic differential resolution of the _de Rham space_ functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the $\infty$-stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic $\infty$-vector bundles with flat connection.

## Models for $\infty$-stacks and their function algebras {#Models}

### Cosimplicial $T$-algebras

A good general notion of [[function algebras]] on generalized [[spaces]] are $T$-algebras, for $T$ a [[Lawvere theory]]. A good general notion of [[function algebras]] on [[internal ∞-groupoid]]s in such spaces are [[cosimplicial object|cosimplicial]] $T$-algebras.

We recall some basics and then discuss a [[model category]] structure on cosimplicial $T$-algebras for the cases that $T$ contains the theory of [[abelian group]]s.

#### $T$-Algebras {#TAlgebras}

A [[Lawvere theory]] may be thought of as a generalization of the theory of ordinary [[associative algebra]]s.

A Lawvere theory is encoded in its [[syntactic category]] $T$, which by definition is a category with finite [[product]]s such that every object is (isomorphic to) a finite cartesian power $x^k$ of a fixed object $x \in T$. We are to think of the [[hom-set]] $T(k,1)$ as the set of $k$-ary operations of the algebras defined by the theory. A **$T$-algebra** is accordingly a product-preserving functor $A : T \to Set$. Its image $U_T(A) \coloneqq A(1) \in Set$ is the underlying set, and its value $A(f) : U_T(A)^k \to U_T(A)$ on an element $f \in T(k,1)$ is the $k$-ary operation $f$ as implemented by $A$.

The category of $T$-algebras is the [[full subcategory]] $T Alg \subset [T, Set]$ of the [[category of presheaves]] on $T^{op}$ on these product-preserving functors.

**Examples**.

* The category $T = \mathcal{Ab}$ of free finitely generated abelian groups is the syntactic category of the Lawvere theory whose algebras are abelian groups.

* For $k$ a field, the category $T = k$ of free finitely generated $k$-algebras is the Lawvere theory whose algebras are $k$-[[associative algebra]]s;

* The category $T = $ [[CartSp]] is the syntactic category whose algebras are [[smooth algebra]]s.

A morphism of Lawvere theories $T_1 \to T_2$ is again a product-preserving functor.

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

An **abelian Lawvere theory** $T$ is a morphism of Lawvere theories $ab_T : \mathcal{Ab} \to T$.

=--

For $T$ abelian, $T$-algebras have an underlying [[abelian group]], given by the functor

$$
  ab_T^* : T Alg \to Ab
  \,.
$$

This functor is a [[right adjoint]].

For example associative algebras and smooth algebras are algebras over an abelian Lawvere theory, and their underlying abelian groups are the evident ones.

Similarly, the forgetful functor $U_T : T Alg \to Set$ has a [[left adjoint]], the free $T$-algebra functor $F_T : Set \to T Alg$. By the [[Yoneda lemma]] this sends the $n$-element set $(n)$ to $F_T(n) : k \mapsto T(n,k)$.

More generally, for any $A \in T Alg$ the copresheaf

$$
  T Alg(F_T(-), A) : T \to Set
$$

is isomorphic to $A$.

The free $T$-algebra $F_T(1)$ on a single generator may be thought of as the $T$-algebra of functions on the  **$T$-line**. For instance

* for $T = k$ we have that $F_T(1) = k[X]$ is the free $k$-algebra on a single generator $X$;

* for $T = CartSp$ we have that $F_T(1) = C^\infty(\mathbb{R})$.

We say more on the canonical $T$-[[line object]] below in [The Line object](#Line)

#### Model structure on cosimplicial $T$-algebras {#ModelTAlg}

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

There is a [[cofibrantly generated model category|cofibrantly generated]] [[model structure on cosimplicial abelian groups]] $Ab^\Delta_{proj}$ whose weak equivalences are the morphisms that induce [[quasi-isomorphism]] under passage to [[Dold-Kan correspondence|normalized cochain complexes]] and fibrations are the degreewise surjections.

With respect to the [canonical sSet-enrichment](#Enrichment) of the [[category of cosimplicial objects]] $Ab^{\Delta}$, this
is a [[simplicial model category]].

For $ab_T : \mathcal{Ab} \to T$ any abelian Lawvere theory, the [[adjunction]]

$$
  ((ab_*)^\Delta \dashv (ab^*)^\Delta )
   : T Alg^{\Delta} \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}} Ab^\Delta
$$

induces a [[transferred model structure]] $T Alg^{\Delta}_{proj}$ on the category of cosimplicial $T$-algebras, whose weak equivalences and fibrations are those morphisms that under $(ab^*)^\Delta$ become weak equivalences and fibrations, respectively, in $Ab^\Delta_{proj}$.

This, too, is a [[simplicial model category]] with respect to its [standard sSet-enrichment](#Enrichment).

=--

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

The proof of the existence of the [[model structure on cochain complexes]] in non-negative degree -- $Ch^\bullet_+(Ab)$ -- whose fibrations are the _degreewise surjections_ (and weak equivalences the usual [[quasi-isomorphism]])s is spelled out here.

By the dual [[Dold-Kan correspondence]] $Ab^\Delta \simeq Ch^\bullet_+(Ab)$ this induces the [[model structure on cosimplicial abelian groups]] whose fibrations are the degreewise surjections (using that the [[Moore complex|normalized cochain complex functor]] sends surjections to surjections).

That with the standard structure of an [[sSet]]-[[enriched category]] on $Ab^\Delta$ this constitutes a [[simplicial model category]]-structure is proven here.

Now we use the basic fact of [[Lawvere theories]] that any morphism $f : T_1 \to T_2$ of such induces a pair of [[adjoint functor]]s

$$
  (f_* \dashv f^*) : T_2 Alg \stackrel{\overset{f_*}{\leftarrow}}{\underset{f^*}{\to}}
  T_1 Alg
$$

between their categories of algebras: the adjunction of relatively free algebras.

Since by assumption that our $T$ is an abelian Lawvere theory we are given a morphism $ab : Ab \to T$ from the theory of [[abelian group]]s, this means that we have an [[adjunction]]

$$
  (ab_* \dashv ab^*) : T Alg \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}}
  Ab
$$

and hence also an adjunction

$$
  (ab_*^\Delta \dashv (ab^*)^\Delta) : T Alg^\Delta \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  Ab^\Delta
  \,.
$$

We need to check that the [[right adjoint]] $(ab^*)^\Delta$ induces the [[transferred model structure]] on $T Alg^\Delta$ from the above model structure $Ab^\Delta_{proj}$.

