foliation

> This entry is about the concept in differential geometry. For [[Bénabou]]'s generalisation of a [[Grothendieck fibration]], see *[[foliated category]]*.

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

#Contents#
* table of contents
{:toc}

## Idea

A _foliation_ of a [[manifold]] $X$ is a decomposition into [[submanifolds]]. These submanifolds are called the _leaves_ of the foliation and one says that $X$ is _foliated by the leaves_. In order to have a useful notion, leaves are required to behave sufficiently well locally. In particular, if all leaves have the same [[dimension]], then one speaks of a _regular foliation_, which is the case discussed here. If the dimension of leaves is allowed to vary, one speaks instead a _[[singular foliation]]_; see there for more details.

For [[smooth manifolds]], smooth foliations are decompositions into [[immersion|immersed]] [[submanifolds]] that locally can be expressed as [[fibers]] of a [[submersion]] (the projection to the [[space of leaves]]).

Historically (as they were introduced in [Ehresmann](#Ehresmann) and [Reeb](#Reeb)), one considered foliations for [[smooth manifolds]] $X$ as arising from subbundles of the [[tangent bundle]] $E \hookrightarrow T X$ that are [[integrable distributions]] in the sense that the [[Lie bracket]] of [[vector fields]] that are [[sections]] of $E$ is again a section of $E$: the leaves are the submanifolds whose tangent vectors are sections of $E$. If one thinks of $E$ as encoding a [[differential equation]], then the leaves are the solution spaces to this equation.

Expressed in terms of [[higher Lie theory]] such an integrable distribution is a sub-[[Lie algebroid]] of the [[tangent Lie algebroid]] of $X$. Accordingly, under [[Lie integration]] of this structure foliations of $X$ are also equivalently encodes as [[Lie groupoids]] whose space of objects is $X$ and whose [[orbits]] are the leaves of the foliation.

Moreover, foliations are classified by [[Čech cohomology]] [[cocycles]] with coefficients in a [[topological groupoid]]/[[Lie groupoid]] called the _[[Haefliger groupoid]]_. These relations make foliation theory of sub-topic of [[Lie groupoid]]-theory. See also at _[[motivation for higher differential geometry]]_.

The Haefliger groupoids in fact classifies structures slightly more general than foliations: _[[Haefliger structures]]_.

## Definition

There are several equivalent definitions of foliations.

### Original definition

Let $M$ be an $n$-[[dimension|dimensional]] [[topological manifold]]. A decomposition of $M$ as a [[disjoint union]] of [[connected topological space|connected]] subsets $V_\alpha$, called **leaves**,

$$ M  = \cup_\alpha V_\alpha $$

is called a **foliation** if there is a [[cover]] of $M$ by a collection of "special" [[charts]] of the form $(U, \phi)$, $\phi = (\phi_1,\ldots,\phi_n) : U \to \mathbb{R}^n$ such that for each "special" chart and each $\alpha$ there is a number $p\leq n$, called the _dimension of the foliation_, such that the intersection of any given leaf $V_\alpha$ with $U$ is one of the level sets, i.e. the solution of the system $\phi_r(x) = const = const(r,U,\alpha)$ for all $r\gt p$.

If the manifold is a [[smooth manifold]], the charts may be required to be smooth too, to obtain the notion of a _smooth foliation_ or _foliation in [[differential geometry]]_. In this case, the $p$-dimensional foliations with underlying manifold $X$ are in 1-1 correspondence with [[integrable distributions]] of hyperplanes of dimension $p$ in the [[tangent bundle]] of $X$.

### Alternative definitions

The following equivalent definitions and their relation are discussed for instance in ([IfLg, 1.2](#MoerdijkMrcun)).

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

A **foliation atlas** of a [[manifold]] $X$ of [[dimension]] $n$ and leaf-[[codimension]] $q$ is an [[atlas]] $\{\phi_i^{-1}: R^n \to X\}_i$ such that the transition functions are globally of the form

$$
  \phi_{i j} : (x,y) \mapsto (g_{i j}(x,y), h_{i j}(y))
$$

with respect to the canonical decomposition $\mathbb{R}^n = \mathbb{R}^{n-q} \times \mathbb{R}^q$.

$$
  \array{
    \mathbb{R}^n &\stackrel{\phi_{i j}}{\to}& \mathbb{R}^n &&
    \\
    \downarrow && \downarrow
    \\
    \mathbb{R}^q &\stackrel{h_{i j}}{\to}& \mathbb{R}^q
  }
  \,.
$$

=--

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

A **foliation atlas** of a [[manifold]] $X$ of [[dimension]] $n$ and leaf-[[codimension]] $q$ is an [[open cover]] $\{U_i \to X\}_i$ of $X$ equipped with [[submersions]] $\{ s_i \colon U_i \to \mathbb{R}^q \}$
such that there exists [[diffeomorphisms]]

$$
   \gamma_{i j} \colon s_j(U_i \cap U_j) \to s_i(U_i \cap U_j)
$$

satisfying on each $U_i \cap U_j$ the condition

$$
  s_i = \gamma_{i j} \circ s_j
  \,.
$$

=--

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

Given a foliation atlas as in def. \ref{FoliationAtlasByLocalSubmersions},
the [[diffeomorphisms]] $\{\gamma_{i j}\}_{i,j}$ satisfy the [[Cech cohomology|Cech]] [[cocycle]] condition

$$
   \gamma_{i j} \circ \gamma_{j k} = \gamma_{i k}
  \,.
$$

This is called the **[[Haefliger cocycle]]** of the foliation atlas.

=--

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

A smooth foliation of a [[smooth manifold]] $X$ is equivalently an [[integrable distribution]] (or an integrable subbundle) $E \hookrightarrow T X$.

=--

### In terms of Lie algebroids and Lie groupoids
 {#InTermsOfLieAlgebroidsAndLieGroupoids}

Definition \ref{AsIntegrableDistribution} above
is immediately reformulated equivalently as the following statement
in [[higher Lie theory]].

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

For $X$ a [[smooth manifold]], a foliation of $X$ is equivalently a [[Lie algebroid]] over $X$ such that the [[anchor map]] is an [[injection]].

=--

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

The [[Lie groupoids]] which under [[Lie differentiation]] give rise to Lie algebroids with injective anchors as in def. \ref{AsLieAlgebroidsWithInjectiveAnchor} are precisely those which are Morita-equivalent to [[étale groupoids]] (hence are the _[[foliation groupoids]]_, see there for more details) ([Crainic-Moerdijk 00, theorem 1](#CrainicMoerdijk)).

=--

One says:

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

A [[Lie groupoid]] **integrates** a given foliation, if it [[Lie integration|Lie integrates]] the coresponding [[Lie algebroid]], according to def. \ref{AsLieAlgebroidsWithInjectiveAnchor}.

=--

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

For a [[simple foliation]] $\mathcal{D}$ of a manifold $X$, example \ref{SimpleFoliation}, hence one where there is a [[submersion]]

$$ p_{\mathcal{D}} \;\colon\; X \to X/\mathcal{D}$$

to the leaf space, that map itself is the [[atlas]] of a Lie groupoid $\mathcal{G}$ which integrates the foliation, which is the [[Cech nerve]]

$$
  \mathcal{G}_\bullet
  =
  \left(
     X \underset{X/\mathcal{D}}{\times} X
     \stackrel{\to}{\to}
     X/\mathcal{D}
  \right)
  \,.
$$

=--

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

Among all Lie groupoids that integrate a given foliation $\mathcal{F}$ of a manifold $X$, the two special extreme

1. [[holonomy groupoid]] $Hol(X,\mathcal{F})_\bullet$

1. [[monodromy groupoid]] $Monod(X,\mathcal{F})_\bullet$

=--

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

Let $\mathcal{G}_\bullet$ be a [[Lie groupoid]] with (for simplicity) [[connected topological space|connected]] source-fibers.

Then there are maps

$$
  hol
  \;\colon\;
  Monod(X,\mathcal{F})_\bullet
  \stackrel{g_{\mathcal{G}}}{\to}
  \mathcal{G}_\bullet
   \stackrel{hol_G}{\to}
  Hol(X,\mathcal{F})
$$

which are surjective [[local diffeomorphisms]] and such that the composite is the holonomy morphism (...).

=--

This is ([Crainic-Moerdijk 00, prop. 1](#CrainicMoerdijk)).

### Of higher smooth spaces

One can consider the generalization of the notion of foliation of manifolds to foliations of structures in [[higher differential geometry]] such as [[Lie groupoids]] and [[Lie algebroids]]. See at

* [[foliation of a Lie algebroid]]

* [[foliation of a Lie groupoid]]

### In terms of differential cohesive higher geometry
 {#InCohesiveHigherGeometry}

The following is a suggestion for an axiomatization of foliations in [[higher differential geometry]] in the formalization of [[differential cohesion|differential]] [[cohesion]], followed by some considerations showing how these axioms reproduce traditional theory.

> Under construction.

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

Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] equipped with [[differential cohesion]].

As usual, we write $(\int \dashv \flat \dashv \sharp)$ for the [[adjoint triple]] of [[modalities]] that defines the [[cohesion]] ([[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]) and we write $(Red \dashv \int_{inf} \dashv \flat_{inf})$ for the adjoint triple of modalities that defines the [[differential cohesion]] ([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]).

=--

Below we are going to axiomatize aspects of the traditional description of foliations by [[Lie groupoids]]/[[foliation groupoids]] as discussed [above](#InTermsOfLieAlgebroidsAndLieGroupoids), so we start by briefly setting up some terminology on [[groupoid object in an (infinity,1)-category|groupoid objects]] in [[differential cohesion]].

+-- {: .num_defn #NotationForGroupoidObjects}
###### Definition (Notation for groupoid objects and atlases)

By the [[Giraud-Rezk-Lurie axioms]]
we may think of a [[1-epimorphism]] $\mathcal{G}_0 \to \mathcal{G}$ in $\mathbf{H}$ as an [[atlas]] of the cohesive $\infty$-groupoid $\mathcal{G} \in \mathbf{H}$, exhibiting equivalently the corresponding [[groupoid object in an (infinity,1)-category|groupoid object]] which we write

$$
  \mathcal{G}_\bullet \coloneqq \mathcal{G}_0^{\times^{\bullet+1}_{\mathcal{G}}}
  \,.
$$

Hence we use notation where omitting the subscript decorationon a groupoid object $\mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}$ refers to its realization

$$
  \mathcal{G} \coloneqq {\underset{\rightarrow}{\lim}}_n \mathcal{G}_{n}
  \;\;\;
  \in \mathbf{H}
  \,.
$$

=--

We have the following "geometricity" constraints on groupoid objects.

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

For $f \colon X \to Y$ any morphism in $\mathbf{H}$, write

$$
  X \stackrel{L(f)}{\to} Y \underset{\int_{inf} Y}{\times} \int_{inf} X
$$

for the canonical morphism induced by the [[natural transformation|naturality]] of the $\int_{inf}$-[[unit of an adjunction|unit]]. We say that

1. $f$ is a **[[formally smooth morphism]]** (or _[[submersion]]_) if $L(f)$ is a [[1-epimorphism]];

1. $f$ is a **[[formally étale morphism]]** (or _[[local diffeomorphism]]_) if $L(f)$ is an [[equivalence in an (infinity,1)-category|equivalence]].

Now if $\pi \colon \mathcal{G}_0 \to \mathcal{G}$ is a [[1-epimorphism]], hence an [[atlas]] for the cohesive $\infty$-groupoid $\mathcal{G}$, then we say about the corresponding [[groupoid object in an (infinity,1)-category|groupoid object]] as in def. \ref{NotationForGroupoidObjects}, that

1. $\mathcal{G}_\bullet$ is an **[[geometric ∞-groupoid]]** if its [[atlas]] $\pi$ is a [[formally smooth morphism]]/[[submersion]].

1. $\mathcal{G}_\bullet$ is an **[[étale ∞-groupoid]]** if its [[atlas]] $\pi$ is a [[formally étale morphism]]/[[local diffeomorphism]].

