higher homotopy van Kampen theorem

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#### Homotopy theory
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#Contents#
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## Idea

One form of a  _higher homotopy van Kampen theorem_ is a theorem that asserts that the [[homotopy type]] of a [[topological space]] can be computed by a suitable [[colimit]] or [[homotopy colimit]] over homotopy types of its pieces. Another form which allows specific computation deals with spaces with certain kinds of structure, for example filtered spaces or $n$-cubes of spaces.

This generalizes the [[van Kampen theorem]], which only deals with the underlying 1-type (the fundamental groupoid).

## Statement

### For topological spaces {#ForTopSpaces}

#### General

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

Let $X$ be a [[topological space]], write $Op(X)$ for its [[category of open subsets]] and let

$$
  \chi : C \to Op(X)
$$

be a [[functor]] out of a [[small category]] $C$ such that

* for each point $x\in X$ the [[full subcategory]] $C_x$ of objects $c$ such that $\chi(x)$ contains $x$ has a [[weak homotopy equivalence|weakly]] [[contractible]] [[nerve]].

Then:

the canonical morphism in [[sSet]] out of the [[colimit]]

$$
   {\lim_\to} Sing \circ \chi \to Sing(X)
$$

into the [[singular simplicial complex]] of $X$ exhibits $Sing(X)$ as the [[homotopy colimit]]  $hocolim Sing \circ \chi$.

=--

This is theorem A.1.1 in ([Lurie](#Lurie)).

#### Strict version

The following is a version of the above general statement restricted to a [[strict ∞-groupoid]]-version of the [[fundamental ∞-groupoid]] and applicable for [[topological spaces]] that are equipped with the extra structure  of a [[filtered topological space]].

Notice that these strict $\infty$-groupoids are equivalent to [[crossed complex]]es.

Suppose $X_*$ is a [[filtered space]] and $X$ is the union of the interiors of sets $U^i$, $i \in I$. Let $U^i_*$ be the filtered space given by the intersections $U^i \cap X_n$ for $n \geq 0$. If $d=(i,j) \in I^2$ we write $U^d$ for $U^i \cap U^j$. We then have a coequaliser diagram of filtered spaces

$$\bigsqcup_{d \in I^2} U^d_* \rightrightarrows ^a_b \bigsqcup _{i \in I} U^i_* \to ^c X_*.$$

+-- {: .un_theorem}
###### Strict Higher van Kampen Theorem

If the filtered spaces $U^f_*$ are [[connected filtered space]]s for all finite intersections $U^f_*$ of the filtered spaces $U^i_*$, then

1. (Conn) The filtered space $X_*$ is connected; and

1. (Iso) The fundamental [[crossed complex]] functor $\Pi$ takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.

A full account is given in ([Brown-Higgins-Sivera](#NAT) and the methodology is discussed in ([Brown](#RBrown)).

=--

**Remarks**

* Note that because $\Pi$ uses groupoids, it obviously takes disjoint unions $\bigsqcup$ of filtered spaces into disjoint unions (= coproducts) $\bigsqcup$ of crossed complexes.

* The proof of the theorem is not direct but goes via the fundamental cubical $\omega$-groupoid with connections of the filtered spaces, as that context allows the notions of _algebraic inverse to subdivision_ and of _commutative cube_. However the proof is a direct generalisation of a proof for the [[van Kampen theorem]] for the [[fundamental groupoid]].

* Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of _singular homology theory_ or of _simplicial approximation_. Also included is a version of the "small simplex theorem", see Theorem 10.4.20 of ([Brown-Higgins-Sivera](#NAT)).

### For objects in a cohesive $(\infty,1)$-topos

In a [[cohesive (∞,1)-topos]] (already in a [[locally ∞-connected (∞,1)-topos]]) higher van Kampen theorems hold in great generality.

See the section cohesive (∞,1)-topos -- van Kampen theorem.

In particular for the cohesive $(\infty,1)$-topos [[?TopGrpd]] of [[topological ∞-groupoid]]s this reproduces the [topological higher van Kampen theorem](#ForTopSpaces) discussed above.

## Examples

Here is one application in dimension 2 not easily obtainable by traditional [[algebraic topology]].

Let $0 \to P \to Q \to R \to 0$ be an [[exact sequence]] of [[abelian groups]]. Let $X$  be the [[mapping cone]] of the induced map $K(P,1) \to K(Q,1)$ of [[Eilenberg-Mac Lane space]]s. Then a [[crossed module]] representing the [[homotopy 2-type]] of $X$ is $\mu: C \to Q$ where $C$ is abelian and is the direct sum $\oplus_{r \in R} P^r$  of copies of $P$ one for each $r \in R$ and the action of $Q$ is via $R$ and permutes the copies by $(p,r)^s=(p,r+s)$. Similar examples for $P,Q,R$ nonabelian are do-able,   more complicated, and certainly **not** obtainable by traditional methods.

## References

The version for topological spaces and the [[fundamental infinity-groupoid]] functor is discussed in  Appendix A of

* [[Jacob Lurie]], _[[Ek-Algebras]]_
 {#Lurie}

The version for filtered topological spaces and the strict homotopy $\infty$-groupoid functor is discussed in

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_ (#NAT)

while the general methodology is discussed in

* [[Ronnie Brown]] "Modelling and computing homotopy types: I" to appear in 2017 in a special issue of Indagationes Math in honour of L.E.J. Brouwer. (https://arxiv.org/abs/1610.07421) (#RBrown)

One area of application of work of Brown and Loday is to a nonabelian tensor product of groups, see:

* [[Ronnie Brown]] Bibliography on the nonabelian tensor product. (http://www.groupoids.org.uk/nonabtens.html) (#tens)

[[!redirects Higher Homotopy van Kampen Theorem]]
[[!redirects higher homotopy van Kampen Theorem]]

[[!redirects higher van Kampen theorem]]


Higher Homotopy van Kampen Theorem > history

< [[Higher Homotopy van Kampen Theorem]]
[[!redirects Higher Homotopy van Kampen Theorem -- history]]


higher inductive type

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### Context
#### Type theory
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[[!include type theory - contents]]
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#### Induction
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[[!include induction - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
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## Idea

*Higher inductive types* (HITs) are a generalization of [[inductive types]] which allow the constructors to produce, not just points of the type being defined, but also elements of its iterated [[identity types]].

While HITs are already useful in [[extensional type theory]], they are most useful and powerful in [[homotopy type theory]], where they allow the construction of [[cell complexes]], [[homotopy colimits]], [[n-truncated|truncations]], [[Bousfield localization of model categories|localizations]], and many other objects from classical [[homotopy theory]].

Defining what a HIT is "in general" is an open research problem.  One mostly precise proposal may be found in [[Shulman & Lumsdaine (2016)](#ShulmanLumsdaine16)].  A more syntactic description of a class of HITs may be found in [[Brunerie (2016)](#Brunerie16), [Vezzosi, Mörtberg & Abel (2019)](#VezzosiMörtbergAbel19)]. A solution to this problem should determine how to define the concept of an [[elementary (∞,1)-topos]].

## Examples

All higher inductive types described below are given together with some pseudo-[[Coq]] code, which would implement that HIT if Coq supported HITs natively.

### The circle
 {#ExamplesTheCircle}

The [[circle type]] is:

    Inductive circle : Type :=
    | base : circle
    | loop : base == base.

Using the [[univalence axiom]], one can prove that the [[loop space]] `base == base` of the circle type is equivalent to the [[integers]].

([Licata & Shulman 13](circle#LicataShulman13), [Bezem, Buchholtz, Grayson & Shulman 19](circle#BezemBuchholtzGraysonShulman19)).

### The interval

The [[homotopy type]] of the [[interval]] can be encoded as

    Inductive interval : Type :=
    | zero : interval
    | one : interval
    | segment : zero == one.

See [[interval type]].  The interval can be proven to be [[contractible type|contractible]].  On the other hand, if the constructors `zero` and `one` satisfy their elimination rules definitionally, then the existence of an interval type implies [[function extensionality]]; see [this blog post](http://homotopytypetheory.org/2011/04/04/an-interval-type-implies-function-extensionality/).
The interval can be defined as the -1-truncation of the booleans; see [here](https://groups.google.com/d/msg/homotopytypetheory/-5mLEi_qMTo/gpNUsmI-ZT4J):

    Inductive interval : Type :=
    | inj : boolean -> interval
    | contr0 : forall (p q : interval) p == q

Or more directly:

    Inductive interval : Type :=
    | zero : interval
    | one : interval
    | contr0 : forall (p q : interval) p == q

### The 2-sphere

Similarly the [[homotopy type]] of the 2-[[dimensional]] [[sphere]]

    Inductive sphere2 : Type :=
    | base2 : sphere2
    | surf2 : idpath base2 == idpath base2.

### Suspension

    Inductive susp (X : Type) : Type :=
    | north : susp X
    | south : susp X
    | merid : X -> north == south.

This is the unpointed [[suspension]].  It is also possible to define the pointed suspension.  Using either one, we can define the $n$-sphere by induction on $n$, since $S^{n+1}$ is the suspension of $S^n$.

### Mapping cylinders
 {#MappingCylinders}

The construction of [[mapping cylinders]] is given by

    Inductive cyl {X Y : Type} (f : X -> Y) : Y -> Type :=
    | cyl_base : forall y:Y, cyl f y
    | cyl_top : forall x:X, cyl f (f x)
    | cyl_seg : forall x:X, cyl_top x == cyl_base (f x).

Using this construction, one can define a (cofibration, trivial fibration) [[weak factorization system]] for types.

### Truncation

    Inductive is_inhab (A : Type) : Type :=
    | inhab : A -> is_inhab A
    | inhab_path : forall (x y: is_inhab A), x == y.

This is the [[(-1)-truncated|(-1)-truncation]] into [[h-propositions]].  One can prove that `is_inhab A` is always a [[proposition]] (i.e. $(-1)$-truncated) and that it is the reflection of $A$ into propositions.  More generally, one can construct the [[(effective epi, mono) factorization system]] by applying `is_inhab` fiberwise to a fibration.

Similarly, we have the [[0-truncated|0-truncation]] into [[h-sets]]:

    Inductive pi0 (X:Type) : Type :=
    | cpnt : X -> pi0 X
    | pi0_axiomK : forall (l : Circle -> pi0 X), refl (l base) == map l loop.

We can similarly define $n$-truncation for any $n$, and we should be able to define it inductively for all $n$ at once as well.

See at _[[n-truncation modality]]_.

### Quotients of types
 {#QuotientsOfTypes}

The [[quotient]] of a pure or Type-valued [[equivalence relation]]:

    Inductive quotient (A : Type) (R : A -> A -> Type) : Type :=
    | proj : A -> quotient A R
    | relate : forall (x y : A), R x y -> proj x == proj y

This definition is translated into Coq from the Cubical Agda library.