By the facts recalled at [[transferred model structure]], we need to check that $T Alg^\Delta_{proj}$

* has a fibrant replacement functor;

* has functorial [[path space object]]s for fibrant objects;

and for the simplicial enrichment that

* $(ab^*)^\Delta$ preserves the [[power]]ing.

The first condition is trivial, since all objects are fibrant. The last condition is evidently satisfied, since

$$
  U_T(A^S)_n = U_T(\prod_{S_n} A_n) = \prod_{S_n} U_T(A_n) = ((U_T(A))^S)_n
  \,.
$$

Using this, we claim that we can take the path space object functor to be given by [[power]]ing with the simplicial [[interval]]

$$
 (-)^I : A \mapsto A^{\Delta[1]}
 \,.
$$

This is because $\Delta[0] \coprod \Delta[0] \hookrightarrow \Delta[1] \to \Delta[0]$ factors the co-diagonal in $sSet_{Quillen}$ by a cofibration followed by a weak equivalence between cofibrant objects. Accordingly the induced

$$
  A \to A^I \to A \times A
$$

factors the digonal, and the morphism on the left is a weak equivalence, since it is the image under the left Quillen functor $A^{(-)}$ of a weak equivalence between cofibrant objects (and by the [[factorization lemma]] such weak equivalences are preserved by left Quillen functors).

=--

### Simplicial presheaves on duals of $T$-algebras

A good notion of a generalized [[space]] modeled on objects in a category $C$ is a [[nLab:sheaf]] on $C$. A good notion of an [[nLab:∞-groupoid]] in such generalized spaces is an [[(∞,1)-sheaf]] on $C$. Such objects are modeled by the [[model structure on simplicial presheaves]] on $C$.

We are interested here in that case that

$$
  T \subset C \hookrightarrow T Alg^{op}
$$

is a [[small category|small]] [[full subcategory]] of the [[nLab:opposite category]] of $T$-algebras, for $T$ an abelian Lawvere theory. In the remainder of this section we assume such a choice to be fixed. Below in the section on [Examples and applications](#Examples) we discuss concrete choices of interest.

Notice that such a choice induces also a full subcategory of (co)simplicial objects

$$
  C^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op}
  \,.
$$

#### The prolonged Yoneda embedding {#ProlongedYoneda}

Write

$$
  j : T Alg^{op} \to [C^{op}, Set]
$$

for the ordinary [[nLab:Yoneda embedding]] and

$$
  j : (T Alg^\Delta)^{op} \to [C^{op}, sSet]
$$

for its degreewise simplicial prolongation

$$
  j(A) : (B \in T Alg)  \mapsto ([n] \mapsto T Alg(A_n, B))
  \,.
$$

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

For $B \in T Alg$ and $(T Alg^\Delta)_s$ denoting the [simplicially enriched](#Enrichment) category of $T$-algebras, we have a natural identification

$$
  j(A) \simeq (T Alg^\Delta)^{op}_s(-, A)
  \,.
$$

=--

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

Using [[nLab:end]]/[[nLab:coend]]-calculus for handling the [canonical enrichment](#Enrichment) of $T Alg^\Delta$, we have
for $B \in T Alg^{op} \hookrightarrow (T Alg \Delta)^{op}$
and $A \in (T Alg^\Delta)^{op}$ [[nLab:natural transformation|natural]] isomorphisms

$$
\begin{aligned}
  (T Alg^\Delta)^{op}_s(B, A)_n
  & \coloneqq
  (T Alg^\Delta)^{op}(B \cdot \Delta^n, A)
  \\
  & \simeq \int_{k \in \Delta} T Alg(A_k, \prod_{\Delta(k,n)} B)
  \\
  & \simeq \int_{k \in \Delta} T Alg(\Delta(k,n)\cdot A_k, B)
  \\
  & \simeq T Alg( \int^{k \in \Delta} \Delta(k,n) \cdot A_k, B)
  \\
  & \simeq T Alg(A_n , B)
  \,,
\end{aligned}
$$

where in the last step we used the isomorphism (described at [[coend]])
$$
  \int^{k \in \Delta} \Delta(k,n) \cdot A_k
  \simeq
 \lim_\to( \Delta/n \to \Delta \stackrel{A}{\to} T Alg)
  \simeq A_n
  \,.
$$

=--

#### The line object {#Line}

The adjunction that we shall be concerned with is essentially [[Isbell conjugation]]. We recall some basics of Function T-algebras on presheaves.

Recall from [above](#TAlgebras) that we write $F_T(*)$ for the free $T$-algebra on a single generator.

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

We call $R \coloneqq j(F_T(*))$ the **line object** in $[C^{op}, sSet]$.

=--

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

As a presheaf, the line object $R$ sends a $T$-algebra $B \in T Alg$ to its underlying set $U_T(B)$

$$
  R : B \mapsto T Alg(F_T(*), B) \simeq Set(*, U_T(B)) \simeq U_T(B)
  \,.
$$

=--

This characterization may look simpler, but does not capture the important fact that homming into $R$ produces _$T$-algebras of functions_ . This is what the following definition deals with.

+-- {: .num_defn}
###### Definition
**($T$-algebras of functions)**

For $X \in [C^{op}, sSet]$, the [[nLab:cosimplicial set]]

$$
  U_T(\mathcal{O}(X)) \coloneqq [C^{op},sSet](X_\bullet, R) \in Set
$$

we call the cosimplicial set of $R$-valued _functions_ on $X$. This is naturally the cosimplical set underlying the cosimplicial $T$-algebra

$$
  \mathcal{O}(X) : (k \in T) \mapsto [C^{op},sSet](X_\bullet, j(F_T(k)))
  \,.
$$

We call $\mathcal{O}(X) \in T Alg^{op}$ the **$T$-algebra of functions** on $X$. This extends to a functor

$$
  \mathcal{O} : [C^{op}, sSet] \to (T Alg^{\Delta})^{op}
  \,.
$$

=--

In the next section we see that $(\mathcal{O} \dashv j)$ forms a [[simplicial Quillen adjunction]].

#### Model structure on simplicial presheaves {#ModelPresheaves}

Write $[C^{op}, sSet]_{proj}$ for the global projective [[model structure on simplicial presheaves]] over $C$. With the simplicial enrichment $[C^{op}, sSet]_s$ this is naturally a [[simplicial model category]].

Let $S \subset mor [C^{op}, sSet]$ be a class of [[split hypercover]]s.

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

Write $[C^{op}, sSet]_{proj,loc}$ for the
[[left Bousfield localization]] $[C^{op}, sSet]_{proj}$ at this class.