=--

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

For $X \in \mathbf{H}$, a **[[foliation]]** of $X$ is
a morphism $\mathcal{D} \colon X \to X//\mathcal{D}$ in $\mathbf{H}$ which is

1. a [[1-epimorphism]];

1. a [[formally smooth morphism]].

Equivalently a foliation of $X$ is a map that exhibits $X$ as an [[atlas]]
for a [[geometric ∞-groupoid]], def. \ref{GeometricAndEtale}.

Given a foliation $\mathcal{D}$ on $X$ we say that the **leaf decomposition** of $X$ induced by the foliation is the [[(∞,1)-pullback]]

$$
  LeafDec(\mathcal{D})
  \coloneqq
  \flat(X//\mathcal{D}) \underset{X//\mathcal{D}}{\times} X
$$

in

$$
  \array{
    LeafDec(\mathcal{D}) &\stackrel{\iota_{\mathcal{D}}}{\to}& X
    \\
    \downarrow && \downarrow^{\mathrlap{\mathcal{D}}}
    \\
    \flat (X//\mathcal{D}) &\to& X//\mathcal{D}
  }
  \,,
$$

where the bottom map is the [[counit of an adjunction|counit]] of the [[flat modality]].

=--

Now let $\mathbb{G} \in Grp_2(\mathbf{H})$ a [[braided ∞-group]]. Write

$$
  \Omega^2_{cl} := \Omega^2_{flat}(-,\mathbb{G})
$$

for the corresponding coefficient object for [[curvature]] forms of $\mathbb{G}$-[[principal ∞-connections]] (as discussed there).

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

Given a closed 2-form

$$
  \omega \;\colon\; X \to \Omega^2_{cl}
$$

a foliation of $X$ by **$\omega$-[[isotropic submanifold|isotropic subspaces]]** is a [[foliation]] $\mathcal{D} \colon X \to X//\mathcal{D}$
as in def. \ref{FoliationInDifferentialCohesion} such that the restriction of $\omega$ to the leaf decomposition is equivalent to the 0-form

$$
  \iota_{\mathcal{D}}^* \omega \simeq 0
  \,,
$$

hence such that the top composite morphism in the diagram

$$
  \array{
    (\flat \mathcal{E}) \underset{\mathcal{E}}{\times} X
    &\to& X &\stackrel{\omega}{\to}& \Omega^2_{cl}
    \\
    \downarrow && \downarrow
    \\
    \flat (X//\mathcal{D}) &\to& X//\mathcal{D}
  }
$$

factors through the point.

=--

We now discuss how low-degree examples of this axiomatics interpreted in $\mathbf{H} \coloneqq $ [[SynthDiff∞Grpd]] reproduces the traditional notions of [[foliations]] and [[isotropic submanifolds]] of [[pre-symplectic manifolds]].

In the following we regard [[smooth manifolds]] canonically under the embedding

[[SmoothMfd]] $\hookrightarrow $ [[Smooth∞Grpd]] $\stackrel{i_!}{\hookrightarrow}$ [[SynthDiff∞Grpd]] $= \mathbf{H}$

as [[reduced object|reduced]] [[synthetic differential ∞-groupoids]].

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

A [[smooth function]] $f \colon X \to Y$ between [[smooth manifolds]] is

1. a [[local diffeomorphism]] in the traditional sense precisely if it is a [[formally étale morphism]] in the sense of def. \ref{GeometricAndEtale};

1. a [[submersion]] in the traditional sense precisely if it is a [[formally smooth morphism]] in the sense of def. \ref{GeometricAndEtale}.

=--

This is discussed at _[[SynthDiff∞Grpd]]_. The idea of the proof is to use the [[∞-cohesive site]] of definition [[CartSp]]${}_{synthdiff}$ and evaluate the [[homotopy pullback]] in def. \ref{GeometricAndEtale} first on all representables of the form $U \times D_1$ where $U$ ranges over [[Cartesian spaces]] and where $D_1$ is the first order ininfitesimal neighbourhood of the origin on $\mathbb{R}^1$ (whose [[smooth algebra]] of fucntions is the [[ring of dual numbers]]). Then the homotopy pullback is represented as an ordinary pullback of sheaves over Cartesian spaces and the naturality diagram in question is the diagram of [[tangent bundles]]

$$
  \array{
    T X &\stackrel{d f}{\to}& T Y
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& Y
  }
  \,.
$$

With this now the claim is reduced to the traditional characterization of [[submersions]] and [[local diffeomorphisms]].

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

Let $\mathcal{G}$ be a [[smooth groupoid]] which has a presentation by a [[simplicial presheaf]] $\mathcal{G}_\bullet$ with values in 2-coskeletak Kan complexes where objects and morphisms are represented by a [[smooth manifold]] each, and consider it equipped with the induced [[atlas]] $\mathcal{G}_0 \to \mathcal{G}$. Then

* if the presentation $\mathcal{G}_\bullet$ is a [[Lie groupoid]] then $\mathcal{G}_0 \to \mathcal{G}$ is a [[geometric ∞-groupoid]]

* if the presentation $\mathcal{G}_\bullet$ is an [[étale groupoid]] then $\mathcal{G}_0 \to \mathcal{G}$ is an [[étale ∞-groupoid]]

in the sense of def. \ref{GeometricAndEtale}.

=--

This follows by the corresponding discussion at [[SynthDiff∞Grpd]]. The idea of the proof is that one presents the atlas in the projective [[model structure on simplicial presheaves]] by the [[décalage]] [[fibration]] [[resolution]], schematically

$$
  \array{
    && g
    \\
    & \swarrow && \searrow & &&& &&& \mathcal{G}_0
    \\
    g_1 &&\to&& g_2
    \\
    \\
    &&&& &&&&&& \downarrow
    \\
    \\
    g_1 &&\to&& g_2 &&&&&& \mathcal{G}
  }
  \,.
$$

Then the [[homotopy pullback]] $\mathcal{G} \underset{\int_{inf}\mathcal{G}} {\times}\int_{inf} X $ is presented by an ordinary pullback and so
example \ref{TraditionalSubmersions} applies degreewise. In degree 0 the above resolution is the target map in the groupoid $\mathcal{G}$ and so by example \ref{TraditionalSubmersions} this is a [[submersion]] or [[local diffeomorphism]], respectively, as claimed.

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

Let $X$ be a smooth manifold and  let $\mathcal{D}$ be a traditional [[foliation]] on $X$ which is a _[[simple foliation]]_, example \ref{SimpleFoliation},
in that the [[leaf space]] $X/\mathcal{D}$ exists as a smooth manifold and the projection map $X \to X/\mathcal{D}$ is a [[submersion]].

Then by the discussion at [[synthetic differential ∞-groupoid]], this projection map is also a [[formally smooth morphism]] in $\mathbf{H}$ according to def. \ref{GeometricAndEtale}. Moreover, being a quotient projection it is a [[1-epimorphism]] and hence exhibits the corresponding [[foliation groupoid]]

$$
  \left(
    X \underset{X/\mathcal{D}}{\times} X
    \stackrel{\to}{\to}
    X/\mathcal{D}
  \right)
$$

as a [[geometric ∞-groupoid]] in the sense of def. \ref{GeometricAndEtale}.

Now $\flat ( X// \mathcal{D}) $ is the underlying set of points of the [[leaf space]], regarded as a [[discrete ∞-groupoid]]. So we have the [[pasting]] diagram of pullbacks

$$
  \array{
    L_{l} &\to& \coprod_{l \in X/\mathcal{D}} L_l &\to& X
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{\mathcal{D}}}
    \\
    {*} &\stackrel{\vdash l}{\to}& \coprod_{l \in D/\mathcal{X}}{*}
    &\to&
    X/\mathcal{D}
  }
$$

for every [[leaf]] $L_l$ labeled by the point $l \in X/\mathcal{D}$ in leaf space, which exhibits the leaf decomposition of $X$ under $\mathcal{D}$ according to def. \ref{FoliationInDifferentialCohesion} as the disjoint union of the leaves of $(X,\mathcal{D})$ in the traditional sense, injected into $X$ in the canonical way.

=--

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

Consider now $\mathcal{G}_\bullet$ any [[Lie groupoid]], hence in particular a [[smooth groupoid]] $\mathcal{G} \in \mathbf{H}$ equipped with an [[atlas]] $\mathcal{G}_0 \to \mathcal{G}$, which hence by example \ref{LieAndEtaleGroupoids} exhibits a [[geometric ∞-groupoid]] in the sense of def. \ref{GeometricAndEtale}, hence a foliation
$\mathcal{D} \;\colon\; \mathcal{G}_0 \to \mathcal{G}$ in the sense of def. \ref{FoliationInDifferentialCohesion}.

Computation of the [[homotopy pullback]]

$$
  \array{
    LeafDec_{\mathcal{D}}(\mathcal{G}_0) &\to& \mathcal{G}_0
    \\
    \downarrow && \downarrow^{\mathrlap{\mathcal{D}}}
    \\
    \flat( \mathcal{G} ) &\to& \mathcal{G} & = \mathcal{G}_0//\mathcal{D}
  }
$$

by the method as in example \ref{LieAndEtaleGroupoids} shows that $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the [[smooth groupoid]] presented by the presheaf of groupoids whose

* smoothyl $U$-parameterized collection of objects are smoothly $U$-parameterized collections of morphisms $\{g_0 \to g(u)\}_{u \in U}$ in $\mathcal{G}_\bullet$ with $g_0$ held fixed;

* morphisms are given by precomposing these collections with a fixed (not varying with $U$) morphism in $\mathcal{G}_\bullet$.

This means that if $\mathcal{G}_\bullet$ is an [[étale groupoid]] to start with, then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the disjoint union of all its [[orbit]] leaves (as smooth manifolds), hence that the abstractly defined $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ reproduces the decomposition of $\mathcal{G}_0$ by the [[foliation]] encoded by the [[foliation groupoid]] $\mathcal{G}_\bullet$ as in traditional theory.

=--

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

We may suggestively summarize example \ref{TraditionalGeneralFoliation} in words as:

"In cohesive higher geometry, every foliation is a [[simple foliation]]."

Because the quotient map to the leaf space of a general foliation is always a [[submersion]]/[[formally smooth morphism]], just not always onto a manifold, but onto a higher space.

=--

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

If the $\mathcal{G}_\bullet$ in example \ref{TraditionalGeneralFoliation} is not an [[étale groupoid]] to start with but a more general [[Lie groupoid]], then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ in general retains information of non-[[discrete group|discrete]] [[isotropy groups]] of $\mathcal{G}_\bullet$.

We might decide to rule out this possibility by adding to the axioms in def. \ref{FoliationInDifferentialCohesion} the clause that $X//\mathcal{G}$ (here $\mathcal{G}_0//\mathcal{D}$) be étale.

However, we might also keep that case and regard it as the first instance of what is certainly a natural phenomenon as we pass to [[higher geometry]], namely that leaves of a foliation no longer need to manifolds but will be (higher) groupoids themselves.

=--

Finally, given the above it is clear how [[isotropic submanifold|isotropic]] appear in the cohesive axiomatics.

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

For $\mathbb{G} = U(1)$ the smooth [[circle group]], $\Omega^2_{cl}$ is the ordinary sheaf of closed [[differential 2-forms]] under the canonical embedding

$$
  Sh(CartSp) \hookrightarrow
  Sh_\infty(CartSp)
  \simeq
  Smooth\infty Grpd
  \stackrel{i_!}{\hookrightarrow}
  SynthDiff\infty\mathrm{Grpd}
  \,.
$$

Then for $X$ a [[smooth manifold]] a morphism $\omega \;\colon\; X \to \Omega^2_{cl}$ is equivalently a [[differential 2-form]].

Then for $\mathcal{D} \;\colon\; X \to X//\mathcal{D}$ a traditional [[foliation]] of $X$ regarded as a foliation in  $SynthDiff\infty Grpd$ by example \ref{TraditionalGeneralFoliation}, it follows with the discussion there that $\iota_{\mathcal{D}}^* \omega$ is precisely the collection of restriction of $\omega$ to each of the leaves of the foliation. Therefore this is a foliation by [[isotropic submanifolds]] in the traditional sense precisely if it is an $\omega$-isotropc foliation in the sense of def. \ref{IsotropicFoliationsInDiffCohe}.

