non-perturbative quantum field theory

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## Idea
 {#Idea}

The description of [[quantum field theory]] is traditionally mostly considered only in the [[infinitesimal neighbourhood]] of the underlying [[classical field theory|classical]] and [[free field theory]] As such these descriptions are referred to as _[[perturbative quantum field theory]]_ (pQFT) since they describe only small (in fact: infinitesimal) "[[perturbations]]" of a (hypothetical) free and classical field, instead of the fully interacting quantum fields.

Since this very coarse (but remarkably successful) perturbative concept of quantum field theory has come to often be considered by default, one speaks of _non-perturbative quantum field theory_ in order to amplify that the full theory is meant to be considered, not just the perturbative approximation.

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    "caption": "(from [SS24, p. 3](#SS24IHH))"
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{#BeyondInfinitesimalNeighbourhood} **Beyond the infinitesimal neighbourhood of perturbation theory.** More concretely, if we think of generic ([[Lagrangian field theory|Lagrangian]]) [[quantum field theories]] as parameterized by

1. [[Planck's constant]]$\;\hbar$,

   expressing the "strength of [[quantum physics|quantum effects]]",

1. [[coupling constants]]$\;g$,

   expressing the strength of [[interactions]] of [[field (physics)|fields]],

then [[perturbative QFT]] is concerned (just) with computing the [[Feynman perturbation series]] for [[scattering amplitudes]] as (just) a [[formal power series]] in these [[variables]], and since this typically has a *vanishing* [[radius of convergence]] (see [there](perturbative+quantum+field+theory#ReferencesNonConvergenceOfThePerturbationSeries)) it concerns just the [[infinitesimal neighbourhood]] of the origin in this parameter space (the infinitesimal [[halo]] of a single point) -- all the rest of the parameter space is the region of non-perturbative quantum field theory.

**As perturbation with non-perturbative effects.**
Passage from the perturbative to the non-perturbative description of QFT involves taking account of *[[non-perturbative effects]]* invisible to perturbation theory.

These arise notably from the presence of [[soliton|solitonic]] field configurations ([[instanton|instantonic]], after [[Wick rotation]]): Such solitonic fields are invisible to perturbation theory because  they are not continuously [[connected topological space|connected]] to the [[vacuum]] field configuration, since they are constituted by topologically non-trivial structures, e.g. [[gauge potentials]] which are [[connection on a bundle|connections]] on a topologically non-trivial [[principal bundle]] on [[spacetime]] (such as [[Dirac monopoles]] or [[Yang-Mills instantons]], see at *[[fiber bundles in physics]]*).

In lack of a general prescription for non-perturbative [[quantization]], authors try to make educated guesses about the would-be non-perturbative effects from inspection of the behaviour of the formal [[Feynman perturbation series]]: popular methods here go by the name *[[Borel resummation]]* and *[[resurgence theory]]*.

**Theory and reality.** Beware the usual habit in theoretical physics jargon of conflating the map with the territory (the field *theory* with the actual fields "in nature" that the theory means to describe): Real quantum fields in nature are always non-perturbative (the value of $\hbar$ and $g$ is not contained in the infinitesimal neightbourhood of the origin -- this is a purely theoretical/computational concept) so that their description by [[perturbative QFT]] is, by design, just an approximation of their complete nature.

**As non-analyticity.** This is just as with [[smooth functions]] and their description by their [[Taylor series]] of their [[derivatives]] at a given origin: The Taylor series is a "perturbative description" of the full smooth function, which in contrast is the full non-perturbative object. Indeed the [[Feynman perturbation series]] in [[pQFT]] are imagined to be just such Taylor series of actual smooth functions (e.g. [[scattering amplitudes]]) in $\hbar$ and $g$, and the goal of non-perturbative QFT is to provide more information on "[[non-perturbative effects]]", hence on the actual functions (scattering amplitudes) at all or at least some positive values of $\hbar$ and/or $g$.

Of course, special among all smooth functions are the [[analytic functions]], whose Taylor series does already capture the full function, at least in some [[open neighbourhood]].
Accordingly, among QFTs that one can write to paper (but which are unlikely to describe anything in the real world) there are interacting examples (cite...) whose perturbative description already exhausts their full non-perturbative content, in that their [[Feynman perturbation series]] actually [[convergence|converges]], or at least has a [[positive number|positive]] [[radius of convergence]].

**Axiomatization and construction.** Among *axiomatizations* of non-perturbative field theory is

1. *[[algebraic quantum field theory]]* (AQFT) in terms of [[local nets of observables]] with values in [[C-star algebra|$C^\ast$-algebras]] (axiomatizing the [[Heisenberg picture]] of [[quantum physics]]),

1. *[[functorial quantum field theory]]* in terms of [[cobordism category]] [[linear representation|representations]] (axiomatizing the [[Schrödinger picture]] of [[quantum physics]]).

Among *construction* methods of AQFT has been "[[constructive field theory]]", which aims to make rigorous [[analysis|analytic]] sense of [[path integrals]] in simple cases. This has worked for [[scalar field theory]] such as [[phi^4 theory|$\phi^4$-theory]]  in spacetime dimension 2 and 3, but (so far) not in dimension 4.

There are also some partial tools available from [[quantum mechanics]] such as [[strict deformation quantization]] ([Rieffel 1989](C-star+algebraic+deformation+quantization#Rieffel89)). Here it may be noteworthy that the established construction of [[perturbative quantum field theory]] has been understood to be an example of the systematic process of [[quantization]] called *[[formal deformation quantization]]*, specifically as an example of [[Fedosov deformation quantization]] ([Collini 16](perturbative+algebraic+quantum+field+theory#Collini16)). Since the non-perturbative version of [[formal deformation quantization]] is _[[strict C*-algebraic deformation quantization]]_ the latter might be a useful perspective on the problem of non-perturbative QFT (cf. e.g. [SS23](#SS23QOQF)).

{#ExamplesInIdeaSection} **Examples of nonperturbative QFT.** Presently, non-perturbative quantum field theories have been constructed only for simple toy examples, notably:

* for [[free field theories]] (no interaction),

* in low [[spacetime]] [[dimension]] (e.g. [[2d CFTs]], see e.g. at *[[conformal net]]*),

* for [[topological quantum field theories]] (see e.g. at *[[quantization of 3d Chern-Simons theory]]*),

* for pure [[scalar field theory]] (most famously [[phi^n-theory|$\phi^4$-theory]], cf. [Serone 2018](#Serone18))

  > (Examples of non-perturbative [[interacting quantum field theory|interacting]] [[scalar field theory]] in _any_ [[spacetime]] [[dimension]] are claimed in [Buchholtz & Fredenhagen 2020](AQFT#BuchholtzFredenhagen20), at least in the guise of [[local nets of observables|local nets]] of [[quantum observables]], pending the construction (or existence) of compatible [[vacuum states]]).

**Key application: Confinement/Mass-gap in QCD/Yang-Mills.** A [[non-perturbative effect]] of paramount importance in [[experiment|experimentally]] observed [[quantum physics]] (and invisible to [[perturbative QFT|perturbative]] [[QCD]]) is the [[confinement]] and the [[mass gap]] of [[quantum chromodynamics]] ([[Yang-Mills theory]] with [[simple Lie group|simple]] [[gauge group]]).

That this experimentally observed confinement mechanism is indeed somehow encoded in the [[Lagrangian density]] for the [[gluons]] and [[quarks]] of [[QCD]] is plausibly verfied currently (only) by computer simulation ([[lattice QCD]]), but an analytic construction of a corresponding non-perturbative [[quantization of Yang-Mills theory]] has remained elusive enough to be pronounced one of the open "Millennium Problems" by the Clay Mathematics Institute (see [there](mass+gap#ReferencesMassGapProblem)).