=--

By general results on left Bousfield localization, this exists always for $S$ a [[small set]], notably for $f$ the set of [[Cech nerve]] projections $C(U) \to X$ for [[cover]]s $\{U_i \to X\}$ of the [[Grothendieck topology]] on $C$. By general results on the [[local model structure on simplicial presheaves]], the localization also exists for $S$ the class of all (split) hypercovers.

### The Yoneda-Quillen-adjunction
  {#YonedaQuillenAdjunction}

We relate now the [model structure on cosimplicial T-algebras](#ModelTAlg) with the [model structure on simplicial presheaves](#ModelPresheaves) over $C \subset T Alg^{op}$ using the [function algebra functor](#Line) $\mathcal{O}$ and the [prolonged Yoneda embedding](#ProlongedYoneda) $j$.

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

The functors $j$ and $\mathcal{O}$ constitute a [[simplicial Quillen adjunction]]

$$
  (\mathcal{O} \dashv j)
  \;\colon\;
  (TAlg^\Delta_{proj})^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\hookrightarrow}}
  [C^{op}, sSet]_{proj}
  \,.
$$

=--

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

We first establish the [[nLab:adjunction]] itself: using [[nLab:end]]-calculus for expressing [[nLab:hom-set]]s in [[nLab:functor category|functor categories]] we have for $X \in [C^{op}, sSet]$ and $A \in T Alg^\Delta$ [[nLab:natural transformation|natural]] isomorphisms

$$
\begin{aligned}
  (T Alg^\Delta)^{op}(\mathcal{O}(X), A)
  & \coloneqq
  T Alg^\Delta (A(-), [C^{op}, sSet](X, j(F_T(-))))
  \\
  & \simeq
  \int_{k \in T} \int_{[n] \in \Delta}
  Set(A_n(k), [C^{op}, Set](X_n, j( F_T(k)) ))
  \\
  & \simeq
  \int_{k \in T} \int_{[n] \in \Delta} \int_{B \in C}
  Set(A_n(k), Set(X_n(B), T Alg(F_T(k), B)))
  \\
  & \simeq
  \int_{k \in T} \int_{[n] \in \Delta} \int_{B \in C}
  Set(A_n(k), Set(X_n(B), B(k)))
  \\
  & \simeq
  \int_{k \in T} \int_{[n] \in \Delta} Set(X_n(B), \int_{k \in T}
   Set(A_n(k), B(k))
  )
  \\
  & \simeq
  \int_{k \in T} \int_{[n] \in \Delta} Set(X_n(B), T Alg(A_n, B))
  \\
  & \simeq
  [C^{op}, sSet](X, j(A))
  \,.
\end{aligned}
 \,,
$$

where the crucial step is the isomorphism $B(-) \simeq T Alg(F_T(-), B)$ for the line object discussed [above](#Line). This computation is just simplicial-degreewise the adjunction discussed at Isbell duality -- Function T-algebras on presheaves.

That this lifts to an $sSet$-enriched adjunction follows with the [prolonged Yoneda lemma](#ProlongedYoneda) $j(A) \simeq (T Alg^\Delta)^{op}_s(-,A)$ and the $sSet$-[[nLab:tensoring]] and [[nLab:cotensoring]] of $[C^{op}, sSet]_s$ and $(T Alg^\Delta)^{op}_s$:

$$
  \begin{aligned}
    (T Alg^\Delta)^{op}_s(\mathcal{O}(X), A)_n
    &
    \coloneqq
    (T Alg^\Delta)^{op}(\mathcal{O}(X), A^{\Delta^n})
    \\
    & \simeq
    [C^{op}, sSet](X, j(A^{\Delta_n}))
    \\
    & \simeq   [C^{op}, sSet](X, (T Alg^\Delta)^{op}_s(-,A^{\Delta^n}))
    \\
    & \simeq   \int_{B \in C} sSet(X(B), (T Alg^\Delta)^{op}_s(B,A^{\Delta^n})))
    \\
    & \simeq   \int_{B \in C} sSet(X(B), (T Alg^\Delta)^{op}_s(B \cdot \Delta^n,A)))
    \\
    & \simeq   \int_{B \in C} sSet(X(B)\times \Delta^n , (T Alg^\Delta)^{op}_s(B,A)))
   \\
    & \simeq
    [C^{op}, sSet](X \cdot \Delta^n, j(A))
    \\
    & =: [C^{op}, sSet]_n(X, j(A))_n
    \,.
  \end{aligned}
$$

By the pushout-product axiom satisfied by the $sSet$-[[nLab:enriched model category]] $(T Alg^\Delta)_s$ and using that in $(T Alg^\Delta_{proj})^{op}$ every object $B$ is cofibrant, we have that for $f : A_1 \to A_2$ a fibration (acyclic fibration) in $(T Alg^\Delta_{proj})^{op}$ and for $B \in C \subset T Alg^{op}$ any object, the morphism $j(A_1 \to A_2)(B) = (T Alg^{\Delta})^{op}_s(B,f)$ is a fibration (acyclic fibration) in $sSet$. Therefore $j(f)$ is a fibration (acyclic fibration) in $[C^{op}, sSet]_{proj}$.

This establishes that $j$ is a right Quillen functor and completes the proof.

=--

The following theorems say that the obstructions to making this Quillen adjunction descent to [[nLab:local model structures on simplicial presheaves]] are mild.

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

Let $J$ be a [[subcanonical coverage]] on $C \subset (TAlg^\Delta)^{op}$, $X \in Ob(C)$ and $f : Y \to j(X)$ a [[split hypercover]] with respect to $J$.

Then for $i \neq  1$ we have that $f$ induces an isomorphism in $R$-cohomology in degree $i$: $H^i(\mathcal{O}(f)) : H^i(\mathcal{O}(X)) \stackrel{\simeq}{\to} H^i(\mathcal{O}(Y))$ .

=--

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

Regard $f$ as a [[simplicial object]] in the [[overcategory]]

$$
  Sh(C)/X \simeq Sh(C/X)
  \,.
$$

Write

$$
  \bar f \in Ab(Sh(C)/X)^{\Delta^{op}}
$$

for the degreewise free abelian group object of that, a simplicial object in the category of abelian group objects in the sheaf topos over $C$. The [[chain homology]] of the corresponding normalized chain complex vanishes in positive degree (as discussed here):

$$
  H_{n \geq 1}(\bar f) = 0
  \,.
$$

Let now by the [[Freyd-Mitchell embedding theorem]]

$$
  i : Ab(Sh(C/Y)) \hookrightarrow R Mod
$$

be a [[full and faithful functor]] from the [[abelian category]] of abelian group object into the category of $R$-[[module]] over some ring $R$.