=--

## Examples

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

For $X \to Y$ a [[submersion]] of [[smooth manifolds]], the connected [[fibers]] of the submersion constitute a foliation of $X$ whose [[codimension]] is the [[dimension]] of $Y$. Foliations of this form are called **[[simple foliations]]**.

=--

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

Every [[Lie groupoid]] gives a folitation on its space of [[objects]]: the leaves are the [[orbits]]. Conversely, every regular foliation gives rise to its [[holonomy groupoid]]. This is a (not necessarily Hausdorff) Lie groupoid whose orbits are the leaves of the original foliation, and which in some sense is minimal with this condition.

=--

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

Every [[Poisson manifold]] has a canonical structure of a foliation whose leaves are its maximal [[symplectic manifold|symplectic]] submanifolds, called *[[symplectic leaves]]*.

=--
## Properties

### Leaf space

The set of components of a foliation is typically non-[[Hausdorff space|Hausdorff]], which is one of the motivations of the [[Alain Connes|Connes]]-style [[noncommutative geometry]].

### Classification

Folitation are classified by the [[Haefliger groupoid]]. See at _[[Haefliger theorem]]_.

### Characteristic classes

There is a theory of [[characteristic classes]] for foliations. A most well known example is the [[Godbillon-Vey characteristic class]] for codimension one foliations.

## Related concepts

* [[Bott connection]]

* [[holonomy groupoid]], [[monodromy groupoid]]

* [[orbifold]]

* [[Frobenius theorem (differential topology)]]

* [[higher differential geometry applied to plain differential geometry]]

## References
 {#References}

The notion of foliated manifolds was introduced in the 1950s, motivated from [[partial differential equation]] theory, in

* [[Charles Ehresmann]], ...

* Reeb, ...

* [[Eli Cartan]], _Sur l'intégration des équations différentiels completement intégrable_, Oeuvres Complètes, Pt. II, Vol. I, 555-561.

A discussion in [[differential geometry]] is in

* Robert Hermann, _On the differential geometry of foliations_, Annals of Mathematics, Second Series, Vol. 72, No. 3 pp. 445-457 ([jstor](http://www.jstor.org/stable/1970226))

A textbook account with a view to the modern formulation in [[Lie groupoid]] theory is

* [[Ieke Moerdijk]], [[Janez Mr?un]], _[[Introduction to foliations and Lie groupoids]]_, Cambridge Studies in Advanced Mathematics __91__, 2003. x+173 pp. ISBN: 0-521-83197-0
 {#MoerdijkMrcun}

Foliations in [[Lie groupoid]] theory are discussed in more detail in

* [[Marius Crainic]], [[Ieke Moerdijk]], _Foliation groupoids and their cyclic homology_ ([arXiv:math/0003119](http://arxiv.org/abs/math/0003119))
 {#CrainicMoerdijk}

The corresponding [[groupoid algebras]] are discussed in chapter 2, section 8 of

* [[Alain Connes]], _[[Noncommutative Geometry]]_

See also

* [wikipedia](http://en.wikipedia.org/wiki/Foliation), Springer Online Enc. of Math.: [foliation](http://eom.springer.de/F/f040740.htm)

A survey by Fuks in Russian Itogi:

* Д. Б. Фукс, _Слоения_, Итоги науки и техн. Сер. Алгебра. Топол. Геом., 1981,  том __18__, стр. 151–-213, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=inta&paperid=93&what=fullt&option_lang=rus)

Cohomology of formal vector fields and characteristic classes of foliations were originally studied in the papers

* D. B. Fuks, _Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations_ (book, Rus. and Eng. versions)

* I. M. Gelʹfand, B. L. Feĭgin, D. B. Fuks, _Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations, Funkcional. Anal. i Priložen.  8  (1974), no. 2, 13--29 (Russian original [mathnet.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=2326&option_lang=eng), [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2326&volume=8&year=1974&issue=2&fpage=13&what=fullt&option_lang=eng))

* Claude Godbillon, _Cohomologies d'algèbres de Lie de champs de vecteurs formels_, Séminaire Bourbaki, 25ème année (1972/1973), Exp. No. 421, pp. 69--87. Lecture Notes in Math. __383__, Springer 1974.

* И. М. Гельфанд, Д. Б. Фукс, _Когомологии алгебры Ли формальных векторных полей_, Изв. АН СССР. Сер. матем., 1970,  __34__, в. 2, стр. 322–-337, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=2418&what=fullt&option_lang=rus)

In a series of works of Connes and Moscovici, the local index formulas in the context of transverse geometry of foliations has been studied in connection to a new cyclic homology of a Hopf algebra arising in this context:

* [[A. Connes]], H. Moscovici, _Modular Hecke algebras and their Hopf symmetry_, Mosc. Math. J., 4:1 (2004), 67–109; [math.QA/0301089](http://arxiv.org/abs/math/0301089), [ams](http://www.ams.org/distribution/mmj/vol4-1-2004/connes_moscovici_1.pdf); _Hopf algebras, cyclic cohomology and the transverse index theory_, [math.DG/9806109](http://arxiv.org/abs/math/9806109), Comm. Math. Phys. __198__, n.1, 1998 [MR99m:58186](http://www.ams.org/mathscinet-getitem?mr=1657389) [doi](http://dx.doi.org/10.1007/s002200050477); _Rankin-Cohen brackets and the Hopf algebra of transverse geometry_, Mosc. Math. J., 4:1 (2004), 111–130; _Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry_, in: Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math. __38__, p. 217–255. Enseignement Math., Geneva, 2001 [MR2003k:58042](http://www.ams.org/mathscinet-getitem?mr=1929328)
* A. Connes, _Cyclic cohomology and the transverse fundamental class of a foliation_, in: Geom. methods in operator algebras (Kyoto, 1983), Pitman Res. Notes in Math. __123__, p. 52–144, 1986 [MR88k:58149](http://www.ams.org/mathscinet-getitem?mr=866491)

More general issues of index theory in noncommutative geometry applied to foliations is in

* Yu. A. Kordyukov, _Noncommutative geometry of foliations_, J. K-Theory, 2:2, Special issue in memory of Y. P. Solovyev, Part 1 (2008), 219–327 [MR2009m:58018](http://www.ams.org/mathscinet-getitem?mr=2456103); _Index theory and non-commutative geometry on foliated manifolds_, Russian Math. Surveys, 64:2 (2009), 273–391 (original: Ю. А. Кордюков, УМН, 64:2(386) (2009), 73–202); _Формула следов для трансверсально-эллиптических операторов на римановых слоениях_, Алгебра и анализ, 12:3 (2000), 81–105 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=1108&what=fullt&option_lang=rus)

* W. P. Thurston, _Existence of codimension-one foliations_,  Ann. of Math. (2) __104__  (1976), no. 2, 249--268 ([doi](http://dx.doi.org/10.1007/BF02566730)); _Foliations and groups of diffeomorphisms_, Bull. Amer. Math. Soc. 80  (1974), 304--307 ([pdf](http://www.ams.org/bull/1974-80-02/S0002-9904-1974-13475-0/S0002-9904-1974-13475-0.pdf)); _The theory of foliations of codimension greater than one_,  Comment. Math. Helv.  49  (1974), 214--231 ([link](http://resolver.sub.uni-goettingen.de/purl?GDZPPN00206135X))

The first of a series on foliations in [[derived geometry]]

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Algebraic foliations and derived geometry: the Riemann-Hilbert correspondence_, ([arXiv:2001.05450](https://arxiv.org/abs/2001.05450))

[[!redirects foliations]]
[[!redirects leaf space]]
[[!redirects leaf spaces]]

[[!redirects space of leaves]]
[[!redirects spaces of leaves]]

[[!redirects foliation theory]]

[[!redirects regular foliation]]
[[!redirects regular foliations]]


foliation of a Lie algebroid

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

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of [[foliation]] of a [[smooth manifold]] from manifolds to [[Lie algebroids]].

## Definition

One of several equivalent definitions of a (regular) foliation of a smooth manifold is

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

A [[regular foliation]] of a [[smooth manifold]] $X$ is a wide sub-[[Lie algebroid]] of its [[tangent Lie algebroid]], hence a [[Lie algebroid]] $\mathcal{P}$ over $X$ with [[injection|injective]] [[anchor map]]

$$
  \array{
    \mathcal{P} &\hookrightarrow & T X
    \\
    \downarrow && \downarrow
    \\
    X &=& X
  }
  \,.
$$

=--

In this spirit there is an evident generalization of the notion to a notion of foliations of Lie algebroids.

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

A (regular) **foliation of a Lie algebroid** $A$ is a sub-[[double Lie algebroid]] of the tangent double Lie algebroid which is wide over $A$

$$
  \array{
    \mathcal{P} &\hookrightarrow& T A
    \\
    \downarrow && {}^{\mathllap{d p_A}}\downarrow & \searrow^{\mathrlap{p_{T A}}}
    \\
    \mathcal{P}_0 &\hookrightarrow& T X && A
    \\
    && & \searrow & \downarrow^{\mathrlap{p_A}}
    \\
    && && X
  }
  \,.
$$

=--

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

Foliations of a Lie algbroid $A \to X$ according to def. \ref{Foliation} are in natural [[bijection]] to the following data:

1. an ordinary [[foliation]] $\mathcal{P}_0 \hookrightarrow T X$

1. $\mathcal{P}_1 \hookrightarrow A$ a sub-[[vector bundle]]

   (this is the joint [[kernel]] $\mathcal{P}_1 = ker(p_{T A}) \cap ker(d p_{A})$ naturally identified as a subspace of $A$)

1. $\nabla$ a [[flat connection]] on the [[quotient]] bundle $A/\mathcal{P}_1$ partially defined over [[vector fields]] in $\mathcal{P}_0$

   (this is the induced linear foliation of the total space $A$ regarded as a horizontal-subspace distribution)

such that over every [[open subset]] $U \hookrightarrow X$

1. The [[sections]] of $A$ that become $\nabla$-[[covariant derivative|constant]] in $A/\mathcal{P}_1$ form a sub-[[Lie algebra]] of $\Gamma_U(A|_U)$ (an [[integrable distribution of subspaces]]);

1. the sections of $\mathcal{P}_1$ are a Lie ideal inside this sub-Lie algebra;

1. the [[image]] of $\mathcal{P}_1$ under the [[anchor map]] is in $\mathcal{P}_0$;

1. the quotiented [[anchor map]] $A/\mathcal{P}_1 \to T X / \mathcal{P}_0$ intertwines $\nabla$ with the $\mathcal{P}_0$-[[Bott connection]].

=--

This is ([EH, theorem 7.2](#EH)).

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

According to prop. \ref{CharacterizationByTriple} the notion of foliation of a Lie algebroid generalizes the notion of _ideal system_ of a Lie algebroid ([Higgins-Mackenzie](#HM), [Mackenzie](#Mackenzie)): A foliation as in def. \ref{Foliation} comes from an ideal system in this sense precisely of $\mathcal{P}_0$ is a [[simple foliation]] (the quotient map exists in smooth manifolds and is a [[surjective submersion]]) and the [[holonomy]] of $\nabla$ is trivial.

=--

## Examples

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

Let $\mathcal{G}_\bullet$ be a [[Lie groupoid]] and let

$$
  \array{
    \mathcal{P} &\hookrightarrow& T \mathcal{G}_1
    \\
    \downarrow\downarrow && {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}}
    \\
    \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0
  }
$$

be a [[foliation of a Lie groupoid]], regarded as an [[internal groupoid]] in [[Lie algebroids]]. Then applying [[Lie differentiation]] yields a foliation of the Lie algebroid $Lie(\mathcal{G}_\bullet)$.