### Disjunctions

The [[disjunction]] of two types $A$ and $B$, yielding an [[hProp]]:

    Inductive disjunction (A B:Type) : Type :=
    | inl : A -> disjunction A B
    | inr : B -> disjunction A B.
    | contr0 : forall (p q : disjunction A B) p == q

### Existential quantifiers

The [[existential quantifier]] of a type $A$ and a type family $B:A \to Type$, yielding an [[hProp]]:

    Inductive existquant (A:Type) (B:A->Type) : Type :=
    | exist : forall (x:A), B x -> existquant A B
    | contr0 : forall (p q : existquant A B) p == q

### Quotients of sets
 {#QuotientsOfSets}

The [[quotient]] of an [[hProp]]-value [[equivalence relation]], yielding an [[hSet]] (a [[0-truncated]] type):

    Inductive quotient (A : Type) (R : A -> A -> hProp) : Type :=
    | proj : A -> quotient A R
    | relate : forall (x y : A), R x y -> proj x == proj y
    | contr1 : forall (x y : quotient A R) (p q : x == y), p == q.

This is already interesting in [[extensional type theory]], where [[quotient types]] are not always included.  For more general homotopical quotients of "[[internal groupoids]]" as in the  [[(∞,1)-Giraud theorem]], we first need a good definition of what such an internal groupoid is.

### Quotient inductive types

A [[quotient inductive type]] is a higher inductive type that includes a "0-truncation" constructor such as `contr1` for a set-quotient.  Many of these are useful in set-based mathematics; in addition to colimits in [[Set]], they can be used to construct free algebras and colimits of algebras of various sorts. Quotient [[inductive-inductive types]] are used to construct sets with [[proposition|propositional]] [[relations]] and various [[countable]] [[completions]] of structures. Examples can be found at [[quotient inductive type]], including:

* multiple definitions of the [[integers]]
* [[polynomial rings]]
* [[Sierpinski space]]
* the [[cumulative hierarchy]]
* the [[Cauchy real numbers]]
* [[partial map classifiers]]
* [[internal type theory]]

### FinSet

Since [[FinSet]] is the initial [[2-rig]], one should be able to construct it as a higher inductive type with a [[1-truncation]] constructor.

### Integers

A definition of the set of [[integers]] as a higher inductive type.

    Inductive int : Type :=
    | zero : int
    | succ : int -> int
    | pred1 : int -> int
    | pred2 : int -> int
    | sec : forall (x : int) pred1 succ x == x
    | ret : forall (x : int) succ pred2 x == x

### Other

* [[circle type]]

* [[sphere type]]

* [[cone type]]

* [[square type]]

* [[pushout type]]

* [[suspension type]]

* [[join type]]

* [[wedge sum type]]

* [[smash product type]]

* [[localization of a type at a family of functions]]

* [[Eilenberg-MacLane space type]]

* [[spectrification of a sequential spectrum type]]

* [[James construction type]]

* [[Rezk completion]]

* [[dependent pushout type]]

## Semantics
See ([Lumsdaine-Shulman17](#LumsdaineShulman17)).

## Related concepts

* [[inductive type]], [[initial algebra of an endofunctor]]

* **higher inductive type**, [[initial algebra of a presentable ∞-monad]]

* [[quotient inductive type]]

## References

Textbook account:

* [[Univalent Foundations Project]], §6 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

Expositions:

* [[Mike Shulman]], _Homotopy type theory IV_ ([blog entry](http://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_vi.html))

* [[Peter LeFanu Lumsdaine]], *Higher inductive types, a tour of the menagerie* (2011) [[blog post](http://homotopytypetheory.org/2011/04/24/higher-inductive-types-a-tour-of-the-menagerie/)]

* [[Peter LeFanu Lumsdaine]], *Higher Inductive Types: The circle and friends, axiomatically* (2011) [[pdf](http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS2011/Lumsdaine_slides.pdf), [[Lumsdaine-HITs.pdf:file]]]

* Kajetan Söhnen, *Higher Inductive Types in Homotopy Type Theory*, Munich (2018) [[pdf](https://www.math.lmu.de/~petrakis/Soehnen.pdf), [[Soehnen-HigherInductiveTypes.pdf:file]]]

Details on the [[categorical semantics]] of HITs:

* {#LumsdaineShulman17} [[Peter LeFanu Lumsdaine]], [[Mike Shulman]], *Semantics of higher inductive types*, Math. Proc. Camb. Phil. Soc. **169** (2020) 159-208 [[arXiv:1705.07088](https://arxiv.org/abs/1705.07088), talk slides [pdf](http://home.sandiego.edu/~shulman/papers/cellcxs.pdf), [doi:10.1017/S030500411900015X](https://doi.org/10.1017/S030500411900015X)]

with precursors in

* {#ShulmanLumsdaine12} [[Mike Shulman]], [[Peter LeFanu Lumsdaine]], *Semantics of higher inductive types* (2012) [[[ShulmanLumsdaine-HITs2012.pdf:file]]]

* {#ShulmanLumsdaine16} [[Mike Shulman]], [[Peter LeFanu Lumsdaine]], _Semantics and syntax of higher inductive types_ (2016) [[slides](http://home.sandiego.edu/~shulman/papers/stthits.pdf)]

Discussion of HITs which arise as homotopy-[[initial algebras of an endofunctor]]:

* [[Kristina Sojakova]], *Higher Inductive Types as Homotopy-Initial Algebras*, ACM SIGPLAN Notices **50** 1 (2015) 31–42 [[arXiv:1402.0761](http://arxiv.org/abs/1402.0761), [doi:10.1145/2775051.2676983](https://doi.org/10.1145/2775051.2676983)]

* [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Homotopy-initial algebras in type theory*, Journal of the ACM **63** 6 (2017) 1–45 [[arXiv:1504.05531](http://arxiv.org/abs/1504.05531), [doi:10.1145/3006383](https://doi.org/10.1145/3006383)]

Further developments:

* [[Henning Basold]], [[Herman Geuvers]], [[Niels van der Weide]], *Higher Inductive Types in Programming*, Journal of Universal Computer Science **23** 1 (2017) 63-88 [[pdf](https://www.jucs.org/jucs_23_1/higher_inductive_types_in/jucs_23_01_0063_0088_basold.pdf), [slides pdf](https://nmvdw.github.io/pubs/Agda25.pdf)]

* {#vanDoorn18} [[Floris van Doorn]], §3 in: *On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory* (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690)]

  > (formalization in [[Lean]])

* [[Niccolò Veltri]], [[Niels van der Weide]], *Constructing Higher Inductive Types as Groupoid Quotients*, Logical Methods in Computer Science **17** 2 (2021)  [[arXiv:2002.08150](https://arxiv.org/abs/2002.08150), doi:10.23638/LMCS-17(2:8)2021]

Implementation of HITs in [[proof assistants]]:

in [[Agda]]:

* {#Brunerie16} [[Guillaume Brunerie]], _Implementation of higher inductive types in HoTT-Agda_ (2016) [[github](https://github.com/HoTT/HoTT-Agda/blob/master/core/lib/types/HIT_README.txt)]

  > (deprecated)

in [[Cubical Agda]]:

* {#VezzosiMörtbergAbel19} [[Andrea Vezzosi]], [[Anders Mörtberg]], [[Andreas Abel]], §4 in: *Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types*, Proceedings of the ACM on Programming Languages **3** ICFP 87  (2019) 1–29 [[doi:10.1145/3341691](https://doi.org/10.1145/3341691), [pdf](https://www.cse.chalmers.se/~abela/icfp19.pdf)]

* [cubical Agda documentation](https://agda.readthedocs.io/en/v2.6.2.2.20221128/language/cubical.html): *[Higher Inductive Types](https://agda.readthedocs.io/en/v2.6.1/language/cubical.html#higher-inductive-types)*

in [[Coq]]:

* Bruno Barras, _Native implementation of Higher Inductive
Types (HITs) in Coq_ [PDF](http://www.crm.cat/en/activities/documents/barras-crm-2013.pdf)

in [[Lean]]: [van Doorn (2018)](#vanDoorn18)

Discussion of [[coinduction]] via HITs:

* Magnus Baunsgaard Kristensen, [[Rasmus Ejlers Møgelberg]], [[Andrea Vezzosi]], *Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks* ([arXiv:2102.01969](https://arxiv.org/abs/2102.01969))

Discussion of impredicative encodings of higher inductive types:

* [[Steve Awodey]], [[Jonas Frey]], and [[Sam Speight]]. “Impredicative Encodings of (Higher) Inductive Types”. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science. LICS ’18. Oxford, United Kingdom: ACM, 2018, pp. 76–85. (ISBN:978-1-4503-5583-4, [doi:10.1145/3209108.3209130](https://doi.org/10.1145/3209108.3209130))

[[!redirects higher inductive types]]

[[!redirects HIT]]
[[!redirects HITs]]


higher Kac-Moody algebra

[[!redirects Higher Kac-Moody algebras]]

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#### Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
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## Idea

* In dimension $1$, we have that $\mathcal{O}(\mathbb{A}^1-\{0\})\otimes\mathfrak{g} = \mathfrak{g}[z,z^{-1}]$ and, thus, we can consider its non-trivial [[central extensions]] by $2$-[[cocycles]]. These are the essential ingredients of an (ordinary) [[Kac-Moody algebra]].

* In higher dimension, [[analytic geometry#hartogs_theorem|Hartogs' extension theorem]] tells us that $\mathcal{O}(\mathbb{A}^n-\{0\})\otimes\mathfrak{g} \cong \mathcal{O}(\mathbb{A}^n)\otimes\mathfrak{g}$, which means that we do not have any interesting central extension. On the other hand, the punctured affine spaces $\mathbb{A}^n-\{0\}$ have non-trivial higher [[cohomology]].

The idea underpinning the definition of higher Kac-Moody algebras in [FHK 19](#FHK19) is that the natural framework to solve this issue and construct a higher-dimensional generalization of [[Kac-Moody algebras]] is [[derived algebraic geometry]]. This is achieved by replacing the algebra $\mathcal{O}(\mathbb{A}^n-\{0\})$ with the dg-algebra $\mathbb{R}\Gamma(\mathbb{A}^{n}-\{0\},\mathcal{O})$ of derived sections, which allows interesting [[central extensions]].

This principle can be applied not only to punctured [[affine spaces]] $\mathbb{A}^{n}-\{0\}$, but also to punctured [[formal disks]] $\mathbb{D}_n^\circ$.

## Details

### Faonte-Hennion-Kapranov higher Kac-Moody algebras

Let $X$ be an $n$-dimensional [[variety]] over $\mathbb{C}$ and $\mathfrak{g}$ a [[Lie algebra]]. We can define the __higher current algebra__ by the [[dg-Lie algebra]]
$$ \mathfrak{g}_n \;=\; \mathfrak{g}\otimes\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X), $$
where $\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X)$ is the [[commutative dg-algebra]] of derived [[global sections]] of the [[structure sheaf]] $\mathcal{O}_X$ on the punctured [[formal disk]] $\mathbb{D}_n^\circ = \mathrm{Spec}(\mathbb{C}[[z_1,\dots,z_n]])-\{0\}$.

A __higher Kac-Moody algebra__ $\widehat{\mathfrak{g}}_{n,\Theta}$ is the [[central extension]] of the higher current algebra $\mathfrak{g}_n$ by an invariant polynomial $\Theta$ on $\mathfrak{g}_n$ of degree $(n+1)$.