However, qualitative understanding of [[confinement]] is given by an argument that separated [[quarks]] become connected by [[flux tubes]] of the [[strong nuclear force]] [[field (physics)|field]], which behave like [[strings]] whose [[string tension|string]]-[[tension]] exerts an attractive [[force]] on any pair of quarks, keeping them together and hence "[[confinement|confined]]" inside a [[hadron]] [[bound state]].

The attempt to understand non-perturbative confined QCD and hence [[quantum hadrodynamics]] as a theory of such [[flux tube]] [[strings]] has been both the origin of [[string theory]] as well as, after some detour through [[GUT]] [[quantum gravity]], its current understanding via the [[holographic principle]]: See at *[[Polyakov gauge-string duality]]*, at *[[AdS/QCD correspondence]]* and at *[confinement via AdS/QCD](confinement#ViaSkyrmions)*.

However, ironically also [[string theory]] has been mostly understood only in [[perturbation theory]], which currently causes the approach of [[holographic QCD]] to require the assumption of a large (in fact: humongous) and hence unrealistic number of [[quark]] [[color charge|colors]] (namely of [[color branes]]).

But for this string theory, at least, there are hints as to its non-perturbative completion, enough so that it has a working title ("[[M-theory]]") albeit remaining elusive. See at *[AdS-CFT correspondence -- small-$N$ corrections](AdS-CFT+correspondence#SmallNCorrections)* for more on this.

## Related concepts

* [[non-perturbative effect]]

* [[Borel resummation]], [[resurgence theory]]

* [[strict deformation quantization]]

* [[conformal bootstrap]]

* [[instanton]]

* [[superselection sector]]

* [[non-perturbative string theory]]

## References

> (See also references at *[[non-perturbative effect]]*, *[[soliton]]* and *[[instanton]]*, particularly at *[[Yang-Mills instanton]]*.)

General introduction:

* Alexander P. Bakulev, [[Dmitry Shirkov]], *Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory*, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23 (2010) Belgrade, Serbia  27--54 [[arXiv:1102.2380](https://arxiv.org/abs/1102.2380), ISBN:978-86-82441-30-4]

Monographs:

* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*,  NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]

* {#Froehlich92} [[Jürg Fröhlich]], _Non-perturbative quantum field theory -- Mathematical Aspects and Applications_, Advanced Series in Mathematical Physics, World Scientific (1992) [[doi:10.1142/1245](https://doi.org/10.1142/1245)]

* [[Yitzhak Frishman]], [[Jacob Sonnenschein]], *Non-Perturbative Field Theory -- From Two Dimensional Conformal Field Theory to QCD in Four Dimensions*, Cambridge University Press (2010) [[doi:10.1017/CBO9780511770838](https://doi.org/10.1017/CBO9780511770838), summary: [arXiv:1004.4859](https://arxiv.org/abs/1004.4859)].
open access (2023) [[doi:10.1017/9781009401654](https://doi.org/10.1017/9781009401654)]

* [[Franco Strocchi]], *An Introduction to Non-Perturbative Foundations of Quantum Field Theory*, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

  > ([[AQFT]] perspective)

Outlook from the point of view of [[causal perturbation theory]]/[[locally covariant perturbative quantum field theory]] is in

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], section 4.1 of _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026))

The case of [[scalar field theory]] (such as [[phi^n-theory|$\phi^n$-theory]]):

* {#Serone18} Marco Serone, from 2:46 on in _A look at $\phi^4_2$ using perturbation theory_ (January 2018) [[recording](https://www.youtube.com/watch?v=J4nxvY1rOhI)&

* {#BuchholtzFredenhagen20} [[Detlev Buchholz]], [[Klaus Fredenhagen]], _A $C^\ast$-algebraic approach to interacting quantum field theories_,  Commun. Math. Phys. **377** (2020) 947–969  [[arXiv:1902.06062](https://arxiv.org/abs/1902.06062), [doi:10.1007/s00220-020-03700-9](https://doi.org/10.1007/s00220-020-03700-9)]

Lectures notes for non-perturbative [[supersymmetry|supersymmetric]] QFT (such as in [[AdS-CFT]]) includes

* John Terning, _TASI-2002 Lectures: Non-perturbative Supersymmetry_ ([arXiv:hep-th/0306119](http://arxiv.org/abs/hep-th/0306119))

On non-perturbative [[BV-BRST formalism]]:

* [[Luigi Alfonsi]], *Derived $n$-plectic geometry: towards non-perturbative BV-BFV quantisation and M-theory*, talk at *[M-Theory and Mathematics 2023](/nlab/show/M-Theory+and+Mathematics#2023)*, NYU Abu Dhabi (2023) [[web](/nlab/show/M-Theory+and+Mathematics#Alfonsi2023)]

On non-perturbative quantization of topological [[flux]] observables in pure [[higher gauge theory]]:

* {#SS23QOQF} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Quantum Observables of Quantized Fluxes]]* [[arXiv:2312.13037](https://arxiv.org/abs/2312.13037)]

* {#SS24IHH} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Introduction to Hypothesis H]]*

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non-projected super Riemann surface moduli -- section

### Non-projected super-moduli of super Riemann surfaces
 {#NonProjectedSuperRiemannSurfaceModuli}

A [[supermanifold]] is called *projected* if it [[retraction|retracts]] onto its bosonic [[body]]. (That's not the wording used in the literature, though.)

Since the computation of [[superstring scattering amplitudes]] involves a [[Berezin integral]] over the [[supergeometry|super]] [[moduli space]] of the given type of super Riemann surfaces, it is of interested to know when this [[moduli space of super Riemann surfaces]] is projected, as that allows to separate the bosonic from the fermionic sectors of this "[[path integral]]".

However, it turns out that the super-[[moduli space of super Riemann surfaces]] $\mathfrak{M}_{g, n_S, n_R}$ is generically *not* projected beyond low [[genus of a surface|genus]] $g$ (the string's [[loop order]]), depending on

* the number $n_S$ of Neveu-punctures

* the number $n_R$ of Ramond-punctures.

Specifically:

* $\mathfrak{M}_{g, 0, 0}$ is not projected for $g \geq 5$,

  $\mathfrak{M}_{g, n_S \geq 1, 0}$ is not projected for $g \geq n_S + 1$

  [[Donagi & Witten 2015](#DonagiWitten15)]

* $\mathfrak{M}_{g, 0, 2r \geq 2}$ is not projected for $g \geq 5r + 1$

  [[Donagi & Ott 2023](#DonagiOtt23)].

On the other hand, $g = 2$ (stringy [[2-loop]]) remains the highest order for which integration over the moduli space has actually been considered/performed, see [D'Hoker & Phong 2002](string+scattering+amplitude#DHokerPhong02).


non-relativistic particle

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## Idea

The [[mechanics]]/[[dynamics]] of a single [[particle]] propagating at low enough [[speed]] that the non-[[relativity|relativistic]] [[mechanics]] applies.

## Definition

For the time being see at _[[1d WZW model]]_.

## Related concepts

* [[relativistic particle]]

* [[rigid body dynamics]]

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non-singular distribution

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## Idea

A [[distribution]], being a [[generalized function]], is called _non-singular_ if it is (defined by) an actual [[smooth function]] (def. \ref{NonSingularCompactlySupportedDistributions} below). The subspace of non-singular distributions is a [[dense subspace]] of that of all distributions.

## Definition

+-- {: .num_defn #NonSingularCompactlySupportedDistributions}
###### Definition
**(non-singular distributions)

For $n \in \mathbb{N}$, a [[smooth function]] $b \in C^\infty(\mathbb{R}^n)$  induces a [[distribution]]

$$
  \int_{\mathbb{R}^n} (-) b dvol_{\mathbb{R}^n}
  \;\colon\;
  C_{cp}^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}
$$

by [[integration]] of smooth functions against $b dvol$.

This construction defines a linear inclusion

$$
  C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n)
$$

of the [[real vector space]] of [[smooth functions]] into that of [[distributions]].

The distributions arising this way are called the _non-singular distributions_.