=--

## Related concepts

* [[foliation of a Lie groupoid]]

* [[exterior differential system]]

## Referemces

Maybe the first discussion of foliations of Lie algebroids appears in

* {#EH} Eli Hawkins, _A Groupoid Approach to Quantization_ ([arXiv:math.SG/0612363](http://arxiv.org/abs/math.SG/0612363))

Ideal systems of Lie algebroids have been introduced and studied in

* {#HM} [[Philip Higgins]], [[Kirill Mackenzie]], _Algebraic constructions in the category of Lie algebroids_ . J. Algebra
129 (1990), no. 1, 194–230.MR1037400

* {#Mackenzie} [[Kirill Mackenzie]], _General theory of Lie groupoids and Lie algebroids_ Cambridge Univ. Press, Cambridge (2005)

Related discussion is in

* [[Cristian Ortiz]], _Multiplicative Dirac structures_ ([arXiv:1212.0176](http://arxiv.org/abs/1212.0176))

[[!redirects foliations of a Lie algebroid]]
[[!redirects foliations of Lie algebroids]]


foliation of a Lie groupoid

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

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of [[foliation]] of a [[smooth manifold]] from manifolds to [[Lie groupoids]].

## Definition

One of several equivalent definitions of a (regular) foliation of a smooth manifold is

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

A [[regular foliation]] of a [[smooth manifold]] $X$ is a wide sub-[[Lie algebroid]] of its [[tangent Lie algebroid]], hence a [[Lie algebroid]] $\mathcal{P}$ over $X$ with [[injection|injective]] [[anchor map]]

$$
  \array{
    \mathcal{P} &\hookrightarrow & T X
    \\
    \downarrow && \downarrow
    \\
    X &=& X
  }
  \,.
$$

=--

In this spirit there is an evident generalization of the notion to a notion of foliations of Lie algebroids.

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

A (regular) **foliation of a Lie groupoid** $\mathcal{G}_\bullet$ is a sub-[[Lie algebroid-groupoid]] of the tangent [[Lie algebroid-groupoid]] which is [[wide subcategory|wide]]

$$
  \array{
    \mathcal{P} &\hookrightarrow& T \mathcal{G}_1
    \\
    \downarrow \downarrow && \downarrow \downarrow
    \\
    \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0
  }
  \,.
$$

=--

## Examples

(...)

## Related concepts

* [[foliation of a Lie algebroid]]

## References

Maybe the first discussion of foliations of Lie groupoids appears in

* Eli Hawkins, _A Groupoid Approach to Quantization_ ([arXiv:math.SG/0612363](http://arxiv.org/abs/math.SG/0612363))
 {#EH}

Related discussion is in

* [[Cristian Ortiz]], _Multiplicative Dirac structures_ ([arXiv:1212.0176](http://arxiv.org/abs/1212.0176))

[[!redirects foliations of a Lie groupoid]]
[[!redirects foliations of Lie groupoids]]


Foling Zou

* [personal page](https://sites.google.com/umich.edu/foling-zou/home)

## Selected writings

On [[equivariant bundles]]:

* {#Zou20} [[Foling Zou]], _Notes on equivariant bundles_ ([arXiv:2008.01268](https://arxiv.org/abs/2008.01268))

category: people


folk theorem

A [[theorem]] whose truth is [[folklore]].

[[!redirects folk theorems]]


folklore

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[mathematics]] (also in [[mathematical physics]] and neighbouring disciplines) a [[theorem]] is sometimes said to be _folklore_ when the community feels that it has been around and generally accepted as true for a long time, without however a [[proof]] of it having been submitted and published in the usual manner.

For instance the [[cobordism hypothesis]] -- which is an intuitively  evident statement, whose formalization and proof, however, is notoriously subtle -- is referred to as "folklore" in [Stolz 14, p. vi](FQFT#Stolz14).

The sociology of folklore theorems can be subtle. In semi-formalized areas such as [[theoretical physics]] folklore convictions, correct ones and incorrect ones, have seriously impeded progress.

The (popular) claim that some statement is "well-known" -- without, however, there being a reference for it -- may signify folklore: If the truth of a statement really is well-known then it must be easy to give a definite reference for it. If it feels "well-known" but just doesn't have a definite reference, then it's folklore.

[[Paul Taylor]] on folklore in [[category theory]] ([Taylor, blog comment Sept 2012](http://math.andrej.com/2012/09/28/substitution-is-pullback/)):

> **folklore**, $[\ldots]$ is a technical term for a method of publication in category theory. It means that someone sketched it on the back of an envelope, mimeographed it (whatever that means) and showed it to three people in a seminar in Chicago in 1973, except that the only evidence that we have of these events is a comment that was overheard in another seminar at Columbia in 1976. Nevertheless, if some younger person is so presumptuous as to write out a proper proof and attempt to publish it, they will get shot down in flames.

[[Clark Barwick]] on folklore in [[homotopy theory]]/[[algebraic topology]] ([Barwick 17, Item 3](homotopy+theory#Barwick17) ):

> We have no good culture of problems and conjectures. The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of a proof -- _and never generate a complete proof_. Then, when others work to develop proofs, they are not said to have "solved a problem of So-and-So", rather, they have "completed the write-up of So-and-So's proof" or "given a proof of So-and-So's theorem". The ossification of a caste system -- in which one group has the general ideas and vision while another toils to realize that vision -- is no way for a subject to flourish. Other subjects have high-status visionaries who are no sketchier in details than those in homotopy theory, but whose unproved insights are nevertheless known as conjectures, problems, and programs.

## Related concepts

* [[list of lost manuscripts in category theory]]

[[!include mathematical statements --- contents]]

## References

Paul Taylor's comment on folklore is quoted in

* {#Aubert2019} Clément Aubert, _Categories for Me, and You?_, 2019 ([arxiv:1910.05172](https://arxiv.org/abs/1910.05172)).

[[!redirects well-known]]
[[!redirects well known]]


foo > history

< [[foo]]


foobar > history

< [[foobar]]


force

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[physics|Physical]] bodies interact. The effects of binary interactions between point particles are known to be summarizable in vector quantities called __forces__ applied to any body in an interaction; namely the forces on one body from each of several other bodies add up as [[vector|vectors]] and such vector sum of forces on the body is proportional to the acceleration of the body (hence [[Newton's laws|Newton's second law]] $F = m a$ is *not* a definition of force but a real law).

Thus force is a vector-like quantity which is a manifestation of the interaction between bodies. If body $A$ acts with force $F$ on body $B$ then $B$ acts on $A$ with the force $-F$ of the opposite vector value. These two forces do not cancel as they act on different bodies and at different points in space, though they are along the same line.

Sometimes one abstracts the forces on a system of bodies from the background by "[[potential energy]]" of the particles. Then the background acts on each particle with a force equal to the negative gradient of the potential energy.

In quantum field theory the forces appear mediated by [[particle|particles]] which get exchanged between the bodies in interaction. For example, the [[strong nuclear force]] is mediated by [[gluon|gluons]]. There are 4 known fundamental forces in nature and all others are derived from them: the [[electromagnetic force|electromagnetic]], [[weak nuclear force|weak]], strong and [[gravity|gravitational]] force; and the first three are unified in the [[standard model of particle physics]].

In nuclei there are also effective forces which are not of vector but of [[tensor]]ial nature, and effective forces involving more than two bodies. But such quantum systems are far from classical mechanical systems.

## Related concepts

* [[gauge field]]

* [[Lorentz force]]

* [[gravity]],

* [[electroweak field]], [[weak nuclear force]], [[strong nuclear force]]

* [[matter]]

* [[entropic force]]

## References

### General

(...)

[[!include classification of long-range forces -- references]]

[[!redirects force]]
[[!redirects forces]]

[[!redirects force field]]
[[!redirects force fields]]


forcing

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

# Forcing
* table of contents
{: toc}

## Idea

In [[set theory]], **forcing** is a way of "adjoining indeterminate objects" to a [[model]] in order to make certain [[axioms]] [[true]] or [[false]] in a resulting new model.

The language of forcing is generally used in [[material set theory]]. From the point of view of [[structural set theory]]/[[categorical logic]] it is more or less equivalent to the construction of [[categories of sheaves]] in [[topos theory]] and [[structural set theory]]: here a [[classifying topos]] of a [[theory]] is a "category of sets" that is  "forced" to contain a generic model of the theory.

## An analogy to polynomial rings

There are three ways in which to describe the construction of forcing, which can be explained by analogy to a situation in algebra.  Suppose that we have a [[ring]] $R$ and we want to know what $R$ would be like if it had a nilsquare element, i.e. an $x$ such that $x^2=0$, although we don't know whether $R$ *actually* has any nilsquare elements.  There are several things we could do.

1. We could consider some larger ring $S$ which contains $R$ as a subring, and also contains some nilsquare element $x$, and then study the subring of $S$ generated by $R$ and $x$, using only our knowledge that it contains $R$ and $x$ and that $x$ is nilsquare.

1. We could work formally in the [[theory]] of $R$ with an extra variable $x$ and the axiom that $x^2=0$, avoiding consideration of any new rings at all.

1. We could construct the polynomial ring $R[x]$ and quotient by the ideal $(x^2)$, resulting in the *universal* ring $R[x]/(x^2)$ containing $R$ and also a nilsquare element.

Obviously these are more or less equivalent ways of doing the same thing.  Arguably, most modern mathematicians would find the third the most natural.  It is certainly the most [[category theory|category-theoretic]] and the most in line with the [[nPOV]].

Now suppose instead that we have a model $V$ of [[ZF]] set theory (for instance), and we want to know what $V$ would be like if it contained a set $G$ satisfying some particular properties.  We can likewise take three approaches.

1. We can assume that $V$ sits inside some larger model of ZF, in such a way that there is a set $G$ outside of $V$ and the "model generated by $V$ and $G$" called $V[G]$ satisfies the desired properties.  The $G$ is called a *generic* set for the desired "notion of forcing."  There is an extra wrinkle in this approach in that such a $G$ will not generally exist unless we assume that $V$ is [[countable set|countable]].  But for purposes of independence proofs in [[classical mathematics]], this is no problem, since the downward [[Löwenheim-Skolem theorem]] guarantees that if ZF is consistent, then it has a countable model.

1. We can define a new notion of "truth" for statements about $V$, which is not the same as the old one, and which will usually take values not in ordinary [[truth values]] but in some more complicated [[Boolean algebra]] (or [[Heyting algebra]]), in such a way so that the statement "there exists a $G$ with the desired properties" is "true."  This is called *forcing semantics*.

1. We can construct a new model of set theory out of whole cloth from $V$, rather than trying to find it as a submodel of some assumed larger model.  In material set theory, this construction is usually called a *Boolean valued model* (or a *Heyting valued model* if it is [[constructive mathematics|intuitionistic]]).  In structural set theory, this construction is simply that of the [[topos of sheaves]] on a suitable [[site]], in the model $V$.  The topos of sheaves is moreover *universal* over $V$ such that it contains an [[object]] $G$ satisfying the desired properties in its [[internal logic]]---in other words, it is a [[classifying topos]] of the [[theory]] of such a $G$. See at _[References -- In terms of classifying toposes](#ReferencesInTermsOfClassifyingToposes)_ for more.

Although by analogy, it would seem that a modern perspective should prefer the third approach ([Shulman 20](#Shulman20)), most material set theorists still seem to prefer one of the first two.

Before [[Paul Cohen|Cohen]] showed how forcing could give rise to models of ZF(C), [[Abraham Fraenkel|Fraenkel]] introduced the method of [[permutation model]]s (later refined by Mostowski and Specker), which gave models of [[ZFA]], a version of ZF with [[urelements|atoms]]. See for example the [[basic Fraenkel model]].