### Gwilliam-Williams higher Kac-Moody factorisation algebras

Let $X$ be a [[complex manifold]] of dimension $n$ equipped with a holomorphic [[principal bundle]] $P\rightarrow X$ with [[structure group]] $G$.

Let $\mathfrak{ad}(P) \coloneqq P\times_G\mathfrak{g}$ be the [[adjoint bundle]] associated to $P$.
Now, let $\mathscr{Ad}(P)$ be the [[sheaf of L-∞ algebras#LocalLieAlgebra|local Lie algebra]] whose sections are $\Omega_c^{0,\ast}(U,\mathfrak{ad}(P))$ on any open set $U\subset X$ and whose differential is the $(0,1)$-connection $\bar{\partial}_P$.

Let $\Theta$ be a degree $1$ [[cocycle]] in the [[sheaf of L-∞ algebras#LocalLieAlgebra|local Chevalley-Eilenberg cochains]] $\mathrm{CE}_{\mathrm{loc}}(\mathscr{Ad}(P))$, which defines a $1$-shifted [[central extension]] $\widehat{\mathscr{Ad}(P)}_\Theta$.

The __higher Kac-Moody factorization algebra__ on $X$ of type $\Theta$ is defined in [GW 21](#GW21) as the twisted [[universal enveloping algebra|enveloping]] [[factorization algebra]] $\mathbb{U}_\Theta(\mathscr{Ad}(P))$ whose sections are

$$ \mathbb{U}_\Theta\left(\mathscr{Ad}(P)\right)(U) \;=\; \Big(\mathrm{Sym}\left(\Omega_c^{0,\ast}(U,\mathfrak{ad}(P))[1]\right), \; \bar{\partial} + \mathrm{d}_\mathrm{CE} + \Theta \Big) $$

on any open set $U\subset X$.

### Relation between the two

Let $r : \mathbb{A}^{n}_{\mathbb{C}} - \{0\} \rightarrow (0,+\infty)$ be the radial projection map sending $(z_1,\dots,z_n)\mapsto \sqrt{|z_1|^2+\dots+|z_n|^2}$.

In [GW 21](#GW21) the following map of [[factorization algebras]] on the positive reals is constructed:

$$ \mathbb{U}(\hat{\pi}_{\mathfrak{g},n,\Theta}) \, : \; \mathbb{U}(\Omega^{0,\ast}_c\otimes\hat{\mathfrak{g}}_{n,\Theta}) \; \longrightarrow \; r_\ast\mathbb{U}_\Theta(\mathscr{Ad}(P)), $$

where on the left-hand side we have the enveloping factorization algebra which encodes the [[universal enveloping algebra|enveloping $A_\infty$-algebra]] of the [FHK 19](#FHK19) higher Kac-Moody algebra $\hat{\mathfrak{g}}_{n,\Theta}$.

This map establishes a relation between [[derived algebraic geometry]] and [[quantum field theory]], formulated the language of [[factorization algebras]]. In particular, the _higher Kac-Moody algebra_ $\hat{\mathfrak{g}}_{n,\Theta}$ "controls" its corresponding _higher Kac-Moody factorization algebra_ just like an affine [[Kac-Moody algebra]] "controls" its corresponding [[vertex algebra]].

## Examples

### Ordinary Kac-Moody algebras

For $n=1$ and $X=\mathbb{A}^1_{\mathbb{C}}$, one recovers the formal current algebra $\mathfrak{g}_1 = \mathfrak{g}((z))$, whose [[central extensions]] $\tilde{\mathfrak{g}}_{1,\Theta}$ are ordinary [[Kac-Moody algebras]].

### Affine space $\mathbb{A}^n_{\mathbb{C}}$

Consider the [[affine space]] $X=\mathbb{A}^n_{\mathbb{C}} = \mathrm{Spec}(\mathbb{C}[z_1,\dots,z_n])$. The cohomology of the complex of derived [[global sections]] $\mathbb{R}\Gamma(\mathbb{D}_n^\circ,\mathcal{O}_X)$ will be the following:

$$ \mathrm{H}^i(\mathbb{D}_n^\circ,\mathcal{O}_X) \;=\; \begin{cases}\mathbb{C}[[z_1,\dots,z_n]], & i=0\\z_1^{-1}\cdots z_n^{-1}\mathbb{C}[z_1^{-1},\dots,z_n^{-1}], & i=n-1\\ 0, & \text{otherwise}\end{cases} $$

## Related concepts

* [[Kac-Moody algebras]]

* [[derived algebraic geometry]]

* [[factorization algebras]]

## References

Higher Kac-Moody algebras were proposed in

* {#FHK19} G. [[Faonte]], B. [[Hennion]] and M. [[Kapranov]], _Higher Kac-Moody algebras and moduli spaces of $G$-bundles_, Advances in Mathematics, Volume 346, 13 April 2019, pages 389-466, [math.AG/1701.01368](https://arxiv.org/abs/1701.01368v3).

* M. [[Kapranov]], _Infinite-dimensional (dg) Lie algebras and factorization algebras in algebraic geometry_, Japanese Journal of Mathematics volume 16, pages 49–80 (2021) [https://doi.org/10.1007/s11537-020-1921-4](https://doi.org/10.1007/s11537-020-1921-4).

Higher Kac-Moody algebras casted in the language of [[factorization algebras]]:

* {#GW21} [[Owen Gwilliam]], B. R. Williams, _Higher Kac-Moody algebras and symmetries of holomorphic field theories_, Adv. Theor. Math. Phys. **25** 1 (2021) 129-239 [[math.QA/1810.06534](https://arxiv.org/abs/1810.06534v2)]

A variant of this construction on the product of a [[worldsheet]] and a [[spectral curve|spectral plane]] (i.e. another copy of
the complex plane $\mathbb{C}$ with marked points which defines any rational [[Gaudin integrable model|Gaudin model]]):

* [[Luigi Alfonsi]], [[Charles A. S. Young]], *Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex*, Journal of Geometry and Physics **191** (2023) 104903 [[doi:10.1016/j.geomphys.2023.104903](https://doi.org/10.1016/j.geomphys.2023.104903), [math.QA/2208.06009](https://arxiv.org/abs/2208.06009)]

Review:

* [[Luigi Alfonsi]], §7 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

[[!redirects higher Kac-Moody algebra]]
[[!redirects higher Kac-Moody algebras]]
[[!redirects higher current algebra]]
[[!redirects higher current algebras]]


higher Klein geometry

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

#Contents#
* table of contents
{:toc}

## Idea

_Higher Klein geometry_ is the generalization of [[Klein geometry]] from traditional ([[differential geometry|differential]]) [[geometry]] to [[higher geometry]]:

where Klein geometry is about ([[Lie group|Lie]]) [[groups]] and their [[quotient]]s, higher Klein geometry is about ([[smooth infinity-group|smooth]]) [[∞-groups]] and their [[groupoid object in an (∞,1)-category|∞-quotients]].

The way that the generalization proceeds is clear after the following observation.

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

Let $G$ be a [[discrete group]] and $H \hookrightarrow G$ a subgroup. Write $\mathbf{B}G$ and $\mathbf{B}H$ for the corresponding [[delooping]] [[groupoid]]s with a single object. Then the [[action groupoid]] $G//H$ is the [[homotopy fiber]] of the inclusion [[functor]]

$$
  \mathbf{B}H \to \mathbf{B}G
$$

in the [[(2,1)-category]] [[Grpd]]: we have a [[fiber sequence]]

$$
  G//H \to \mathbf{B}H \to \mathbf{B}G
$$

that exhibits $G//H$ as the $G$-[[principal bundle]] over $\mathbf{B}H$ which is classified by the [[cocycle]] $\mathbf{B}H \to \mathbf{B}G$.

Moreover, the [[decategorification]] of the [[action groupoid]] (its [[0-truncated|0-truncation]]) is the ordinary [[quotient]]

$$
  \tau_0 (G//H) = G/H
  \,.
$$

=--

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

This should all be explained in detail at [[action groupoid]].

The fact that a [[quotient]] is given by a [[homotopy fiber]] is a special case of the general theorem discussed at [∞-colimits with values in ∞Grpd](http://ncatlab.org/nlab/show/limit+in+a+quasi-category#WithValInooGrpd)

=--

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

That fiber sequence continues to the left as

$$
  H \to G \to G//H \to \mathbf{B}H \to \mathbf{B}G
  \,.
$$

=--

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

The above statement remains true _verbatim_ if [[discrete group]]s are generalized to [[Lie group]]s -- or other [cohesive groups](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#InfinGroups) --  if only we pass from the [[(2,1)-topos]] [[Grpd]] of [[discrete groupoid]]s to the [[(2,1)-topos]] [[Smooth∞Grpd|SmoothGrpd]] of _smooth groupoids_ .

=--

This follows with the discussion at [[smooth ∞-groupoid -- structures]].

Since the quotient $G/H$ _is_ what is called a _[[Klein geometry]]_ and since by the above observations we have analogs of these quotients for [higher cohesive groups](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#InfinGroups), there is then an evident definition of _higher Klein geometry_  :

## Definition

Let $\mathbf{H}$ be a choice of [[cohesive (∞,1)-topos|cohesive structure]]. For instance choose

* $\mathbf{H} = $ [[Disc∞Grpd]] for discrete higher Klein geometry (no actual geometric structure);

* $\mathbf{H} = $ [[ETop∞Grpd]] for continuous higher Klein geometry (with [[topological structure]]);

* $\mathbf{H} = $ [[Smooth∞Grpd]] for higher Klein geometry based on [[differential geometry]];

* $\mathbf{H} = $ [[SuperSmooth∞Grpd]] for the [[supergeometry]] version of higher Klein geometry

* and so on.

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

An **$\infty$-Klein geometry** in $\mathbf{H}$ is a [[fiber sequence]] in $\mathbf{H}$

$$
  G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G
$$

for $i$ any morphism between two [[connected]] objects, as indicated, hence $\Omega i : H \to G$ any morphism of [[∞-group]] objects.

=--

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

For $X$ an object equipped with a $G$-[[action]] and $f : Y \to X$ any morphism, the higher Klein geometry induced by "the shape $Y$ in $X$" is given by taking $i : H \to G$ be the [[stabilizer ∞-group]] $Stab(f) \to G$ of $f$ in $X$. See there at _[Examples -- Stabilizers of shapes / Klein geometry](stabilizer+group#KleinGeometry)_.

=--

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

* By the discussion at [[looping and delooping]], and using that a [[cohesive (∞,1)-topos]] has [[homotopy dimension]] 0 it follows that  every connected object indeed is the delooping of an [[∞-group]] object.

* The above says that $G//H$ is the [[principal ∞-bundle]] over $\mathbf{B}H$ that is classified by the [[cocycle]] $i$.

  Continuing this [[fiber sequence]] further to the left yields the long fiber sequence

  $$
    H \to G \to G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G
  $$

  This exhibits $G$ indeed as the [[fiber]] of $G//H \to \mathbf{B}H$.

=--

## Examples

### Higher super Poincaré Klein geometry

Let $\mathbf{H} = $ [[SuperSmooth∞Grpd]] be the context for [[SynthDiff∞Grpd|synthetic]] [[higher geometry|higher]] [[supergeometry]].