Write $K \in R Mod$ for the canonical $T$-[line object](#Line) regarded first as the abelian group object $U_T(-) \times X \in Ab(Sh(C/X))$ and then injected with $i$ into $R Mod$.

Using this, the cochain cohomology $H^i(\mathcal{O}(Y_\bullet))$ that we are after is equivalently the cohomology of

$$
  \begin{aligned}
    \mathcal{O}(Y)
    & \simeq Sh(C)^\Delta(Y_\bullet, U_T(-))
    \\
    & \simeq Sh(C)/X(f_\bullet , U_T(-) \times X)
    \\
    & \simeq Ab(Sh(C)/X)( \bar f_\bullet, U_T(-)\times X)
    \\
    & \simeq R Mod( i(\bar f_\bullet), i(U_T(-) \times X) )
    \\
    & \simeq R Mod( i(\bar f_\bullet), K )
  \end{aligned}
  \,.
$$

To compute this, we use the [[universal coefficient theorem]], which says that we have an [[exact sequence]]

$$
  0 \to Ext^1(H_{n-1}(i(\bar f_\bullet), K))
  \to H^n(R Mod(i(\bar f_\bullet), K))
  \to Ab(H_n(i(\bar f_\bullet)), C)
  \to 0
  \,.
$$

By the above fact that the homology $H_n(i(\bar f))$ vanishes in positive degree, this gives finally that

$$
  H^n(\mathcal{O}(Y))
$$

vanishes in degree $n \geq 2$. That it also vanishes in degree 0 is seen to be equivalent to the sheaf condition on $X$, which is true by the assumption that we are working with a [[subcanonical coverage]].

=--

+-- {: .num_theorem}
###### Theorem
**(passage to local model structure)**
{#PassageToLocalTheorem}

If for all split (hyper-)covers $f \in S$ we have that $H^1(\mathcal{O}(f))$ is an isomorphisms then $(\mathcal{O} \dashv j)$ is a [[simplicial Quillen adjunction]] to the [[local model structure on simplicial presheaves]].

$$
  (\mathcal{O} \dashv j) :
  (TAlg^\Delta_{proj})^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\hookrightarrow}}
  [C^{op}, sSet]_{proj, loc}
  \,.
$$

=--

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

By the previous proposition we have that under the given assumptions every (hyper-)cover $f : Y\to X $ in $[C^{op}, sSet]$ is taken by $\mathcal{O}$ to a weak equivalence.

Using this we can follow the remainder of the argument of [Toën, prop. 2.2.2](#Toen):

Since the [[model structure on simplicial presheaves]] is a [[left proper model category]] and since [[left Bousfield localization]] preserves left properness, we have that $[C^{op}, sSet]_{proj,loc}$ is [[proper model category|left proper]]. Since moreover left Bousfield localization does not change the class of cofibrations, we know that $\mathcal{O}$ still preserves cofibrations.

Then by the recognition theorem for simplicial Quillen adjunction it is sufficient to check that $j$ sends fibrant objects $A \in (T Alg^\Delta_{proj})^{op}$ to [[local object]]s with respect to the morphisms $f$.

Since by definition of hypercovers, their domain and codomain is cofibrant (codomain because it is a representable, domain by assumption that it is a degreewise coproduct of representables with disjoint degeneracies, see the discussion of cofibrancy in the projective structure at [[model structure on simplicial presheaves]]), this means that it is sufficient to check that for all $f$ and fibrant $c$ we have that $[C^{op}, sSet]_s(f, j(c))$ is a weak equivalence. But by the adjunction $(\mathcal{O} \dashv j)$ this is isomorphically $(T Alg^\Delta)^{op}(\mathcal{O}(f), c)$.

Now by the above propositions and assumptions, we have that $\mathcal{O}(f)$ is a weak equivalence. Since all objects in $(T Alg^\Delta_{proj})^{op}$ are cofibrant, it is a weak equivalence between cofibrant objects.  With the [[factorization lemma]] it follows that in an [[enriched model category]] the enriched hom of a weak equivalence into a fibrant object is a weak equivalence.

=--

The following proposition asserts that the Quillen adjunction that we have established is very special, in that it is the model-category theoretic analog of a [[reflective subcategory]]. Below in the section [Localization of the (∞,1)-topos at R-cohomology](#Intrinsic) we see that this indeed presents such a reflective inclusion in [[(∞,1)-category theory]].

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

When restricted along $C^{\Delta^{op}} \subset (T Alg^\Delta)^{op}$ the functor $j$ is _homotopy full and faithful_ in that for all $A \in C^{\Delta^{op}}$ we have that the canonical morphism

$$
  A \to \mathbb{L}\mathcal{O} \; \mathbb{R}j \; A
$$

into the image of the [[derived functor]]s of $j$ and $\mathcal{O}$ is an [[isomorphism]] in the [[homotopy category]] $Ho(T Alg^\Delta_{proj})$.

=--

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

With the above results, this follows verbatim as the proof of the analogous  ([Toën, corollary 2.2.3](#Toen)).

=--

##  Localization of the $(\infty,1)$-topos at $R$-cohomology {#Intrinsic}

We consider now the [[cohomology localization]] of
$Sh_{(\infty,1)}(C)$ at the canonical [[line object]].

In this section we discuss that in terms of the [[(∞,1)-category theory]]
that is [[presentable (∞,1)-category|presented]] by the model category theoretic structures [above](#Models), these serve to establish the following intrinsic statement.

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

The Quillen adjunction $(\mathcal{O} \dashv j)$ is a [[nLab:presentable (∞,1)-category|presentation]] that models $C$-small objects (...) in the [[reflective sub-(∞,1)-category]]

$$
  \mathbf{L}_C \stackrel{\stackrel{\mathcal{O}}{\leftarrow}}{\hookrightarrow}
  \mathbf{H} \coloneqq
  Sh_{(\infty,1)}(C)
$$

of the [[nLab:(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(C)$, where $\mathbb{L}_C$ is  the [[nLab:localization of an (∞,1)-category|localization]] at those morphisms that induce isomorphisms in intrinsic $R$-[[cohomology]], for $R$ the [canonical T-line object](#Line).

=--

We obtain a proof of this after the following discussions.

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

The resulting [[localization]] [[modality]] $Spec \mathcal{O}$ we might call the _[[affine modality]]_. It is similar to exhibiting $C$ as a [[total category]].

=--

### $R$-Cohomology

Since $T$ is assumed to be an abelian Lawvere theory, the [T-line object](#Line) $R \in [C^{op}, sSet]$ canonically has the structure of an abelian [[group object]] in $[C^{op}, sSet]$. As such it presents a 0-[[truncated]] [[∞-group]] in the $Sh_{(\infty,1)}(C)$, and so we may consider its [[Eilenberg-MacLane object]]s $\mathbf{B}^n R$ for $n \in \mathbb{N}$.