## Related concepts

* [[Kripke-Joyal semantics]]

* [[double negation topology]]

* [[homotopy type theory FAQ]] -- [What is homotopy type theory? -- For set theorists](http://ncatlab.org/nlab/show/homotopy+type+theory+FAQ)

## References

Exposition is in

* [[Timothy Chow]], _A beginner's guide to forcing_, Contemp. Math ([arXiv:0712.1320](https://arxiv.org/abs/0712.1320))

Another good introduction is in [Schoenfield 71](#Schoenfield71)

### General

* [[Paul Cohen]], _Set theory and the [[continuum hypothesis]]_, Benjamin, New York 1966

* [[Gonzalo Reyes]], _Typical and generic in a Baire space for relations_, thesis 1967

* [[Abraham Robinson]], _Infinite forcing in model theory_,  in Proc. 2nd Scand. Logic Synp. pp. 317-340, ed. J. E. Fenstad

* Jon Barwise ed. _Handbook of mathematical logic_, 1977, in chapters by Macintyre, Burgess and Keisler

* {#Schoenfield71} J. R. Schoenfield, _Unramified forcing_, pp. 383--395 in: Axiomatic set theory, vol. 1, ed. D. S. Scott, Proc. Symp. Pure Math. __13__ (1971) (([pdf](https://www.math.ucsd.edu/~sbuss/CourseWeb/Math260_2012F2013W/Shoenfield_UnramifiedForcing.pdf)))

* S. Shelah, _Proper and Improper Forcing_ , Perspectives in Math. Logic vol. 5 Springer Heidelberg 1998. ([toc](http://projecteuclid.org/euclid.pl/1235419814#toc))

* [[Paul Cohen]], _The Discovery of Forcing_, Rocky Mountain J. Math.
Volume 32, Number 4 (2002), 1071-1100. ([Euclid](http://projecteuclid.org/euclid.rmjm/1181070010))

* math overflow, what is the generic poset used in forcing really?, [web](http://mathoverflow.net/questions/51187/what-is-the-generic-poset-used-in-forcing-really)

### In terms of classifying toposes
 {#ReferencesInTermsOfClassifyingToposes}

On the interpretation of forcing via [[classifying toposes]] (see also at [[Kripke-Joyal semantics]]):

* [[William Lawvere]], p. 5 of: _Quantifiers and sheaves_, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 ([pdf](https://pdfs.semanticscholar.org/6630/846a00261a071b71e264e0f532e29cd9152f.pdf))

* [[Myles Tierney]], _Forcing Topologies and Classifying Topoi_, pages 211-219 in: [[Alex Heller]], [[Myles Tierney]] (eds.) _Algebra, Topology, and Category Theory: A Collection of Papers in Honor of Samuel Eilenberg_, Academic Press 1976  ([doi:10.1016/C2013-0-10841-0](https://doi.org/10.1016/C2013-0-10841-0))

* {#MakkaiReyes77} [[Michael Makkai]], [[Gonzalo Reyes]], _First Order Categorical Logic_,  Lecture Notes in Mathematics Volume 611, Springer 1977 ([doi:10.1007/BFb0066201](https://doi.org/10.1007/BFb0066201))

* {#Johnstone77} [[Peter Johnstone]], pages 270, 273, 348, 353 of: _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)

* [[Andreas Blass]], [[Andrej Ščedrov]], *Classifying topoi and finite forcing*, Journal of Pure and Applied Algebra **28** (1983) 111-140 [doi:10.1016/0022-4049(83)90085-3]

* {#Scedrov84} [[Andrej Ščedrov]], *Forcing and classifying topoi*, Memoirs of the American Mathematical Society (1984) [[AMS:memo-48-295](https://bookstore.ams.org/memo-48-295/5)]

* J. Benavides, _The logic of sheaves, sheaf forcing and the independence of the Continuum Hypothesis_ ([arXiv:1111.5854](https://arxiv.org/abs/1111.5854))

* {#Roberts15} [[David Roberts]], _Class forcing and topos theory_, talk at _[Topos at l'IHES](https://indico.math.cnrs.fr/event/747/)_ 2015 ([talk notes](https://doi.org/10.4225/55/5b2252e3092af),  [video recording](https://youtu.be/4AaSySq8-GQ))

* {#Shulman20} [[Mike Shulman]], _Answer to 'A better way to explain forcing?'_, ([MathOverflow](https://mathoverflow.net/a/370135/447))

On [Scedrov 84](#Scedrov84), [[Peter Johnstone]] writes in his review ([The Journal of Symbolic Logic Vol. 50, No. 3 (Sep., 1985), pp. 852-85]( http://www.jstor.org/stable/2274338 )):

>  Until recently, logicians could have been forgiven for treating topos theory as a closed book, whose applications to logic (however interesting they might appear from the outside) were accessible only to insiders well schooled in the mysteries of category theory $[...]$ $[$ Ščedrov's book makes $]$ a very clear case for the advantages of the topos-theoretic approach to forcing namely, that it provides a clear conceptual view of what generic structures really are, as well as a workable mean $[...]$ Although categorical ideas and terminology are inevitably present, they are handled so skilfully that the categoriphobic reader need have few qualms about them $[...]$ Ščedrov's work ought to go a long way towards demolishing the wall of incomprehension that has hitherto prevented many logicians from appreciating what topos theory has to offer their subject.

[[!redirects forcing]]


forgetful functor

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

#Contents#
* table of contents
{:toc}

## Idea

A **forgetful functor** is a [[functor]] which is defined by 'forgetting' some [[mathematical structure|structure]].  For example, the forgetful functor from [[Grp]] to [[Set]] forgets the group structure of a [[group]], remembering only the underlying [[set]].

In common parlance, the term 'forgetful functor' has no precise definition, being simply used whenever a functor is obviously defined by forgetting something.  Many forgetful functors of this sort have left or right [[adjoint functor|adjoints]] (and many are actually [[monadic functor|monadic]] or [[comonadic functor|comonadic]]), leading to the paradigmatic adjunction "free $\dashv$ forgetful."

On the other hand, from the perspective of [[stuff, structure, property]], _every_ functor is regarded as a forgetful functor and classified by how much it forgets (namely, stuff, structure, or properties).  From this perspective, the forgetful functor from $Grp$ to $Set$ forgets the structure of a group and the property of admitting a group structure, as usual; but its left adjoint (the [[free group]] functor) is *also* forgetful: if you identify $Set$ with the category of free groups with specified generators, then it forgets the structure of a set of free generators and the property of being free.

## Related concepts

* [[free construction]]

## References

See [[stuff, structure, property]] and also section 2.4 [page 15](http://arxiv.org/PS_cache/math/pdf/0608/0608420v2.pdf#page=15) of

* John C. Baez, Michael Shulman, _Lectures on n-Categories and Cohomology_ ([arXiv](http://arxiv.org/abs/math.CT/0608420)).

See also

* Wikipedia, *[Forgetful functor](https://en.wikipedia.org/wiki/Forgetful_functor)*

[[!redirects forgetful functors]]

[[!redirects underlying]]

[[!redirects forgetful]]


fork

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

#Contents#
* table of contents
{:toc}

## Idea

A **fork** is a [[diagram]] of the form

$$
A\underset{\quad e \quad}{\to}B\underoverset{\quad g \quad}{f}{\rightrightarrows}C
$$

such that $f e=g e$. An example of a special type of a fork is an [[equalizer]]. Another example is a _reflexive fork_, where $C = A$ and $f e = 1_A$.

A dual notion is also called a fork,
but some people distinguish forks and coforks.

## Related concepts

* [[parallel pair]]
* [[monadicity theorem]]

[[!redirects fork]]
[[!redirects forks]]
[[!redirects cofork]]
[[!redirects coforks]]
[[!redirects reflexive fork]]
[[!redirects reflexive forks]]


forking

The notion of __forking extension__ is introduced by Shelah, see

* [[Saharon Shelah]], _Classification theory and the number of non-isomorphic models_, revised edition, North-Holland (1990)

* Artem Chernikov, Itay Kaplan, Saharon Shelah, _On non-forking spectra_, [arxiv/1205.3101](http://arxiv.org/abs/1205.3101)

* Will Boney, Rami Grossberg, _Forking in short and tame AECs_, [arxiv/1306.6562](http://arxiv.org/abs/1306.6562)

category: model theory


form

> Substance **is** form. -- Eduardo Dubuc (from the Introduction to Kan Extensions in Enriched Category Theory).

A **form** in mathematics might refer to one of the following:

* [[bilinear form]],

* [[differential form]],

* [[modular form]],

* [[quadratic form]],

* a multiary generalisation of enriched natural transformations between enriched profunctors ([Street–Day](#StreetDay97)),

etc., with some special versions being notable enough to have entries of their own in the nLab, e.g., [[topological modular form]], [[Maurer-Cartan form]], [[Killing form]], etc.

"Form" has also of course philosophical meanings, which sometimes come to the fore when speaking of the relation of category theory and its abstracting principles to the rest of mathematics. For example, the quote of Dubuc at the top of this page is part of a larger remark: "The author hopes that it will not be totally incorrect to say that this paper is a testimony of two basic mathematico-philosophical principles. First, 'the relevant properties of mathematical objects are those which can be stated in terms of their abstract structure rather than in terms of the elements which the objects were thought to be made of (Lawvere)' coupled with 'the relevant facts of category theory hold because of formal interconnections between the concepts involved rather than because of their substantial content (which is none)'. This, because of that peculiar characteristic of the mind which leads every human being to the convinction that abstract ideas are real, can be pushed forward (extrapolated) into a simple purely philosophical principle, namely, 'substance *is* form'. Second, 'everything in mathematics that can be categorized is trivial (Freyd)' which should be understood. 'Category Theory is good ideas rather than complicated techniques'." See also [[nPOV]].

## References

* {#StreetDay97} [[Ross Street]] and [[Brian Day]]. _Monoidal bicategories and Hopf algebroids_. Adv. Math 129 (1997): 99-157.


form factor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[particle physics]] and specifically within [[effective field theories]] such as [[quantum hadrodynamics]], the _form factor_ of a given [[bound state]] is the [[ratio]] of its actual [[differential cross section]] for a given [[scattering]] process, by that of an actually pointlike particle

## Related concepts

* [[vector meson dominance]]

## References

### General

Review:

* {#Brambilla14} Brambilla et al., Section 3.2.5 of: _[[QCD and strongly coupled gauge theories - challenges and perspectives]]_, Eur Phys J C Part Fields. 2014; 74(10): 2981  ([arXiv:1404.3723](https://arxiv.org/abs/1404.3723), [doi:10.1140/epjc/s10052-014-2981-5](https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-014-2981-5))

* M. Kunkel, slide 6 of: _Dalitz Decays of Pseudo-Scalar Mesons_ ([[KunkelDalitzDecay.pdf:file]])

See also

* Wikipedia, _Form_factor_(quantum_field_theory)_

### In relation to flavour anomalies

[[hadron|Hadronic]] form factors in relation to the [[flavour anomalies]]:

* {#DGKV19} Sébastien Descotes-Genon, Alexander Khodjamirian, [[Javier Virto]], _Light-Cone Sum Rules for $B \to K \pi$ Form Factors and Applications to Rare Decays_ ([arXiv:1908.02267](https://arxiv.org/abs/1908.02267))

* {#DescotesGenon19b} Sebastien Descotes-Genon, _Light-cone sum rules for $B \to K \pi$ form factors and applications to rare decays_, talk at [bsll2019](https://indico.in2p3.fr/event/18646/) ([pdf](https://indico.in2p3.fr/event/18646/contributions/74316/attachments/54770/71918/slides_bslllyon.pdf), [[DescotesGenonBssl19.pdf:file]])

### Via holographic QCD

via [[holographic QCD]]:

for [[nucleons]]:

* Kiminad A. Mamo, [[Ismail Zahed]],  _Nucleon mass radii and distribution: Holographic QCD, Lattice QCD and GlueX data_ ([arXiv:2103.03186](https://arxiv.org/abs/2103.03186))

[[!redirects form factors]]


form factor (QFT)

## Idea

(...)

## Related concepts

* [[quantum field theory]]

* [[scattering amplitude]]

## References

* Wikipedia, _Form factor (Quantum Field Theory)_


form of an algebraic group

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[field]] $K$ and an [[algebraic group]] $G$ over $K$, and given a [[field extension]] $k \hookrightarrow K$, then a _$k$-form_ of $G$ is an algebraic group $G_k$ over $k$ such that its [[base change]] to $K$ yields $G$:

$$
  Spec(K)\times_{Spec (k)} G_k \simeq G
  \,.
$$

Typically one requires that $G$ is also defined over $k$, hence one considers $k$-forms of the $K$-ification of a given $k$-form.

## Examples

### Multiplicative group and circle group
 {#MultiplicativeAndCircleGroup}

Consider the inclusion $\mathbb{R} \hookrightarrow \mathbb{C}$ of the [[real numbers]] into the [[complex numbers]] and let $G = \mathbb{G}_m = \mathbb{C}^\times$ be the [[multiplicative group]] hence the [[group of units]] of $\mathbb{C}$.