=--

+-- {: .num_defn #SingularSupport}
###### Definition
**([[singular support of a distribution]])**

For $n \in \mathbb{N}$ let $\phi \in \mathcal{D}'(\mathbb{R}^n)$ be a [[distribution]]. Then the _[[singular support of a distribution|singular support]]_  $supp_{sing}(\phi) \subset X$ is the [[subset]] of points such that for every [[open neighbourhood]] $U_x \subset X$ the [[restriction of distributions|restriction]] $\phi\vert_{U_x}$ is singular, hence not a non-singular distribution (def. \ref{NonSingularCompactlySupportedDistributions}).

=--

## Properties

### Non-singular distributions are dense in all distributions

+-- {: .num_prop #NonSingularDistributionsAreDenseInAllDistributions}
###### Proposition
**(non-singular distributions are [[dense subspace|dense]] in all distributions)**

The inclusion of non-singular distributions into all distributions

$$
  C^\infty(\mathbb{R}^n) \hookrightarrow \mathcal{D}'(\mathbb{R}^n)
$$

from def. \ref{NonSingularCompactlySupportedDistributions} exhibits a [[dense subspace]] of $\mathcal{D}'(\mathbb{R}^n)$ (equipped with the [[dual space]] topology, [this def.](dual+vector+space#LinearDualOfATopologicalVectorSpace) or [Hörmander 90, p. 38](#Hoermander90)):

Every [[distribution]] $u \in \mathcal{D}'(\mathbb{R}^n)$ is the [[limit of a sequence|limit]] of a [[sequence]] $\{f_n \in C^\infty(\mathbb{R}^n)\}_{n \in \mathbb{N}}$ of smooth functions in that for every [[compactly supported function|compactly supported]] [[smooth function]] $\phi \in C^\infty_{cp}(\mathbb{R}^n)$ we have that the value $u(\phi)$ is the [[limit of a sequence|limit]] of [[integrals]]

$$
  u(\phi)
    \;=\;
  \underset{n \to \infty}{\lim} \int_{\mathbb{R}^n} \phi(x) f_n(x) dvol(x)
  \,.
$$

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([Hörmander 90, theorem 4.1.5](#Hoermander90))

Analogous statements hold for smaller classes of distributions: Also the inclusion

$$
  C^\infty_{cp}(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)
$$

of [[compactly supported function|compactly supported]] [[smooth functions]] ([[bump functions]]) into the space of [[tempered distributions]] is [[dense subspace|dense]].

(e.g. [Hörmander 90, lemma 7.1.8](#Hoermander90))

+-- {: .num_defn #DistributionsAsGeneralizedFunctions}
###### Remark

Proposition \ref{NonSingularDistributionsAreDenseInAllDistributions} justifies to think of [[distributions]] as "[[generalized functions]]", and to seek generalizations of standard operations on functions, such as concepts of _[[product of distributions]]_ and _[[composition of distributions]]_.

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## References

* {#Hoermander90} [[Lars Hörmander]], section 2.3 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

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non-unital operad

## Idea

A **non-unital operad** is a structure like an [[operad]] but without a unit or operadic identity. However, in contrast to operads where the multiplication operation substitutes $m$ operations simultaneously into a single $m$-ary operation, the primary operations for a non-unital operad are ones which substitute only one operation at a time.

It is possible to package these individual substitution operations into a single operation, using not the [[plethysm|plethystic]] or substitution monoidal product (usually denoted $\circ$) but a certain lax monoidal product introduced below, called the *graft product*. At all events, it must be understood that the desired notion is not simply a species equipped $F$ equipped with an associative product $F \circ F \to F$, because that is not expressive enough.

## Definitions

There are various possible notions of non-unital operad; here we focus on the permutative and non-permutative cases.

### Graft product of (permutative) species

Let $FB$ be the monoidal groupoid of finite sets and bijections, where the monoidal product is obtained by restricting the coproduct on the category of finite sets. Let $F, G: FB \to V$ be (Joyal) species valued in a bicomplete symmetric monoidal closed category $V$. The **graft product** $F \ast G$ is defined by the formula

$$(F \ast G)[S] = \sum_{T \subseteq S} F[S/T] \otimes G[T]$$

where $S/T$ denotes the pushout of $T \to 1$ along the inclusion $T \subseteq S$ in the category of finite sets. This includes the degenerate case where $T = 0$; in this case $S/0$ is the result of freely adjoining a point to $S$. In the theory of species, there is a fundamental differentiation functor $F \mapsto F'$ defined by the formula $F'[S] = F[S/0]$.

The set $S/T$ has $T/T$ as distinguished basepoint, and if $U$ is complementary to $T$, we have natural bijections $S/T \cong U + \{T/T\} \cong U/0$. It follows that

$$\array{
(F \ast G)[S] & \cong & \sum_{U + T = S} F[U + *] \otimes G[T] \\
 & \cong & \sum_{U + T = S} F'[U] \otimes G[T] \\
 & \cong & (F' \otimes G)[S]
}$$

where $\otimes$ refers to the usual Day convolution product on species, induced from the tensor product on $FB$. Since differentiation of species,

$$V^{FB} \stackrel{- + *}{\to} V^{FB},$$

is cocontinuous, and since Day convolution is cocontinuous in each of its arguments, we see that

$$F \ast G \cong F' \otimes G$$

is separately cocontinuous in each of its arguments $F$, $G$. (Contrast with the plethystic or substitution product on species $F \circ G$, which is cocontinuous in the first argument $F$ but not in the second $G$.)

For $V = Set$, a structure of species $F \ast G$ is given by three data:

* A tree obtained by grafting the root of a sprout (aka corolla) with leaf set $\tau$ to a leaf of another sprout with leaf set $\sigma$;

* An element of $F[\sigma]$;

* An element of $G[\tau]$.

### Lax monoidal structure of graft product

The graft poduct is not associative up to isomorphism, but there is a lax associativity. Specifically, we calculate (using a categorified [[Leibniz rule]])

$$\array{
(F \ast G) \ast H & \cong & (F' \otimes G)' \otimes H \\
 & \cong & F'' \otimes G \otimes H + F' \otimes G' \otimes H \\
 & \cong & F'' \otimes G \otimes H + F \ast (G \ast H)
}$$

so that there is a noninvertible associativity (an inclusion)

$$\alpha_{F G H}: F \ast (G \ast H) \to (F \ast G) \ast H$$

which is natural in each of its arguments $F$, $G$, and $H$, and which satisfies an evident pentagon coherence condition. Additionally (although we won't really need this here), there is a lax monoidal unit, defined as the species $X$ for which $X[S]$ is the monoidal unit of $V$ if $card(S) = 1$, else $X[S]$ is initial, for which we have evident natural maps

$$\lambda_F: X \ast F \to F, \qquad \rho_F: F \ast X \to F$$

The first map $\lambda_F$ is invertible, but the second is not: its component at a finite set $S$ is the codiagonal

$$\nabla: S \cdot F[S] \to F[S]$$

However, the standard unit coherence conditions for monoidal categories hold. We may call such a structure, relaxing the condition of invertibility of $\alpha$ and $\rho$ but retaining the usual naturality and coherence conditions, a **lax monoidal category**.

