geometry of physics -- manifolds and orbifolds

> this entry contains one chapter of _[[geometry of physics]]_.

> previous chapters: _[[geometry of physics -- smooth sets|smooth sets]]_, _[[geometry of physics -- smooth homotopy types|smooth homotopy types]]_

> next chapters: _[[geometry of physics -- G-structure and Cartan geometry|G-structure and Cartan geometry]]_, _[[geometry of physics -- supergeometry|supergeometry]]_, _[[geometry of physics -- BPS charges|BPS charges]]_

#Contents#
* table of contents
{:toc}

## **Manifolds and orbifolds**
 {#ManifoldsAndOrdbifolds}

In the discussion at _[[geometry of physics -- smooth sets]]_ we have started with very simple model spaces for [[differential geometry]], namely just the [[Cartesian spaces]], and have obtained from these, by passing to the [[sheaf topos]] over the [[site]] that they form, very general [[smooth spaces]]. We remarked that _[[smooth manifolds]]_ form a [[full subcategory]] for these general [[smooth sets]] and hence sit in between the very simple model spaces and the very general smooth sets.

$$
  CartSp \hookrightarrow SmoothMfd \hookrightarrow Smooth0Type
$$

In making this statement, we assumed that we already know what smooth manifolds are. Here now we consider the question of _characterizing_ smooth manifolds as special smooth sets, hence (re-)defining them, but in a way that is slightly (though not much) different from the traditional textbook description, a way that is true to the spirit of regarding smooth sets as the good ambient context for doing differential geometry.

Or rather, as throughout these notes, by characterizing smooth manifolds among smooth sets in a neat general abstract way, we may carry this characterization verbatim from [[smooth sets]] to the more general context of [[smooth homotopy types]], where its interpretation becomes something that goes beyond what traditional textbook methods allow to cover: the concept of smooth manifold thus embedded into [[smooth groupoids]] comes out as the concept of _[[étale groupoid]]_ ([[differentiable stack|differentiable]] [[étale stack]]). A traditionally fairly familiar special class of these are _[[orbifolds]]_ the direct analog in [[differential geometry]] of what in [[algebraic geometry]] is famous as [[Deligne-Mumford stacks]].

Fully generally, the interpretation of the abstract concept of manifold in [[smooth homotopy types]] is a concept of _smooth [[étale ∞-groupoids]]_.

In fact more is true. The terminology _smooth [[étale groupoid]]_ without further qualification refers to [[smooth groupoids]] that are _locally diffeomorphic_ to a [[Cartesian space]] $\mathbb{R}^n$, in a suitable way. But more generall one may replace $\mathbb{R}^n$ here with any other [[group object]] $V$ in [[smooth sets]], hence with any [[smooth group]], and in fact with any [[∞-group]] object in [[smooth homotopy types]], hence with any [[smooth ∞-group]]: given any such, then a _[[V-manifold]]_ is a smooth homotopy type $X$ which is locally diffeomorphic to $V$ in a suitable sense. One may say that $X$ is _étalé_ (spread out), but over $V$.

For instance in [[higher Cartan geometry|higher]] [[super Cartan geometry]] $V$ is typically an [[extended super Minkowski spacetime]] (some super [[n-group]]) and $X$ is a [[superspacetime]] locally diffeomorphic to that.

A key aspect featured by $V$-manifolds, in this general sense, but not necessarily by more general smooth homotopy types, is that they carry a _[[frame bundle]]_, a [[principal bundle]] for the [[general linear group]] $GL(V)$ -- which in the generality of higher differential geometry we define to be the [[automorphism ∞-group]] of the [[infinitesimal neighbourhood]] of the neutral element in $V$. The class of this frame bundle characterizes how the local diffeomorphic identifications of $X$ with $V$ are twisted with respect to each other as one moves around on $X$.

This is the key to formulating genuine geometric structure on smooth spaces: following an insight that goes back to [[Élie Cartan]], most every sort of geometric structure on a smooth space $X$ is a ([[integrable G-structure|infinitesimally integrable]]) _[[G-structure]]_, hence a [[reduction of the structure group]] of the [[frame bundle]] of $X$, subject to some local rigidity constraint. This subsumes notably ([[pseudo-Riemannian geometry|pseudo]]-)[[Riemannian geometry|Riemannian geometric structure]], [[complex structure]], [[symplectic structure]], [[conformal structure]] and much more. These geometric structures on higher manifolds are the topic of the chapter _[[geometry of physics -- G-structure and Cartan geometry]]_.

### Model Layer
 {#ModelLayer}

#### Manifolds
 {#ModelLayerManifolds}

We give first an account of the traditional definition of smooth manifolds, but (re-)phrasing it equivalently in category-theoretic terms that lend themselves to generalization.

##### Formal smooth Cartesian spaces
 {#FormalSmoothCartesianSpaces}
1.

In order to speak about [[tangent vectors]] and [[local diffeomorphisms]] in a way that generalizes to [[higher differential geometry]], it is useful to introduce structure that allows to _co-represent_ tangent vectors: where a smooth path in a Cartesian space $X$ is a smooth function out of the [[real line]] $\mathbb{R}$ into $X$, a [[tangent vector]] as traditionally defined is intuitively the restriction of such a map to the first order [[infinitesimal neighbourhood]] of the origin in $\mathbb{R}$: an "infinitesimal path".

The idea now (which is the idea of [[synthetic differential geometry]]) is to make that intuition into a mathematical fact by passing to a context of slightly generalized Cartesian spaces where that [[infinitesimal neighbourhood]] exists _literally_ as a [[subobject]] $\mathbb{D} \hookrightarrow \mathbb{R}$, such that the a tangent vector is _literally_ a morphism $\mathbb{D} \to X$.

In order to see what these generalized Cartesian spaces might be, recall a basic fact about [[smooth functions]]:

Write $CAlg_{\mathbb{R}}$ for the [[category]] of [[commutative algebras]] over the [[field]] of [[real numbers]]. Write [[SmoothCartSp]] for the [[category]] of [[Cartesian spaces]] and [[smooth functions]] between them.

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

The [[functor]]

$$
  C^\infty(-) \colon SmoothCartSp \longrightarrow CAlg_{\mathbb{R}}^{op}
$$

which sends a [[Cartesian space]] to (the [[formal dual]] of) its $\mathbb{R}$-[[commutative algebra|algebra]] of [[smooth functions]] is a [[full and faithful functor]].

In other words, for two [[Cartesian spaces]] $X,Y$ there is a [[natural bijection]] between the [[smooth functions]] $X \to Y$ and the algebra homomorphisms $C^\infty(X)\leftarrow C^\infty(Y)$.

=--

See at _[[embedding of smooth manifolds into formal duals of R-algebras]]_ for more on this.

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

One has to be careful that prop. \ref{EmbeddingOfSmoothManifoldsIntoFormalDualsOfRAlgebras} might seem to imply more than it does. In order that all constructions on all commutative algebras have the desired dual effect on formally dual smooth spaces (e.g. construction of [[products]]/[[coproducts]], or construction of [[Kähler differentials]]) one needs to refine plain commutative algebras over $\mathbb{R}$ to _[[smooth algebras]]_. See there for more on this point, which however for our purposes here is not of further concern.

=--

With Cartesian spaces embedded into the larger context of formal duals of commutative algebras by prop. \ref{EmbeddingOfSmoothManifoldsIntoFormalDualsOfRAlgebras}, we may pass to a larger subcategory of the latter to obtain a notion of generalized smooth spaces. Since we are after [[infinitesimal spaces]] of sorts, and since, via the embedding of prop. \ref{EmbeddingOfSmoothManifoldsIntoFormalDualsOfRAlgebras}, a space is equivalently encoded in the [[algebra of functions]] it carries, we need to specify those commutative algebras that behave like algebras of functions on infinitesimal Cartesian spaces. The intuition is that these spaces are so small, that the canonical coordinate functions $x^i$ on them take values which are so very small $\llt 1$ that when taken to some power they vanish not only approximately, but identically. But this intuition is easily formalized:

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

Write

$$
  InfPoint \hookrightarrow CAlg_{\mathbb{R}}^{op}
$$

for the [[full subcategory]] of the [[opposite category]] of [[commutative algebras]] over $\mathbb{R}$ on [[formal duals]] of [[commutative algebras]] over the [[real numbers]] of the form $\mathbb{R}\oplus V$ with $V$ a finite-dimensional  [[nilpotent ideal]] ([[local Artin algebras]] over the real numbers). We call this the category of _[[infinitesimally thickened points]]_.

Write moreover

$$
  FormalSmoothCartSp
  \coloneqq
  CartSp \rtimes InfPoint \hookrightarrow CAlg_{\mathbb{R}}^{op}
$$

for the [[full subcategory]] on [[formal duals]] of those algebras which are [[tensor products]] of commutative $\mathbb{R}$-algebras of the form

$$
  C^\infty(\mathbb{R}^n) \otimes C^\infty(D)
$$

of algebras $C^\infty(\mathbb{R}^p)$ of [[smooth functions]] $\mathbb{R}^n$ as in def. \ref{EmbeddingOfSmoothManifoldsIntoFormalDualsOfRAlgebras} with algebras corresponding to infinitesimally thickened points $D$ as above.

=--

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

The basic idea of prop. \ref{FormalSmoothCartesianSpaces} is familiar in [[algebraic geometry]], going back to [[Alexander Grothendieck]]'s work in the 1950s, related to [[crystalline cohomology]]. Traditionally the idea is often embodied in the concept of [[formal schemes]], which is however considerably more involved ([[locally ringed spaces]], [[ind-objects]]) than necessary for capturing the basic phenomenon. The point that the simple idea of modelling infinitesimals by nilpotent element in functions rings is not at all restricted to [[algebraic geometry]] but also works in [[differential geometry]] has been highlighted by [[William Lawvere]] in the 1960s, who phrased it in terms of what is now known as the [[Kock-Lawvere axioms]] of [[synthetic differential geometry]]. In this context textbooks tend to amplify the usefulness of [[smooth algebras]] with infinitesimals, but as long as one does not insist on all geometric operations to have algebraic duals, see remark \ref{CarefulWithSmoothAlgebras} above, then using just plain real algebras with nilpotent elements as done here is sufficient.

=--

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

The evident [[full subcategory]] inclusion
 of [[smooth Cartesian spaces]] into [[formal smooth manifold|formal smooth Cartesian spaces]], def. \ref{FormalSmoothCartesianSpaces} is [[coreflective
subcategory|coreflective]], hence has a [[right adjoint]] $p$

$$
  (i \dashv p)
  \;\colon\;
  SmoothCartSp
   \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\longleftarrow}}
  FormalSmoothCartSp
  \,.
$$

On formal dual algebras, $p$ is given by quotienting out the [[nilpotent ideal]].

=--

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

We need to check that for all $n_1, n_2 \in \mathbb{N}$ and for all $D \in InfPoint$, there is a [[natural bijection]] between the set of morphisms

$$
  \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}\times D
$$

in $FormalSmoothCartSp$, and the set of morphisms

$$
  \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}
$$

in $SmoothCartSp$. By definition, the former set is equivalently that of $\mathbb{R}$-algebra homomorphisms $\phi$ of the form

$$
  C^\infty(\mathbb{R}^{n_1})
   \longleftarrow
  C^\infty(\mathbb{R}^{n_2})\otimes_{\mathbb{R}} V
  \,,
$$

where $V$ is the [[nilpotent ideal]] of the formal dual algebra of functions on $D$.

Now for $v$ an element in the algebra on the right which is nilpotent, hence for which there is $p \in \mathbb{N}$ such that $v^p = 0$, then any algebra homomorphism $\phi$ needs to preserve this condition, hence also $\phi(v)^p = 0$ and hence also $\phi(v)$ is nilpotent. But the only nilpotent element in $C^\infty(\mathbb{R}^{n_1})$ is zero. Therefore every algebra homomorphism $\phi$ as above has to have $V$ in its [[kernel]], hence it has to factor through the quotient projection $C^\infty(\mathbb{R}^{n_2})\otimes_{\mathbb{R}} V \to C^\infty(\mathbb{R}^{n_2})$.