The following proposition provides a model for these Eilenberg-MacLane objects.

Write $\Xi : Ch^\bullet_+ \to Ab^\Delta$ for the dual [[Dold-Kan correspondence]] map. Notice that the free $\mathcal{Ab}$-algebra is $F_{Ab}(*) = \mathbb{Z}$, the free abelian group on a single generator, the [[integers]]. Write $F_{Ab}(*)[n] = \mathbb{Z}[n]$ for the [[cochain complex]] concentrated in degree $n$ on $F_{Ab}(*)$. For $ab_* : Ab \to T Alg$ the left adjoint to the underlying abelian group functor $ab^* : T Alg \to Ab$ we have then that $ab_* \Xi (F_{Ab}(*)[n])$ is the cosimplicial $T$-algebra which in degree $k$ is a product of copies of the free $T$-algebra corresponding to the product of copies $\mathbb{Z}$ in $\Xi \mathbb{Z}[n]$.

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

For $n \in \mathbb{N}$ the object $\mathbf{B}^n R \in Sh_{(\infty,1)}(C)$ is presented in $[C^{op}, sSet]_{proj,loc}$ by

$$
  \mathbf{B}^n R_{chn} \coloneqq j(ab_* \Xi(F_{Ab}(*)[n])
  \,.
$$

=--

Every [[(∞,1)-topos]] such as $\mathbf{H} = Sh_{(\infty,1)}(C)$ comes with its [[cohomology|intrinsic notion of abelian cohomology]]: for $X \in \mathbf{H}$ any object and for $A \in \mathbf{H}$ a [[∞-group]] object with arbitrary [[delooping]]s $\mathbf{B}^n A$, the $n$th cohomology group of $X$ with coefficients in $A$ is

$$
  H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X,\mathbf{B}^n A)
  \,.
$$

In terms of the [[model category]] presentation by $[C^{op}, sSet]_{proj,loc}$ and writing $X \in [C^{op}, sSet]$ for a representative of $X \in \mathbf{H}$ this is the [[hom-set]] in the [[homotopy category]]

$$
  \cdots \simeq Ho_{[C^{op}, sSet]_{proj,loc}}(X, \mathbf{B}^n A_{chn})
  \,.
$$

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

For $X \in [C^{op}, sSet]$ representing an object $X \in \mathbf{H}$, the intrinsic $R$-cohomology of $X$ coincides with the [[cochain cohomology]] of its cosimplicial function algebra $\mathbb{L}\mathcal{O}(X) \in T Alg^\Delta$:

$$
  H^n(X,R) \simeq H^n( \mathbb{L} \mathcal{O}(X))
  \,.
$$

=--

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

Notice that $ab_* \Xi(\mathbb{Z}[n])$, being the image of a cofibrant object in $Ab^\Delta$, is cofibrant in $T Alg^\Delta_{proj}$, hence fibrant in
$(T Alg^\Delta_{proj})^{op}$.

Using this, we compute as follows

$$
  \begin{aligned}
    H(X,\mathbf{B}^n R)
    & =
    Ho_{[C^{op}, sSet]_{proj}}(X, j(ab_* \Xi(\mathbb{Z}[n]))
    \\
    & \simeq
    Ho_{(T Alg^\Delta_{proj})^{op}}(\mathbb{L}\mathcal{O}(X),
     ab_* \Xi(\mathbb{Z}[n])
    \\
    & \simeq
    Ho_{(T Alg^\Delta_{proj})}(
        ab_* \Xi(\mathbb{Z}[n]),
       \mathbb{L}\mathcal{O}(X)
    )
    \\
    & \simeq
    Ho_{(Ab^\Delta_{proj})}(
        \Xi(\mathbb{Z}[n]),
       ab^* \mathbb{L}\mathcal{O}(X)
    )
    \\
    & \simeq
    Ho_{Ch^\bullet}(
        \mathbb{Z}[n],
       N^\bullet ab^* \mathbb{L}\mathcal{O}(X)
    )
    \\
    & \simeq
    H^n(\mathbb{L}\mathcal{O}(X))
  \end{aligned}
$$

=--

This is essentially the argument of ([Toën, corollary 2.2.6](#Toen)).

### $R$-Local objects

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

We say a morphism $f : X \to Y$ in $[C^{op}, sSet]$ is an **$R$-equivalence** if it induces isomorphisms in $R$-cohomology.

$$
  H^i(f,R) : H^i(Y,R) \stackrel{\simeq}{\to} H^i(X,R)
  \,.
$$

By prop. \ref{RCohomologyByO} this is equivalent to
saying that the [[derived functor]] $\mathbb{L}\mathcal{O}$ takes
$f$ to a weak equivalence.

We say an object $K \in [C^{op}, sSet]_{proj,loc}$ is an **$R$-[[local object]]** if for all $R$-equivalences $f$ we have that

$$
  \mathbf{H}(f,K) : \mathbf{H}(Y,K) \to \mathbf{H}(X,K)
$$

is an equivalence, equivalently if the [[derived hom-space]]
functor produces a weak equivalence
$\mathbb{R}Hom_{[C^{op}, sSet]_{proj,loc}}(f,K)$
(of [[Kan complex]]es).

=--

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

The $R$-local objects of $[C^{op}, sSet]_{proj,loc}$ that are equivalent to those in the image of $C^{\Delta^{op}} \hookrightarrow [C^{op}, sSet]$ span precisely the homotopy-essential image of the restriction of $\mathbb{R}j$ to $C^{\Delta^{op}}$

$$
  C^{\Delta^{op}} \hookrightarrow (T Alg^{\Delta})^{op}
  \stackrel{\mathbb{R}j}{\to}
  [C^{op}, sSet]_{proj,cov}
  \,.
$$

=--

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

We may explicitly see this by observing that the proof of
([Toën, theorem 2.2.9](#Toen)) goes through verbatim: it only uses the general properties of the $(\mathcal{O} \dashv j)$-adjunction that we have established above, as well as the fact that $T Alg^{\Delta}_{proj}$ is a
[[cofibrantly generated model category]] for $T$ the theory of ordinary
commutative algebras. But by our [result on the model structure on TAlg](#ModelTransferToTAlg) (theorem \ref{ModelTransferToTAlg}) we have that for general $T$ this is the [[transferred model structure]] of the [[model structure on cosimplicial abelian groups]], which is cofibrantly generated. Hence by the general properties of transferred model structures, also $T Alg^\Delta_{proj}$ is.