In the sequel, $F^{\ast n}$ will denote the iterated graft product defined recursively by

$$F^{\ast 0} = X, \qquad F^{\ast (n+1)} = F^{\ast n} \ast F$$

so that all parentheses in $F^{\ast n}$ are to the left. We state without proof the following partial coherence theorem:

+-- {: .num_prop}
###### Proposition
Any two maps
$$F^{\ast m} \ast F^{\ast n} \to F^{\ast (m+n)}$$
definable in the language of lax monoidal categories are equal. We denote this map by $\alpha_{m n}$.
=--

### Permutative non-unital operads

+-- {: .num_defn}
###### Definition
A (permutative) _non-unital operad_ in a [[cosmos]] $V$ is a [[species]] $F: FB^{op} \to V$ equipped with a multiplication $m: F \ast F \to F$, satisfying the following two associativity axioms:

1. $$\array{
F \ast (F \ast F) & \stackrel{1 \ast m}{\to} & F \ast F & \\
\alpha \downarrow & & & \searrow m \\
(F \ast F) \ast F & \underset{m \ast 1}{\to} F \ast F & \underset{m}{\to} & F
}$$
commutes;

1. The two composites $F'' \otimes F^{\otimes 2} \to F$ named in
$$F'' \otimes (F \otimes F) \stackrel{\overset{\sigma}{\to}}{\underset{1}{\to}} F'' \otimes (F \otimes F) \stackrel{i}{\to} (F \ast F) \ast F \stackrel{m(m \ast 1)}{\to} F$$
commute. Here $i$ denotes the inclusion complementary to $\alpha: F \ast (F \ast F) \to (F \ast F) \ast F$, and $\sigma$ is the involution
$$\sigma_1 \otimes \sigma_2: F'' \otimes F^{\otimes 2} \to F'' \otimes F^{\otimes 2}$$
where $\sigma: F'' \to F''$ is the natural involution on the second derivative
$$(-)'' = (- + 2)*: V^{FB} \to V^{FB}$$
induced by the nonidentity involution $2 \to 2$, and $\sigma_2$ is the symmetry isomorphism $F^{\otimes 2} \to F^{\otimes 2}$.

=--

Of course the notion can be reformulated perfectly well in a general symmetric monoidal category; the use of coproducts just makes for a more efficient packaging.

### Graft product of (non-permutative) species

A non-permutative species in a [[monoidally cocomplete category]] $V$ is simply an $\mathbb{N}$-[[graded object]] in $V$, which we may think of as equivalent to a functor $\mathbb{N} \to V$ from the monoidal groupoid of finite [[linear orders]], with the monoidal product given by restriction of the [[ordinal sum]] to the [[core]] groupoid. If $S$ is a finite linear order and $T = [a, b] \subseteq S$ is a [[subinterval]] of $S$, then we may again define $S/T$ via a pushout construction in the category of finite linear orders, and define a corresponding graft product by the formula

$$(F \ast G)[S] = \sum_{subintervals T} F[S/T] \otimes G[T].$$

This graft product shares certain formal properties with the permutative graft product defined above, such as the fact that it is a lax monoidal product. A triple graft product $(F \ast G) \ast H$ cab be decomposed as a coproduct of parts, the first being

$$F \ast (G \ast H)[S] = \sum_{subintervals T_1 \subseteq T_2} F[S/T_2] \otimes G[T_2/T_1] \otimes H[T_1]$$

and the second being

$$\sum_{subint. T_1, T_2: T_1 \cap T_2 = \emptyset} F[S/(T_1, T_2)] \otimes G[T_1] \otimes H[T_2]$$

where $S/(T_1, T_2)$ denotes the pushout of the inclusion $T_1 + T_2 \hookrightarrow S$ along the quotient $! + !: T_1 + T_2 \to 1 + 1$. By interchanging the roles of $T_1$ and $T_2$, we get a nontrivial involution on the second part.

To be continued...


Nonabelian Algebraic Topology

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

This entry is about the book

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]]:

  \linebreak

  **Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids**

  \linebreak

  Tracts in Mathematics **15**

  European Mathematical Society (2011)

  ISBN 978-3-03719-083-8

  [doi:10.4171/083](https://doi.org/10.4171/083)

  [webpage](http://groupoids.org.uk/nonab-a-t.html)

This treats [[algebraic topology]] using tools of [[strict ∞-groupoid]]-theory: notably the traditional [[homological algebra]] use of [[chain complex]]es of abelian groups is generalized to [[crossed complex]]es, and emphasis is put on the notion of [[fundamental groupoid]] and its strict [[higher category theory|higher categorical]] generalizations to the cubical [[fundamental omega-groupoid]] of a filtered space over the bare [[homotopy group]]s of a space.

One of the main motivations for the development of Nonabelian Algebraic Topology was the observation that the [[Seifert-van Kampen theorem]] is most naturally understood as being not about [[homotopy group]]s, but about the cubical [[fundamental omega-groupoid]] of a filtered space and may be generalized to a [[higher homotopy van Kampen theorem]] this way.

The restriction to [[strict ∞-groupoid]]s/[[crossed complex]]es is still a severe restriction as compared to the full [[homotopy theory]] of [[topological space]]s but already more general than the strict and strictly abelian $\infty$-groupoids used in traditional [[algebraic topology]] in the guise of [[chain complex]]es of abelian groups. In terms of the [[cosmic cube]] of [[higher category theory]] the approach of _Nonabelian algebraic topology_ used here is somewhere half way between homology and homotopy theory; it in this border area that traditional accounts seem to most lacking, and are unable to cope well with the nonabelian second relative homotopy group of a pair of spaces. The start of the new approach is to replace this by a homotopy double groupoid of a pair of spaces, which allows an _algebraic inverse to subdivision_. The philosophy behind the work is also given in the paper [Modelling and Computing Homotopy Types: I] (https://groupoids.org.uk/pdffiles/brouwer-hon.pdf}

#Contents#
* automatic table of contents goes here
{:toc}

## History

Some comments from [[Ronnie Brown]] himself:

> I hope it is helpful to relate my experiences from the 1960s and later with [[nonabelian cohomology]].

> In writing my book on [[topology]] in the 1960s, I got offended by having to make a detour to get the [[fundamental group]] of the circle, and then was attracted by [[Paul Olum]]'s paper referenced below. I extended Olum's work to a [[Mayer-Vietoris sequence|Mayer?Vietoris type sequence]] in the second paper below, and this enabled one to compute the fundamental group of, for example,  a [[wedge product|wedge]] of circles.

> (I use an MV sequence in _Topology and Groupoids_ in connection with pullbacks of [[covering space]]s.)

> So I decided to use this account for the book, thus giving students the advantage, it seemed,  of an introduction to [[cohomology|cohomological]] ideas.

> The problem was that the account when written in detail came to 30 pages (or maybe 40) and when looked at  in the cold light of day seemed incredibly boring (a full account is different from Olum's research account).

> I was at the time looking for exercises and came across [[Philip Higgins]]' paper on [[presentation]]s of groupoids, which used free products with amalgamation of groupoids. So I decided to give an exercise on the fundamental groupoid of a union. Then I felt I ought to write out a solution. When I had done this, it seemed streets ahead in exposition of all that [[nonabelian cohomology]] stuff  and moreover, when souped up to the _[[fundamental groupoid]] on a set of base points_ , gave results not reachable by the MV sequence; for example you could not with the MV sequence deduce the _precise calculation_ of the fundamental group of a union of two open sets whose intersection had say 150 path components. (This anomaly is also significant, in illustrating  the limitations of exact sequences.)

> So I decided to switch to an exposition of groupoids in 1-dimensional [[homotopy theory]]  (also spurred by a meeting with [[George Mackey]] in 1967 where he told me of his work on ergodic groupoids, which is now seen as a preliminary to [[noncommutative geometry|Noncommutative Geometry]]).

> It occurred to me that if one could come to the groupoid idea from two distinct directions, then there was likely to be more in this than met the eye. At the same time, an examination of the proof of the [[van Kampen theorem]] for groupoids, suggested that the theorem should have an extension to all dimensions, if one could define homotopy gadgets with the right properties. Another stimulus was the proof (used in the book) by [[Frank Adams]] (circulated in handwritten lecture notes) of the [[cellular approximation theorem]], which had analogies to  parts of the van Kampen proof, but failed to get algebraic results because, apparently, of the lack of an appropriate algebraic gadget in dimension $n \gt 1$.

> It took 9 years to find such a gadget  in dimension $2$, and another 3 to get them in all dimensions, in work with [[Philip Higgins]].