With this the statement follows by prop. \ref{EmbeddingOfSmoothManifoldsIntoFormalDualsOfRAlgebras}.

(A similar argument gives that $p$ is a functor in the first place.)

=--

##### Formal smooth sets
 {#FormalSmoothSets}

Following closely the theory at _[[geometry of physics -- smooth sets]]_, we are now to pass to the [[generalized smooth spaces]] which are _locally probe-able_ by the formal smooth Cartesian spaces of def. \ref{FormalSmoothCartesianSpaces}, and these are the [[sheaves]] on the [[site]] $FormalSmoothCartSp$ that the latter form.

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

Regard the category $FormalSmoothCartSp$ of def. \ref{FormalSmoothCartesianSpaces} as a [[site]] by equipping it with the [[coverage]] whose covering families are of the form

$$
  \left\{
    U_i \times D \stackrel{(\phi_i,id_D)}{\longrightarrow} U \times D
  \right\}
$$

for $U, U_i \in SmoothCartSp \hookrightarrow FormalSmoothCartSp$, $D \in InfPoint \hookrightarrow FormalSmoothCartSp$, and $\{U_i \stackrel{\phi_i}{\to} U\}$ a covering family in $SmoothCartSp$, hence a traditional [[open cover]].

=--

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

A _formal smooth set_ is a [[sheaf]] on the site $FormalSmoothCartSp$ of def. \ref{FormalSmoothCartesianSpacesSite}. Write

$$
  FormalSmooth0Type \coloneqq Sh(FormalSmoothCartSp)
$$

for the [[category of sheaves]] on that site.

=--

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

The category of def. \ref{ToposOfFormalSmoothSets}
is traditionally known as the _[[Cahiers topos]]_.

=--

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

The co-reflective inclusion of sites of prop. \ref{SmoothCartSpCoreflectiveInFormalSmoothCartSp}

$$
  SmoothCartSp \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\longleftarrow}} FormalSmoothCartSp
$$

extends via [[Kan extension]] to an [[adjoint quadruple]] of functors

$$
  Smooth0Type
   \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\stackrel{\overset{i_\ast}{\hookrightarrow}}{\underset{i^!}{\longrightarrow}}}}
  FormalSmooth0Type
$$

between [[smooth sets]] and [[formal smooth sets]].

=--

We agree that when in the following we write an unlabeled inclusion

$$
  Smooth0Type \hookrightarrow FormalSmooth0Type
$$

then we always mean inclusion by $i_!$ in the above, not by $i_\ast$. On representables this is the inclusion $i$ of smooth Cartesian spaces into Cartesian spaces.

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

We write

$$
  (\Re \dashv \Im \dashv \&)
  \;\colon\;
  FormalSmooth0Type
   \stackrel{\overset{i^\ast}{\longrightarrow}}{\stackrel{\overset{i_\ast}{\longleftarrow}}{\underset{i^!}{\longrightarrow}}}
   Smooth0Type
   \stackrel{\overset{i_!}{\longrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}}
  FormalSmooth0Type
$$

for the [[adjoint triple]] of ([[comonad|co]]-)[[monads]] induced from the functors in prop. \ref{AdjointQuadrupleOnFormalSmoothSets}. At times we call these, respectively

* $\Re$ -- [[reduction modality]];

* $\Im$ -- [[infinitesimal shape modality]];

* $\&$ -- [[infinitesimal flat modality]].

=--

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

The single key fact to understand the operations in def. \ref{AdjointTripleOnFormalSmoothSets} is the [[left Kan extension]] of presheaves along a functor between sites restricts to that functor on representables. (See at _[Kan extension -- Left Kan extension on representables](Kan+extension#LeftKanOnRepresentables)_).

This means that for $U \in SmoothCartSp \hookrightarrow FormalSmoothCartSp$ and $D \in InfPoint \hookrightarrow FormalSmoothCartSp$ then

* $\Re( U \times D) \simeq D$;

* $Hom(U \times D, \Im(X)) \simeq Hom(U, X)$.

=--

##### Tangent vectors
 {#TangentVectors}

The following is the key observation regarding generally the role of nilpotent elements in function algebras and specifically as to the nature of formal Cartesian spaces.

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

Write $\mathbb{D}$ for the [[formal dual]] of the [[algebra of dual numbers]] $(\mathbb{R}[\epsilon])/(\epsilon^2)$. Then morphisms

$$
  \mathbb{R}^n \times \mathbb{D}\longrightarrow \mathbb{R}^n
$$

which are the identity after restriction along $\mathbb{R}^n \to \mathbb{R}^n \times \mathbb{D}$, are equivalently algebra homomorphisms of the form

$$
  (C^\infty(\mathbb{R}^n) \oplus \epsilon C^\infty(\mathbb{R}^n))
  \longleftarrow
  C^\infty(\mathbb{R}^n)
$$

which are the identity modulo $\epsilon$. Such a morphism has to take any function $f \in C^\infty(\mathbb{R}^n)$ to

$$
  f + (\partial f) \epsilon
$$

for some smooth function $(\partial f) \in C^\infty(\mathbb{R}^n)$. The condition that this assignment makes an algebra homomorphism is equivalent to the statement that for all $f_1,f_2 \in C^\infty(\mathbb{R}^n)$

$$
  (f_1 f_2 + (\partial (f_1 f_2))\epsilon )
  =
  (f_1 + (\partial f_1) \epsilon)
  (f_2 + (\partial f_2) \epsilon)
  \,.
$$

Multiplying this out and using that $\epsilon^2 = 0$ this in turn is equivalent to

$$
  \partial(f_1 f_2) = (\partial f_1) f_2 + f_1 (\partial f_2)
  \,.
$$

This in turn means equivalently that $\partial\colon C^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ is a [[derivation]]. But derivations of algebras of [[smooth functions]] are equivalent to [[vector fields]]. (See at _[[derivations of smooth functions are vector fields]]_).

In particular one finds that maps

$$
  \mathbb{D} \longrightarrow \mathbb{R}^n
$$

are equivalently single [[tangent vectors]].

=--

Based on this, the following is a classical fact of [[synthetic differential geometry]].

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

For $X \in SmoothMfd \hookrightarrow Smooth0Type \hookrightarrow FormalSmooth0Type$ a [[smooth manifold]], then the [[internal hom]] $[\mathbb{D},X]$ out of the [[infinitesimal interval]] is equivalent to the total space manifold of the [[tangent bundle]] of $X$:

$$
  T X \simeq [\mathbb{D},X]
  \,.
$$

=--

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

By definition of [[internal hom]], for every object $U$ then morphisms

$$
  U \longrightarrow [\mathbb{D},X]
$$

are in [[natural bijection]] with morphisms

$$
  U \times \mathbb{D}\longrightarrow X
  \,.
$$

By example \ref{TangentVectorsByAlgebra} these are smoothly $U$-parameterized tangent vectors in $X$, hence are naturally isomorphism to morphisms

$$
  U \longrightarrow T X
  \,.
$$

Hence $[\mathbb{D},X]$ and $T X$ represent isomorphic [[representable functors]]. Then the [[Yoneda lemma]] implies that they are themselves [[isomorphism|isomorphic]].

=--

##### Local diffeomorphisms

The traditional definition of [[local diffeomorphisms]] says

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

A [[smooth function]] $f \colon X \longrightarrow Y$
between [[smooth manifolds]] is a _local diffeomorphism_ if for all points $x\in X$ its [[derivatives]] at that point induce an [[isomorphism]] of [[tangent bundle|tangent]] [[vector spaces]]

$$
  \underset{x \in X}{\forall} \;\; d f \;\colon\; T_x X \stackrel{\simeq}{\longrightarrow} T_{f(x)}Y
  \,.
$$

=--

A more [[category theory|category theoretic]] way to say this, which may still be found in some traditional textbooks, is this

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

A [[smooth function]] $f \colon X \longrightarrow Y$
between [[smooth manifolds]] is a _local diffeomorphism_, def. \ref{LocalDiffeomorphismTraditional}, equivalently if the [[diagram]] of [[tangent bundles]] that it induces

$$
  \array{
    T X &\stackrel{}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{d f}} && \downarrow
    \\
    T Y &\stackrel{}{\longrightarrow}& Y
  }
$$

is a [[pullback]] diagram.

=--

Now this in turn we may further reformulate as follows:

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

A [[smooth function]] $f \colon X \longrightarrow Y$
between [[smooth manifolds]] is a _local diffeomorphism_, def. \ref{LocalDiffeomorphismTraditional}, equivalently if after regarding in under the embedding $Smooth0Type \hookrightarrow FormalSmooth0Type$, the [[unit of a monad|unit]] [[natural transformation|naturality square]] of the [[infinitesimal shape modality]], def. \ref{AdjointTripleOnFormalSmoothSets},

$$
  \array{
    X &\stackrel{}{\longrightarrow}& \Im X
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\stackrel{}{\longrightarrow}& \Im Y
  }
$$

is a [[pullback]]

=--

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

Pullbacks of sheaves are detected by giving pullbacks of sets after evaluating on all possible objects of the [[site]].

Consider first those objects of the form $U \in SmoothCartSp \hookrightarrow FormalSmoothCartSp \hookrightarrow FormalSmooth0Type$. By example \ref{NatureOfTheModalitiesOnCahiersTopos} for these the diagram in question becomes

$$
  \array{
    Hom(U,X) &\stackrel{id}{\longrightarrow}& Hom(U,X)
    \\
    \downarrow^{\mathrlap{Hom(U,f)}} && \downarrow^{\mathrlap{Hom(U,f)}}
    \\
    Hom(U,Y) &\stackrel{id}{\longrightarrow}& Hom(U,Y)
  }
  \,,
$$

which is trivially a pullback. Consider then objects of the form $U \times \mathbb{D}$ for $\mathbb{D}$ the [[infinitesimal interval]]. By example \ref{NatureOfTheModalitiesOnCahiersTopos} and by prop. \ref{TangentBundlesAsInternalHoms} they all giving pullbacks is equivalent to the diagram

$$
  \array{
    T X &\stackrel{}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{d f}} && \downarrow
    \\
    T Y &\stackrel{}{\longrightarrow}& Y
  }
$$

being a pullback. By prop. \ref{LocalDiffeosDetectedByPullbackOfTangentBundle} this is already equivalent to $f$ being a local diffeomorphism.

But local diffeomorphisms of [[smooth manifolds]] (as opposed to [[formally étale morphisms]] of [[schemes]]!) are already necessarily [[étale morphisms]], i.e. they diffeomorphically identify not just an [[infinitesimal neighbourhood]] around each point, but the whole [[germ]]. This implies that for $D \in InfPoint \hookrightarrow FormalSmoothCartSp \hookrightarrow FormalSmooth0Type$ any object, also the image of the diagram under $Hom(U \times D,-)$ gives a pullback.

=--

Given prop. \ref{TraditionalLocalDiffeosByModality} it makes sense to say

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

A morphism $f \colon X\to Y$ in $FormalSmooth0Type$ is a _[[local diffeomorphism]]_ (or [[formally étale morphism]]) if the [[natural transformation|naturality sqare]] of the [[unit of a monad|unit]] of its [[infinitesimal shape modality]]

$$
  \array{
    X &\stackrel{}{\longrightarrow}& \Im X
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\stackrel{}{\longrightarrow}& \Im Y
  }
$$

is a [[pullback]] diagram.

=--

##### Smooth manifolds
  {#SmoothManifold}

With the (re-)formulation of local diffeomorphisms between smooth manifolds and their generalization to maps of smooth sets by prop. \ref{LocalDiffeosGenerallyInCahiersTopos}, we now have the following neat way to characterize smooth manifolds among [[smooth sets]].