Write $\mathfrak{sugra}_11$ for the [[super L-∞ algebra]] called the [[supergravity Lie 6-algebra]]. This has a sub-super $L_\infty$-algebra of the form

$$
  \mathbf{B}(\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R})
  \hookrightarrow
  \mathbf{B}\mathfrak{sugra}_11
  \,,
$$

where

* $\mathfrak{so}(d,1)$ is the [[special orthogonal Lie algebra]];

* $b^{n-1} \mathbb{R}$ is the [[line Lie n-algebra]].

The quotient

$$
  \mathfrak{sugra}_11 /
  ((\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}))
$$

is the [[super translation Lie algebra]] in 11-dimensions.

This higher Klein geometry is the local model for the [[higher Cartan geometry]] that describes [[11-dimensional supergravity]]. See [[D'Auria-Fre formulation of supergravity]] for more on this.

## Related concepts

[[!include local and global geometry - table]]

## References

That there ought to be a systematic study of higher Klein geometry and [[higher Cartan geometry]] has been amplified by [[David Corfield]] since 2006.

Such a formalization is offered in

* [[schreiber:differential cohomology in a cohesive topos]]

For more on this see at _[[higher Cartan geometry]]_ and _[[schreiber:Higher Cartan Geometry]]_.

[[!redirects higher Klein geometries]]


higher linear algebra

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

#Contents#
* table of contents
{:toc}

## Idea

The [[higher algebra]] refinement of ordinary [[linear algebra]]:

for $E \in CRing_\infty$ an [[E-∞ ring]], higher linear algebra over $E$ is the study of the [[tensor (∞,1)-category]] $E Mod$ of [[∞-modules]] over $E$.

## Related concepts

* [[higher structures]]

* [[(∞,1)-module bundle]]

* [[(∞,n)-module]]


higher linear algebra - contents

**[[linear algebra]]**, **[[higher linear algebra]]**

## Ingredients

* [[algebra]], [[higher algebra]],

* [[homological algebra]], [[stable homotopy theory]]

## Basic concepts

* [[ring]], [[A-∞ ring]]

* [[commutative ring]], [[E-∞ ring]]

* [[module]], [[∞-module]], [[(∞,n)-module]]

* [[field]], [[∞-field]]

* [[vector space]], [[2-vector space]]

  [[rational vector space]]

  [[real vector space]]

  [[complex vector space]]

* [[topological vector space]]

* [[linear basis]],

  [[orthogonal basis]], [[orthonormal basis]]

* [[linear map]], [[antilinear map]]

* [[matrix]] ([[square matrix|square]], [[invertible matrix|invertible]], [[diagonal matrix|diagonal]], [[hermitian matrix|hermitian]], [[symmetric matrix|symmetric]], ...)

* [[general linear group]], [[matrix group]]

* [[eigenspace]], [[eigenvalue]]

* [[inner product]], [[Hermitian form]]

  [[Gram-Schmidt process]]

* [[Hilbert space]]

## Theorems

(...)


higher local field

[[!redirects higher local fields]]

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

#Contents#
* table of contents
{:toc}

## Idea

The concept of _higher local field_ is the generalization of that of _[[local field]]_ to higher _[[dimension]]_. They are the subject of [[higher arithmetic geometry]].

In particular two-dimensional local field are the analog of [[loop spaces]] in [[arithmetic geometry]].

Their examples are $\mathbb{C}((t))$, $\mathbb{R}((t))$, $\mathbb{Q}_p((t))$, $\mathbb{F}_q((t))((u))$, etc. All these are naturally associated to a full flag of subschemes on algebraic and arithmetic surfaces.

These fields are not locally compact with respect to any reasonable topology on them.  But they are topologically self-dual, similar to the classical local fields.
For a collection of articles on all the main aspects of higher local fields known by 2000, see ([Invitation to Higher Local Fields](#HLF00)). For a survey of higher local fields and associated algebraic theories see ([Morrow12](#Morrow12)).

One of the main new developments is the theory translation invariant measure and integration on higher local fields, a higher Haar measure, see ([Fesenko03](#Fesenko03)), ([Fesenko06](#Fesenko06)), ([Morrow10](#Morrow10)). This measure takes values not in $\mathbb{C}$ but in $\mathbb{C}((X))$. Fourier transform with respect to this rigorously defined measure has many features similar to those of the Feynman [[path integral]], the latter still does not have a rigorous math justification, see ([Fesenko 06, sect. 18](#Fesenko06)).

## References

* Wikipedia, _[Higher local field](http://en.wikipedia.org/wiki/Higher_local_field)_

* {#HLF00} [[Ivan Fesenko]], M. Kurihara (eds.), _Invitation to Higher Local Fields_, Geometry and Topology Monographs vol 3, Warwick 2000, 304 pp. ([web](https://www.maths.nottingham.ac.uk/personal/ibf/volume.html))

* {#Fesenko03}[[Ivan Fesenko]], _Analysis on arithmetic schemes. I_,
Docum. Math. Extra Volume Kato (2003) 261-284;
([web](https://www.maths.nottingham.ac.uk/personal/ibf/a1.pdf))

* {#Fesenko06} [[Ivan Fesenko]], _Measure, integration and elements of harmonic analysis on generalized loop spaces_,Proceed. St. Petersburg Math. Soc., Vostokov Festschrift, vol. 12 (2005), 179199;
English translation in AMS Transl. Series 2, vol. 219, 149–164, 2006 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/aoh.pdf))

* {#Morrow10} M. Morrow, Integration on valuation fields
over local fields, arXiv:0712.2172, Tokyo J. Math. 33(2010),  235-281

* {#Morrow12} M. Morrow, An introduction to higher local fields and adeles, arXiv:1204.0586

For more on applications of higher local fields to fundamental open problems in arithmetic geometry see also

* {#Fesenko08} [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))


higher monadic descent

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Higher monadic descent is the generalization of the notion of [[monadic descent]] from [[category theory]] to [[higher category theory]]. It relates to the [[descent]] of [[∞-stacks]] as ordinary monad descent relates to [[stack]]s.

See also [[cohomological descent]].

## Examples

### Amitsur complex, Sweedler corings, Hopf algebroids
 {#AmitsurComplex}

For $\phi \,\colon\, B \longrightarrow A$ a [[homomorphism]]
of suitable [[monoids]], there is the corresponding pull-push [[adjunction]]
([[extension of scalars]] $\dashv$ [[restriction of scalars]]) on [[categories of modules]]

$$
  \big(
    (- )\otimes_B A \dashv \phi^\ast
  \big)
  \;\colon\;
   Mod_A
    \underoverset
      {\underset{\;\; \phi^\ast \;\;}{\longrightarrow}}
      {\overset{(-)\otimes_B A}{\longleftarrow}}
      {\bot}
   Mod_B
  \,.
$$

The [[bar construction]] of the corresponding [[monad]] is the corresponding [[Amitsur complex]].

(e.g. [Hess 10, section 6](#Hess10))

## Related concepts

* [[descent]]

  * [[cover]]

  * [[cohomological descent]]

  * [[monadic descent]],

    * [[Sweedler coring]]

    * **higher monadic descent**

    * [[descent in noncommutative algebraic geometry]]

  * [[descent spectral sequence]]

* [[sheaf]], [[(2,1)-sheaf]], [[2-sheaf]] [[(∞,1)-sheaf]]

## References

### 2-categorical monadic descent

A pseudomonadicity theorem for [[2-monad|pseudomonads]] is proved in Theorem 3.6 of:

* I. J. Le Creurer, [[Francisco Marmolejo]], [[Enrico M. Vitale]], _Beck's theorem for pseudo-monads_, J. Pure Appl. Algebra **173** 3 (2002) 293-313 [doi:10.1016/S0022-4049(02)00038-5]

and sharpened in Theorem A.2 of:

* [[Claudio Hermida]], *Descent on 2-fibrations and strongly 2-regular 2-categories*, Appl. Categ. Structures **12** 5-6 (2004)  427-459 [[doi:10.1023/B:APCS.0000049311.17100.da](https://doi.org/10.1023/B:APCS.0000049311.17100.da)]

(see Remark A.3 for a comparison with the work of Creurer et al.).

More generally, this paper discusses a notion of 2-[[fibered category|fibered categories]], originally defined in Gray's work, and then again introduced and discussed at length in this paper. The appendix also discusses a 2-categorical version of Beck-Chevalley condition needed for a comparison with 2-monadic descent.

### $(\infty,1)$-categorical monadic descent

A comprehensive treatment in the context of [[(∞,1)-category]]-theory,  general theory of [[(∞,1)-monad]]s and their [[monadicity theorem]] is in

* [[Jacob Lurie]], _Derived algebraic geometry II: Noncommutative algebra_, [math.CT/0702299](http://arxiv.org/abs/math/0702299), Section 3.

later absorbed as

* [[Jacob Lurie]], section 6.2.4 of _[[Higher Algebra]]_

Unfortunately, [[Maxim Kontsevich|Kontsevich]]'s [[monadicity theorem]] (July 2004) which is in the setup of [[A-infinity category|A-∞-categories]], still remains unpublished. The [[triangulated category|triangulated]] version is in Rosenberg's lectures

* [[Alexander Rosenberg|A. L. Rosenberg]], _Topics in noncommutative algebraic geometry, homological algebra and K-theory_, preprint MPIM Bonn 2008-57 [pdf](http://www.mpim-bonn.mpg.de/preblob/3589), page 36-37.

Its proof is based on [[Verdier's abelianization functor]].

See also, for another point of view,

* [[Paul Balmer]], _Descent in triangulated categories_, [pdf](http://www.math.ucla.edu/~balmer/research/Pubfile/Descent.pdf)

Discussion of monadic descent for [[simplicially enriched category|simplicially enriched categories]] is in

* [[Kathryn Hess]], _A general framework for homotopic descent and codescent_, [arXiv/1001.1556](http://arxiv.org/abs/1001.1556)
 {#Hess10}

* [[Kathryn Hess]], [[Brooke Shipley]], _The homotopy theory of coalgebras over a comonad_ ([arXiv:1205.3979](http://arxiv.org/abs/1205.3979))

and for [[quasi-categories]] in

* [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_ ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279))
 {#RiehlVerity13}


higher noncommutative geometry

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

Higher noncommutative geometry refers to the version of [[higher geometry]] where the spaces are locally in some sense represented by [[higher algebra]]s without commutativity constraints, or by higher categories of [[quasicoherent sheaves]].

The standard case is the [[derived noncommutative geometry]] where spaces are represented by nice $A_\infty$-categories or by [[spectral categories]].


higher observational type theory

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

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

Where the idea of (non-higher) [[observational type theory]] is to equip all [[type formation]] rules (notably [[dependent functions]], [[dependent pairs]] and [[W-types|inductive constructions]]) with their dedicated notion of *structure preserving* [[definitional equality]] --- namely: component-wise, ie. [[homomorphism|homo-morphic]], hence: "observational" ---; the idea of *higher* observational type theory is to do the same for [[propositional equality]] hence for [[identification types]] used in [[homotopy type theory]] (where [[types]] behave like [[homotopy n-type|*higher* homomotopy types]], whence the qualifier "higher").