> It seemed to me  unfortunate that this work aroused the opposition, for reasons never explained to me, of Frank Adams,  who told people the whole programme was "ridiculous". His opinion became the opposite only when I told him (1985?) of the extension to the non simply connected case of the Blakers-Massey description of $\pi_3$ of a triad, using the nonabelian tensor product (work with [[Jean-Louis Loday]]).

> The higher order van Kampen theorems, and the often nonabelian calculations which result,  have not been obtained by cohomological methods, but only by working directly  with structures appropriate to the geometry of higher homotopies, i.e. forms of strict [[n-fold category|multiple groupoids]]. This confirms the comment of [[Philip Hall]], Philip Higgins' supervisor, that one should not try to force the geometry into a given algebraic mode, but search for the algebra which models the geometry. So it seems to me that algebraic topology has  been mainly restricted to, or not got out of,  the single base point and "group", not "groupoid", mode, nor appreciated the possibilities of [[colimit]] type theorems in algebraic (and geometric?) topology -- no algebraic or geometric topology text (except mine!) mentions the higher order van Kampen work with Philip Higgins.

> You can also see this restriction in the contrast between the unsymmetrical, choice laden,  definition of the second relative homotopy group, with its compositions in one direction (recall the limitations of "Lineland" described in "Flatland") and the definition of the fundamental [[double groupoid]] of a pointed pair of spaces $\rho_2(X,A)$, with its compositions in $2$ directions.  This contrast gets more significant in higher dimensions.

> For all these reasons, my inclination is to look for the applications of the "appropriate" (whatever that is!) structures rather than cohomology with coefficients in such structures, where lots of detail is likely to get lost. Also, in making calculations it is convenient to work with strict algebraic structures, where the notion of colimit is more comprehensible. Even there, it has been a problem to make say colimit calculations with [[crossed module]]s into a symbolic computer algebra format. See the work by [[Chris Wensley]] listed below.

> These results could not have been obtained without the intuitions on multiple compositions easily allowed by a cubical approach.

> One of the key observations for this programme was that one could define a strict homotopy double groupoid for a _pointed pair of spaces_, and that this was closely related to the well known fundamental crossed module of a pair of spaces, first considered by J.H.C. [[Whitehead]]. His paper listed below was a key source of ideas.

> The natural extension of this observation is to construct a [[strict cubical omega-groupoid|strict cubical ∞-groupoid]] $\rho X_*$ of a _[[filtered space]]_ $X_*$, and find its relation to the quite classical homotopically defined fundamental crossed complex functor $\Pi: (filtered spaces) \to (crossed complexes)$. The proofs here are non trivial. By proving using $\rho$ a colimit theorem for $\Pi$ one can shortcut [[singular homology]], and obtain old and new results in algebraic topology, including some explicit calculations of homotopy groups, even  as modules over the fundamental group. This working with filtered space is not unreasonable since they abound. For example, classifying spaces often come with convenient filtrations, as do [[geometric realisation]]s of [[simplicial set|simplicial]] or [[cubical set|cubical]] sets. These ideas generalise of course to [[multifiltered space]]s or $n$-cubes of spaces. It is not so clear that one _must_ work with a kind of bare topological space, and so have little handle on which to construct invariants, except say by first taking a singular complex, or using multipaths.

> The main idea of the [[higher homotopy van Kampen Theorem]]s is to model algebraically the gluing of homotopy types, or limited models of such.

> An indication of a beginnings of a Čech type approach to nonabelian cohomology using groupoids and crossed complexes is given in the new book, Chapter 12. This has not been developed in terms of [[sheaf theory]].

> Another big gap in comparison with traditional algebraic topology is [[intersection theory]] and [[Poincare duality]], although the (quite complicated) machinery of tensor products is available in the crossed complex context.

> An obvious gap is also that of extending [[Grothendieck]]'s work on the fundamental group!

## Contents

### I 1 and 2-dimensional results

* [[groupoid]]

* [[2-groupoid]]

* [[2-group]]

* [[crossed module]]

* [[fundamental groupoid]]

### Crossed complexes

#### 7 The basics of crossed complexes

* [[crossed complex]]

##### 7.1 Our basic categories and functors

###### 7.1.i The category of filtered topological spaces

###### 7.1.ii Modules over groupoids

A _[[module over a groupoid]]_ is a collection of [[abelian group]]s equipped with a linear  [[action]] by a [[groupoid]].

+-- {: .un_def}
###### Definition
**(module over a groupoid)**

Let $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ be a [[groupoid]]. A **module over the groupoid** $\mathcal{G}$ is a collection $\{N_x\}_{x \in \mathcal{G}_0}$ of [[abelian group]]s equipped with a collection of maps

$$
  N_x \times \mathcal{G}(x,y) \to N_y
$$

that are linear and respect the groupoid composition in the obvious way.

=--

###### 7.1.iii The category of crossed complexes

* [[crossed complex]]

###### 7.1.iv Homotopy and homology groups of crossed complexes

(...)

###### 7.1.v The fundamental crossed complex functor {#FundamentalCrossedComplex}

The notion of [[fundamental groupoid]] of a [[topological space]] generalizes to a notion of [[fundamental ∞-groupoid]]. There is a strict version of this (which loses some information): the fundamental strict $\infty$-groupoid. In a [[filtered space]] $X_*$ one can consider the variant where the [[k-morphism]]s of the fundamental $\infty$-groupoid are constrained to lie in $X_k$. The _fundamental crossed complex_ of a filtered space is the equivalent [[crossed complex]] incarnation of the fundamental strict $\infty$-groupoid of a filtered space.

+-- {: .un_def}
###### Definition
**(fundamental crossed complex)**

Let $X_\bullet$ be a [[filtered space]].

Write $\Pi_1(X_1,X_0)$ for the subgroupoid of the [[fundamental groupoid]] $\Pi_1(X_1)$ of $X_1$ on objects that are in $X_0$.

The **fundamental crossed complex** $\Pi X_*$ of $X$ is the [[crossed complex]]

with

$$
  (\Pi X_*)_1 = \Pi_1(X_1,X_0)
$$

and

$$
  (\Pi X_*)_n
  :=
  \coprod_{x \in X_0}
  \pi_n(X_n, X_{n-1}, x)
  \;\;\;\;
  for n \geq 2
  \,,
$$

where $\pi_n(X_n, X_{n-1}, x)$ is the [[relative homotopy group]] obtained by equivalence classes of maps from the pointed $n$-disk into $X$ such that the disk lands in $X_n$, its boundary in $X_{n-1}$ and its basepoint on $x$.

=--

See also section 1 of

* [[Ronnie Brown]], [[Rafael Sivera]], _Normalization of the fundamental crossed complex of a simplicial set_ ([arXiv:math/0611728](http://arxiv.org/abs/math/0611728), [pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.63.9955&rep=rep1&type=pdf))

+-- {: .un_example}
###### Example
**(fundamental crossed complex of the $n$-simplex)**

The topological $n$-[[simplex]] $\Delta^n$ is canonically a [[filtered space]] with $(\Delta^n)_k$ being the union of its $k$-faces.

Then we have that $\Pi_1((\Delta^n)_1, (\Delta^n)_0)$ is the groupoid whose objects are the $n+1$ vertices of $\Delta^n$ and which has precisely one morphism $x_i \to x_j$ for each ordered pair $x_i,x_j \in (\Delta^n)_0$
(all of them being [[isomorphism]]s)

$$
  \Pi_1((\Delta^2)_1,(\Delta^2)_0) =
  \left\{
     \array{
     && x_1
     \\
     & \nearrow\swarrow && \searrow \nwarrow
     \\
    x_0 &&\stackrel{\leftarrow}{\to}&& x_2
    }
  \right\}
  \,.
$$

At any $x_i$ the relative homotopy group $\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i)$ is a group on the set of 2-faces that have $x_i$ as a 0-face: there is a unique homotopy class of disks in $\Delta^n$ that sits in the 2-faces $(\Delta^n)_2$, whose base point is at $x_j$ and whose boundary runs along the boundary of a given 2-face of $\Delta^n$.