Then one $\mathbb{R}$-form, hence a [[real form]] of $\mathbb{G}_m$ is given by $\mathbb{R}^\times$, and another is given by the [[circle group]] $S^1 = U(1)$.

To see this, realize $\mathbb{C}^\times$ as the group of 2x2 [[matrices]] with entries in the [[complex numbers]] which are diagonal and of unit [[determinant]]:

$$
  \left(
    \array{
       x & 0
       \\
       0 & y
    }
  \right)
  \;\;\; with
  \;
  x y = 1
  \,.
$$

The same prescription over the real numbers yields $\mathbb{R}^\times$ and exhibits it as a real form of $\mathbb{C}^\times$.

On the other hand, realize the circle group as the group of 2x2 real matrices of the form

$$
  \left(
    \array{
       x & y
       \\
       -y & x
    }
 \right)
  \;\;\;\;\;
  with
  \;
  x^2 + y^2 = 1
  \,.
$$

One checks that over the complex numbers this is isomorphic to the previous group of diagonal matrices, with the [[isomorphism]] being given by

$$
  \left(
    \array{
       x & y
       \\
       -y & x
    }
 \right)
  \mapsto
  \left(
    \array{
       x + i y & 0
       \\
       0 & x- i y
    }
  \right)
  \,.
$$

(e.g. [eom](#eom))

## Related concepts

* [[relation between compact Lie groups and reductive algebraic groups]]

## References

* David Herzog, _Forms of algebraic groups_, Proceedings of the American Mathematical Society Vol. 12, No. 4 (Aug., 1961), pp. 657-660 ([JSTOR](http://www.jstor.org/stable/2034264), [[BuzzardForms.pdf:file]])

* {#eom} [[eom]], _[Form of an algebraic group](http://www.encyclopediaofmath.org/index.php/Form_of_an_algebraic_group)_

[[!redirects forms of an algebraic group]]

[[!redirects form of a reductive algebraic group]]
[[!redirects forms of a reductive algebraic group]]


formal (infinity,1)-category theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

**Formal $(\infty,1)$-category theory** is to [[(infinity,1)-category theory|$(\infty,1)$-category theory]] what [[formal category theory]] is to [[category theory|1-category theory]], that is: a [[synthetic mathematics|synthetic]] approach to $(\infty,1)$-categories standing in relation to [[synthetic homotopy theory]] as $(\infty, 1)$-category theory stands to [[homotopy theory]]. Consequently, formal $(\infty,1)$-category theory has also been called **synthetic $(\infty,1)$-category theory**.

## Approaches

There have been two main styles of approaches to formal $(\infty,1)$-category theory: *categorical*, which are $(\infty,2)$-categorical in nature; and *type theoretic*, which propose [[deductive systems]] for reasoning about $(\infty,1)$-categories.

### Category theoretic approaches

* Following [Joyal 2008](#Joyal08) [p. 158](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=10), [Riehl & Verity 2013](#RiehlVerity13) observe that the [[homotopy 2-category|homotopy 2-category of $(\infty,1)$-categories]] $2Ho\big( Cat_{(\infty,1)}\big)$ retains much of the interesting structure of the $(\infty,2)$-category $Cat_{(\infty,1)}$. Thus, one may use the techniques of [[formal category theory]] to study certain aspects of the formal theory of $(\infty,1)$-categories. For instance, this includes the theory of [[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]] (which are automatically homotopy coherent), but not the theory of [[(infinity,1)-monads|$(\infty,1)$-monads]] (which are not). This is developed in [[Elements of ∞-Category Theory]] in the setting of a [[virtual equipment]].

* For [[presentable (infinity,1)-categories|presentable $(\infty,1)$-categories]], formal $(\infty,1)$-category theory is directly accessible from "abstract [[homotopy theory]]" in the sense of [[combinatorial model category|combinatorial]] [[model category]]-theory, in that $2Ho\big( PresCat_{(\infty,1)}\big) \,\simeq\,$ [[2Ho(CombModCat)]] ([Pavlov 2021](HoCombModCat#Pavlov21)).
- [Ruit 2023](#Ruit23) has introduced an $\infty$-double categorical generalisation of [[fibrant double categories]].

### Type theoretic approaches

* [Riehl & Shulman 2017](#RiehlShulman17) propose a type theory called *[[simplicial type theory]]*, which have [[categorical semantics]] in [[bisimplicial sets]]. This is further developed by [Buchholtz & Weinberger 2021](#BuchholtzWeinberger21).

* [Weaver & Licata 2020](#WeaverLicata20) propose a type theory, with semantics in [[bicubical sets]].

## Related pages

* [[formal category theory]]

* [[(infinity,2)-topos|$(\infty,2)$-topos]]

* [[internal category in homotopy type theory]]

* [[synthetic mathematics]], [[synthetic homotopy theory]]

* [[(infinity,1)-category theory|$(\infty,1)$-category theory]]

* [[directed homotopy type theory]]

* [[simplicial type theory]], [[cubical type theory]]

## References

### Category theoretic approaches

With ([[small (infinity,1)-category|small]]) [[(∞,1)-categories]] modeled as [[quasi-categories]], their [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category]] was considered first in

* {#Joyal08} [[André Joyal]], [p. 158](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf#page=10) in: _The theory of quasicategories and its applications_, lectures at _[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)_, CRM 2008 ([pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [[JoyalTheoryOfQuasiCategories.pdf:file]])

and then developed further (in the generality of [[homotopy 2-categories]] of [[∞-cosmoi]]) in:

* {#RiehlVerity13} [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 ([arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021))

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]],  _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))

* {#RiehlVerity21} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$

* {#Riehl21} [[Emily Riehl]], *The formal theory of ∞-categories*, talk at *[Categories and Companions Symposium June 8–12, 2021](http://web.science.mq.edu.au/groups/coact/seminar/CaCS2021/)* ([video](https://youtu.be/iX5Fxen3K-U))

In an [[(∞,1)-category theory|(∞,1)-category theoretic]] version of [[proarrow equipments]]:

* {#Ruit23} Jaco Ruit, _Formal category theory in ∞-equipments I_ ([arXiv:2308.03583](https://arxiv.org/abs/2308.03583))

### Type theoretic approaches

For synthetic (infinity, 1)-category theory in [[cubical type theory]] via [[bicubical sets]]:

* {#WeaverLicata20} [[Matthew Weaver]], [[Daniel Licata]], *A Constructive Model of Directed Univalence in Bicubical Sets*, in: *Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’20*, Association for Computing Machinery (2020) 915–928 [[doi:10.1145/3373718.3394794](https://doi.org/10.1145/3373718.3394794)]

Exposition in

* [[Matthew Weaver]], *A Constructive Model of Directed Univalence in Bicubical Sets*, talk at HoTTEST (April 2020) [[pdf](https://www.cs.princeton.edu/~mzweaver/pdfs/HoTTEST.pdf), [video](https://youtu.be/kkfNjqSx4Nw)]

For synthetic (infinity, 1)-category theory in [[simplicial type theory]]:

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$

* {#BuchholtzWeinberger21} [[Ulrik Buchholtz]], [[Jonathan Weinberger]], *Synthetic fibered $(\infty,1)$-category theory* $[$[arXiv:2105.01724](https://arxiv.org/abs/2105.01724), [talk slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Weinberger-2022-01-20-HoTTEST.pdf)$]$

* [[Jonathan Weinberger]], *A Synthetic Perspective on $(\infty,1)$-Category Theory: Fibrational and Semantic Aspects* $[$[arXiv:2202.13132](https://arxiv.org/abs/2202.13132)$]$

* Fredrik Bakke, *Segal Spaces in Homotopy Type Theory*. Master thesis $[$[no.ntnu:inspera:99217069:14871483	](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/2995704)$]$

* César Bardomiano Martínez, *Limits and colimits of synthetic $\infty$-categories* $[$[arXiv:2202.12386](https://arxiv.org/abs/2202.12386)$]$

* [[Jonathan Weinberger]], *Strict stability of extension types* $[$[arXiv:2203.07194](https://arxiv.org/abs/2203.07194)$]$

* [[Jonathan Weinberger]], *Two-sided cartesian fibrations of synthetic $(\infty,1)$-categories* $[$[arXiv:2204.00938](https://arxiv.org/abs/2204.00938)$]$

* [[Jonathan Weinberger]], *Internal sums for synthetic fibered $(\infty,1)$-categories* $[$[arXiv:2205.00386](https://arxiv.org/abs/2205.00386)$]$

A talk on [[synthetic (infinity,1)-category theory]] in [[simplicial type theory]] and [[infinity-cosmos | infinity-cosmos theory]]:

* [[Emily Riehl]], *The synthetic theory of infinity-categories vs the synthetic theory of infinity-categories*, [[Homotopy Type Theory Electronic Seminar Talks]], 1 March 2018, ([video](https://www.youtube.com/watch?v=ge-9m1SsEmc), [slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Riehl-2018-03-01-HoTTEST.pdf))

[[!redirects formal (∞,1)-category theory]]
[[!redirects formal infinity1-category theory]]

[[!redirects synthetic (infinity,1)-category theory]]
[[!redirects synthetic (∞,1)-category theory]]
[[!redirects synthetic infinity1-category theory]]


formal adjoint differential operator

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

#Contents#
* table of contents
{:toc}

## Definition

Over a [[smooth manifold]] $\Sigma$ of [[dimension]] $p+1$, let $E \overset{fb}{\to} \Sigma$ be a [[smooth vector bundle]] and $\tilde E^\ast \coloneqq E^\ast \otimes_{\Sigma} \wedge_\Sigma^{p+1} T^\ast \Sigma$ the [[tensor product of vector bundles]] of the [[dual vector bundle]] with the [[differential n-form|differential (p+1)-form bundle]].

+-- {: .num_defn #FormallyAdjointDifferentialOperators}
###### Definition
**(formally adjoint differential operators)**

Two [[differential operators]]

$$
  P, P^\ast
   \;\colon\;
  \Gamma_\Sigma(E)
    \longrightarrow
  \Gamma_\Sigma(\tilde E^\ast)
$$

are called _formally adjoint_ if there exists a [[bilinear map|bilinear]] [[differential operator]]

$$
  \label{FormallyAdjointDifferentialOperatorWitness}
  K \;\colon\;
  \Gamma_\Sigma(E) \otimes \Gamma_\Sigma(E)
    \longrightarrow
   \Gamma_\Sigma(\wedge^{p} T^\ast \Sigma)
$$

such that for all $\Phi_1, \Phi_2 \in \Gamma_\Sigma(E)$ we have

$$
  P(\Phi_1) \cdot \Phi_2
  -
  \Phi_1 \cdot P^\ast(\Phi_2)
   \;=\;
  d K(\Phi_1, \Phi_2)
$$

This implies by [[Stokes' theorem]], in the case of [[compact support]], that under an [[integral]] $P$ and $P^\ast$ are related via [[integration by parts]].

=--

([Khavkine 14, def. 2.4](#Khavkine14))

See also ([Vinogradov-Krasilshchik 99, chapter 5, §2.3](#VinogradovKrasilshchik99))

## Examples
 {#Examples}

+-- {: .num_example #FormallySelfAdjointKleinGordonOperator}
###### Example
**([[Klein-Gordon operator]] is [[formally self-adjoint differential operator]])**

Let $\Sigma = \mathbb{R}^{p,1}$ be [[Minkowski spacetime]] with [[Minkowski metric]] $\eta$ and let $E \coloneqq \Sigma \times \mathbb{R}$ be the [[trivial line bundle]]. The canonical [[volume form]] $dvol_\Sigma$ induces an [[isomorphism]] $\tilde E^\ast \simeq E$.