+-- {: .num_defn #ManifoldsAsEtaleGroupoids}
###### Definition/Proposition

For $n\in \mathbb{N}$, a [[smooth manifold]] of [[dimension]] $n$ is an object $X \in FormalSmooth0Type$ such that there exists a morphisms of the form

$$
  \underset{i }{\coprod} U_i
  \stackrel{=}{\to}
  \underset{i }{\coprod} \mathbb{R}^n
  \longrightarrow
  X
$$

which is

1. an [[epimorphism]],

1. a [[local diffeomorphism]], def. \ref{LocalDiffeosGenerallyInCahiersTopos};

=--

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

A quick abstract way to prove this, for which however at this point we have not provided all the ingredients yet, is to observe that the map being an [[1-epimorphism]] makes it equivalent to its [[Cech nerve]] [[groupoid object in an (infinity,1)-category|groupoid object]], and it being a local diffeomorphism makes that an [[étale groupoid]]. This being 0-truncated, its underlying object (stack) is equivalent to a smooth manifold, in fact as a groupoid object it is the [[Cech groupoid]] of that smooth manifold

$$
  \underset{i,j}{\coprod}  U_i \underset{X}{\times} U_j
  \stackrel{\longrightarrow}{\longrightarrow}
  \underset{i}{\coprod} U_i
$$

with respect to the cover that we started with above.

=--

##### Frame bundles
 {#FrameBundleTraditional}

> the following is some semi-traditional basic discussion. Need to see what to do with this here.

On each overlap $U_i \cap U_j$ of two charts, the [[partial derivatives]] of the corresponding [[coordinate transformations]]

$$
  \phi_j\circ \phi_i^{-1}
  :
  U_i \cap U_j \subset \mathbb{R}^n \to \mathbb{R}^n
$$

form the [[Jacobian matrix]] of [[smooth functions]]

$$
  ((\lambda_{i j})^{\mu}{}_{\mu})
  \coloneqq
  \left[\frac{d}{d x^\nu} \phi_j \circ \phi_i^{-1} (x^\mu)
  \right]
  :
  U_i \cap U_j
  \to
  GL_n
$$

with values in invertible [[matrices]], hence in the
[[general linear group]] $GL(n)$. By construction (by the [[chain rule]]), these functions satisfy on triple overlaps of coordinate charts the [[matrix product]] equations

$$
  (\lambda_{i j})^\mu{}_\lambda (\lambda_{j k})^\lambda{}_{\nu}
  =
  (\lambda_{i k})^\mu{}_{\nu}
  \,,
$$

(here and in the following sums over an index appearing upstairs and downstairs are explicit)

hence the equation

$$
  \lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}
$$

in the [[group]] $C^\infty(U_i \cap U_j \cap U_k, GL(n))$ of smooth $GL(n)$-valued functions on the chart overlaps.

This is the _[[cocycle]] condition_ for a smooth [[Cech cohomology|Cech cocycle]] in degree 1 with coefficients in $GL(n)$ (precisely: with coefficients in the [[sheaf]] of [[smooth functions]] with values in $GL(n)$ ). We write

$$
  [(\lambda_{i j})] \in H^1_{smooth}(X, GL_n)
  \,.
$$

Formulated as [[smooth groupoids]]

* $X$ itself is a [[Lie groupoid]] $(X \stackrel{\longrightarrow}{\longrightarrow} X)$ with trivial morphism structure;

* from the atlas $\{U_i \to X\}$ we get the corresponding [[Cech groupoid]]
  $$
    C(\{U_i\}) =
    (\coprod_{i, j} U_i \cap U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_i U_i)
    =
    \left\{
      \array{
         && (x,j)
         \\
         & \nearrow &=& \searrow
         \\
        (x,i) &&\to&& (x,k)
      }
      \;\;\;
      for\, x \in U_i \cap U_j \cap U_k
    \right\}
    \,,
  $$

  whose objects are the points in the atlas, with morphisms identifying lifts of a point in $X$ to different charts of the atlas;

The above situation is neatly encoded in the existence of a [[diagram]] of Lie groupoids of the form

$$
  \array{
     C(\{U_i\}) &\stackrel{\tau_X}{\longrightarrow}& \mathbf{B} GL(n).
     \\
     {}^{\mathllap{\simeq}}\downarrow
     \\
     X
  }
  \,,
$$

where

* the left morphism is [[stalk]]-wise (around small enough [[neighbourhoods]] of each point) an [[equivalence of groupoids]] (we make this more precise in a moment);

* the horizontal functor $\tau_X$ has as components the functions $\lambda_{i j}$ and its functoriality is the cocycle condition $\lambda_{i j} \cdot \lambda_{j k} = \lambda_{i k}$.

A [[natural transformation|transformation]] of smooth functors $\lambda_1 \Rightarrow \lambda_2 : C(\{U_i\}) \to \mathbf{B} GL(n)$ is precisely a [[coboundary]] between two such cocycles.

This defines a morphism of [[smooth groupoids]]

$$
  \tau_X \;\colon\; X \to \mathbf{B}GL(n)
  \,.
$$

The [[homotopy fiber]] of this map is a $GL(n)$-[[principal bundle]] called the [[frame bundle]] of $X$, while  the canonically [[associated bundle]] via the canonical [[representation]] of $GL(n)$ on $\mathbb{R}^n$ is the [[tangent bundle]]

$$
  T X \to X
  \,.
$$

#### Étale groupoids

We now lift the [above](#ModelLayerManifolds) discussion of smooth manifolds inside [[smooth sets]]. phrased via an [[infinitesimal shape modality]] from the context of [[smooth sets]] to that of [[smooth groupoids]]. For background on [[smooth groupoids]] see the relevant discussions at _[[geometry of physics -- smooth homotopy types]]_.

##### Formal smooth groupoids

In view of the refinement of [[smooth sets]] on the one hand to [[formal smooth sets]] and on the other hand to [[smooth groupoid]], we now consider the joint refinement to [[formal smooth groupoids]].

This proceeds essentially verbatim to the previous definitions

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

A _pre-formal smooth groupoid_ is a [[presheaf]] of [[groupoids]] on $FormalSmoothCartSp$.

Write

$$
  FormalSmooth1Type \coloneqq L_{lwhe} PreFormalSmooth1Type
$$

for the [[(2,1)-category]] which is the [[simplicial localization]] at the local weak equivalences.

=--

(...)

$$
  \array{
    Smooth1Type &
      \stackrel{\hookrightarrow}{\stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}}}
    & FormalSmooth1Type
    \\
    \downarrow && \downarrow
    \\
    Smooth0Type
    &
      \stackrel{\hookrightarrow}{\stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}}}
    &
    FormalSmooth0Type
  }
$$

We may now directy adapt the definition of local diffeos as in def. \ref{LocalDiffeosGenerallyInCahiersTopos} simply by generalizing [[0-types]] to [[1-types]] and [[pullbacks]] to [[homotopy pullbacks]]:

+-- {: .num_defn #LocalDiffeosGenerallyInFormalSmooth1Types}
###### Definition

A morphism $f \colon X\to Y$ in $FormalSmooth1Type$, def. \ref{FormalSmoothGroupoids}, is a _[[local diffeomorphism]]_ (or [[formally étale morphism]]) if the [[natural transformation|naturality sqare]] of the [[unit of a monad|unit]] of its [[infinitesimal shape modality]]

$$
  \array{
    X &\stackrel{}{\longrightarrow}& \Im X
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\stackrel{}{\longrightarrow}& \Im Y
  }
$$

is a [[homotopy pullback]] diagram.

=--

##### Étale Lie groupoids
 {#EtaleLieGroupoids}

The traditional definition of smooth [[étale groupoids]] is the following:

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

A [[Lie groupoid]] $\mathcal{G}_\bullet$ is _[[étale groupoid|étale]]_ if its target map

$$
  t \colon \mathcal{G}_1 \longrightarrow \mathcal{G}_0
$$

is a [[local diffeomorphism]].

=--

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

The condition in def. \ref{EtaleGroupoidTraditional} implies that all the other structure maps are local diffeomorphisms, too:

Inversion $(-)^{-1}\colon \mathcal{G}_1\stackrel{\simeq}{\to} \mathcal{G}_1$ is even a global [[diffeomorphism]]. Hence if the target map is a local diffeomorphism then the source map $s = t \circ (-)^{-1}$ is the composite of two local diffeos and as such itself a local diffeo.

The identity-assigning map $i \colon \mathcal{G}_0 \to \mathcal{G}_1$ is a [[section]] of a local diffeo (either $s$ or $t$) and hence itself a local diffeo.

The two projections out of $\mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1$ are pullbacks of local diffeos, and hence themselves local diffeos.

=--

We now rephrase this more intrinsically.

Given a [[Lie groupoid]]
we write $\mathcal{G}_0 \longrightarrow \mathcal{G}$ for the corresponding [[differentiable stack]] with [[atlas]].

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

A [[Lie groupoid]] $\mathcal{G}_\bullet$ is an [[étale groupoid]], def. \ref{EtaleGroupoidTraditional}, precisely if $\mathbb{G}_0 \to \mathcal{G}$ is a [[local diffeomorphism]] of smooth groupoids, def. \ref{LocalDiffeosGenerallyInFormalSmooth1Types} in that

$$
  \array{
    \mathcal{G}_0 &\longrightarrow& \Im \mathcal{G}_0
    \\
    \downarrow && \downarrow
    \\
    \mathcal{G} &\longrightarrow& \Im \mathcal{G}
  }
$$

is a [[homotopy pullback]] diagram of [[formal smooth groupoids]]

=--

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

We compute the homotopy pulback by fibrant presentation of the morphism on the right, via the [[factorization lemma]].

$$
  \array{
    \mathcal{G}_0 \underset{\mathcal{G}_\bullet}{\times} \mathcal{G}^I_\bullet  &\longrightarrow& \Im \mathcal{G}_0 \underset{\Im\mathcal{G}_\bullet}{\times} \Im\mathcal{G}^I_\bullet
    \\
    \downarrow && \downarrow
    \\
    \mathcal{G}_\bullet &\longrightarrow& \Im \mathcal{G}_\bullet
  }
$$

Inspection shows that in degree 0 this diagram is just the naturality square of the $\Im$-unit on the target map $t\colon \mathcal{G}_1 \to \mathcal{G}_0$. Hence by prop. \ref{TraditionalLocalDiffeosByModality} this is a pullback in degree-0 precisely already if $\mathcal{G}_\bullet$ is étale. By remark \ref{ImplicitConditionsInEtaleGroupoidDef} this implies that the projection $p_2 \colon \mathcal{G}_1\underset{\mathcal{G}_0}{\times}\mathcal{G}_1 \to \mathcal{G}_1$ is a local diffeo, and that is equivalent to the above diagram being a pullback in degree 1.

=--

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

In view of def./prop. \ref{ManifoldsAsEtaleGroupoids},
prop. \ref{EtaleGroupoidByInfinitesimalShape} gives that [[étale groupoids]] are precisely the "1-higher manifolds". For historical reasons this kind of statement is more prominent in the context of [[algebraic geometry]], where it essentially corresponds to regarding [[Artin stacks]] (in particular [[Deligne-Mumford stacks]]) as higher or [[derived schemes]].

=--

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

Another point that prop. \ref{EtaleGroupoidByInfinitesimalShape} makes maybe more manifest is that the traditional definition \ref{EtaleGroupoidTraditional} makes a silent assumption that one may want to make explicit: it characterizes a Lie groupoid as [[étale Lie groupoid|étale]] _relative to a manifold_. A priori it may seem that the characterization is relative specifically to its maniold of objects. However, if $\mathcal{G}_0 \to \mathcal{G}$ is an étale atlas of a [[differentiable stack]], and if $\{\mathbb{R}^n \simeq U_i \to \mathcal{G}_0\}$ is an [[open cover]] of the smooth manifold $\mathcal{G}_0$, then also the composite

$$
  U \coloneqq \underset{i}{\coprod} \mathbb{R}^n \longrightarrow \mathcal{G}_0
  \longrightarrow
  \mathcal{G}
$$

is an étale atlas. This means that the étale nature of $\mathcal{G}_\bullet$ is really with respect to the chosen model space

$$
  V \coloneqq \mathbb{R}^n
  \,.
$$

One way to make this explicit in the more intrinsic diagrammatic formulation that we are discussing here is to say that we have a [[correspondence]]

$$
  \array{
      && U
      \\
      & \swarrow && \searrow
      \\
     V && && X
  }
$$

where

1. both morphisms are [[local diffeomorphisms]] of smooth groupoids, def. \ref{LocalDiffeosGenerallyInFormalSmooth1Types};

1. the right morphism in addition is an [[effective epimorphism]] ([[1-epimorphism]]).