Concretely:

* the "observational" principle for identification of [[dependent functions]] is to say that these are dependent functions of identifications of arguments and values (a statement otherwise known as [[function extensionality]]),

* the "observational" principle for identifications of [[dependent pairs]] is to say that these are dependent pairs of identifications of factors,

and in ordinary [[univalence axiom|univalent]] [[homotopy type theory]] this form of the "[[structure identity principle]]" follows as a [[type equivalence]] between [[identification types]]:

The idea of higher observational type theory is to make these and analogous structural characterizations of [[identification types]] be part of their definitional [[inference rules]], thus building the [[structure identity principle]] right into the rewrite rules of the type theory.

In such a higher observational theory, in particular also the [[univalence axiom]] would be a [[definitional equality]] and hence would "compute".

This most desirable property of any [[homotopy type theory]] has previously been accomplished only by [[cubical type theories]].  However, cubical type theories achieve this only by adding syntax for auxiliary/spurious [[interval types]] with rewrite rules which encode technical detail that has no abstract motivation other than making the univalence axiom compute and which one would rather keep out of the syntactic logic and instead relegate to the construction of [[categorical semantics]].

The hope is therefore that higher observational type theory would provide a type system which achieves both:

1. its syntax is as logically clean as that of [[Martin-Löf dependent type theory]] equipped with the [[univalence axiom]];

1. its [[inference rules]] for [[identification types]] make the [[univalence axiom]] be a computable function as it is in [[cubical type theories]].

To which extent this hope is being realized would ideally be elucidated by the references [below](#References).

## Details

> This section does not quite get to the key points across and may need to be largely rewritten from scratch.

We are working in a [[dependent type theory]] with [[judgmental equality]].

### Telescopes ###

[[telescope types]] contain lists of iteratively dependent terms

Rules for the empty telescope

$$\frac{}{\epsilon\ \mathrm{tel}}$$

Rules for telescopes given a telescope and a type

$$\frac{\Delta\ \mathrm{tel} \quad \Delta \vdash A \mathrm{type}}{(\Delta,x:A)\ \mathrm{tel}}$$

Rules for the empty context in a telescope

$$\frac{}{():\epsilon}$$

Rules for ...

$$\frac{\delta:\Delta \quad \Delta \vdash A \mathrm{type} \quad a:A[\delta]}{(\delta,a):(\Delta,x:A)}$$

...

$$\frac{\Delta \vdash a:A \quad \delta:\Delta}{a[\delta]:A[\delta]}$$

### Identity telescopes ###

$$\frac{\Delta\ \mathrm{tel} \quad \delta:\Delta \quad \delta^{'}:\Delta}{\delta =_\Delta \delta^{'}\ \mathrm{tel}}$$

...

$$() =_\epsilon () \equiv \epsilon$$

...

$$(\delta,a) =_{\Delta,x:A} (\delta^{'},a^{'}) \equiv \left(\varsigma:\delta =_\Delta \delta^{'}, \alpha:a =_{\Delta.A}^\varsigma a^{'}\right)$$

### Dependent identity types ###

$$\frac{\varsigma:\delta =_\Delta \delta^{'} \quad \delta \vdash A\ \mathrm{type} \quad a:A[\delta] \quad a^{'}:A[\delta^{'}]}{a =_{\Delta.A}^\varsigma a^{'}\ \mathrm{type}}$$

The [[identity types]] in higher observational type theory is defined as

$$a =_A a^{'} \equiv a =_{\epsilon.A}^{()} a^{'}$$

Computation rules are defined for pair types:

$$(s =_{A \times B}^\varsigma t) \equiv (\pi_1(s) =_A^\varsigma \pi_1(t)) \times (\pi_2(s) =_B^\varsigma \pi_2(t))$$

Computation rules are defined for function types:

$$(f =_{A \to B}^\varsigma g) \equiv \prod_{a:A} \prod_{b:A} \prod_{q:(a =_A b)} (f(a) =_B^\varsigma g(b))$$

Computation rules are defined for dependent function types:

$$(f =_{\prod_{x:A} B(x)}^\varsigma g) \equiv \prod_{a:A} \prod_{b:A} \prod_{p:(a =_{A}^\varsigma b)} (f(a) =_{B}^{\varsigma,p} g(b))$$

Computation rules are defined for dependent pair types:

$$(s =_{\sum_{x:A} B(x)}^\varsigma t) \equiv \sum_{p:(\pi_1(s) =_A^\varsigma \pi_1(t))} (\pi_2(s) =_{B}^{\varsigma,p} \pi_2(t))$$

### Dependent Ap ###

...

### With universes ###

We are working in a [[dependent type theory]] with Tarski-style [[universes]].

The [[identity types]] in a universe in higher observational type theory have the following formation rule:

$$\frac{A:\mathcal{U} \quad a:\mathcal{T}_\mathcal{U}(A) \quad b:\mathcal{T}_\mathcal{U}(A)}{\mathrm{id}_A(a, b):\mathcal{U}}$$

We define a general congruence term called ap

$$\frac{x:A \vdash f:B \quad p:\mathrm{id}_A(a, a^{'})}{\mathrm{ap}_{x.f}(p):\mathrm{id}_B(f[a/x], f[a^{'}/x])}$$

and the reflexivity terms:

$$\frac{a:A}{\mathrm{refl}_{a}:\mathrm{id}_A(a, a)}$$

and computation rules for identity functions

$$\mathrm{ap}_{x.x}(p) \equiv p$$

and for constant functions $y$

$$\mathrm{ap}_{x.y}(p) \equiv \mathrm{refl}_{y}$$

Thus, ap is a higher dimensional explicit substitution. There are definitional equalities

$$\mathrm{ap}_{x.f}(\mathrm{refl}_{a}) \equiv \mathrm{refl}_{f[a/x]}$$

$$\mathrm{ap}_{y.g}(\mathrm{ap}_{x.f}(p)) \equiv \mathrm{ap}_{x.g[f/y]}(p)$$

$$\mathrm{ap}_{x.t}(p) \equiv \mathrm{refl}_{t}$$

for constant term $t$.

#### Identity types for universes ####

Let $A \cong_\mathcal{U} B$ be the type of [[one-to-one correspondences]] between two terms of a universe $A:\mathcal{U}$ and $B:\mathcal{U}$, and let $\mathrm{id}_\mathcal{U}(A, B)$ be the identity type between two terms of a universe $A:\mathcal{U}$ and $B:\mathcal{U}$. Then there are rules

$$\frac{R:A \cong_\mathcal{U} B}{\Delta(R):\mathrm{id}_\mathcal{U}(A, B)} \qquad \frac{P:\mathrm{id}_\mathcal{U}(A, B)}{\nabla(P):A \cong_\mathcal{U} B} \qquad \frac{R:A \cong_\mathcal{U} B}{\nabla(\Delta(R)) \equiv R}$$

#### Identity types in universes and singleton contractibility ####

Given a term of a universe $A:\mathcal{U}$

$$\mathrm{id}_{\mathcal{T}_\mathcal{U}(A)} \equiv \pi_1(\nabla(\mathrm{refl}_A))$$

with terms representing **singleton contractibility.

$$\pi_1(\pi_2(\nabla(\mathrm{refl}_A)):\prod_{a:\mathcal{T}_\mathcal{U}(A)} \mathrm{isContr}\left(\sum_{b:\mathcal{T}_\mathcal{U}(A)} \mathrm{id}_{\mathcal{T}_\mathcal{U}(A)}(a, b)\right)$$

$$\pi_2(\pi_2(\nabla(\mathrm{refl}_A))):\prod_{b:\mathcal{T}_\mathcal{U}(A)} \mathrm{isContr}\left(\sum_{a:\mathcal{T}_\mathcal{U}(A)} \mathrm{id}_{\mathcal{T}_\mathcal{U}(A)}(a, b)\right)$$

#### Dependent identity types in universes ####

Given a term of a universe $A:\mathcal{U}$, a judgment $z:\mathcal{T}_\mathcal{U}(A) \vdash B:\mathcal{U}$, terms $x:\mathcal{T}_\mathcal{U}(A)$ and $y:\mathcal{T}_\mathcal{U}(A)$, and an identity $p:\mathrm{id}_{\mathcal{T}_\mathcal{U}(A)}(x,y)$, we have

$$\mathrm{ap}_{z.B}(p):\mathrm{id}_\mathcal{U}(B(x),B(y))$$

and

$$(u,v):\mathcal{T}_\mathcal{U}(B(x)) \times \mathcal{T}_\mathcal{U}(B(y)) \vdash \pi_1(\nabla(\mathrm{ap}_{z.B}(p)))(u,v):\mathcal{U}$$

We could define a dependent identity type as

$$\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{p}(u, v) \coloneqq \pi_1(\nabla(\mathrm{ap}_{z.B}(p)))(u, v)$$

There is a rule

$$\frac{A:\mathcal{U} \quad z:\mathcal{T}_\mathcal{U}(A) \vdash B:\mathcal{U} \quad a:\mathcal{T}_\mathcal{U}(A)}{\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{\refl_{a}}(u, v) \equiv \mathrm{id}_{\mathcal{T}_\mathcal{U}(B[a/z])}(u, v)}$$

and for constant families $B:\mathcal{U}$

$$\mathrm{id}_{\mathcal{T}_\mathcal{U}(z.B)}^{p}(u, v) \equiv \mathrm{id}_{\mathcal{T}_\mathcal{U}(B)}(u, v)$$

## See also

* [[observational type theory]]

* [[homotopy type theory]]

* [[Martin-Löf type theory]]

* [[cubical type theory]]

## References
 {#References}

Higher observational type theory was introduced as joint work of [[Thorsten Altenkirch]], [[Ambrus Kaposi]] and [[Michael Shulman]], first presented in:

* [[Michael Shulman]], *Towards a Third-Generation HOTT:*, talk at *[Homotopy Type Theory at CMU](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (2022)  [part 1: [slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-04-28.pdf), [video](https://www.youtube.com/watch?v=FrxkVzItMzA); part 2: [slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-05-05.pdf), [video](https://www.youtube.com/watch?v=5ciDNfmvMdU); part 3: [slides](https://www.cmu.edu/dietrich/philosophy/hott/slides/shulman-2022-05-12.pdf), [video](https://www.youtube.com/watch?v=9pDddxB7LbE)]

* {#AgdaCode} Agda code: [github.com/mikeshulman/ohtt](https://github.com/mikeshulman/ohtt)

The following article is about a fragment of higher observational type theory:

* [[Thorsten Altenkirch]], [[Yorgo Chamoun]], [[Ambrus Kaposi]], [[Michael Shulman]], *Internal parametricity, without an interval* ([arXiv:2307.06448](https://arxiv.org/abs/2307.06448))

The authors of the above article write:

> We presented a type theory with internal parametricity, a presheaf model and a canonicity proof. It can be seen as a baby version of higher observational type theory (HOTT). To obtain HOTT, we plan to add the following additional features to our theory:

> * a bridge type which can be seen as an indexed version of $\forall$,
> * Reedy fibrancy, which replaces spans by relations,
> * a strictification construction which turns the isomorphism for $\Pi$ types into a definitional equality (in case of bridge, we also need the same for $\Sigma$),
> * Kan fibrancy, which adds transport and turns the bridge type into a proper identity type. This would also change the correspondence between $\forall U$ and spans into $\forall U$ and equivalences.