So (using the equivalence of crossed complexes with strict $\omega$-groupoids) for instance $\Pi \Delta^2$ is generated from $\Pi_1((\Delta^2)_1,(\Delta^2)_0)$ as above and a 2-cell

$$
  \array{
     && x_1
     \\
     & \swarrow &\Downarrow& \nwarrow
     \\
    x_0 &&\to&& x_2
    }
$$

under [[whiskering]] and composition. For instance [[whiskering]] this with $x_1 \to x_2$ yields the 2-morphism

$$
  \array{
     && x_1
     \\
     & \swarrow &\swArrow& \searrow
     \\
    x_0 &&\to&& x_2
    }
  \,.
$$

One sees that $\Pi \Delta^2$ is the strict groupoidification of the second [[oriental]].

Generally, $\Pi \Delta^n$ is the $n$-groupoid freely generated from $k$-morphisms for each $k$-face of $\Delta^n$.

=--

##### 7.4 Crossed complexes and chain complexes {#CrsdAndChainCplx}

+-- {: .un_def}
###### Definition
**(groupoid module chain complexes)**

Write $Chn$ for the [[category]] of [[chain complex]]es of modules over a [[groupoid]].

=--

This is Def. 7.4.1.

+-- {: .un_def}
###### Definition
**(groupoid module chain complexes)**

Write $Crs$ for the [[category]] of [[crossed complex]]es.

=--

###### 7.4.i Adjoint module and augmentation module {#AdjointModule}

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

Given a [[module over a groupoid]] $(N,\mathcal{G})$, the **semidirect product groupoid** $\mathcal{G} \ltimes N$ has the same objects as $\mathcal{G}$ and morphisms

$$
  (\mathcal{G} \ltimes N)(p,q) = \mathcal{G}(p,q) \ltimes N(q)
$$

with composition given by the action of $\mathcal{G}$ on $N$.

=--

This is def. 7.4.5

+-- {: .un_def}
###### Definition
**(covering morphism)**

For $(t : N \to \mathcal{G}_0,\mathcal{G})$ a [[module over a groupoid]], write $P(N,\mathcal{G})$ for the groupoid $\mathcal{G}$ pulled back to the underlying set of $N$:

an object of $P(N,\mathcal{G})$ is an element in $N$ and a morphism $n_1 \to n_2$ is a morphism $\mathcal{G}(t(n_1),t(n_2))$.

=--

This is def. 7.4.9.

###### 7.4.iii The derived chain complex of a crossed complex {#ChainOfCrossed}

+-- {: .un_def}
###### Definition
**(chain complex from a crossed complex)**

Define a functor $\nabla : Crs \to Chn$ from [[crossed complex]]es to [[modules over groupoids]] as follows:

For $C$ a [[crossed complex]] we set for $n \geq 3$

$$
  (\nabla C)_n := C_n \;\;\;\; for n \geq 3
$$

and for $n \leq 2$ it is given by ...

=--

This is definition 7.4.20.

###### 7.4.v The right adjoint of the derived functor {#CrsFromCh}

We describe a construction of a [[crossed complex]] from a [[chain complex]] of [[modules over a groupoid]] $(A_n, \mathcal{H})$. As a special case it in particular gives an map of ordinary [[chain complex]]es of [[abelian group]]s into the category of [[crossed complex]]es, and hence into [[strict ∞-groupoid]]s.

Recall the definition of the [semidirect product groupoid](#AdjointModule)  $\mathcal{H} \ltimes A_n$.

+-- {: .un_def}
###### Definition
**(crossed complex from a chain complex)**

For $A$ a [[chain complex]] of [[modules over a groupoid]] $\mathcal{H}$, let $\Theta A \in Crs$ be the [[crossed complex]]

$$
  \Theta A := \kappa^* \Theta' A
  \,,
$$

where

$$
  \Theta' A :=
  \left[
    A_n
      \stackrel{\partial_n}{\to}
    A_{n-1}
     \stackrel{}{\to}
     \cdots
     \stackrel{}{\to}
    A_{3}
    \stackrel{\partial_3}{\to}
    A_2
    \stackrel{(0,\partial_2)}{\to}
    \mathcal{H}\ltimes A_1
  \right]
$$

and where

$$
  \kappa : P(A_0, \mathcal{H}) \to \mathcal{H} \ltimes A_0
$$

is the [canonical covering morphism](#AdjointModule) from above.

$$
  \array{
    \cdots \to &
    (\Theta A)_3
    &\to&
    (\Theta A)_2
    &\to&
    (\Theta A)_1
    &\to&
    P(A_0, \mathcal{H})
    \\
    & \downarrow && \downarrow && \downarrow && \downarrow
    \\
    \cdots \to &
    A_3
    &\stackrel{\partial_3}{\to}&
    A_2
    &\stackrel{(0,\partial_2)}{\to}&
    \mathcal{H} \ltimes A_1
    &\stackrel{(1, \partial_1)}{\to}&
    \mathcal{H} \ltimes A_0
  }
  \,.
$$

Here $\mathcal{H} \ltimes A_1$ acts on $A_n$ for $n \geq 2$ via the projection $\mathcal{H} \ltimes A_1 \to \mathcal{H}$, i.e. $A_1$ acts trivially.
(...)

Finally set $\Theta(A)_0 := A_0$.

=--

We spell out what this boils down to explicitly.

**Explicit description**

Let $A_\bullet$ be a [[chain complex]] of [[modules over a groupoid|modules over the groupoid]] $\mathcal{H}$. Then the [[crossed complex]] $\Theta(A)$ is the following.

* Its set of objects is $\Theta(A)_0 = A_0$.

  Remember that $A_0$ itself is a module over $\mathcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0)$, so that $A_0 = \coprod_{p \in \mathcal{H}_0} (A_0)_p$.

* For $x \in (A_0)_p$ and $y \in (A_0)_q$ a morphism in $\Theta(A)_1$ from $x$ to $y$ is labeled by $h \in \mathcal{H}_1$ and $a \in (A_1)_q$

  $$
    x \stackrel{(h,a)}{\to} (y = \rho(h)(x) - \partial a)
    \,,
  $$

  where $\rho$ denotes the [[action]] of $\mathcal{H}$ on $A_0$.

  The composition law is given by

  $$
    \array{
       && y
       \\
       & {}^{\mathllap{(h_1, a_1)}}\nearrow && \searrow^{\mathrlap{(h_2,a_2)}}
       \\
       x &&\stackrel{(h_1 \circ h_2, \rho(h_2)(a_1) + a_2)}{\to}&& z
    }
    \,.
  $$

* For $k \geq 2$ the family of groups $\Theta(A)_k$ is over $x \in (A_0)_p$ the group $(A_k)_q$

  $$
    \Theta(A)_{k \geq 2} = \coprod_{p \in \mathcal{H}_0} \coprod_{x\in (A_0)_q} (A_k)q
  $$

* The boundary maps and actions are the obvious ones...

+-- {: .un_example}
###### Example
**(ordinary abelian chain complex as crossed complex)**

Let $C_\bullet$ be an ordinary [[chain complex]] of abelian groups, i.e. a chain complex of [[module over a groupoid|modules over the trivial groupoid]].

Then $(\Theta C)_1$ is the groupoid with objects $C_0$ and morphisms $\{x \stackrel{b}{\to} (x + \partial b)\}$. And for $n \geq 2$ we have that $(\Theta C)_n$ is $\coprod_{x \in C_0} C_n$.

=--

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

These form a pair of [[adjoint functor]]s

$$
  (\nabla \dashv \Theta)
  :
  Chn
  \stackrel{\overset{\nabla}{\leftarrow}}{\underset{\Theta}{\to}}
  Crs
$$

where...