Consider then the [[Klein-Gordon operator]]

$$
  (\Box - m^2)
    \;\colon\;
  \Gamma_\Sigma(\Sigma \times \mathbb{R})
     \longrightarrow
  \Gamma_\Sigma(\Sigma \times \mathbb{R}) \otimes \langle dvol_\Sigma\rangle
  \,.
$$

This is its own formal adjoint (def. \ref{FormallyAdjointDifferentialOperators}) witnessed by the bilinear differential operator (eq:FormallyAdjointDifferentialOperatorWitness) given by

$$
  K(\Phi_1, \Phi_2)
  \;\coloneqq\;
  \left(
    \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
    -
    \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
  \right)
  \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
  \,.
$$

=--

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

$$
  \begin{aligned}
    d K(\Phi_1, \Phi_2)
    & =
    d
    \left(
      \frac{\partial \Phi_1}{\partial x^\mu} \Phi_2
      -
      \Phi_1 \frac{\partial \Phi_2}{\partial x^\mu}
    \right)
    \eta^{\mu \nu}\iota_{\partial_\nu} dvol_\Sigma
    \\
    &=
    \left(
    \left(
      \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
      +
      \eta^{\mu \nu} \frac{\partial \Phi_1}{\partial x^\mu} \frac{\partial \Phi_2}{\partial x^\nu}
    \right)
    -
    \left(
      \eta^{\mu \nu}
      \frac{\partial \Phi_1}{\partial x^\nu} \frac{\partial \Phi_2}{\partial x^\mu}
      +
      \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
    \right)
    \right)
    dvol_\Sigma
    \\
    & =
    \left(
      \eta^{\mu \nu}\frac{\partial^2 \Phi_1}{\partial x^\mu \partial x^\nu} \Phi_2
      -
      \Phi_1 \eta^{\mu \nu} \frac{\partial^2 \Phi_2}{\partial x^\nu \partial x^\mu}
    \right)
    dvol_\Sigma
    \\
    & =
    \Box(\Phi_1) \Phi_2 - \Phi_1 \Box (\Phi_2)
  \end{aligned}
$$

=--

+-- {: .num_example #DiracOperatorOnDiracSpinorsIsFormallySelfAdjointDifferentialOperator}
###### Example
**([[Dirac operator]] on [[Dirac spinors]] is [[formally self-adjoint differential operator|formally anti-self adjoint]])**

The [[Dirac operator]] on [[Dirac spinors]] is a [[formally self-adjoint differential operator|formally anti-self adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}):

$$
  D^\ast = - D
  \,.
$$

=--

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

In brief, the point is that when the Clifford generators themselves are formally self-adjoint, as they are ([this Prop.](geometry+of+physics+--+supersymmetry#CliffordRepresentationIsDiracSelfConjugate)) with respect to the [[Dirac conjugate]] ([this Def. ](geometry+of+physics+--+supersymmetry#DiracConjugate)), then (only) the single [[derivative]] operator picks up a sign under passing to adjoints (i.e. under [[integration by parts]]).

In more formal detail:

Regard the Dirac operator as taking values in the [[dual vector bundle|dual]] [[spin bundle]] by using the [[Dirac conjugate]] $\overline{(-)}$:

$$
  \array{
    \Gamma_\Sigma(\Sigma \times S)
      &\overset{D}{\longrightarrow}&
    \Gamma_\Sigma(\Sigma \times S^\ast)
    \\
   \Psi &\mapsto&
   \overline{(-)} \gamma^\mu \partial_\mu  \Psi
  }
$$

Then we need to show that there is $K(-,-)$ such that for all [[pairs]] of [[spinor]] [[sections]] $\Psi_1, \Psi_2$ we have

$$
  \overline{\Psi_2}\gamma^\mu (\partial_\mu \Psi_1)
  -
  \overline{\Psi_1}\gamma^\mu (-\partial_\mu \Psi_2)
  \;=\;
  d K(\psi_1, \psi_2)
  \,.
$$

But the spinor-to-vector pairing is symmetric (see at _[[spin representation]]_),  hence this is equivalent to

$$
  \overline{\partial_\mu \Psi_1}\gamma^\mu \Psi_2
  +
  \overline{\Psi_1}\gamma^\mu (\partial_\mu \Psi_2)
  \;=\;
  d K(\psi_1, \psi_2)
  \,.
$$

By the [[product law]] of [[differentiation]], this is solved, for all $\Psi_1, \Psi_2$, by

$$
  K(\Psi_1, \Psi_2)
    \;\coloneqq\;
  \left( \overline{\Psi_1} \gamma^\mu \Psi_2\right)
  \,
  \iota_{\partial_\mu} dvol
  \,.
$$

=--

## Related concepts

* [[generalized solution of a differential equation]]

* [[Green hyperbolic differential operator]]

## References

* [[Peter Olver]], chapter 5.3, around p. 328-330 of _Applications of Lie groups to differential equations_, Springer; _Equivalence, invariants, and symmetry_, Cambridge Univ. Press 1995.

* {#VinogradovKrasilshchik99} [[Alexandre Vinogradov]], I. S. Krasilshchik (eds.) _Symmetries and Conservation Laws for Differential Equations of Mathematical Physics_, vol. 182 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1999. ([pdf](ftp://softbank.iust.ac.ir/MathBooks/V/Vinogradov%20-%20Symmetries%20and%20Conservation%20Laws%20for%20Differential%20equations%20of%20mathematical%20physics.pdf))

* {#Khavkine14} [[Igor Khavkine]], _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv:1402.1282](https://arxiv.org/abs/1402.1282))

[[!redirects formal adjoint differential operators]]

[[!redirects formally adjoint differential operator]]
[[!redirects formally adjoint differential operators]]

[[!redirects formally self-adjoint differential operator]]
[[!redirects formally self-adjoint differential operators]]


formal Brauer group

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

#Contents#
* table of contents
{:toc}

## Idea

A [[formal group]]-analog of the [[Brauer group]]. The special case of the concept of [[Artin-Mazur formal group]] for $n = 2$. The $n = 1$-analog is the _[[formal Picard group]]_.

## Properties

### Relation to $K3$-cohomology

In higher dimensional analogy of how the [[formal Picard group]] of an [[elliptic curve]] gives the [[formal group]] of an [[elliptic spectrum]] representing an [[elliptic cohomology theory]], so the formal Brauer group of a [[K3 surface]] gives the formal group of an [[complex oriented cohomology theory]] given by a [[spectrum]] hence called a _[[K3-spectrum]]_ [[Brown representability theorem|representing]] _[[K3-cohomology]]_.

## Related concepts

* [[Brauer stack]]

[[!include moduli of higher lines -- table]]

## References

The original account of the construction of formal Picard groups is

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  [numdam](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663)

Modern reviews include

* {#Szymik} [[Markus Szymik]], section 3 of _K3 spectra_ ([pdf](http://www.math.ku.dk/~xvd217/szymik.K3spectra.pdf))

* {#Liedtke14} Christian Liedtke, around p. 40 of _Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem_ ([arXiv.1403.2538](http://arxiv.org/abs/1403.2538))

[[!redirects formal Brauer groups]]


formal category theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

There are many flavours of category:

- ordinary [[categories]]
- [[enriched categories]]
- [[internal categories]]
- [[fibred categories]]
- [[indexed categories]]
- [[monoidal categories]]

and so on. In each flavour of category theory, one finds essentially the same definitions and theorems:

- [[presheaves]] and the [[Yoneda lemma]]

- [[adjunctions]] and [[adjoint functor theorems]]

- [[monads]] and [[monadicity theorems]]

- theories of [[accessible category|accessible]] and [[locally presentable categories]] and [[Gabriel--Ulmer duality|duality theorems]]

and so on. Rather than define these concepts and prove these theorems in each of the different flavours of category theory independently (as has been done traditionally), one would rather be able to define these concepts and prove these theorems abstractly, and then specialise to each of these flavours, obtaining for instance the statements for enriched category theory as a corollary of the more abstract version. This is the motivation for formal category theory.

**Formal category theory** may be thought of as applying the philosophy of category theory to category theory itself. Typically, this takes the form of applying 2-dimensional category theories (e.g. [[2-category|2-category theory]] or [[double category|double category theory]]) to study 1-dimensional category theory. This may be viewed as a [[synthetic mathematics|synthetic]] approach to category theory.

This may be seen as a [[categorification]] of the approach of [[structural set theory]], which is a [[synthetic mathematics|synthetic]] approach to [[set theory]]. (In fact, just as a [[topos]] is intended as an axiomatisation of the logical nature of the archetypical topos [[Set]], [Street & Walters](Yoneda+structure#SW78) introduced [[Yoneda structures]], one of the earliest approaches to formal category theory, as an axiomatisation of the "logical" nature of the archetypical [[2-topos]] [[Cat]].)

## Approaches

An important question is: what is the appropriate setting in which to study the formal theory of categories? There is not one obvious answer, and several approaches have been proposed in the literature.

Roughly speaking, there are four main approaches (listed in chronological order of the first time they were proposed).

1. [[2-categories]] with [[property-like structure]]

   (cf. [Gray 1974](#Gray74))

1. [[Yoneda structures]]

   (cf. [Street & Walters 1978](Yoneda+structure#SW78))

1. [[proarrow equipments]]/[[framed bicategories]]/(augmented) [[virtual equipments]]

   (cf. [Wood 1982](2-category+equipped+with+proarrows#Wood82), [1985](2-category+equipped+with+proarrows#Wood85); [Shulman 2008](framed+bicategory#Shulman08); [Cruttwell & Shulman 2010](generalized+multicategory#CruttwellShulman10); [Koudenburg 2022](#Koudenburg22))

1. [[lax-idempotent 2-monads]]

   (cf. [Bunge & Funk 1999](lax-idempotent+2-monad#BF99), Trimble's [epistemologies](https://ncatlab.org/toddtrimble/published/Epistemologies))

The relationship between the approaches, for the most part, is not laid out clearly in the literature, but a basic intuition is that:

1. [[2-categories]] may be equipped with various [[property-like structure]] possessed by the 2-category [[Cat]], e.g. [[2-limits]] and [[2-colimits]], [[exponentials]], etc.

1. Yoneda structures are based on formalising the [[category of presheaves|presheaf construction]].

1. Proarrow equipments and related structures are based on formalising [[distributors]]/[[profunctors]].

1. Lax-idempotent pseudomonads (also called **KZ-doctrines**) are based on formalising the [[free cocompletion]]-construction.

The first, purely 2-categorical, approach is unsufficient to capture [[enriched categories]] (though captures many aspects of ordinary and internal category theory), and is therefore not viable as a general approach to formal category theory.

On the other hand, in nice settings, e.g. in [[enriched category theory]] for a well-behaved [[base of enrichment]] (eg. a [[Bénabou cosmos]]), approaches (2 -- 4) are all viable, and are distinguished primarily by their methodology. However, in weaker settings, some approaches are no longer viable.

For instance, one may not in general have a [[presheaf category]]-construction or [[free cocompletion]], and so Yoneda structures and [[lax-idempotent pseudomonads]] are not applicable in general. Without enough [[coequalisers]] in the base, [[distributors]] do not compose, and so proarrow equipments and fibrant double categories are not applicable. For instance, for a general [[monoidal category]] $V$, only the formalisms of [[virtual equipments]] and [[augmented virtual equipments]] capture the structure of $V$-[[enriched categories]]. These are thus the most general approaches to formal category theory. Here it is possible to formalise [[presheaves]] as in [Koudenburg 2022](#Koudenburg22), thus recovering [[Yoneda structure]]; as well as free cocompletions (though this had not been developed in the literature): these approaches consequently strictly subsume the others.

## Theorems in formal category theory

Below, we list some fundamental theorems in category theory that have been formulated in formal category theory.

* General results about [[weighted limits]] and [[weighted colimits]] are developed in [Street & Walters 1978](Yoneda+structure#SW78) (for [[Yoneda structures]]); [Wood 1982](2-category+equipped+with+proarrows#Wood82) (for [[proarrow equipments]]); [Arkor & McDermott 2023a](#AM23b) (for [[virtual equipments]]).

* [[adjoint functor theorem|Adjoint functor theorems]] are proven in Proposition 21 of [Street & Walters 1978](Yoneda+structure#SW78); Corollary 10 of [Wood 1982](2-category+equipped+with+proarrows#Wood82); Theorem 2.16 of [Arkor, Di Liberti, Loregian](#ALL23).