[[!redirects higher observational type theories]]


Higher Operads, Higher Categories

A book by [[Tom Leinster]].

London Mathematical Society Lecture Note Series 298 (2004).

Cambridge University Press.

ISBN 0 521 53215 9.

[doi:10.1017/cbo9780511525896](http://dx.doi.org/10.1017/cbo9780511525896).

[arXiv:math/0305049](https://arxiv.org/abs/math/0305049).


higher order frame bundle

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

#Contents#
* table of contents
{:toc}

## Idea

The _higher order frame bundle_ is higher order [[jet]] version of the [[frame bundle]].

Its [[structure group]] is the [[jet group]] $GL^k$ of the given order.

In [[synthetic differential geometry]]/[[differential cohesion]] the higher order frame bundle is the bundle of [[infinitesimal disks]] of the given order. It is the [[principal bundle]] ([[principal infinity-bundle]]) to which the [[infinitesimal disk bundle]] is the [[associated bundle]] ([[associated infinity-bundle]])

See at _[differential cohesion -- Frame bundles](differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles)_.

## References

* [[Demeter Krupka]], Josef Janyška,  _Lectures on differential invariants_, Univerzita JEP, Brno, 1990.

* [[Ivan Kolář]], [[Peter Michor]], [[Jan Slovák]], Section 12.12 of _[[Natural operations in differential geometry]]_, Springer 1993
([doi:10.1007/978-3-662-02950-3](https://link.springer.com/book/10.1007/978-3-662-02950-3), [pdf](http://www.emis.de/monographs/KSM/kmsbookh.pdf))

* [[Ivan Kolář]], _Connections on higher order frame bundles and their gauge analogies_ [pdf](http://www.math.muni.cz/~kolar/KolKRU.pdf)

* [MO discussion](http://mathoverflow.net/a/55322/381)

[[!redirects higher order frame bundles]]


higher orientifold

higher [[orientifold]]

[[!redirects higher orientifolds]]


higher parallel transport

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

#Contents#
* table of contents
{:toc}

## Idea

A [[connection on a bundle]] induces a notion of [[parallel transport]] over _paths_ . A [[connection on a 2-bundle]] induces a generalization of this to a notion of parallel transport over _surfaces_ . Similarly a [[connection on a 3-bundle]] induces a notion of parallel transport over 3-dimensional volumes.

Generally, a [[connection on an ∞-bundle]] induces a notion of parallel transport in arbitrary dimension.

## Definition

The higher notions of [[differential cohomology]] and [[Chern-Weil theory]] make sense in any [[cohesive (∞,1)-topos]]

$$
  (\Pi \dashv Disc \dashv \Gamma) :
   \mathbf{H}
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd \simeq Top
  \,.
$$

In every such there is a notion of [[connection on an ∞-bundle]] and of its higher parallel transport.

A typical context considered (more or less explicitly) in the literature is $\mathbf{H} = $ [[Smooth∞Grpd]], the cohesive $(\infty,1)$-topos of [[∞-Lie groupoid|smooth ∞-groupoids]]. But other choices are possible. (See also the [Examples](#Examples).)

### Higher parallel transport

Let $A$ be an [[∞-Lie groupoid]] such that morphisms $X \to A$ in [[Smooth∞Grpd]] classify the $A$-[[principal ∞-bundle]]s under consideration. Write $A_{conn}$ for the differential refinement described at [[∞-Lie algebra valued form]], such that lifts

$$
  \array{
     && A_{conn}
     \\
     & {}^{\mathllap{\nabla}}\nearrow & \downarrow
     \\
     X &\stackrel{g}{\to}& A
  }
$$

describe [[connections on ∞-bundles]].

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

For $n \in \mathbb{N}$ say that $\nabla$ **admits parallel $n$-transport** if for all [[smooth manifold]]s $\Sigma$ of [[dimension]] $n$ and all morphisms

$$
  \phi : \Sigma \to X
$$

we have that the pullback of $\nabla$ to $\Sigma$

$$
  \phi^* \nabla : \Sigma \stackrel{\phi}{\to} X \stackrel{\nabla}{\to} A_{conn}
$$

**flat** in that it factors through the canonical inclusion $\mathbf{\flat}A \to A_{conn}$.

In other words: if all the lower [[curvature]] $k$-forms, $1 \leq k \leq n$ of $\phi^* \nabla$ vanish (the higher ones vanish automatically for dimensional reasons).

=--

Here $\mathbf{\flat}A = [\mathbf{\Pi}(-),A]$ is the coefficient for [[schreiber:path ∞-groupoid|flat differential A-cohomology]].

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

This condition is automatically satisfied for ordinary [[connections on bundles]], hence for $A = \mathbf{B}G$ with $G$ an ordinary [[Lie group]]: because in that case there is only a single curvature form, namely the ordinary [[curvature]] 2-form.

But for a [[principal 2-bundle]] with connection there is in general a 2-form curvature and a 3-form curvature. A 2-connection therefore admits parallel transport only if its 2-form curvature is constrained to vanish.

Notice however that if the coefficient object $A$ happens to be $(n-1)$-[[connected]] -- for instance if it is an [[Eilenberg-MacLane object]] in degree $n$, then there is no extra condition and _every_ connection admits parallel transport. This is notably the case for [[circle n-bundles with connection]].

=--

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

For $\nabla : X \to A_{conn}$ an $\infty$-connection that admits parallel $n$-transport, this is for each $\phi : \Sigma \to X$ the morphism

$$
  \mathbf{\Pi}(\Sigma) \to A
$$

that corresponds to $\phi^* \nabla$ under the equivalence

$$
  \mathbf{H}(\Sigma, \mathbf{\flat}A ) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), A)
  \,.
$$

=--

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

The objects of the path ∞-groupoid $\mathbf{\Pi}(\Sigma)$ are points in $\Sigma$, the morphisms are paths in there, the 2-morphisms surfaces between these paths, and so on. Hence a morphism $\mathbf{\Pi}(\Sigma) \to A$ assigns fibers in $A$ to points in $X$, and equivalences between these fibers to paths in $\Sigma$, and so on.

=--

### Higher holonomy

We now define the higher [[analogs]] of [[holonomy]] for the case that $\Sigma$ is closed.

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

Let $\nabla : X \to A_{conn}$ be a connection with parallel $n$-transport and $\phi : \Sigma \to X$ a morphism from a _closed_ $n$-[[manifold]].

Then the **$n$-holonomy** of $\nabla$ over $\Sigma$ is the image $[\phi^* \nabla]$ of

$$
  \phi^* \nabla : \Pi(\Sigma) \to \Gamma(A)
$$

in the [[homotopy category]]

$$
  [\phi^* \nabla] \in [\Pi(\Sigma), \Gamma(A)]
$$

=--

## Examples {#Examples}

### For trivial circle $n$-bundles / for $n$-forms

The simplest example is the parallel transport in a [[circle n-bundle with connection]] over a [[smooth manifold]] $X$ whose underlying $\mathbf{B}^{n-1}U(1)$-bundle is trivial. This is equivalently given by a degree $n$-[[differential form]] $A \in \Omega^n(X)$. For $\phi : \Sigma_n \to X$ any [[smooth function]] from an $n$-dimensional manifold $\Sigma$, the corresponding parallel transport is simply the [[integral]] of $A$ over $\Sigma$:

$$
  \tra_A(\Sigma) = \exp(i \int_\Sigma \phi^* A)
  \;\;\; \in \;\; U(1)
  \,.
$$

One can understand higher parallel transport therefore as a generalization of integration of differential $n$-forms to the case where

* the $n$-form is not globally defined;

* the $n$-form takes values not in $\mathbb{R}$ but more generally is an [[∞-Lie algebroid valued differential form]].

### For circle $n$-bundles with connection

We show how the $n$-holonomy of [[circle n-bundles with connection]] is reproduced from the above.

Let $\phi^* \nabla : \mathbf{\Pi}(\Sigma) \to \mathbf{B}^n U(1)$ be the parallel transport for a [[circle n-bundle with connection]] over a $\phi : \Sigma \to X$.

This is equivalent to a morphism

$$
  \Pi(\Sigma) \to \mathcal{B}^n U(1)
  ,.
$$

We may map this further to its $(n-dim \Sigma)$-[[nLab:truncated|truncation]]

$$
  :\infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1))
  \to
  \tau_{n-dim \Sigma} \infty Grpd(\Pi(X), \mathcal{B}^n U(1))
  \,.
$$

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

We have

$$
  \tau_{n-dim\Sigma} \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1))
  \simeq
  \mathbf{B}^{n-dim \Sigma} U(1)
  \,.
$$

=--

(This is due to an observation by [[nLab:Domenico Fiorenza]].)

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

By general abstract reasoning (recalled at [[nLab:cohomology]] and [[nLab:fiber sequence]]) we have for the [[nLab:homotopy group]]s that

\[
  \pi_i \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))
  \simeq
  H^{n-i}(\Sigma, U(1))
  \,.
\]

Now use the [[nLab:universal coefficient theorem]], which asserts that we have an [[nLab:exact sequence]]

\[
  0
  \to
  Ext^1(H_{n-i-1}(\Sigma,\mathbb{Z}),U(1))
  \to
  H^{n-i}(\Sigma,U(1))
  \to
  Hom(H_{n-i}(\Sigma,\mathbb{Z}),U(1))
  \to 0
  \,.
\]

Since $U(1)$ is an [[nLab:injective object|injective]] $\mathbb{Z}$-[[nLab:module]] we have

$$
  Ext^1(-,U(1))=0
  \,.
$$

This means that we have an [[nLab:isomorphism]]

\[
  H^{n-i}(\Sigma,U(1))
  \simeq
  Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1))
\]

that identifies the [[nLab:cohomology group]] in question with the [[nLab:internal hom]] in [[nLab:Ab]] from the integral [[nLab:homology]] group of $\Sigma$ to $U(1)$.

For $i\lt (n-dim \Sigma)$, the right hand is zero, so that

$$
  \pi_i \infty Grpd(\Pi(\Sigma),\mathbf{B}^n U(1)) =0 \;\;\;\;
  for i\lt (n-dim \Sigma)
  \,.
$$

For $i=(n-dim \Sigma)$, instead, $H_{n-i}(\Sigma,\mathbb{Z})\simeq \mathbb{Z}$, since $\Sigma$ is a closed $dim \Sigma$-manifold and so

$$
  \pi_{(n-dim\Sigma)} \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))\simeq U(1)
  \,.
$$

=--

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

The resulting morphism

$$
  \mathbf{H}(\Sigma, A_{conn})
  \stackrel{\exp(i S(-))}{\to}
  \mathbf{B}^{n-dim\Sigma} U(1)
$$

in [[nLab:∞Grpd]] we call the **$\infty$-Chern-Simons action** on $\Sigma$.

=--

Here in the language of [[nLab:quantum field theory]]

* the [[nLab:object]]s of $\mathbf{H}(\Sigma,A_{conn})$ are the [[nLab:gauge field]] on $\Sigma$;

* the [[nLab:morphism]]s in $\mathbf{H}(\Sigma, A_{conn})$ are the [[nLab:gauge transformation]]s;.