But more abstractly, we can also simply use the general theory of [[reflective sub-(∞,1)-categories]] and their characterization as the reflective [[localization of an (∞,1)-category]] at a set of weak equivalences: from the above we know that on the full sub-$(\infty,1)$-category of $((T Alg^\Delta_{proj})^{op})^\circ$ on the objects in $C^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op}$ is a reflective sub-$(\infty,1)$-category

$$
  \mathbf{L}
   \stackrel{\overset{\mathbb{L} \mathcal{O}}{\longleftarrow}}{\underset{\mathbb{R} i}{\hookrightarrow}}
  \mathbf{H} \coloneqq Sh_{(\infty,1)}(C)
$$

and that the left adjoint to the embedding inverts precisely the $R$-equivalences. Hence $\mathbf{L}$ is the full sub-$(\infty,1)$-category of $\mathbf{H}$ on $R$-local objects.

=--

## In derived geometry {#DerivedGeometry}

We now discuss [[function algebras]] on $\infty$-stacks more generally in the context of [[derived geometry]], meaning that we we pass in the above from sites inside the opposite of a 1-category of $T$-algebras to an [[(∞,1)-site]] inside the opposite of an [[(∞,1)-category]] of [[∞-algebras over an (∞,1)-algebraic theory]].

### Over ordinary associative algebras

Let $k$ be a [[field]] of characteristic $0$.

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

Write $(cdgAlg_k^{op})^\circ$ for the [[(∞,1)-category]] that is
[[presentable (∞,1)-category|presented]] by the [[model structure on dg-algebras|model structure on unbounded commutative cochain dg-algebras]] over $k$.

Write

$$
  i : (cdgAlg_k^{op})^\circ_- \subset (cdgAlg_k^{op})^\circ
$$

for the full [[sub-(∞,1)-category]] on cochain dg-algebras concentrated in non-positive degree.

=--

Let $C \subset (cdgAlg_k^{op})^\circ_-$ be a [[small (∞,1)-category|small]] full sub-$(\infty,1)$-category equipped with the structure of a [[subcanonical coverage|subcanonical]] [[(∞,1)-site]].

Set

$$
  \mathbf{H} \coloneqq Sh_{(\infty,1)}(C)
  \,.
$$

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

Write

$$
  \mathcal{O} : \mathbf{H} \to (cdgAlg_k^{op})^\circ
$$

for the $(\infty,1)$-[[Yoneda extension]] of the inclusion $C \hookrightarrow (cdgAlg_k^{op})^\circ_- \hookrightarrow (cdgAlg_k^{op})^\circ$.

=--

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

By the [[(∞,1)-co-Yoneda lemma]] we may express any $X \in Sh_{(\infty,1)}(C)$ by an [[(∞,1)-colimit]] over representables

$$
  X \simeq {\lim_\to}_i U_i \;\; \in Sh(C)
  \,.
$$

The functor $\mathcal{O}$ simply evaluates this colimit in $(cdgAlg_k^{op})^\circ$, which is the [[(∞,1)-limit]] in the [[opposite (∞,1)-category]]

$$
  \mathcal{O}X \simeq {\lim_\leftarrow}_i \mathcal{O}(U_i) \;\; \in (cdgAlg_k)^\circ
  \,,
$$

where we write $\mathcal{O}(U_i)$ simply for the object $U_i$ regarded in the opposite category.

=--

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

By construction $\mathcal{O}$ is a colimit-preserving $(\infty,1)$-functor between [[locally presentable (∞,1)-categories]]. Accordingly, by the [[adjoint (∞,1)-functor theorem]] is has a [[right adjoint|right]] [[adjoint (∞,1)-functor]].

$$
  j : (cdgAlg_k^{op})^\circ \to Sh_{(\infty,1)}(C)
  \,.
$$

This is given by

$$
  Spec(A) : U \mapsto (cdgAlg_k^{op})^\circ(U,A)
  \,.
$$

=--

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

This follows by the general yoga of [[Kan extension]]s. Explcitly, we check the hom-equivalence

$$
  \begin{aligned}
    Sh_C(X, Spec A) & \simeq \mathbf{H}({\lim_{\to}}_i U_i, Spec A)
    \\
    & \simeq {\lim_\leftarrow}_i \mathbf{H}(U_i, Spec A)
    \\
    & \simeq
    {\lim_\leftarrow}_i C(U_i, A)
    \\
     & \simeq
    {\lim_\leftarrow}_i (cdgAlg_k)^\circ(A, \mathcal{O}(U_i))
    \\
    & \simeq
    (cdgAlg_k)^\circ(A, {\lim_\to}_i  \mathcal{O}(U_i))
    \\
    & \simeq
    (cdgAlg_k^{op})^\circ({\lim_\to}_i  \mathcal{O}(U_i), A)
  \end{aligned}
  \,.
$$

=--

This is considered in ([Ben-Zvi/Nadler, prop. 3.1](#BenZviNadler)).

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

The above [Yoneda-Quillen adjunction](#YonedaQuillenAdjunction) for $T$ the theory of commutative $k$-algebras is compatible with this in that it also does  model the $(\infty,1)$-Yoneda extension of the inclusion

$$

  T Alg_k^{op} \hookrightarrow (T Alg_k^{\Delta})^{op}
$$

=--

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

By the general discussion of cofibrant replacement in the projective [[model structure on simplicial presheaves]] we have that every $X \in [C^{op}, sSet]_{proj,loc}$ has a cofibrant resolution of the form $\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \coprod_{i_n} U_{i_n}$, where the integrand the integrand we have the [[fat simplex]] tensored degreewise with a coproduct of representables such that the degenerate cells split off as direct summands (a [[split hypercover]]). This makes $[n] \mapsto \coprod_{i_n} U_{i_n}$ [[Reedy model structure|Reedy cofibrant]] an therefore the whole coend is a model for its [[homotopy colimit]].

Since both the simplex as well as the [[fat simplex]] $\mathbf{\Delta}$ are Reedy cofibrant cosimplicial simplicial sets, this is moreover equivalent to $\int^{[k]} \mathbf{\Delta}[k] \cdot \coprod_{i_n} U_{i_n}$ and this is still cofibrant. Now the left Quillen functor $\mathcal{O}$ takes this to $\int^{[k]} \mathbf{\Delta}[k] \cdot \coprod_{i_n} \mathcal{O}(U_{i_n})$. Since every object in $(T Alg^{\Delta})^{op}$ is cofibrant, this coend is still a homotopy colimit.

This shows that the [[derived functor]] of the left Quillen functor $\mathcal{O}$ sends the decomposition of any $\infty$-stack as the $(\infty,1)$-colimit over representable to the $(\infty,1)$-colimit of the images of these representables.