Jointly this says that $U \to X$ is an [[atlas]] which is étale relative to the model space $V$.

This perspective leads to the general abstract definition of [[V-manifolds]] in [[higher differential geometry]], hence of [[étale ∞-groupoids]], [below](#VManifolds).

=--

### Semantics Layer

#### Differential cohesion

In the above discussion we found an [[adjoint quadruple]], prop. \ref{AdjointQuadrupleOnFormalSmoothSets}, on the category of [[formal smooth sets]], and ended up expressing the theory of [[smooth manifolds]] entirely in terms of the induced [[monad]] which we called the [[infinitesimal shape modality]] $\Im$.

One may hence turn this around and regard any [[cohesive (∞,1)-topos]] that is equipped with an adjoint quadruple inclusion in this way as a formal context for manifold theory ("[[elasticity]]").

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

Given a [[cohesive (∞,1)-topos]] $\mathbf{H}$ then the structure of
_[[differential cohesion]]_ on it is a choice of sub-topos $i_! \colon \mathbf{H}_{\Re} \hookrightarrow \mathbf{H}$ such that the inclusion extends to an [[adjoint quadruple]]

$$
  \mathbf{H}_{\Re}
   \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\stackrel{\overset{i_\ast}{\hookrightarrow}}{\underset{i^!}{\longrightarrow}}}}
  \mathbf{H}
$$

and such that $i_!$ preserves finite products. Given this then we write

$$
  (\Re \dashv \Im \dashv \&)
  \;\colon\;
  \mathbf{H}
   \stackrel{\overset{i^\ast}{\longrightarrow}}{\stackrel{\overset{i_\ast}{\longleftarrow}}{\underset{i^!}{\longrightarrow}}}
   \mathbf{H}_{\Re}
   \stackrel{\overset{i_!}{\longrightarrow}}{\stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}}
  \mathbf{H}
$$

for the induced [[adjoint triple]] of ([[(∞,1)-comonad|co]]-)[[(∞,1)-monads]] induced from the four [[(∞,1)-functors]]. At times we call these, respectively

* $\Re$ -- [[reduction modality]];

* $\Im$ -- [[infinitesimal shape modality]];

* $\&$ -- [[infinitesimal flat modality]].

=--

#### $V$-Manifolds
 {#VManifolds}

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

Given $X,Y\in \mathbf{H}$ then a morphism $f \;\colon\; X\longrightarrow Y$ is a _[[local diffeomorphism]]_ if its naturality square of the [[infinitesimal shape modality]]

$$
  \array{
    X &\longrightarrow& \Im X
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\longrightarrow& \Im Y
  }
$$

is a [[homotopy pullback]] square.

=--

Let now $V \in \mathbf{H}$ be given, equipped with the structure of a [[group]] ([[∞-group]]).

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

A _[[V-manifold]]_ is an $X \in \mathbf{H}$ such that there exists a _$V$-[[atlas]]_, namely a [[correspondence]] of the form

$$
  \array{
    && U
    \\
    & \swarrow && \searrow
    \\
    V && && X
  }
$$

with both morphisms being [[local diffeomorphisms]], def. \ref{LocalDiffeomorphismsInDifferentialCohesion}, and the right one in addition being an [[epimorphism]], hence an [[atlas]].

=--

#### Frame bundles
 {#FrameBundles}

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

For $X \in \mathbf{H}$ an object and $x \colon \ast \to X$ a point, then we say that the _[[infinitesimal neighbourhood]] of_, or the _[[infinitesimal disk]] at_ $x$ in $X$ is the [[homotopy fiber]] $\mathbb{D}^X_x$ of the [[unit of a monad|unit]] of the [[infinitesimal shape modality]] at $x$:

$$
  \array{
    \mathbb{D}^X_x &\longrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    \ast &\stackrel{x}{\longrightarrow}& \im X
  }
  \,.
$$

=--

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

For $X$ any object in differential cohesion, its _infinitesimal disk bundle_ $T_{inf} X \to X$ is the [[homotopy pullback]]

$$
  \array{
    T_{inf} X &\stackrel{ev}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{p}} && \downarrow
    \\
    X &\longrightarrow& \Im X
  }
$$

of the [[unit of a monad|unit]] of its [[infinitesimal shape modality]] along itself.

=--

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

By the [[pasting law]], the [[homotopy fiber]] of the infinitesimal disk bundle, def. \ref{FormalDiskBundle},  over any point $x \in X$ is the infinitesimal disk $\mathbb{D}^X_x$ in $X$ at that point, def.\ref{InfinitesimalDisk}. Nevertheless, for general $X$ the infinitesimal disk bundle need not be an [[fiber ∞-bundle]] with typical fiber (the infinitesimal disks at different points need not be equivalent, and even if they are, the bundle need not be locally trivial). Below in prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} we see that for $X$ a $V$-manifold modeled on a group object $V$, then its infinitesimal disk bundle is indeed an [[fiber ∞-bundle]], and hence is the [[associated ∞-bundle]] to some [[principal ∞-bundle]]. That principal bundle is the [[frame bundle]] of $X$.

=--

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

The [[Atiyah groupoid]] of $T_{inf} X$ is the [[jet groupoid]] of $X$.

=--

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

If $\iota \colon U \to X$ is a local diffeomorphism, def. \ref{LocalDiffeomorphismsInDifferentialCohesion}, then

$$
  \iota^\ast T_{inf} X \simeq T_{inf}U
  \,.
$$

=--

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

By the definition of local diffeos and using the [[pasting law]] we have an equivalence of [[pasting diagrams]] of [[homotopy pullbacks]] of the following form:

$$
  \array{
     \iota^\ast T_{inf} X &\longrightarrow& T_{inf} &\longrightarrow& X
     \\
     \downarrow && \downarrow && \downarrow
     \\
     U &\longrightarrow& X &\longrightarrow& \Im X
  }
  \;\;\;\;
  \simeq
  \;\;\;\;
  \array{
      T_{inf} U &\longrightarrow& U &\longrightarrow& X
       \\
       \downarrow && \downarrow && \downarrow
       \\
       U &\longrightarrow& \Im U &\longrightarrow& \Im X
  }
$$

=--

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

For $V$ an object, a **[[framing]]** on $V$ is a trivialization of its infinitesimal disk bundle, def. \ref{FormalDiskBundle}, i.e. an object $\mathbb{D}^V$ -- the typical [[infinitesimal disk]] or [[formal disk]], def. \ref{InfinitesimalDisk}, -- and a (chosen) [[equivalence]]

$$
  \array{
    T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n
    \\
    & \searrow && \swarrow_{\mathrlap{p_1}}
    \\
    && V
  }
  \,.
$$

=--

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

For $V$ a framed object, def. \ref{Framing}, we write

$$
  GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V)
$$

for the [[automorphism ∞-group]] of its typical [[infinitesimal disk]]/[[formal disk]].

=--

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

When the [[infinitesimal shape modality]] exhibits first-order infinitesimals, such that $\mathbb{D}(V)$ is the first order [[infinitesimal neighbourhood]] of a point, then $\mathbf{Aut}(\mathbb{D}(V))$ indeed plays the role of the [[general linear group]]. When $\mathbb{D}^n$ is instead a higher order or even the whole [[formal neighbourhood]], then $GL(n)$ is rather a [[jet group]]. For order $k$-jets this is sometimes written $GL^k(V)$ We nevertheless stick with the notation "$GL(V)$" here, consistent with the fact that we have no index on the [[infinitesimal shape modality]]. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here.

=--

This class of examples of framings is important:

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

Every differentially cohesive [[∞-group]] $G$ is canonically framed (def. \ref{Framing}) such that the horizontal map in def. \ref{FormalDiskBundle} is given by the left action of $G$ on its [[infinitesimal disk]] at the neutral element:

$$
  ev \;\colon\; T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G
  \,.
$$

=--

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

By the discussion at _[[Mayer-Vietoris sequence]]_ in the section _[Over an ∞-group](Mayer-Vietoris%20sequence#OverAGroupObject)_ and using that the [[infinitesimal shape modality]] preserves group structure, the defining [[homotopy pullback]] of $T_{inf} G$, def. \ref{FormalDiskBundle}, is equivalent to the pasting of pullback diagrams

$$
  \array{
    T_{inf} G &\stackrel{}{\longrightarrow}&
    \mathbb{D}^G_e
    &\stackrel{}{\longrightarrow}& \ast
    \\
    \downarrow &&  \downarrow && \downarrow
    \\
    G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
    &\stackrel{}{\longrightarrow}&
    \Im G
  }
$$

where the right square is the defining pullback for the [[infinitesimal disk]] $\mathbb{D}^G$. Finally for the left square we find by [this proposition](Mayer-Vietoris%20sequence#HTTArgumentForPullback) that $T_{inf} G \simeq G\times \mathbb{D}^G$ and that the top horizontal morphism is as claimed.

=--

By lemma \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} it follows that:

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

For $V$ a framed object, def. \ref{Framing}, let $X$ be a $V$-manifold, def. \ref{VManifold}. Then the infinitesimal disk bundle, def. \ref{FormalDiskBundle}, of $X$ canonically trivializes over any $V$-cover $V \leftarrow U \rightarrow X$ , i.e. there is a [[homotopy pullback]] of the form

$$
  \array{
     U \times \mathbb{D}^V &\longrightarrow& T_{inf} X
     \\
     \downarrow && \downarrow
     \\
     U &\longrightarrow& X
  }
  \,.
$$

This exhibits $T_{inf} X\to X$ as a $\mathbb{D}^V$-[[fiber ∞-bundle]].

=--

+-- {: .num_prop #ModulatingMapOfFormalDiskBundle}
###### Remark

By [this discussion](fiber+infinity-bundle#Properties) this fiber [[fiber ∞-bundle]] is the [[associated ∞-bundle]] of an essentially uniquely determined $\mathbf{Aut}(\mathbb{D}^V)$-[[principal ∞-bundle]] $Fr(X)$, i.e. there exists a [[homotopy pullback]] diagram of the form

$$
  \array{
     T_{inf} X \simeq
     &
   Fr(X) \underset{\mathbf{Aut}(\mathbb{D}^V)}{\times} \mathbb{D}^V
     &\longrightarrow&
     V//\mathbf{Aut}(\mathbb{D}^V)
     \\
     & \downarrow && \downarrow
     \\
     & X &\stackrel{}{\longrightarrow}& \mathbf{B}\mathbf{Aut}(\mathbb{D}^V)
  }
  \,.
$$

=--

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

Given a $V$-manifold $X$, def. \ref{VManifold}, for framed $V$, def. \ref{Framing}, then its _[[frame bundle]]_

$$
  \array{
     Fr(X)
     \\
     \downarrow
     \\
     X
  }
$$

is the $GL(V)$-[[principal ∞-bundle]] given by prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} via remark \ref{ModulatingMapOfFormalDiskBundle}.

=--

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

As in remark \ref{OrderOfInfinitesimalDisks}, this really axiomatizes in general [[higher order frame bundles]] with the order implicit in the nature of the [[infinitesimal shape modality]].

=--

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

By prop. \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} the construction of frame bundles in def. \ref{FrameBundleMap} is functorial in [[formally étale maps]] between $V$-manifolds.

=--

This provides all the necessary structure to now set up an axiomatic theory of [[G-structure]] and [[higher Cartan geometry]]. This is discussed further at _[[geometry of physics -- G-structure and Cartan geometry]]_.

### Syntax Layer

(...) formulation of the above in [[homotopy type theory]] in [Wellen 17](#Wellen17) (...)

## References

The above formulation of [[V-manifolds]] in [[differential cohesion]] is due to [[schreiber:differential cohomology in a cohesive topos|dcct]] and

* {#KhavkineSchreiber17} [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

and the formalization in [[homotopy type theory]] is in

* {#Wellen17} [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, PhD Thesis, 2017


geometry of physics -- modules

[[!redirects geometry of physics - modules]]

> This entry contains one chapter of _[[geometry of physics]]_. See there for background and context.