=--

This is proposition 7.4.29.

(...)

#### 8 The Higher Homotopy van Kampen Theorem and its applications

* [[higher homotopy van Kampen theorem]]

##### 8.4 The chain complex of a filtered space and of a CW-complex {#ChainCplxOfFilteredSpace}

Let all [[topological space]]s $X$ in the following by [[Hausdorff space]]s that admit a [[universal cover]]. $\hat X$.

+-- {: .un_def}
###### Definition
**(homology chain complex of a filtered space)**

For $X = (X_\bullet)$ a [[filtered space]] define a [[chain complex]] of [[modules over a groupoid]] $\mathcal{C}_\bullet(X)$ as follows.

The [[groupoid]] $\mathcal{G} := \Pi_1(X,X_0)$ is the full subgroupoid of the [[fundamental groupoid]] of $X$ on points in $X_0$.

For $x_0 \in X$ let $\hat X(x_0) := \coprod_y \Pi_1(y,x_0)$ be the standard model for the [[universal cover]] of $X$ in terms of homotopy classes of paths into $x_0$.

For all $x \in X_0 = Obj(\Pi_1(X,X_0))$ take the [[module over a groupoid|modules over]] $\Pi_1(X,X_0)$ to be the [[relative homology group]]s

$$
  (\mathcal{C}_0 X)_x := H_0(\hat X_0(x))
$$

and for $n \geq 1$

$$
  (\mathcal{C}_n X)_x := H_n(\hat X_n(x),\hat X_{n-1}(x) )
  \,.
$$

The action of $\Pi_1(X,X_0)$ on this is the evident one induced by composition of paths.

This extends to a [[functor]]

$$
  \mathcal{C}_\bullet : FTop \to Chn
  \,.
$$

=--

This is _def 8.4.1_

The next proposition asserts that this notion of chain complex of a filtered topological space is reproduced by the combination of

* the [fundamental crossed complex](#FundamentalCrossedComplex) $\Pi X_\bullet$

* and the [chain complex of a crossed complex](#ChainOfCrossed) $\nabla \Pi X_\bullet$.

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

If the [[filtered space]] $X_\bullet$ is [[connected space|connected]] then there is a [[natural isomorphism]]

$$
  \mathcal{C}_\bullet X \simeq \nabla \Pi X
  \,.
$$

=--

This is _proposition 8.4.2_ . Use the [[Hurewicz theorem|relative Hurewicz theorem]] to translate from [[homotopy group]]s to [[homology group]]s.

###### Example: Chains on the $n$-simplex

+-- {: .un_example}
###### Example
**(chains on the $n$-simplex)**

Consider $X = \Delta^n$, the standard topological $n$-[[simplex]] regarded as a [[filtered space]] with the union of its $k$-faces in degree $k$.

Notice that since $\Delta^n$ is a [[simply connected space]] in this case we have that for each basepoint $x \in (\Delta^n)_0$ the universal cover $\hat X_{x} = X$ coincices with $X$.

We have that

$$
  \mathcal{C}_\bullet \Delta^n \simeq \nabla \Pi \Delta^n
  \simeq
   N_\bullet \Delta[n]
$$

is, over each vertex $x \in (\Delta^n)_0$, the [[Dold-Kan correspondence|normalized chain complex]] of [[cochain on a simplicial set|chains on the simplicial set]] $\Delta[n]$

$$
  \mathcal{C}_0 \Delta^n = \mathbb{Z}^{n+1}
  \;\;\;\;\;
  (\mathcal{C}_0 \Delta^n)_x = \mathbb{Z}$$

$$
  (\mathcal{C}_1 \Delta^n)_{x} = \mathbb{Z}^n
$$

etc.

$$
  (\mathcal{C}_n \Delta^n)_x = \mathbb{Z}
  \,.
$$

We have moreover that $\Pi_1(\Delta^n, (\Delta^n)_0)$ is the [[codiscrete groupoid]] on $n+1$ objects. It acts on the $\mathcal{C}_k(\Delta^n)$ by identity maps

$$
  (x_i \to x_j) : (\mathcal{C}_{k} \Delta^n)_{x_i}
  \stackrel{=}{\to}
  (\mathcal{C}_{k} \Delta^n)_{x_j}
  \,.
$$

It follows in particular that for $D_\bullet$ an ordinary [[chain complex]] of [[abelian group]]s regarded as a complex of [[modules over a groupoid]] in the trivial way, morphisms of modules over groupoids

$$
  \mathcal{C}_\bullet \Delta^n \to D
$$

are canonically identified with morphisms of ordinary chain complexes of abelian groups

$$
  N_\bullet \Delta[n] \to D
  \,.
$$

For more on this see [Dold-Kan map and omega-nerve](#DoldKanMap).

=--

#### 9 Tensor products and homotopies of crossed complexes

##### 9.9 The homotopy addition lemma for a simplex {#HomotopyAdditionSimplex}

For $\Delta^n$ the topological $n$-simplex regarded as a [[filtered space]] in the canonical way, the fundamental crossed complex $\Pi X^n$ is a groupoid-version of the $n$-[[oriental]]: the free [[strict ∞-groupoid]] on a single $n$-simplex.

##### 9.10 Simplicial sets and crossed complexes {#SimpSetAndCrs}

By the discussion at [The homotopy addition lemma for a simplex](#HomotopyAdditionSimplex) the [fundamental crossed complex](#FundamentalCrossedComplex) $\Pi \Delta^n$ plays the role of the free strict $n$-groupoid on the $n$-[[simplex]].

The [[cosimplicial object|cosimplicial]] $\infty$-groupoid

$$
  \Pi \Delta^\bullet :
  \Delta \to Crs \simeq Str \infty Grpd
$$

induced by the discussion at [[nerve and realization]] a simplicial [[nerve]] operation on [[strict ∞-groupoid]] -- an [[∞-nerve]]:

+-- {: .un_def}
###### Definition
**(simplicial nerve)**

Let $C$ be a [[crossed complex]]. Its **simplicial [[nerve]]** $N^\Delta C \in  $ [[sSet]] is

$$
  (N^\Delta C)_n := Crs(\Pi \Delta^n, C)
$$

=--

This is definition 9.10.2.

###### Remark 9.10.6 (Dold-Kan map and $\omega$-nerve) {#DoldKanMap}

+-- {: .un_prop}
###### Proposition
**(Dold-Kan map)**

For $D \in Chn$ a [[chain complex]] (of [[abelian groups]]) regarded as a chain complex of  [[modules over a groupoid|modules over]] the trivial groupoid, we may regard it as a [[crossed complex]] $\Theta D$ as described at [Crossed complex from chain complex](#CrsFromCh), hence as a [[strict ∞-groupoid]].

The [[∞-nerve]] $N^\Delta \Theta D \in $ [[sSet]] (described in [Crossed complexes and simplicial sets](#SimpSetAndCrs)) of this is the [[Kan complex]] underlying the image of $D$ under the [[Dold-Kan correspondence]] $Chn \to sAb$.

=--

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

By definition we have

$$
  N^\Delta (\Theta D)  := Crs(\Pi \Delta^\bullet, \Theta D)
  \,.
$$

By [[adjunction]] $(\Pi \dashv \Theta)$ with the [Theta-map](#CrsFromCh) this is equivalently

$$
  \cdots \simeq Chn( \nabla \Pi \Delta^\bullet, D)
$$

Using the propositions and examples discussed at [Chain complex of a filtered space](#ChainCplxOfFilteredSpace) we have that $\nabla \Pi \Delta^n$ is standard normalized [[chain complex]] $N_\bullet \Delta[n]$ of chains on the simplicial $n$-[[simplex]] as discussed at [[cochain on a simplicial set|chains on a simplicial set]] and [[Dold-Kan correspondence]], but regarded as a complex of [[modules over a groupoid|modules over the groupoid]] $\Pi_1(\Delta^n, (\Delta^n)_0)$. But since the groupoid action on $D$ is trivial, the above is equivalent to

$$
  \cdots \simeq Chn( N_\bullet \Delta^\bullet , D)
  \,.
$$

=--

This appears as _remark 9.10.6_ together with its _footnote 116_ .