=--

## Examples and applications {#Examples}

+-- {: .num_prop}
###### Proposition (Examples)

The conditons of the above theorem are satisfied for instance for

* $T$ the theory of ordinary commutative algebras over a field $k$ and $J$ the [[fpqc topology]].

  In this case the adjunction is that considered in ([Toën](#Toen)).

* $T$ the theory of [[nLab:smooth algebra]]s and $C \hookrightarrow T Alg^{op}$ the [[site]] of the [[Cahiers topos]]. This is what we discuss in more detail below.

=--

### Rational homotopy theory

$T$ the Lawvere theory of $\mathbb{Q}$-algebras. Then $(\mathcal{O} \dashv j)$ reproduces the setup discussed at [[rational homotopy theory in an (∞,1)-topos]].

### $\infty$-Lie theory in the $\infty$-Cahiers topos

In this section we study the general theory for the case that

* $T \coloneqq $ [[CartSp]] is the ([[syntactic category]] of the) Lawvere theory of [[smooth algebra]]s.

Write $Smooth Alg \coloneqq T Alg$ for the category of smooth algebras. Sheaf toposes on sub-sites $C \subset Smooth Alg^{op}$ are well known to provide [[smooth topos]]es that are [[Models for Smooth Infinitesimal Analysis|well adapted models]] for [[synthetic differential geometry]].

We consider here the choice

* $C \subset Smooth Alg^{op}$ is the [[site]] for the [[Cahiers topos]].

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

The [[Cahiers topos]] is the [[sheaf topos]] $Sh(ThCartSp)$ on the [[site]] [[ThCartSp]] $\subset CartSp Alg^{op}$ with [[coverage]] given by the families $\{U_i \times S \stackrel{(p,Id)}{\to} X \times S\}$, where $U \in $ [[CartSp]], $S$ is an [[infinitesimal space]] (the dual of a Weil algebra) and where $\{U_i \to X\}$ is a [[good open cover]] in [[CartSp]].

The **$(\infty,1)$-Cahiers-topos** is the [[(∞,1)-category of (∞,1)-sheaves]] on [[ThCartSp]] with respect to the good open cover coverage.

=--

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

The good open cover [[coverage]] generates the [[Grothendieck topology]] of all [[open cover]]s on [[CartSp]].  Therefore the sheaf toposes on $ThCartSp$ with covering families coming from all open covers of Cartesian spaces is equivalent to the sheaf topos on $ThCartSp$ with only good open covering.

By the discussioin at Cech localization of simplicial presheaves at a coverage, the analogous statement holds true for the [[(∞,1)-topos]]es over these sites.

Therefore we may model $Sh_{(\infty,1)}(ThCartSp_{good-open})$ by the [[left Bousfield localization]] of $[ThCartSp^{op}, sSet]_{proj}$ at the [[Cech nerve]]s of all good open cover. Notice that the construction of [[good open cover]]s (see there) on [[paracompact space]]s (such as [[Cartesian space]]s) by geodescally convex regions shows that we may always find a good open cover all whose finite non-empty intersections are [[diffeomorphism|diffeomorphic]] to an open ball, hence to a Cartesian space. We shall adopt for the present purposes therefore that a cover $\{U_i \to X\}$ is _good_ if all finite intersections are isomorphic to Cartesian spaces.

The point is that with this definition, the [[Cech nerve]] $C(U) \in [ThCartSp^{op}, sSet]_{proj}$ is cofibrant, by the characterization of cofibrant objects in the projective model structure.

As a consequence of this, we have the following useful technical result.

=--

+-- {: .num_lemma}
###### Definition/Observation

Write $[ThCartSp^{op}, sSet]_{proj,cov}$ for the [[left Bousfield localization]] of the global projective model structure $[ThCartSp^{op}]_{proj}$ at the [[Cech nerve]]s $C(U) \to X\times S$ of [[good open cover]]s $\{U_i \times S \to X \times S\}$ in [[ThCartSp]].

We have that

* this presents the $(\infty,1)$-Cahiers topos $Sh_{(\infty,1)}(ThCartSp) \simeq ([ThCartSp^{op}, sSet]_{proj,cov})$;

* the fibrant objects of $[ThCartSp^{op}, sSet]_{proj,cov}$ are precisely those fibrant objects  $A \in [ThCartSp^{op}, sSet]_{proj}$ such that for all goop open covers $\{ U_i \times S \to X \times S\}$ with Cech nerve $p_U : C(U) \to X \times S$ we have that

  $$
    [ThCartSp^{op}, sSet]( p_U , A )
  $$

  is a weak equivalence (of [[Kan complex]]es).

=--

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

The Cech nerves projections $p_U : C(U) \to X \times S$ induce isomorphisms on the cohomology of their cosimplicial function algebras: $H^p(\mathcal{O}(p_U))$ is an isomorphism, for all $p \in \mathbb{N}$.

=--

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

This is a standard fact about [[Cech cohomology]]. An explicit way to see it is to choose a smooth [[partition of unity]] subordinate to the cover. See Coboundaries for Cech cocycles.

=--

This means that the assumptions of the [Theorem on passage to the local model structure](#PassageToLocalTheorem) are satisfied.

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

We have a [[simplicial Quillen adjunction]]

$$
  (Smooth Alg^\Delta_{proj})^{op}
  \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}}
  [ThCartSp^{op}, sSet]_{proj,cov}
  \,.
$$

=--

#### $\infty$-Lie algebroids

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

The objects of the $(\infty,1)$-Cahiers topos we call **synthetic differential** [[∞-Lie groupoid]]s.

The objects of the reflective sub-$(\infty,1)$-category of $R$-local objects in the $(\infty,1)$-Cahiers topos

$$
  \mathbf{L} \stackrel{\leftarrow}{\hookrightarrow}
  \mathbf{H} = Sh_{(\infty,1)}(ThCartSp)
$$

we call **[[∞-Lie algebroid]]**s.

A [[connected]] $\infty$-Lie algebroid we call an **[[∞-Lie algebra]]**.

=--

(...)

Passing along the embedding $\mathbf{L} \hookrightarrow \mathbf{H}$ we may compute [[∞-Lie algebra cohomology]] in $\mathbf{H}$.

(...)

#### The infinitesimal path $\infty$-groupoid of a manifold

(...)

For $U \in CartSp$ let

$$
   U^{\Delta^\bullet_{inf}} \in C^{\Delta^{op}}
$$

be the simplicial object of infinitesimal simplices in $U$.