> previous chapters: _[[geometry of physics -- representations and associated bundles|representations and associated bundles]]_, _[[geometry of physics -- stable homotopy types|stable homotopy types]]_

> next chapters: _[[geometry of physics -- geometric quantization with KU-coefficients|geometric quantization with KU-coefficients]]_

***

#Contents#
* table of contents
{:toc}

## **Modules**

In the [previous section](geometry+of+physics#AssociatedNBundle) we have seen [[∞-actions]] of [[∞-groups]] on objects in the ambient [[(∞,1)-topos]]. Here we specialize this to an important class of cases and then generalize from [[∞-actions]] to $(\infty,n)$-actions for $n \in \mathbb{N}$. That special case is the one where all objects involved are required to be equipped with _additive structure_ (in generalization of the sense in which [[abelian groups]] are additive) and where all actions are _linear_ (in generalization of the sense of abelian group [[homomorphisms]], hence  [[linear functions]]). Moreover, we equip groups with additional (linear) monoidal structure such as to become higher analogs of [[rings]]. Linear [[actions]] of a [[ring]] are called _[[modules]]_ and hence here we discuss modules and their higher analogs.

Notice the different role of the "$(r,n)$" index:

| $(r,n)$-ring | presentation  |
|--|--|
| [[ring]] = 1-ring = $(1,1)$-ring |  compatibly monoidal [[abelian group]] |
| [[2-ring]] = $(2,2)$-ring | compatibly [[monoidal category]] |
| commutative $(\infty,1)$-ring  | [[E-∞ ring]] = compatibly monoidal [[∞-group]] / [[spectrum]] |
| $(\infty,2)$-ring | compatibly [[monoidal (∞,1)-category]]

A special case of modules are [[vector spaces]], which are the modules over a [[ring]] that is a [[field]]. For them the theory of modules is the theory of [[linear algebra]]. But as opposed to the notion of _[[ring]]_, the more specialized notion of _[[field]]_ has in general no natural or useful lift to [[higher category theory]]. Therefore we generally speak here of modules over rings and their higher analogs. But the reader who feels more at home with vector spaces may find it helpful to think of higher modules as being higher analogs of vector spaces.

### 1-Modules

We briefly disucss here some basics of ordinary [[linear algebra]] with an emphasis on the generalization from [[vector spaces]] over [[fields]] to [[modules]] over [[rings]].

#### Physics motivation: Superposition of states and fields

One key aspect of [[quantum mechanics]], that distinguishes it from [[classical mechanics]], is that there is _linear structure_ on [[quantum states]], as follows (the following applies to quantum states in the [[Schrödinger picture]] only, not to states in the sense of [[state on an operator algebra]], which are really the [[expectation values]] of [[observables]] in the states as discussed here):

* The **[[superposition principle]]** in quantum mechanics says that for $\psi_1$ and $\psi_2$ two [[quantum states]] one can form their [[sum]] $\psi_1 + \psi_2$ as well as their difference $\psi_1 - \psi_2$ such that this is again a quantum state. Moreover there is a 0-state such that $\psi + 0 = \psi$ for all $\psi$. Together this means there is the structure of an additive [[abelian group]] on the set of quantum states.

* Quantum states have **complex phases**. This means that for $\psi$ a quantum state and for $c \in \mathbb{C}$ a [[complex number]], there is a new quantum state $c \psi$ and this is such that for two such complex numbers we have  $c_1(c_2 \psi) = (c_1 c_2) \psi$ and for two states we have $c(\psi_1 + \psi_2) = c \psi_1 + c \psi_2$.  This means that multiplying by phases is a linear [[action]] of the [[ring]] of [[complex numbers]] on quantum states.

Together this says equivalently that the additive group of quantum states is a _[[module]]_ over the [[complex numbers]]. Since the ring of complex numbers is special in that it is a [[field]], such a module is equivalently called a _[[complex vector space]]_.

This linear structure is a crucial aspect of quanum theory. It is at the heart of phenomena such [[quantum interference]] and [[entanglement]]

In macroscopic physics similar behaviour is known in [[wave mechanics]] for freely propagating waves. Not unrelated to this is the term _[[wave function]]_ for a quantum state. However, apart from wave mechanics, linearity is not manifest in macoscopic physical phenomena, even though it is fundamentally present; this is called _[[decoherence]]_.

It is this linear structure of quantum theory that, since shortly after its conception, led to a close relaton to [[group]] [[representation theory]] which was unknown in classical physics. (And this came unexpected to physicist at the beginning of the 20th century and was not embraced by all theoreticians at first, see at _[[Gruppenpest]]_.) If a [[symmetry group]] acts on [[field (physics)|field configurations]] of a [[physical system]] and is preserved by [[quantization]], then it still acts on the quantum states and does so in a linear way; hence then the [[space of quantum states]] forms a [[linear representation]] space of the [[symmetry group]]. This relation between quantum mechanical systems with symmetry and linear [[representation theory]] goes so far that under natural assumptions _every_ representation of a group arises as the action of a group of symmetries on a quantum system  -- a relation known as the "[[orbit method]]" in [[geometric quantization]]. Deep results about abstract linear [[representation theory]] have been proven by considering systems [[quantum mechanics]] this way.

But the relation between quantum physics and linear representation theory goes deeper still. Not only do the quantum states form modules over the complex numbers, but already the [[matter]] [[field (physics)|fields]] in fundamental physics form modules over the [[algebra of functions]] on [[spacetime]]. This means in turn that matter fields form an additive [[abelian group]] where for $\phi_1$ and $\phi_2$ two field configurations also $\phi_1 + \phi_2$ is a field configuration, such that with $\phi$ a matter field for each [[smooth function]] $f$ on [[spacetime]] there is also a matter field denoted $f \phi$ and such that $f_1 (f_2 \phi) = (f_1 f_2) \phi$ and $f(\phi_1 + \phi_2) = f \phi_1 + \phi_2$.

There is a more geometric expression of this: in fact the matter fields form a module with special properties (they are what is called _[[projective module|projective]]_ and _[[finitely generated module|finitely generated]]_) and the [[Serre-Swan theorem]] asserty that modules with these special properties consist of [[sections]] of a [[fiber bundle]] whose typical fiber is a [[module]] over the [[complex numbers]], hence a [[complex vector space]]: a complex _[[vector bundle]]_. Moreover, these vector bundles are [[associated bundles]] via [[linear representations]] of the [[gauge group]] of the [[force]] [[field (physics)|fields]] under which the given matter fields are [[charged particle|charged]]. So in addition to being a linear module over the [[algebra of functions]] on [[spacetime]], matter fields also form a [[linear representation]] of the relevant [[gauge group]]. Finally all this combines with the [[symmetry group]] actions as for the quantum states themselves: is a group acts by symmetries on [[spacetime]] -- notably for instance the action of the [[Poincaré group]] on [[Minkowski spacetime]] -- then this translates into a [[linear representation]] of the group on the module of [[matter]] [[field (physics)|fields]], via [[pullback of functions]].  This goes as far as that under suitable conditions one can classify/identify [[particle]] species with [[unitary representation|unitary]] [[irreducible representations]] of the spacetime symmetry group, a relation known as _[[Wigner classification]]_, discussed in more detail at _[[unitary representation of the Poincaré group]]_. A closely related relation is the [[Doplicher-Roberts reconstruction theorem]] which shows that [[superselection sectors]] of a quantum field theory form the [[representation category]] of the global [[gauge group]] of a [[local net of observables]] of a quanutm field theory on spacetime -- a special case of the deep principle of [[Tannaka duality]] in [[representation theory]].

In summary this means that the relation between quantum theory and [[module]] theory/[[linear algebra]]/[[representation theory]] is intimate.

#### Modules

* [[module]]

#### Vector bundles

* [[vector bundle]]

### $(\infty,n)$-Modules

We turn now to the discussion of the generalization of the notion of [[module]] in [[higher category theory]]. After a motivation from [[physics]]

* [Physics motivation: Higher order states in extended QFT](#PhysicsMotivationHigherOrderStatesInQFT)

we first discuss how the [[2-category]] (i.e. [[(n,r)-category|(2,2)-category]]) of [[2-modules]] and their 2-linear maps is presented by the classical 2-category of [[associative algebras]] with [[bimodules]] between them and [[intertwiners]] between those.

* [2-Modules](#2Modules).

This serves as an instructive blueprint for all the following generalizations and is in itself a useful theory that neatly subsumes and organizes classical constructions and results such as [[Tannaka duality]] for associative algebras, the [[Eilenberg-Watts theorem]], and the basic [[Morita equivalence|Morita]] [[invariant theory]] of [[commutative rings]] given by [[Brauer group]], [[Picard group]] and [[group of units]].

Next we pass to [[homotopy theory]] proper and
discuss the construction of the [[(∞,2)-category]] of [[A-∞ algebras]] and [[∞-bimodules]] internal to any compatibly cocomplete [[monoidal (∞,1)-category]].

* [(∞,2)-modules](#?2Modules).

By iterating this construction in analogy to the iterative construction of [[(∞,n)-categories]] as $n$-fold [[categories in an (∞,1)-category]], we obtain a notion of [[(∞,n)-modules]]

* [(∞,n)-modules](#?nModule)

#### Physics motivation: Higher order states in extended QFT
 {#PhysicsMotivationHigherOrderStatesInQFT}

There two key classes of examples of [[(∞,n)-categories]] of relevance in [[physics]]. One is clearly the [[(∞,n)-categories of cobordisms]] $Bord_n^S$ (with some specified extra structure $S$) which by the theory of [[extended quantum field theory]] are the [[domain]] of [[monoidal (∞,n)-functors]]

$$
  Z \;\colon\; Bord_n^S \to \mathcal{C}
  \,.
$$

In the abstract mathematical discussion the [[codomain]] $\mathcal{C}$ here is usually allowed to be any [[symmetric monoidal (∞,n)-category]]. But we expect that for those quantum field theories of actual relevance in [[physics]] (notably those obtained by [[quantization]] from [[extended prequantum field theories]]) the [[codomain]] here is special. Notably in [[codimension]] 1 for $\Sigma_{n-1}$ a [[closed manifold]] of [[dimension]] $n-1$ the value $Z(\Sigma_{n-1})$ should be the [[space of quantum states]] over $\Sigma_{n-1}$ and thus be a _$\mathbb{C}$-[[module]]_ (a complex [[vector space]]) possibly with some extra properties and structure (e.g. a [[topological vector space]], a [[Hilbert space]], etc., depending on the precise details).

Therefore we expect that for relevant [[theory (physics)|theories of physics]] in [[dimension]] $n$, $\mathcal{C}$ is something that deserves to be called an _[[(∞,n)-category]] of [[(∞,n)-modules]]_ (maybe: [[(∞,n)-vector spaces]])

$$
  \mathcal{C}
  \simeq
  n Mod_{R}
$$

over some base [[E-∞ ring]] $R$.

Here we discuss such $n$-modules.

#### 2-Abelian groups

An [[abelian group]] [[semigroup]] is a set equipped with a commutative  and hence "additive" and unital pairing. A natural [[categorification]] of addition is the [[coproduct]] operation in a category, hence more generally the operation of taking [[colimits]]. Hence a sensible categorification of the notion of additive semigroup is that of _[[category]] with all [[colimits]]_. But in the spirit of [[category theory]] we should not try just to categorify abelian semigroups, but rather the category that these form. The category [[Ab]] of [[abelian groups]] is notably a [[closed monoidal category|closed]] [[symmetric monoidal category]] and accordingly we demand that the [[2-category]] of 2-abelian semigroups to replace it is similarly a closed [[symmetric monoidal 2-category]]. This is achived by restricting attention among all categories with colimits to the _[[presentable categories]]_.
Moreover, since the [[homomorphisms]] of abelian groups are [[linear functions]], hence addition-respecting functions, so the 2-linear maps between these 2-abelian semigroups should be [[colimit]]-preserving [[functors]]. This way we arrive at the following definition.