### Nonabelian parallel transport in low dimension

At least in low categorical dimension one has the definition of the [[path n-groupoid]] $\mathbf{P}_n(X)$ of a smooth manifold, whose $n$-morphisms are [[thin homotopy]]-classes of smooth functions $[0,1]^n \to X$. Parallel $n$-transport with only the $(n+1)$-curvature form possibly nontrivial and all the lower curvature degree 1- to $n$-forms nontrivial may be expressed in terms of smooth $n$-functors out of $\mathbf{P}_n$ ([SWI](#SWI), [SWII](#SWII), [MartinsPickenI](#MartinsPickenI), [MartinsPickenII](#MartinsPickenII)).

#### 1-Transport

See [[parallel transport]].

#### 2-Transport

We work now concretely in the category $2DiffeoGrpd$ of [[2-groupoid]]s [[internalization|internal to]] the category of [[diffeological space]]s.

Let $X$ be a [[smooth manifold]] and write $\mathbf{P}_2(X) \in 2DiffeoGrpd$ for its [[path 2-groupoid]]. Let $G$ be a [[Lie 2-group]] and $\mathbf{B}G \in 2DiffeoGrpd$ its [[delooping]] 1-object 2-groupoid. Write $\mathfrak{g}$ for the corresponding [[Lie 2-algebra]].

Assume now first that $G$ is a [[strict 2-group]] given by a [[crossed module]] $(G_1 \to G_0)$. Corresponding to this is a [[differential crossed module]] $(\mathfrak{g}_1 \to \mathfrak{g}_0)$.

We describe now how smooth 2-functors

$$
  tra : \mathbf{P}_2(X) \to \mathbf{B}G
$$

i.e. morphisms in $2DiffeoGrpd$ are characterized by [[Lie 2-algebra valued differential forms]] on $X$.

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

Given a morphism $F : \mathbf{P}_2(X) \to \mathbf{B}G$ we construct a $\mathfrak{g}_1$-valued 2-form  $B_F \in \Omega^2(X, \mathfrak{g}_1)$ as follows.

To find the value of $B_F$ on two vectors $v_1, v_2 \in T_p X$ at some point, _choose_ any smooth function

$$
  \Gamma : \mathbb{R}^2 \to X
$$

with

* $\Gamma(0,0) = p$

* $\frac{d}{d s}|_{s = 0} \Gamma(s,0) = v_1$

* $\frac{d}{d t}|_{t = 0} \Gamma(0,t) = v_2$.

Notice that there is a canonical 2-parameter family

$$
  \Sigma_{\mathbb{R}} : \mathbb{R}^2 \to 2Mor \mathbf{P}_2(\mathbb{R}^2)
$$

of classes of bigons on the plane, given by sending $(s,t) \in \mathbb{R}^2$ to the class represented by any bigon (with sitting instants) with straight edges filling the square

$$
  \Sigma_{\mathbb{R}}(s,t) =
  \left(
    \array{
      (0,0) &\to& (0,t)
      \\
      \downarrow && \downarrow
      \\
      (s,0) &\to& (s,t)
    }
  \right)
  \,.
$$

Using this we obtain a smooth function

$$
  F_\Gamma :
  \mathbb{R}^2 \stackrel{\Sigma_{\mathbb{R}}}{\to}
  2Mor \mathbf{P}_2(\mathbb{R}^2)
  \stackrel{\Gamma_*}{\to}
  2Mor \mathbf{P}_2(X)
  \stackrel{F}{\to}
  G_0 \times G_1
  \stackrel{p_2}{\to}
  G_1
  \,.
$$

Then set

$$
  B_F(v_1, w_1) :=
   \frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(0,0)}
  \,.
$$

=--

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

This is well defined, in that $B_F(v_1,v_2)$ does not depend on the choices made. Moreover, the 2-form defines this way is smooth.

=--

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

To see that the definition does not depend on the choice of $\Gamma$, proceed as follows.

For given vectors $v_1,v_2 \in \T_X X$ let $\Gamma, \Gamma' : \mathbb{R}^2 \to X$ be two choices of smooth maps as in the defnition.  By restricting, if necessary, to a neighbourhood of the origin of $\mathbb{R}^2$, we may assume without restriction that these maps land in a single coordinate patch in $X$. Using the vector space structure of $\mathbb{R}^n$ defined by such a patch, define a smooth homotopy

$$
  \tau : [0,1]^3 \to X : (x,y,z) \mapsto (1-z)\Gamma(x,y) + z \Gamma'(x,y)
$$

Let

$$
  Z = \{(x,y,w) \in [0,1]^3 | 0 \leq w \leq \frac{1}{2}(x^2 + y^2) \}
$$

and consider the map $f : [0,1]^3 \to Z$ given by

$$
  f : (x,y,z) \mapsto (x,y, \frac{1}{2}(x^2 + y^2) z)
$$

and the map $g : Z \to X$ given away from $(x^2 + y^2) = 0$ by

$$
  g : (x,y,w) \mapsto \tau(x,y, 2 \frac{w}{x^2 + y^2})
  \,.
$$

Using [[Hadamard's lemma]] and the fact that by constructon $\tau$ has vanishing 0th and 1st order differentials at the origin it follows that this is indeed a [[smooth function]].

We want to similarly factor the smooth family of bigons $[0,1]^3 \to 2Mor(\mathbf{P}_2(X))$ given by

$$
  [0,1]^3 \times [0,1]^2 \to X
$$

$$
  ((x,y,z),(s,t)) \mapsto \tau(s x, t y, z)
$$

as $[0,1]^3 \times [0,1]^2 \to Z \times [0,1]^2 \to Z \to X$

$$
  ((x,y,z),(s,t))
    \mapsto
  ((x, y, \frac{1}{2}(x^2 + y^2)), (s,t))
    \mapsto
  (s x , t y, \frac{1}{2}((s x)^2 + (t y)^2)z)
    \mapsto

  \tau(s x, s y, z)
  \,.
$$

As before using Hadamard's lemma this is a sequence of smooth functions. To make this qualify as a family of bigons (which are maps from the square that are constant in a [[neighbourhood]] of the left and right boundary of the square) furthermore precompose this with a suitable smooth function $[0,1]^2 \to [0,1]^2$ that realizes a square-shaped bigon.

Under the hom-adjunction it corresponds to a factorization of $G_\Gamma : [0,1]^3 \to 2 Mor(\mathbf{P}_2(X))$ into

$$
 G_\Gamma :  [0,1]^3 \stackrel{f}{\to} Z \to 2 Mor(\mathbf{P}_2(X))
  \,.
$$

By the above construction we have the the push-forwards

$$
  f_* : \frac{\partial}{\partial x}(x=0,y=0,z)
    \mapsto
    \frac{\partial}{\partial x}(x= 0, y = 0, w = 0)
$$

and similarly for $\frac{\partial}{\partial y}$ are indendent of $z$. It follows by the [[chain rule]] that also

$$
  \frac{\partial^2 G_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}
$$

is independent of $z$. But at $z = 0$ this equals $\frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}$, while at $z = 1$ it equals $\frac{\partial^2 F_{\Gamma'}}{\partial x \partial y}|_{(x=0,y=0)}$. Therefore these two are equal.

=--

#### 3-Transport

see [[3-groupoid of Lie 3-algebra valued forms]]

### Flat $\infty$-parallel transport in $Top$ {#FlatInTop}

Even though it is a degenerate case, it can be useful to regard the [[(∞,1)-topos]] [[Top]] explicitly a [[cohesive (∞,1)-topos]]. For a discussion of this see [[discrete ∞-groupoid]].

For $\mathbf{H} = $ [[Top]] lots of structure of cohesive $(\infty,1)$-topos theory  degenerates, since by the [[homotopy hypothesis]]-theorem here the [[global section]] [[(∞,1)-geometric morphism]]

$$
  (\Pi \dashv \Delta \dashv \Gamma) : Top
   \stackrel{\overset{\Pi}{\leftarrow}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \in
 \infty Grpd
$$

an [[equivalence in an (∞,1)-category|equivalence]]. The abstract [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\Pi$ is here the ordinary [[fundamental ∞-groupoid]]

$$
   \Pi : Top \stackrel{\simeq}{\to} \infty Grpd
  \,.
$$

If both [[(∞,1)-topos]]es here are [[presentable (∞,1)-category|presented]] by their standard [[model category]] models, the standard [[model structure on simplicial sets]] and the standard [[model structure on topological spaces]], then $\Pi$ is presented by the [[singular simplicial complex]] functor in a [[Quillen equivalence]]

$$
  (|-| \dashv Sing) : Top \stackrel{\leftarrow}{\overset{\simeq_{Quillen}}{\to}}
  Top
  \,.
$$

This means that in this case many constructions in [[topology]] and classical [[homotopy theory]] have equivalent reformulations in terms of $\infty$-parallel transport.

For instance: for $F \in Top$ and $Aut(F) \in Top$ its [[automorphism ∞-group]], $F$-fibrations over a base space $X \in Top$ are classfied by morphisms

$$
  g : X \to B Aut(F)
$$

into the [[delooping]] of $Aut(F)$. The corresponding fibration $P \to X$ itself is the [[homotopy fiber]] of this cocycles, given by the [[homotopy pullback]]

$$
  \array{
     P &\to& *
     \\
     \downarrow && \downarrow
     \\
     X &\stackrel{g}{\to}&  B Aut(F)
  }
$$

in [[Top]], as described at [[principal ∞-bundle]].

Using the [[fundamental ∞-groupoid]] functor we may send this equivalently to a [[fiber sequence]] in [[∞Grpd]]

$$
  \Pi(P) \to \Pi(X) \to B Aut(\Pi(F))
  \,.
$$

One may think of the morphism $\Pi(X) \to B Aut(\Pi(F))$ now as the $\infty$-parallel transport coresponding to the original fibration:

* to each point in $X$ it assigns the unique object of $B Aut(\Pi(F))$, which is the fiber $F$ itself;

* to each path $(x \to y)$ in $X$ it assigns an equivalence between the fibers $F_x to F_y$ etc.

If one presents $\Pi$ by $Sing : Top \to sSet_{Quillen}$ as above, then one may look for explicit simplicial formulas that express these morphisms. Such are discussed in [Stasheff](#Stasheff).

We may embed this example into the smooth context by regarding $Aut(F)$ as a [[discrete space|discrete]] [[∞-Lie groupoid]] as discussed in the section
[Flat ∞-Parallel transport in Smooth∞Grpd](#InLieGrpd).

For that purpose let

$$
  (\Pi_{smooth} \dashv Disc_{smooth} \dashv \Gamma_{smooth}) :
   \infty LieGrpd
    \stackrel{\overset{\Pi_{smooth}}{\to}}{\stackrel{\overset{Disc_{smooth}}{\leftarrow}}{\underset{\Gamma_{smooth}}{\to}}}
  \infty Grpd \simeq Top
$$

be the [[global section]] [[(∞,1)-geometric morphism]] of the  [[cohesive (∞,1)-topos]] [[Smooth∞Grpd]].