* An analogue of the [[bijective-on-objects functor|bijective-on-objects]]/[[fully faithful functor|fully faithful]] factorisation system is established in [Wood 1985](2-category+equipped+with+proarrows#Wood85).

* General results about [[monads]] are developed in [Street & Walters 1978](Yoneda+structure#SW78) (for [[Yoneda structures]]); [Wood 1985](2-category+equipped+with+proarrows#Wood85) (for [[proarrow equipments]]); [Arkor & McDermott 2023a](#AM23b) in the additional generality of [[relative monads]] (for [[virtual equipments]]).

* The theory of [[total categories]] is developed in [Street & Walters 1978](Yoneda+structure#SW78) (for [[Yoneda structures]]); [Koudenburg 2022](#Koudenburg22) (for [[augmented virtual equipments]]).

* A general duality theorem subsuming [[Gabriel–Ulmer duality]] (and many others) is proven in [Di Liberti & Loregian 2023](#DL18) (for [[lax-idempotent pseudomonads]]).

* General results about [[presheaves]] are developed in [Koudenburg 2022](#Koudenburg22) (for [[augmented virtual equipments]]).

* A ([[relative monad|relative]]) [[monadicity theorem]] is proven in [Arkor & McDermott 2023a](#AM23b) (for [[virtual equipments]]).

For instance, one motivation for the development of this theory in the setting of (augmented) [[virtual equipments]] is to develop [[enriched category theory]] without the assumptions taken, for instance, by [[Max Kelly|Kelly]] in [Basic concepts of enriched category theory](enriched+category#Kelly82). This therefore gives a development of enriched category theory for categories enriched in a nonsymmetric [[monoidal category]], a [[bicategory]], or even a [[virtual double category]].

## Intramural and extramural formal category theory

There are two distinct aspects to the [[category theory|theory of categories]]. One aspect is concerned with the structure of an individual category, with regards to its [[objects]] and [[morphisms]]. For instance, the study of [[monomorphisms]] and [[epimorphisms]], [[limits]] and [[colimits]], and so on. We shall refer to this aspect as **intramural category theory**.

The other aspect is concerned with the relation between different categories, constructions on categories (e.g. [[product categories]], [[functor categories]], [[adjoint functors]]), and so on. In essence, this is the theory of the [[2-category]] [[Cat]]. It is important to note that this latter aspect is not the more general theory of objects in an arbitrary [[2-category]] (which would be the study of formal category theory). We shall refer to this aspect as **extramural category theory**.

In the same way, there are both intramural and extramural aspects to formal category theory. **Intramural formal category theory** is what we have meant by "formal category theory" above, i.e. developing category theory inside some 2-dimensional structure (e.g. a [[Yoneda structure]] or [[proarrow equipment]]). On the other hand, **extramural category theory** is the theory of such 2-dimensional structures, e.g. the theory of Yoneda structures, or the theory of proarrow equipments. Note that, just as extramural category theory is distinct from formal category theory, so too is extramural formal category theory distinct from what might be called "formal 2-category theory", "formal 2-dimensional category theory", or "formal formal category theory".

Extramural formal category theory is motivated by formal category theory. For instance, an important construction in extramural formal category theory is the [[bimodule|monads and bimodules construction]], which constructs a new [[proarrow equipment]] $Mod(X)$ from an existing [[proarrow equipment]] (there are analogues for other approaches to formal category theory). This provides a family of new examples of formal category theories, in which one can perform intramural formal category theory. For instance, proarrow equipments of the form $Mod(X)$ are very well behaved, which equips them with a rich intramural category theory (as studied in [Wood 1985](2-category+equipped+with+proarrows#Wood85)): for instance, the existence of a factorisation system, and the existence of [[Kleisli objects]] and [[Eilenberg–Moore objects]].

## Related pages

* [[enriched category theory]]

* [[ETCC]]

* [[2-topos]]

* [[formal (infinity,1)-category theory]]

## References

An axiomatization of (just) the ([[very large category|very large]]) [[1-category]] of all categories, but fully in [[formal logic]] is the *[[elementary theory of the category of categories]]* (ETCC) due to:

* {#Law66} [[William Lawvere]], *The Category of Categories as a Foundation for Mathematics*, pp. 1-20 in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

The undertaking of laying these foundations instead for the [[2-category]] of all categories is famously associated with:

* {#Gray74} [[John Gray]], *[[Adjointness for 2-Categories|Formal category theory: adjointness for $2$-categories]]*, Lecture Notes in Mathematics, **391**, Springer 1974 ([doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280))

Review and further discussion, including on the relationship between the different approaches:

* [[Ivan Di Liberti]], [[Simon Henry]], [[Mike Lieberman]], [[Fosco Loregian]], _Formal category theory_, course notes (2017) [[pdf](https://tetrapharmakon.github.io/stuff/course_muni-formal.pdf), [[DLHLL-FormalCategoryTheory.pdf:file]]]

* [[Ivan Di Liberti]], [[Fosco Loregian]], _On the unicity of formal category theories_ [[arXiv:1901.01594](https://arxiv.org/abs/1901.01594)]

The relation between [[lax-idempotent pseudomonads]] and [[Yoneda structures]] is due to:

* Charles Walker, *Yoneda Structures and KZ Doctrines*, [arxiv](https://arxiv.org/abs/1703.08693)
 {#WalkerYS}

Discussion in [[double category]]-theory:

* [[Michael Shulman]], _Framed bicategories and monoidal fibrations_.

* {#CruttwellShulman10} [[Geoff Cruttwell]] and [[Mike Shulman]], _A unified framework for generalized multicategories_, Theory Appl. Categ. **24** 21 (2010) 580-655 [[arxiv/0907.2460](http://arxiv.org/abs/0907.2460), [TAC:24-21](http://www.tac.mta.ca/tac/volumes/24/21/24-21abs.html)]

* {#Koudenburg15} [[Seerp Roald Koudenburg]], *A double-dimensional approach to formal category theory* ([arXiv:1511.04070](http://arxiv.org/abs/1511.04070)) 2015.

* {#Koudenburg19} [[Seerp Roald Koudenburg]], *Augmented virtual double categories*, Theory and Applications of Categories, Vol. 35, 2020, No. 10, pp 261-325 ([arXiv:1910.11189](https://arxiv.org/abs/1910.11189), [tac:35-10](http://www.tac.mta.ca/tac/volumes/35/10/35-10abs.html))

* {#Koudenburg22} [[Seerp Roald Koudenburg]], *Formal category theory in augmented virtual double categories* [[arXiv:2205.04890](https://arxiv.org/abs/2205.04890)]

  > (On [[augmented virtual double categories]], a expanded version of Sec. 1-3 of [the previous reference](#Koudenburg15).)

A domain-specific [[type theory]] for formal category theory:

* [[Max S. New]], [[Daniel Licata]], _A Formal Logic for Formal Category Theory_ ([arXiv:2210.08663](https://arxiv.org/abs/2210.08663))

On the intuition for [[Yoneda structures]] as [[2-toposes]]:

* [[Ross Street]], *An Australian conspectus of higher category theory*, talk at Institute for Mathematics and its Applications Summer Program: *$n$-Categories: Foundations and Applications* at the University of Minnesota (Minneapolis, 7–18 June 2004), in: *[[Towards Higher Categories]]*, The IMA Volumes in Mathematics and its Applications **152**, Springer (2010) 237-264 [[pdf](http://www.math.uchicago.edu/~may/IMA/Street.pdf), [[Street-Conspectus.pdf:file]], [doi:10.1007/978-1-4419-1524-5](https://link.springer.com/book/10.1007/978-1-4419-1524-5)]

Papers developing general theorems in (intramural) formal category theory:

* {#DL23} [[Ivan Di Liberti]], [[Fosco Loregian]], _Accessibility and Presentability in 2-Categories_, Journal of Pure and Applied Algebra 227.1 (2023)([arXiv:1804.08710](https://arxiv.org/abs/1804.08710))

* {#ALL23} [[Nathanael Arkor]], [[Ivan Di Liberti]], [[Fosco Loregian]], _Adjoint functor theorems for lax-idempotent pseudomonads_, [[arXiv:2306.10389](https://arxiv.org/abs/2306.10389)]

* {#AM23a} [[Nathanael Arkor]], [[Dylan McDermott]], _The formal theory of relative monads_ (2023) [[arXiv:2302.14014](https://arxiv.org/abs/2302.14014)]

* {#AM23b} [[Nathanael Arkor]], [[Dylan McDermott]], _Relative monadicity_ (2023) [[arXiv:2305.10405](https://arxiv.org/abs/2305.10405)]

Papers developing general theorems in (extramural) formal category theory:

* [[Ross Street]], "Fibrations in bicategories", [Numdam](http://www.numdam.org/item/?id=CTGDC_1980__21_2_111_0), and [correction](http://www.ams.org/mathscinet-getitem?mr=903151).

* [[Aurelio Carboni]] and [[Scott Johnson]] and [[Ross Street]] and [[Dominic Verity]], "Modulated bicategories" (MB) [MR](http://www.ams.org/mathscinet-getitem?mr=1285544).

* [[Bob Rosebrugh]] and [[Richard Wood]], "Proarrows and cofibrations" (PC), [MR](http://www.ams.org/mathscinet-getitem?mr=961365)

* [[Bob Rosebrugh]] and [[Richard Wood]], "Gamuts and cofibrations". Cahiers de topologie et geometrie differentielle categoriques 31.3 (1990): 197-211. [Numdam](http://www.numdam.org/item/CTGDC_1990__31_3_197_0/)

* [[Bob Rosebrugh]] and [[Richard Wood]], "Pullback preserving functors". Journal of Pure and Applied Algebra 73.1 (1991): 73-90.

* [[Aurelio Carboni]], Scott Johnson, [[Ross Street]], and [[Dominic Verity]], _Modulated bicategories_

* Renato Betti, [[Aurelio Carboni]], [[Ross Street]], and Robert Walters, _Variation through enrichment_

* [[Richard Garner]] and [[Mike Shulman]], *Enriched categories as a free cocompletion*, [arXiv](http://arxiv.org/abs/1301.3191)

Another approach, related to Yoneda structures:

* [[Michał Przybyłek]], _Analysis and construction of logical systems: a category-theoretic approach_ (2014), PhD thesis ([pdf](https://www.mimuw.edu.pl/~mrp/phd_mrp.pdf)).


formal cohesive infinity-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _[[cohesive ∞-groupoid]]_ is **formal** if all its spaces of [[k-morphism]] have [[infinitesimal object|infinitesimal extension]]. If they have first order infinitesimal extension, the formal cohesive $\infty$-groupoid is an [[∞-Lie algebroid]].

## Definition

For the moment see the section _Formal cohesive ∞-groupoids_ at _[[cohesive (∞,1)-topos]]_ .

## Examples

### Special cases

A [[1-truncated]] formal cohesive $\infty$-groupoid is a [[formal groupoid]]. If the space of objects is the terminal object, this is the [[delooping]] of a [[formal group]].

More generally, if the space of objects of a formal cohesvive $\infty$-groupoid is the terminal object, it is the delooping of a **formal [[∞-group]]** .

[[!redirects formal cohesive ∞-groupoid]]

[[!redirects formal cohesive ∞-group]]
[[!redirects formal cohesive infinity-group]]

[[!redirects formal cohesive ∞-groupoids]]
[[!redirects formal cohesive infinity-groupoids]]

[[!redirects formal cohesive ∞-groups]]
[[!redirects formal cohesive infinity-groups]]

[[!redirects formal ∞-groupoid]]
[[!redirects formal infinity-groupoid]]

[[!redirects formal ∞-group]]
[[!redirects formal infinity-group]]

[[!redirects formal ∞-groupoids]]
[[!redirects formal infinity-groupoids]]

[[!redirects formal ∞-groups]]
[[!redirects formal infinity-groups]]


formal completion

## Definitions

* [[formal completion of a ring]]

* [[formal completion of a module]]

* [[p-completion]]

* [[formal scheme]]

* [[nilpotent completion]]

## Related concepts

* [[formal neighbourhood of the diagonal]]

[[!redirects formally complete]]

[[!redirects formal completion]]
[[!redirects formal completions]]

[[!redirects adic completion]]
[[!redirects adic completions]]