+-- {: .num_defn #2Category2Ab}
###### Definition

Write

$$
  2 Ab \in 2Cat
$$

for the [[2-category]] of [[presentable categories]] and [[colimit]]-preserving [[functors]] between them.

=--

([CJF, def. 2.1.8](#CJF))

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

By the [[adjoint functor theorem]] this is equivalently the 2-category of presentable categories and [[left adjoint]] functors between them.

=--

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

Given a [[small category]] $\mathcal{C}$, the [[presheaf category]] $Set^{\mathcal{C}}$ is a [[presentable category]].

=--

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

Forming presheaves on $\mathcal{C}$ is the [[free cocompletion]] of $\mathcal{C}$. Under the interpretation of colimits as the categorification of additive sums, this means that $Set^{\mathcal{C}}$ is the 2-[[free abelian group]] generated by the "2-set" $\mathcal{C}$.

Moreover, every [[presentable category]] (as discussed there) is a [[reflective subcategory]] hence a left [[localization]] of a category of presheaves. This means that not every 2-abelian group is 2-free, but is possibly a localization of a 2-free abelian group.

=--

+-- {: .num_example #CategoryOfModulesAs2AbelianGroup}
###### Example

Given an ordinary [[ring]] $R$, its [[category of modules]] $Mod_R$ is presentable, hence may be regarded as a 2-abelian group.
This example we discuss in more detail below in _[Bases for 2-modules: Tannaka duality for associative algebras](BasesFor2ModulesAndTannakaDualityForAssociativeAlgebras)_.

=--

([CJF, example 2.1.5](#CJF))

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

These two examples are directly analogous from the perspective of [[enriched category theory]]. For $A$ an $R$-algebra write $\mathbf{B}A$ for the corresponding 1-object $Mod_R$-[[enriched category]] with $A$ as its $R$-module of morphisms. Then

$$
  Mod_A \simeq [\mathbf{B}A, Mod_R]
$$

is the [[enriched functor category]].

Compare oth situation to how an ordinary [[free module]] $N$ (of finite [[rank]]) over a commutative [[ring]] is equivalently an [[algebra of functions]]

$$
  N \simeq [B,R]
$$

for $B$ a [[basis]] [[set]].

=--

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

The 2-category $2Ab = PrCat$ of def. \ref{2Category2Ab} is a [[closed monoidal 2-category|closed]] [[symmetric monoidal 2-category]] with respect to the [[tensor product]] $\boxtimes \colon 2Ab \times 2Ab \to 2Ab$ which corepresents "bi-2-linear" functors; in
that for $A,B, C \in 2Ab$ the [[hom-category]] $Hom_{2Ab}(A \boxtimes B, C)$ is equivalently the full [[subcategory]] of the [[functor category]] $Hom_{Cat}(A \times B, C)$ on those that preserve [[colimits]] in each argument separately.

=--

See also at [[Pr(∞,1)Cat]] for more on this.

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

This is analous to the [[Deligne tensor product of abelian categories]].

=--

+-- {: .num_example #SetPresheafCategorisAs2AbelianGroups}
###### Example

For $\mathcal{C}$ a [[small category]], the [[category of presheaves]] $Set^{\mathcal{C}}$ is [[presentable category|presentable]] and

$$
  Set^{\mathcal{C}_1} \boxtimes
  Set^{\mathcal{C}_2}
  \simeq
  Set^{\mathcal{C}_1 \times \mathcal{C}_2}
  \,.
$$

=--

+-- {: .num_prop #TensorProductModuleCategoriesAsOf2AbelianGroups}
###### Proposition

For $R$ a [[ring]] the [[category of modules]] $Mod_R$ is presentable and

$$
  Mod_{R_1} \boxtimes Mod_{R_2}
  \simeq
  Mod_{R_1 \otimes R_2}

  \,,
$$

=--

([CJF, example 2.2.7](#CJF))

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

As we discuss below
in _[Bases for 2-modules: Tannaka duality for associative algebras](BasesFor2ModulesAndTannakaDualityForAssociativeAlgebras)_, a category of modules over an $R$-algebra is a [[presheaf category]] in $Mod_R$-[[enriched category theory]]. Notice that in this language a plain presheaf over a [[locally small category]] is a presheaf in [[Set]]-[[enriched category theory]].

Therefore comparison of
example \ref{SetPresheafCategorisAs2AbelianGroups} with example \ref{TensorProductModuleCategoriesAsOf2AbelianGroups} shows that in the context of 2-abelian semigroups plain sets play a role of a "nonlinear generalization" of linear algebra. We see this in a more pronounced way once we have introduced the notion of [[2-modules]] [below](#2Modules). (It has become fashionable to speak of sets regarded as non-linear module categories as being modules over the "[[field with one element]]".)

=--

#### 2-Rings

With the ambient [[monoidal 2-category]] $2Ab$ chosen, it is now straightforward to define [[2-rings]] as [[monoids]] [[internalization|internal]] to that.

+-- {: .num_defn #2RingAsCompatiblyMonoidalPresentableCategory}
###### Definition

Write

$$
  2Ring \in 2Cat
$$

for the [[2-category]] of [[monoid objects]] [[internalization|internal]] to $2 Ab$. An [[object]] of this 2-category we call a **2-ring**.

Equivalently, a 2-ring in this sense is a [[presentable category]] equipped with the structure of a [[monoidal category]] where the [[tensor product]] preserves [[colimits]].

=--

([CJF, def. 2.1.8](#CJF))

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

The category [[Set]] with its [[cartesian product]] is a 2-ring and it is the [[initial object]] in $2Ring$.

=--

([CJF, example 2.3.4](#CJF))

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

The category [[Ab]] of [[abelian groups]] with its standard [[tensor product of abelian groups]] is a 2-ring.

=--

+-- {: .num_example #CommutativeRingGivesCommutative2Ring}
###### Example

For $R$ an ordinary [[commutative ring]], $Mod_R$ equipped with its usual [[tensor product of modules]] is a commutative 2-ring.

=--

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

For $R$ an ordinary [[commutative ring]] and $Mod_R$ its ordinary [[category of modules]], regarded as a 2-abelian group by example \ref{CategoryOfModulesAs2AbelianGroup}, the structure of a 2-ring on $Mod_R$ is equivalently the structure of a [[sesquiunital sesquialgebra]] on $R$.

=--

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

By prop. \ref{TensorProductModuleCategoriesAsOf2AbelianGroups}, a 2-linear map

$$
  Mod_A \boxtimes Mod_A \to Mod_A
$$

is equivalently a 2-linear map

$$
  Mod_{A \otimes A} \to Mod_A
  \,.
$$

By the [[Eilenberg-Watts theorem]] (which we discuss in more detail as theorem \ref{EilenbergWattsTheorem} [below](#MatrixCaclulusFor2Modules)) this is equivalently an $A \otimes A$-$A$-[[bimodule]]. Similarly the unit map

$$
  Mod_R \to Mod_A
$$

is equivalently an $R$-$A$-bimodule.

=--

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

For $R$ a [[commutative ring]], $Mod_R$ is a commutative 2-ring and is canonically an [[2-algebra]] over the [[2-ring]] $Ab$ in that we have a canonical 2-ring homomorphism

$$
  Ab \simeq Mod_{\mathbb{Z}} \to Mod_R
  \,.
$$

=--

([CJF, example 2.3.7](#CJF))

#### 2-Modules
 {#2Modules}

+-- {: .num_defn #2CategoryOf2Modules}
###### Definition

For $\mathcal{R}$ a 2-ring, def. \ref{2RingAsCompatiblyMonoidalPresentableCategory}, write

$$
  2Mod_{\mathcal{R}} \in 2Cat
$$

for the [[2-category]] of [[module objects]] over $A$ in $2Ab$.

This means that a 2-module over $A$ is a [[presentable category]] $N$ equipped with a functor

$$
  A \boxtimes N \to N
$$

which satisfies the evident action property.

=--

([CJF, def. 2.3.3](#CJF))

+-- {: .num_example #2ModulesOverThe2RingOfAnOrdinaryRing}
###### Example

Let $R$ be an ordinary [[commutative ring]] and $A$ an ordinary $R$-[[associative algebra|algebra]]. Then by example \ref{CategoryOfModulesAs2AbelianGroup} $Mod_A$ is a 2-abelian group and by example
\ref{CommutativeRingGivesCommutative2Ring} $Mod_R$ is a commutative ring. By example \ref{TensorProductModuleCategoriesAsOf2AbelianGroups}
$Mod_R$-[[2-module]] structures on $Mod_A$

$$
  Mod_R \boxtimes \Mod_A \to Mod_A
$$

correspond to colimit-preserving functors

$$
  Mod_{R \otimes_{\mathbb{Z}} A} \to Mod_{A}
$$

that satisfy the action property. Such as presented under the [[Eilenberg-Watts theorem]], prop. \ref{EilenbergWattsTheorem}, by $R \otimes_{\mathbb{Z}} A$-$A$ [[bimodules]]. $A$ itself is canonically such a bimodule and it exhibits a $Mod_R$-[[2-module]] structure on $Mod_A$.

=--

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

Coming back to remark \ref{SetsAsNonlinearModuleCategories} we observe that in [[2-module]] theory plain [[presentable categories]] which do not "look linear" in the ordinary sense are naturally regarded as [[Set]]-[[2-modules]], hence as being "[[Set]]-linear", whereas the categories of modules over an algebra that are traditionally regarded as being "$R$-linear categories" are $Mod_R$-[[2-modules]]. This unification of sets as "nonlinear linear structure" has become fashionable as "linear algebra over the [[field with one element]]".

=--

##### Bases for 2-modules: Tannaka duality for associative algebras
 {#BasesFor2ModulesAndTannakaDualityForAssociativeAlgebras}

The construction in example \ref{2ModulesOverThe2RingOfAnOrdinaryRing}
of 2-modules over the 2-ring induced by an ordinary commutative ring $R$ as categories of 1-modules over some $R$-algebra may be understood as presenting a 2-[[free module]] by a 2-[[basis]].

To see this, write

$$
  \mathcal{V} \coloneqq Mod_R
$$

for the [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]] [[category of modules]] and regard it as an [[cosmos|enriching context]] for $\mathcal{V}$-[[enriched category theory]].

So a $\mathcal{V}$-[[enriched category]] $\mathcal{A}$ serves as a _2-[[basis]]_ for a free $\mathcal{V}$-[[2-module]]. Notice that an $R$-algebra $A$ is equivalently a (pointed) $\mathcal{V}$-enriched category $\mathcal{A} = \mathbf{B}A$ with a single object (the [[delooping]] of $A$). Generally, $\mathcal{A}$ may be called  _$R$-[[algebroid]]_.

Since [[colimits]] in the category underlying a [[2-abelian group]] [[categorification|categorify]] the sum operation in an [[abelian group]] the 2-analog of the free module on a [[basis]] [[set]] is the [[free cocompletion]] of $\mathcal{A}$ as a $\mathcal{V}$-[[enriched category]]. This is the $\mathcal{V}$-[[presheaf category]], hence the [[enriched functor category]]

$$
  [\mathcal{A}, \mathcal{V}]
  \in
  \mathcal{V}Cat
  \,.
$$

Compare this to how for a (finite dimensional) vector space a presentation by a basis is a set $B$ such that the vector space is

$$
 Maps(B,k)
  \in
  Vect_k
$$

Indeed, for $\mathcal{A} = \mathbf{B}A$ this is the ordinary [[category of modules]] over $A$:

$$
  [\mathbf{B}A, \mathcal{V}]
  \simeq
  Mod_A
  \,.
$$

In fact in [[enriched category theory]] it is customary generally to call $\mathcal{V}$-[[enriched functors]] into $\mathcal{V}$ "[[modules]]" (see there).

Without further assumptions on $\mathcal{A}$, the category $[\mathcal{A}, \mathcal{V}]$ is just a $\mathcal{V}$-[[2-module]]. But if $\mathcal{A}$ is equipped with further [[extra structure]] and/or [[property]], also $[\mathcal{A}, \mathcal{V}]$ inherits further property. This relation between types of algebras and types of monoidal categories is known as _[[Tannaka duality]]_.