We may reflect the [[∞-group]] $Aut(F)$ into this using the [[constant ∞-stack]]-functor $Disc$ to get the discrete [[∞-Lie group]] $Disc Aut(F)$. Let then $X$ be a [[paracompact topological space|paracompact]] [[smooth manifold]], regarded naturally as an object of [[Smooth∞Grpd]]. Then we can consider [[cocycle]]s/classifying morphisms

$$
  X \to \mathbf{B} Disc Aut(F)
  \,,
$$

now in the smooth context of $\infty LieGrpd$.

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

The [[∞-groupoid]] of $F$-fibrations in [[Top]] is equivalent to the $\infty$-groupoid of $Disc Aut(F)$-[[principal ∞-bundle]]s in [[Smooth∞Grpd]]:

$$
  \infty LieGrpd(X, \mathbf{B} Disc Aut(F))
  \simeq
  Top(X, B Aut(F))
  \,.
$$

Moreover, all the [[principal ∞-bundle]]s classified by the morphisms on the left have canonical extensions to Flat differential cohomology in $\infty LieGrpd$, in that the flat parallel $\infty$-transport $\nabla_{flat}$ in

$$
  \array{
    X &\stackrel{g}{\to}& \mathbf{B} Disc Aut(F)
    \\
    \downarrow & \nearrow_{\nabla_{flat}}
    \\
    \mathbf{\Pi}(X)
  }
$$

always exists.

=--

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

The first statement is a special case of that spelled out at [[Smooth∞Grpd]] and [[nonabelian cohomology]]. The second follows using that in a [[∞-connected (∞,1)-topos|connected]] [[locally ∞-connected (∞,1)-topos]] the functor $Disc$ is a [[full and faithful (∞,1)-functor]].

=--

### Flat $\infty$-parallel transport in $\infty LieGrpd$ {#InLieGrpd}

(...)

### $\infty$-Parallel transport from flat differential forms with values in chain complexes

A typical choice for an [[(∞,1)-category]] of "$\infty$-vector spaces" is that [[presentable (∞,1)-category|presented]] by the a [[model structure on chain complexes]] of modules. In a geometric context this may be replaced by some stack of complexes of vector bundles over some site.

If we write $Mod$ for this stack, then the $\infty$-parallel transport for a flat $\infty$-vector bundle on some $X$ is a morphism

$$
  \mathbf{\Pi}(X) \to Mod
  \,.
$$

This is typically given by differential form data with values in $Mod$.

A discussion of how to integrate flat differential forms with values in chain complexes -- a representation of the [[tangent Lie algebroid]] as discussed at [[representations of ∞-Lie algebroids]] --
to flat $\infty$-parallel transport $\mathbf{\Pi}(X) \to Mod$ is in ([AbadSchaetz](#abadSchaetz)), building on a construciton in ([Igusa](#Igusa)).

## Applications

### In physics

In [[physics]] various [[action functional]]s for [[quantum field theories]] are nothing but higher parallel transport.

* The gauge interaction part of the action functional for the particle charged under a background [[electromagnetic field]], which is a [[circle n-bundle with connection|circle bundle with connection]] $\nabla$, is the parallel 1-transport of $\nabla$.

* The gauge interaction part of the action functional for the string charged under a background [[Kalb-Ramond field]], which is a [[circle n-bundle with connection|circle 2-bundle with connection]] $\nabla$, is the parallel 2-transport of $\nabla$.

* The gauge interaction part of the action functional for the membrane charged under a background [[supergravity C-field]], which is a [[circle n-bundle with connection|circle 3-bundle with connection]] $\nabla$, is the parallel 3-transport of $\nabla$.

* The action functional of [[Chern-Simons theory]] is the parallel 3-transport of a [[Chern-Simons circle 3-bundle]].

## Related concepts

* [[connection on a bundle]], [[connection on a 2-bundle]], [[connection on an infinity-bundle]],

* [[parallel transport]], **higher parallel transport**

* [[holonomy]]

  * [[holonomy group]]

  * [[special holonomy]]

* [[Wilson surface]]

## References
 {#References}

For references on ordinary 1-dimensional parallel transport see [[parallel transport]].

For references on parallel 2-transport in [[bundle gerbes]] see [[connection on a bundle gerbe]].

The description of parallel $n$-transport in terms of $n$-functors on the [[path n-groupoid]] for low $n$ is in

* [[Urs Schreiber]],  [[Konrad Waldorf]], _Smooth Functors and Differential Forms_, Homology, Homotopy Appl., 13(1), 143-203 (2011) ([arXiv:0802.0663](http://arxiv.org/abs/0802.0663))

* {#MartinsPickenI} [[João Faria Martins]], [[Roger Picken]], _On 2-dimensional holonomy_ ([arXiv](http://arxiv.org/abs/0710.4310))

* {#MartinsPickenII} [[João Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_ ([arXiv](http://arxiv.org/abs/0907.2566))

The description of [[connections on a 2-bundle]] in terms of such parallel 2-transport

* [[John Baez]], [[Urs Schreiber]], _Higher gauge theory_, in A. Davydov et al. (eds.), _Categories in Algebra, Geometry and Mathematical Physics_, Contemp Math 431, AMS, Providence, Rhode Island (2007) pp 7-30 ([arXiv:0511710](http://arxiv.org/abs/math/0511710), [arXiv:hep-th/0412325](http://arxiv.org/abs/hep-th/0412325))

* {#SWIII} [[Urs Schreiber]], [[Konrad Waldorf]], _Connections on nonabelian gerbes and their holonomy_, Theory Appl. Categ., Vol. 28, 2013, No. 17, pp 476-540 ([arXiv:0808.1923](http://arxiv.org/abs/0808.1923), [TAC](http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html))

Much further discussion and illustration and relation to [[tensor networks]] is in

* {#Parzygnat18} [[Arthur Parzygnat]], _Two-dimensional algebra in lattice gauge theory_, Journal of Mathematical Physics 60, 043506 (2019) ([arXiv:1802.01139](https://arxiv.org/abs/1802.01139), [doi:10.1063/1.5078532](https://doi.org/10.1063/1.5078532))

Applications are discussed in

* [[Arthur Parzygnat]], _Gauge invariant surface holonomy and monopoles_, Theory and Applications of Categories, Vol. 30, 2015, No. 42, pp 1319-1428 ([TAC](http://www.tac.mta.ca/tac/volumes/30/42/30-42abs.html))

Discussion via [[adjusted Weil algebras]]:

* {#KimSaemann20} [[Hyungrok Kim]], [[Christian Saemann]], *Adjusted Parallel Transport for Higher Gauge Theories*, J. Phys. A **52** (2020) 445206 [[arXiv:1911.06390](https://arxiv.org/abs/1911.06390), [doi:10.1088/1751-8121/ab8ef2](https://doi.org/10.1088/1751-8121/ab8ef2)]

Parallel transport for [[circle n-bundles with connection]] is discussed generally in

* [[Kiyonori Gomi]], [[Yuji Terashima]], _Higher dimensional parallel transport_ Mathematical Research Letters 8, 25–33 (2001) ([pdf](http://mrlonline.org/mrl/2001-008-001/2001-008-001-004.pdf))

* David Lipsky, _Cocycle constructions for topological field theories_ (2010) ([[LipskyThesis.pdf:file]])

see also the discussion at _[[fiber integration in ordinary differential cohomology]]_.

Realization of this as an [[extended TQFT]] is discussed in

* [[Nersés Aramyan]], Research statement ([pdf](http://math.illinois.edu/~aramyan2/research.pdf))

Parallel transport with [[coefficients]] in [[crossed complexes]]/[[strict infinity-groupoids]] is discussed in

* {#Kapranov15} [[Mikhail Kapranov]], _Membranes and higher groupoids_ ([arXiv:1502.06166](http://arxiv.org/abs/1502.06166))

The integration of flat differential forms with values in chain complexes to flat $\infty$-parallel transport on $\infty$-vector bundles is in

* {#AbadSchaetz} [[Camilo Arias Abad]], [[Florian Schaetz]], _The $A_\infty$ de Rham theorem and integration of representations up to homotopy_ ([arXiv](http://arxiv.org/abs/1011.4693))

based on

* Jonathan Block, Aaron Smith, _A Riemann Hilbert correspondence for infinity local systems_ ([arXiv](http://arxiv.org/abs/0908.2843))
{#BlockSmith}

in turn based on constructions in

* [[Kiyoshi Igusa]], _Iterated integrals of superconnections_ ([arXiv](http://arxiv.org/abs/0912.0249))
{#Igusa}

Remarks on $\infty$-parallel transport in [[Top]] are in

* {#Stasheff}[[Jim Stasheff]], _[[StasheffParallelTransportv02.pdf:file]]_

Understanding [[higher parallel transport]] of [[circle n-bundles with connection]] as an [[extended functorial field theory|extended]] [[homotopy field theory]]:

* [[Lukas Müller]], [[Lukas Woike]], *Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory*,  J. Homotopy Relat. Struct. **15** (2020) 113–142  [[arXiv:1802.10455](https://arxiv.org/abs/1802.10455), [doi:10.1007/s40062-019-00242-3](https://doi.org/10.1007/s40062-019-00242-3)]

See also:

* {#LeeOberhauser23} Darrick Lee, Harald Oberhauser, *Random Surfaces and Higher Algebra* [[arXiv:2311.08366](https://arxiv.org/abs/2311.08366)]

[[!redirects surface holonomy]]
[[!redirects surface holonomies]]

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higher Poisson structure

There are various proposals for and definitions of higher order generalization of the concept of _[[Poisson brackets]]_.

Dedicated $n$lab entries exist so far for the following cases:

* _[[Poisson n-algebra]]_ (rationalied [[En-algebra]] stucture)

* _[[Poisson bracket Lie n-algebra]]_ ([[L-infinity algebra]] structure)

Definitons analogous to [[Nambu brackets]] are discussed in

* [[José de Azcárraga]], [[José M. Izquierdo]], J. C. Perez Bueno, *On the generalizations of Poisson structures*, J. Phys. A **30** (1997) L607-L616 [[arXiv:hep-th/9703019](http://arxiv.org/abs/hep-th/9703019), [doi:10.1088/0305-4470/30/18/001](https://doi.org/10.1088/0305-4470/30/18/001)]

* Raúl Ibáñez, Juan C. Marrero, and David Martn de Diego, _Dynamics of generalized Poisson and Nambu–Poisson brackets_, Journal of Mathematical Physics 38, 2332 (1997); doi: 10.1063/1.531960

 [pdf](http://digital.csic.es/bitstream/10261/102443/1/RIbañez.pdf)

Definitions that yield Filippov [[n-Lie algebras]] include

* [[Janusz Grabowski]], Giuseppe Marmo, _On Filippov algebroids and multiplicative Nambu-Poisson structures_, Differ. Geom. Appl. 12:35-50,2000 ([arXiv:math/9902127](http://arxiv.org/abs/math/9902127))

[[!redirects higher Poisson structures]]

[[!redirects higher Poisson bracket]]
[[!redirects higher Poisson brackets]]

[[!redirects higher-ary Poisson bracket]]
[[!redirects higher-ary Poisson brackets]]