**Definition**

We call

$$
  \mathbf{\Pi}_{inf}(U)
  \coloneqq
  \mathbb{R}j\; (U^{\Delta^\bullet_{inf}})
  \in
  [C^{op}, sSet]
$$

the infinitesimal path $\infty$-Lie groupoid of $U$.

Or the **path $\infty$-Lie algebroid** .

(...)

#### The tangent category of smooth algebras

(...)

The [[tangent category]] of the category of [[smooth algebra]]s is the category of modules over $C^\infty$-rings.

**Proposition** This abstract definition of module over $C^\infty$-rings reproduces the definition given by Kock.

The tangent category of the category of _simplicial_ $C^\infty$-rings is ...

This serves the purpose of presenting the $\infty$-stack of $\infty$-vector bundles on $T Alg^{op}$.

(...)

## Appendix

### Enrichment of categories of simplicial objects {#Enrichment}

We make use of the canonical structure of an [[sSet]]-[[enriched category]] on any [[category of cosimplicial objects]] in a category with all limits and colimits (see there).

+-- {: .num_example}
###### Example

For $A \in (T Alg^{\Delta})^{op}$ and $S \in sSet $ we have that the tensoring is given by

$$
  (A \cdot S)_n
  =
  \prod_{S_s} A \in T Alg
  \,,
$$

with the [[nLab:product]] taken in $T Alg$.

=--

### Model structure on cosimplicial abelian groups

We use the [[model category]] structure on $Ab^\Delta$ whose fibratin are the degreewise surjections, and whose weak equivalences are the usual [[quasi-isomorphism]]s under the dual [[Dold-Kan correspondence]] $Ab^\Delta \simeq Ch^\bullet_+(Ab)$.

The model structure is described in detail at model structure on chain complexes - the projective structure.

The structure of a [[simplicial model category]] is described in detail at [[model structure on cosimplicial abelian groups]].

## Related concepts

* [[geometric stack]], [[geometric ∞-stack]]

* [[Lie integration]]

## References

The Quillen adjunction over abelian $T$-algebras that we consider generalizes that discussed in

* {#Toen} [[Bertrand Toën]], _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))

over ordinary commutative $k$-algebras. See also [[rational homotopy theory in an (infinity,1)-topos]].

The generalization to arbitrary abelian $T$-algebras and the application to synthetic differential geometry is the content of

* {#Stel} [[Herman Stel]], _$\infty$-Stacks and their function algebras -- with applications to $\infty$-Lie theory_, MSc thesis, Utrecht 2010 ([dspace:1874/203187](https://dspace.library.uu.nl/handle/1874/203187), [[schreiber:master thesis Stel|web]])

* {#Stel13} [[Herman Stel]], _Cosimplicial $C^\infty$-rings and the de Rham complex of Euclidean space_ ([arXiv:1310.7407](http://arxiv.org/abs/1310.7407))

on which this entry here is based.

The considerations in

* {#Spivak} [[David Spivak]], _Derived smooth manifolds_, Duke Math. J. Volume 153, Number 1 (2010), 55-128. ([pdf](http://www.uoregon.edu/~dspivak/derived-smooth-manifolds.pdf))

on [[derived smooth manifold]]s may be considered as complementary to the approach taken here: there simplicial $C^\infty$-rings are considered, instead of cosimplicial ones. A fully comprehensive treatment of _derived synthetic differential geometry_ would consider the combination of both aspects: simplicial presheaves on duals of simplicial $C^\infty$-rings with a functor $\mathcal{O}$ taking them to cosimplicial-simplicial $C^\infty$-rings.

For ordinary commutative algebras the generalizaton of Toen's setup to geometry over duals of simplicial algebras is used for instance in

* {#BenZviNadler} [[David Ben-Zvi]], [[David Nadler]], _Loop spaces and connections_ ([arXiv:1002.3636](http://arxiv.org/abs/1002.3636))

[[!redirects function algebras on ∞-stacks]]

[[!redirects rational homotopy theory in an (∞,1)-topos]]
[[!redirects rational homotopy theory in an (infinity,1)-topos]]


function application

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[function]] $f$ is defined by its association to each input value $x$ (belonging to some allowable [[domain]] of values) of an output value, usually denoted $f(x)$ or $f x$.  The process of passing from $f$ and $x$ to $f(x)$ is called **function application**, and one speaks of **applying** $f$ to $x$ to produce $f(x)$.

The determination of the allowable domain for $x$, given $f$, depends a bit on [[foundations|foundational]] choices.  In [[type theory]] and [[structural set theory]], all [[functions]] have a [[type]] (a [[function type]], naturally) which specifies their domain and codomain.  In [[material set theory]], a function is sometimes defined to be simply a particular sort of set of ordered pairs, with its domain specified implicitly as the set of elements occurring as first components of some such pair.  (However, even in material set theory it is sometimes important for a function to come with a specified domain and/or codomain, in which case it can be defined to be an ordered triple.)

## Syntax versus semantics

In formalized [[logic]] and [[type theory]], $f$, $x$, and $f(x)$ are [[terms]] (or more precisely, [[metavariable|metavariables]] standing for terms), and the process of function application is a rule of term formation.  This is something which belongs to the realm of [[syntax]]. On [[propositions]] ([[(-1)-truncation|(-1)-truncated]] types) this is the [[modus ponens]] [[deduction]] rule.

Under a [[denotational semantics|denotational]] [[semantics]], each of these terms denotes a particular object, and we also refer to the object denoted by $f(x)$ as the result of applying the object denoted by $f$ to the object denoted by $x$.  For instance, in a [[material set theory|material set-theoretic]] semantics, $f$ would denote a set of [[ordered pairs]] such that for any $a$, there is at most one $b$ such that $(a,b)\in f$, and $x$ would denote some $a$ such that there *does* exist such a $b$, and $f(x)$ would denote that uniquely specified $b$.  The distinction between the terms $f$, $x$, and $f(x)$ and what they denote is usually (and harmlessly) blurred in ordinary mathematical practice, but when studying logic and type theory formally it becomes important.

Under an [[operational semantics|operational]] [[semantics]], by contrast, the "meaning" of the term $f(x)$ lies in how it is "evaluated".  Usually this proceeds by [[beta-reduction]] and related rules.  For instance, if $f$ is the [[lambda-calculus|term]] $\lambda x. x*x$ and $x$ is the term $s(s(0))$ (the [[numeral]] [[two]]), then $f(x)$ is $(\lambda x.x*x)(s(s(0)))$ which beta-reduces to $s(s(0))*s(s(0))$.  The definition of $*$ can then be invoked to cause futher beta-reductions resulting in $s(s(s(s(0))))$ (the numeral four).

## Related concepts

* [[evaluation map]]

[[!redirects application]]