The following table lists the main items of the disctionary of Tannaka duality for associtive algebras that are used in the literature.

[[!include structure on algebras and their module categories - table]]

##### Matrix calculus for 2-modules: Bimodules and Eilenberg-Watts theorem
 {#MatrixCaclulusFor2Modules}

If a $\mathcal{V} \coloneqq Mod_R$-[[enriched category]] serves as a 2-[[basis]] for a $Mod_R$-[[2-module]] by the [above](#BasesFor2ModulesAndTannakaDualityForAssociativeAlgebras) discussion, there should be an analog of [[matrix calculus]] to present 2-linear maps between [[2-modules]] equipped with such a 2-basis.

Such a higher [[matrix]] is what in [[enriched category theory]] is called a _$\mathcal{V}$-[[profunctor]]_ (or "[[distributor]]"): a profunctor between $\mathcal{V}$-enriched categories $\mathcal{A}_1, \mathcal{A}_2 \in \mathcal{V}Cat$ is a $\mathcal{V}$-[[enriched functor]]

$$
  K
  \;\colon\;
  \mathcal{A}_1^{op}
  \times
  \mathcal{A}_2
  \to
  \mathcal{V}
  \,.
$$

Compare this to how a matrix between two vector spaces equipped with basis sets $B_1, B_2$ is a function

$$
  K \colon B_1 \times B_2 \to k
$$

to the [[ground field]]. Notice that if the vector space is given by $[B_1,k]$, then the action of this matrix on a vector $v \colon B_1 \to k$ is given by the vector $K v \colon B_2 \to k$

$$
  b_2 \mapsto \sum_{b_1 \in B_1} v(b_1) \cdot K(b_1,b_2)
  \,.
$$

If we replace in this formula the sum by a [[coend]], then we get the action of the [[profunctor]] $K$ on a $\mathcal{V}$-functor $v \colon \mathcal{A}_1 \to \mathcal{V}$ to yield $K \otimes v \colon \mathcal{A}_2 \to \mathcal{V}$ with

$$
  a_2 \mapsto \int^{a_1 \in A_1} v(a_1) \otimes K(a_1, a_2)
  \,.
$$

For the case at hand where $\mathcal{V} = Mod_R$ and in the special case that $\mathcal{A}_1 = \mathbf{B}A_1$ and $\mathcal{A}_2 = \mathbf{B}A_2$, such a profunctor $\mathbf{B}A_1 \times \mathbf{B}A_2 \to \mathcal{V}$ is equivalently an $A_1$-$A-2$-[[bimodule]] in the traditional sense of associative algebra. Moreover, the [[coend]]-action above is equivalently the traditional [[tensor product of modules]] over the given algebra. Motivated by this example one also generally calls profunctors "bimodules".

Now one should ask to which extent these "2-matrices" given by profunctors capture all the 2-linear maps between [[2-modules]]. This is what the classical **[[Eilenberg-Watts theorem]]** solves:

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

There is a [[natural equivalence|natural]] [[equivalence of categories]]

$$
  (-) \otimes_{A_1} (-)
  \;\colon\;
  {}_{A_1} Mod_{A_2}
    \stackrel{\simeq}{\to}
  Hom_{2Ab}(Mod_{A_1}, Mod_{A_2})
$$

between [[colimit]]-preserving functors $Mod_{A_1} \to Mod_{A_2}$ between [[categories of modules]] and $A_1$-$A-2$-[[bimodules]], given by sending a bimodule $N$ to the [[tensor product]] functor

$$
  (-) \otimes_{A_1} N \;\colon\; Mod_{A_1} \to Mod_{A_2}
  \,.
$$

=--

In summary we this find that for $R$ a [[commutative ring]], the [[2-category]] $Mod_{Mod_R}$ of $Mod_R$-[[2-modules]] with 2-basis is equivalent to the 2-category $Prof(Mod_R)$ of $Mod_R$-enriched [[profunctors]]. Indeed, another common notation for [[Prof]] is [[Mod]], and so we have the unambiguous notation

$$
  2 Mod_R \simeq Mod(Mod_R)
  \,.
$$

Notice how this is analogous to the identification

$$
  (\infty,2)Cat \simeq Cat(Cat_{(\infty,1)})
$$

discussed at _[[internal (∞,1)-category]]_.

#### 2-Module bundle

* [[2-module bundle]]

#### 2-Lines

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

For $\mathcal{R}$ a commutative [[2-ring]], def. \ref{2RingAsCompatiblyMonoidalPresentableCategory}, a **[[2-line]]** over $\mathcal{R}$ is an $\mathbb{R}$-[[2-module]], def. \ref{2CategoryOf2Modules} which is invertible under the [[tensor product]] of $\mathbb{R}$-2-modules.

=--

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

Write

$$
  Line^\simeq_\mathbb{R} \to Mod_{\mathbb{R}}
$$

for the maximal [[2-groupoid]] of $\mathbb{R}$ 2-lines with 2-basis inside all $\mathbb{R}$-2-modules. This is the **[[Picard 2-groupoid]]** of $\mathcal{R}$. Moreover, by construction it in inherits the structure of a [[3-group]] from the tensor products of 2-modules: as such it is the **[[Picard 3-group]]**.

=--

##### Classification: Azumaya algebras, Brauer group, Picard group

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

For $R$ an ordinary [[commutative ring]] and $\mathcal{R} \coloneqq Mod_R$ the [[homotopy groups]] of the [[Picard 2-groupoid]] are

* $\pi_0(2Line^\simeq_R) \simeq Br(R)$ the [[Brauer group]] of $R$;

* $\pi_1(2Line^\simeq_R) \simeq Pic(R)$ the [[Picard group]] of $R$;

* $\pi_2(2Line^\simeq_R) \simeq R^\times$ the [[group of units]] of $R$.

=--

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

By the [[Eilenberg-Watts theorem]], \ref{EilenbergWattsTheorem}, an [[equivalence]] in $2Mod_R$ between 2-modules with 2-basis given by $R$-algebras is a [[Morita equivalence]] of $R$-algebras.

By the prop. \ref{TensorProductModuleCategoriesAsOf2AbelianGroups} and the [[Eilenberg-Watts theorem]], the invertible 2-modules with 2-basis are therefore precisely the [[Azumaya algebras]]. This shows that the connected components of $2Line^\simeq_R$ form the [[Brauer group]] of $R$.

Next, an [[isomorphism class]] of an [[automorphism]] of the canonical base point $Mod_R \in 2Line^\simeq_R$ is by Eilenberg-Watts equivalently an isomorphism class of an $R$-$R$-[[bimodule]] which is invertible under horizontal composition of bimodules. Since $R$ is commutative this just means that it is an isomorphism class of an $R$-module which is invertible under the ordinary [[tensor product of modules]]. But this just means that it is the isomorphism class of an ordinary $R$-[[line]]. This shows that the [[fundamental group]] of $2Line^\simeq_R$ is the ordinary [[Picard group]] of $R$.

Finally an [[automorphism]] of the $R$ regarded as the identity $R$-$R$-[[bimodule]] is an invertible $R$-[[linear function]] $R \to R$, hence is given by multiplication with an invertible element of $R$. This shows that $\pi_2(2Line^\simeq_R)$ is the [[group of units]] of $R$.

=--

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

Below in [2-line bundles and their sections](#2LineBundles) we discuss
how forming a [[fiber infinity-bundle|fiber 2-bundle]] of [[2-lines]] as above produces a structure whose [[sections]] are [[twisted bundles]] with typical fiber $Mod_R$, hence for instance [[twisted unitary bundles]] if $R \simeq \mathbb{C}$. This means that the [[characteristic classes]] of $Mod_R$-[[2-line bundles]] are the _twists_ for the [[twisted cohomology]] classifying these twisted bundles. After [[stabilization]] the latter are the [[cocycles]] of [[twisted K-theory]], hence the Brauer group, Picard group and group of units of $R$ are the degree 0, 1, 2-twists of [[twisted K-theory]] over $R$-resectively. Below we discuss two examples of this phenomenon.

=--

##### 2-Line bundles and their sections: twisted K-theory
 {#2LineBundles}

* [[2-line bundle]]

* [[twisted bundle]] / [[Chan-Paton gauge field]] / [[twisted K-theory]]

#### $(\infty,2)$-Abelian groups

(...)

#### $(\infty,2)$-rings

* [[(∞,2)-ring]]

#### $(\infty,2)$-modules
 {#?2Modules}

* [[(∞,2)-module]]

##### Bases for $(\infty,2)$-modules: $A_\infty$-algebras

* [[A-∞ algebra]]

##### Matrix calculus for $(\infty,2)$-modules: $\infty$-Bimodules

* [[∞-bimodule]]

#### $(\infty,n)$-Abelian groups

(...)

#### $(\infty,n)$-rings

* [[(∞,n)-ring]]

#### $(\infty,n)$-modules
 {#?2Modules}

* [[(∞,2)-module]]

##### Bases for $(\infty,n)$-modules

(...)

##### Matrix calculus for $(\infty,n)$-modules

(...)

## References

The notions of [[2-rings]] and [[2-modules]] are nicely set up in

* [[Alexandru Chirvasitu]], [[Theo Johnson-Freyd]], _The fundamental pro-groupoid of an affine 2-scheme_, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. ([DOI](http://dx.doi.org/10.1007/s10485-011-9275-y), [arXiv:1105.3104](http://arxiv.org/abs/1105.3104))
 {#CJF}

The $(\infty,2)$-category of $A_\infty$-algebras and $\infty$-bimodules between them is constructed in section 3.4 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

For further references see behind the relevant links.


geometry of physics -- perturbative quantum field theory

These notes ([[Schreiber_pQFT20211103.pdf:file]], 323 pages) mean to give an expository but rigorous introduction to the basic concepts of [[relativistic quantum field theory|relativistic]] [[perturbative quantum field theories]], specifically those that arise as the [[perturbative quantum field theory|perturbative]] [[quantization]] of _[[Lagrangian field theories]]_ -- such as [[quantum electrodynamics]], [[quantum chromodynamics]], and [[perturbative quantum gravity]] appearing in the [[standard model of particle physics]].

$\,$

> This is one chapter of _[[geometry of physics]]_.

> Previous chapters: _[[geometry of physics -- smooth sets|smooth sets]]_, _[[geometry of physics -- supergeometry|supergeometry]]_.

$\,$

#Contents#
* table of contents
{:toc}

$\,$

For broad introduction of the idea of the topic of _[[perturbative quantum field theory]]_ see [[perturbative quantum field theory|there]] and see

* PhysicsForums-Insights: _[Introduction to Perturbative Quantum Field Theory](https://www.physicsforums.com/insights/paqft-idea-references/)_

Here, first we consider [[classical field theory]] (or rather [[pre-quantum field theory]]), complete with [[BV-BRST formalism]];
then its [[deformation quantization]] via [[causal perturbation theory]] to [[perturbative quantum field theory]].
This mathematically rigorous (i.e. clear and precise) formulation of the traditional informal lore
has come to be known as _[[perturbative algebraic quantum field theory]]_.

We aim to give a _fully [[local field theory|local]]_ discussion, where all structures arise
on the "[[jet bundle]] over the [[field bundle]]" (introduced [below](#FieldVariations)) and "[[transgression|transgress]]" from there to the [[spaces of field histories]] over [[spacetime]] (discussed [further below](#Observables)).
This "[[schreiber:Higher Prequantum Geometry]]" streamlines traditional constructions and serves the conceptualization in the theory.
This is joint work with [[Igor Khavkine]].

In full beauty these concepts are extremely general and powerful; but the aim here is to give a first precise idea of the subject, not
a fully general account.
Therefore we concentrate on the special case where [[spacetime]] is [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime} below), where the [[field bundle]] (def. \ref{FieldsAndFieldBundles} below) is an ordinary [[trivial vector bundle]] (example \ref{TrivialVectorBundleAsAFieldBundle} below) and hence the [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime} below) is [[locally variational field theory|globally defined]].
Similarly, when considering [[gauge theory]] we consider just the special case that the [[gauge parameter]]-bundle
is a [[trivial vector bundle]]  and we concentrate on the case that the gauge symmetries are "closed irreducible" (def. \ref{GaugeParametersClosed} below).
But we aim to organize all concepts such that the _structure_ of their generalization to [[AQFT on curved spacetime|curved spacetime]]
and non-trivial [[field bundles]] is immediate.

This comparatively simple setup already subsumes what is considered in traditional texts on the subject; it captures the established [[perturbative quantum field theory|perturbative]] [[BV formalism|BRST-BV quantization]] of [[gauge fields]] coupled to [[fermions]] [[AQFT on curved spacetime|on curved spacetimes]]
-- which is the state of the art.
Further generalization, necessary for the discussion of global topological effects, such as [[instanton]] configurations
of [[gauge fields]], will be discussed elsewhere (see at _[[homotopical algebraic quantum field theory]]_).

Alongside the theory we develop the concrete examples of the [[real scalar field]], the [[electromagnetic field]] and the [[Dirac field]]; eventually combining these to a disussion of [[quantum electrodynamics]].

**running examples**
{#RunningExamples}

| [[field (physics)|field]] |  [[field bundle]] |  [[Lagrangian density]] | [[equation of motion]] |
|---------------------------|------------------|-------------------------|------------------------|
| [[real scalar field]] |  expl. \ref{RealScalarFieldBundle} |  expl. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime} | expl. \ref{FreeScalarFieldEOM} |
| [[Dirac field]] | expl. \ref{DiracFieldBundle}  | expl. \ref{LagrangianDensityForDiracField}  |  expl. \ref{EquationOfMotionOfDiracFieldIsDiracEquation}  |
| [[electromagnetic field]] | expl. \ref{Electromagnetism} | expl. \ref{ElectromagnetismLagrangianDensity} | expl. \ref{ElectromagnetismEl}  |
| [[Yang-Mills field]] | expl. \ref{YangMillsFieldOverMinkowski},
 expl. \ref{YangMillsFieldInInstantonSector} | expl. \ref{YangMillsLagrangian}  | expl. \ref{YangMillsOnMinkowskiEl} |
| [[B-field]] | expl. \ref{BField} | expl \ref{BFieldLagrangianDensity} | expl. \ref{EulerLagrangeFormBField} |

$\,$

| [[field (physics)|field]] | [[Poisson bracket]]  |  [[causal propagator]] | [[Wightman propagator]] | [[Feynman propagator]] |
|---------------------------|----|----|-----|-----|
| [[real scalar field]] | expl. \ref{PeierlsBracketEistsForScalarFieldAndDiracField},
 expl. \ref{PoissonBracketForRealScalarField} |  prop. \ref{ModeExpansionOfCausalPropagatorForKleinGordonOnMinkowski} | def. \ref{StandardHadamardDistributionOnMinkowskiSpacetime} | def. \ref{FeynmanPropagatorForKleinGordonEquationOnMinkowskiSpacetime}  |
| [[Dirac field]] | expl. \ref{PeierlsBracketEistsForScalarFieldAndDiracField},
  expl. \ref{PoissonBracketForDiracField}   |  prop. \ref{DiracEquationOnMinkowskiSpacetimeAdvancedAndRetardedPropagators} | def. \ref{HadamardPropagatorForDiracOperatorOnMinkowskiSpacetime} | def. \ref{FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime} |
| [[electromagnetic field]]  |  | prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge}  |   |  prop. \ref{PhotonPropagatorInGaussianAveragedLorenzGauge}  |

$\,$

| [[field (physics)|field]] | [[gauge symmetry]] | [[local BRST complex]] | [[gauge fixing]] |
|---------------------------|--------------------|------------------------|------------------|
| [[electromagnetic field]] | expl. \ref{InfinitesimalGaugeSymmetryElectromagnetism}  | expl. \ref{LocalBRSTComplexForFreeElectromagneticFieldOnMinkowskiSpacetim} |  expl. \ref{NLGaugeFixingOfElectromagnetism}  |
| [[Yang-Mills field]]      |   expl. \ref{InfinitesimalGaugeSymmetryOfYangMillsTheory}      |  expl. \ref{YangMillsLocalBRSTComplex}   | ... |
| [[B-field]]    |   expl. \ref{InfinitesimalGaugeSymmetryOfTheBField} |  expl. \ref{LocalBRSTComplexBFieldMinkowskiSpacetime}  |  ... |

$\,$

| [[interacting field theory]] | [[interaction]] [[Lagrangian density]] | [[interaction]] [[Wick algebra]]-element |
|------------------------------|------------------------|---------------|
| [[phi^n theory]] | exp. \ref{phintheoryLagrangian} | expl. \ref{InWickAlgebraphinInteraction} |
| [[quantum electrodynamics]] | expl. \ref{LagrangianQED} | expl. \ref{InWickAlgebraElectronPhotonInteraction} |
$\,$

**References**
 {#References}

Pointers to the literature are given in each chapter, alongside the text. The following is a selection of these references.

The discussion of [[spinors]] in chapter _[2. Spacetime](#Spacetime)_ follows [Baez-Huerta 09](spin+representation#BaezHuerta09).

The [[functorial geometry]] of [[supergeometry|supergeometric]] [[spaces of field histories]] in _[3. Fields](#Fields)_ follows [Schreiber 13](higher+prequantum+geometry#Schreiber13), further developed by [Giotopoulos & Sati 2023](smooth+set#GiotopoulosSati23).

For the [[jet bundle]]-formulation of [[variational calculus]] of [[Lagrangian field theory]] in _[4. Field variations](#FieldVariations)_, and _[5. Lagrangians](#Lagrangians)_ we follow [Anderson 89](variational+bicomplex#Anderson89) and [Olver 86](#Olver86), further developed by [Giotopoulos & Sati 2023](smooth+set#GiotopoulosSati23); for _[6. Symmetries](#Symmetries)_ augmented by [Fiorenza-Rogers-Schreiber 13b](#higher+prequantum+geometry#FRS13b).

The identification of [[polynomial observables]] with [[distributions]] in _[7. Observables](#Observables)_ was observed by [Paugam 12](distributions+are+the+smooth+linear+functionals#Paugam12).

The discussion of the [[Peierls-Poisson bracket]] in _[8. Phase space](#PhaseSpace)_ is based on [Khavkine 14](Peierls+bracket#Khavkine14).

The derivation of [[wave front sets]] of [[propagators]] in _[9. Propagators](#Propagators)_ takes clues from [Radzikowski 96](Hadamard+distribution#Radzikowski96) and uses results from [Gelfand-Shilov 66](Cauchy+principal+value#GelfandShilov66).

For the general idea of [[BV-BRST formalism]] a good review is [Henneaux 90](BV-BRST+formalism#Henneaux90).

The [[Lie algebroid]]-perspective on [[BRST complexes]] developed in chapter _[10. Gauge symmetries](#GaugeSymmetries)_, may be compared to [Barnich 10](BRST+complex#Barnich10).

For the [[local BV-BRST complexes|local BV-BRST theory]] laid out in chapter _[11. Reduced phase space](#ReducedPhaseSpace)_ we are following [Barnich-Brandt-Henneaux 00](local+BRST+cohomology#BarnichBrandtHenneaux00).

For the [[gauge fixing|BV-gauge fixing]] developed in _[12. Gauge fixing](#GaugeFixing)_ we take clues from [Fredenhagen-Rejzner 11a](BV-BRST+formalism#FredenhagenRejzner11a).

For the free quantum [[BV-operators]] in _[13. Free quantum fields](#FreeQuantumFields)_ and the interacting [[quantum master equation]] in _[15. Interacting quantum fields](#InteractingQuantumFields)_ we are following [Fredenhagen-Rejzner 11b](BV-BRST+formalism#FredenhagenRejzner11b), [Rejzner 11](BV-BRST+formalism#Rejzner11), which in turn is taking clues from [Hollands 07](BV-BRST+formalism#Hollands07).

The discussion of [[quantization]] in _[13. Quantization](#Quantization)_ takes clues from [Hawkins 04](geometric+quantization+of+symplectic+groupoids#EH), [Collini 16](star+product#Collini16) and spells out the derivation of the [[Moyal star product]] from [[geometric quantization of symplectic groupoids]] due to [Gracia-Bondia & Varilly 94](Moyal+deformation+quantization#GBV).

The perspective on the [[Wick algebra]] in _[14. Free quantum fields](#FreeQuantumFields)_ goes back to [Dito 90](Wick+algebra#Dito90) and was revived for [[pAQFT]] in [Dütsch-Fredenhagen 00](Wick+algebra#DuetschFredenhagen00). The proof of the folklore result that the perturbative [[Hadamard vacuum state]] on the [[Wick algebra]] is indeed a [[state on a star-algebra|state]] is cited from  [Dütsch 18](Wick#algebra#Duetsch18).

The discussion of [[causal perturbation theory]] in _[15. Interacting quantum fields](#InteractingQuantumFields)_ follows the original [Epstein-Glaser 73](causal+perturbation+theory#EpsteinGlaser73). The relevance here of the [[star product]] induced by the [[Feynman propagator]] was highlighted in [Fredenhagen-Rejzner 12](perturbative+algebraic+quantum+field+theory#FredenhagenRejzner12). The proof that the [[interacting field algebra of observables]] defined by [[Bogoliubov's formula]] is a [[causally local net]] in the sense of the [[Haag-Kastler axioms]] is that of [Brunetti-Fredenhagen 00](pAQFT#BrunettiFredenhagen00).

Our derivation of [[Feynman diagrams|Feynman diagrammatics]] follows [Keller 10, chapter IV](Feynman+diagram#Keller10), our derivation of the [[quantum master equation]] follows [Rejzner 11, section 5.1.3](BV-BRST+formalism#Rejzner11), and our discussion of [[Ward identities]] is informed by [Dütsch 18, chapter 4](Ward+identity#Duetsch18).

In chapter _[16. Renormalization](#Renormalization)_ we take from [Brunetti-Fredenhagen 00](renormalization#BrunettiFredenhagen00) the perspective of [[Epstein-Glaser renormalization]] via [[extension of distributions]] and from [Brunetti-Dütsch-Fredenhagen 09](renormalization#BrunettiDuetschFredenhagen09) and [Dütsch 10](renormalization#Duetsch10) the rigorous formulation of [[Gell-Mann-Low renormalization cocycle|Gell-Mann & Low renormalization group flow]], [[UV-regularization]], [[effective quantum field theory]] and [[Polchinski's flow equation]].

$\,$

**Acknowledgement**

These notes profited greatly from discussions with [[Igor Khavkine]] and [[Michael Dütsch]].

Thanks also to [[Marco Benini]], [[Klaus Fredenhagen]], [[Arnold Neumaier]] and [[Kasia Rejzner]] for helpful discussion.

$\,$

$\,$

[[!include A first idea of quantum field theory -- Geometry]]

[[!include A first idea of quantum field theory -- Spacetime]]

[[!include A first idea of quantum field theory -- Fields]]

[[!include A first idea of quantum field theory -- Field variations]]

[[!include A first idea of quantum field theory -- Lagrangians]]

[[!include A first idea of quantum field theory -- Symmetries]]

[[!include A first idea of quantum field theory -- Observables]]

[[!include A first idea of quantum field theory -- Phase space]]

[[!include A first idea of quantum field theory -- Propagators]]

[[!include A first idea of quantum field theory -- Gauge symmetries]]

[[!include A first idea of quantum field theory -- Reduced phase space]]

[[!include A first idea of quantum field theory -- Gauge fixing]]

[[!include A first idea of quantum field theory -- Quantization]]

[[!include A first idea of quantum field theory -- Free quantum fields]]

[[!include A first idea of quantum field theory -- Interacting quantum fields]]

[[!include A first idea of quantum field theory -- Renormalization]]

[[!redirects geometry of physics -- A first idea of quantum field theory]]
[[!redirects A first idea of quantum field theory]]