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

In the [[cosmic cube]] of [[higher category theory]] this realizes two edges

$$
  \array{
    ChainCplx &\stackrel{\Theta}{\hookrightarrow}&
    CrossedCplx
    &\stackrel{N^\Delta}{\hookrightarrow}&
    KanCplx
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    StrAb Str\infty Grpd
    &\hookrightarrow&
    Str \infty Grpd
    &\hookrightarrow&
    \infty Grpd
  }
$$

including [[strict ∞-groupoid]]s with strict abelian [[∞-group]]-structure -- modeled as [[chain complex]]es of [[abelian group]]s -- into [[strict ∞-groupoid]]s -- modeled as [[crossed complex]]es -- into all [[∞-groupoid]]s
-- modeled as [[Kan complex]]es. The composite is the map $Ch_\bullet \to sAb \to KanCplx$ to [[simplicial group]]s from the [[Dold-Kan correspondence]].

=--

#### 10 Resolutions

#### 11 The cubical classifying space of a crossed complex

#### 12 Nonabelian cohomology: spaces, groupoids

* [[nonabelian cohomology]]

### Cubical $\omega$-groupoids

#### 13 The algebra of crossed complexes and cubical $\omega$-groupoids

#### 14 Cubical homotopy groupoid

##### 14.8 The cubical Dold-Kan theorem {#CubicalDoldKan}

See also [[Dold-Kan correspondence]].

## References ##

For an extensive list of relevant publications see

* [[Ronnie Brown]], _publication list_ ([web](http://groupoids.org.uk/publicfull.html))

Some selected references are:

1. [[Paul Olum]], _Non-abelian cohomology and van Kampen's theorem_, Ann. Math. 68 (1958) 658--667.

1.  Brown, R., _On a method of P. Olum_, J. London Math. Soc. 40 (1965) 303--304.

1.  Brown, R.,  _Elements of Modern Topology_, McGraw Hill, Maidenhead, 1968.

1. Brown, R., _Topology and Groupoids_, Booksurge, 2006.

1. Higgins, P.J., _Presentations of groupoids, with applications to groups_, Proc. Camb. Phil. Soc., 60 (1964) 7--20.

1. Brown, R. and Higgins, P.J., _On the connection between the second relative homotopy groups of some related spaces_, Proc. London Math.  Soc.(3) 36 (1978) 193--212.

1. Brown, R. and Higgins, P.J., _Colimit theorems for relative homotopy groups_, J. Pure Appl. Algebra 22 (1981) 11--41.

1. Whitehead, J.H.C., _Combinatorial Homotopy II_, Bull.
Amer. Math. Soc., 55 (1949), 453--496.

1. Brown, R. _Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems_, in Handbook of Algebra 6, Edited M. Hazewinkel, Elsevier, 2009.

1. Wensley, C.D. and Alp, M., XMOD, a GAP share package for
computation with crossed modules, _GAP Manual_, (1997), 1355--1420.

1. Brown, R., Higgins, P.J., and Sivera, R., _Nonabelian Algebraic Topology: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids_, EMS Tracts in Mathematics, Vol. 15, (Autumn 2010).

[[!redirects nonabelian algebraic topology]]
[[!redirects Nonabelian algebraic topology]]
[[!redirects Nonabelian Algebraic Topology]]

category: reference


Nonabelian algebraic topology > history

< [[Nonabelian algebraic topology]]
[[!redirects Nonabelian algebraic topology -- history]]


nonabelian bundle 2-gerbe

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## Idea

A _nonabelian bundle 2-gerbe_ is a model for a [[principal infinity-bundle|principal 3-bundle]].

The notion is

* the next step in [[higher category theory]] of the notion of [[nonabelian bundle gerbe]];

* a generalization of the notion of [[bundle 2-gerbe]] from abelian to nonabelian structure [[infinity-group|3-groups]].

## Related concepts

* [[principal 3-bundle]]

## References

* [[Branislav Jurčo]], *Nonabelian bundle 2-gerbes*,  International Journal of Geometric Methods in Modern Physics **08** 01 (2011) 49-78 [[arXiv:0911.1552](http://arxiv.org/abs/0911.1552), [doi:10.1142/S0219887811004963](https://doi.org/10.1142/S0219887811004963)]

A model for the [[supergravity C-field]] in terms of nonabelian bundle 2-gerbes:

* [[Paolo Aschieri]], [[Branislav Jurčo]], *Gerbes, M5-Brane Anomalies and $E_8$ Gauge Theory*, JHEP 0410:068 (2004) [[arXiv:hep-th/0409200](https://arxiv.org/abs/hep-th/0409200), [doi:10.1088/1126-6708/2004/10/068](https://doi.org/10.1088/1126-6708/2004/10/068)]

[[!redirects nonabelian bundle 2-gerbes]]


nonabelian bundle gerbe

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## Idea

A _nonabelian bundle gerbe_ (as studied by Aschieri-Cantini-Jurco) is a model for the [[Lie groupoid]] which is the total space of a smooth $AUT(H)$-[[principal 2-bundle]] for $AUT(H)$ the [[Lie 2-group]] that is the [[automorphism 2-group]] of a [[Lie group]] $H$.

Specifically, a nonabelian bundle gerbe on a [[smooth manifold]] $X$ is given by a [[surjective submersion]] $Y \to X$ and an $H$-[[bibundle]] $P \to Y\times_X Y$ together with a morphism of $H$-bibundles

$$
  \mu : \pi_0^* P \otimes \pi_2^* P \to \pi_1^* P
$$

that is associative in the evident sense. This construction serves to model [[pullback]]s of [[Lie 2-groupoid]]s of the form

$$
  \array{
    \tilde P &\to& \mathbf{E}AUT(H)
    \\
    \downarrow && \downarrow
    \\
    C(Y) &\stackrel{g}{\to}& \mathbf{B}AUT(H)
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
  \,,
$$

where on the right we have the [[universal principal infinity-bundle|universal principal 2-bundle]].

The resulting [[Lie groupoid]] $\tilde P$ is an extension of the [[Cech groupoid]] $C(Y)$ by $AUT(H)$.
This generalizes the case of ordinary [[bundle gerbe]]s, which are models for $\mathbf{B}U(1)$-principal 2-bundles, for $\mathbf{B}U(1)$ the circle 2-group.

This can all be extended to [[internal category|topological groupoids]], and to structure [[2-groups]] given by more general [[crossed modules]] than $H\to Aut(H)$.

## References

* Paolo Aschieri, Luigi Cantini, [[Branislav Jurco]], _Nonabelian Bundle Gerbes, their Differential Geometry and Gauge Theory_ ([arXiv:hep-th/0312154](http://arxiv.org/abs/hep-th/0312154))

[[!redirects nonabelian bundle gerbes]]


nonabelian cocycle

A nonabelian cocycle is, generally, a cocycle in [[nonabelian cohomology]].

This means usually that it is modeled in either of the following two ways:

* a descent datum, i.e. an object in the category of [[descent]] data (sometimes briefly called a descent category);
* an $\omega$-[[anafunctor]].

Notice that for $n\geq 2$, descent data in a setup of strict $n$-categories still form a strict $(n+1)$-category, while $n$-anafunctors form weak $(n+1)$-category.


nonabelian cohomology

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## Idea

The notion of [[cohomology]] finds its natural general formulation in terms of [[hom-spaces]] in an [[(∞,1)-topos]], as described at [[cohomology]]. Many of the cohomologies which have been traditionally considered, such as [[abelian sheaf cohomology|sheaf cohomology]], turn out to be just a special case of the general situation, for objects which are sufficiently abelian in the sense of [[stable (infinity,1)-category|stable (∞,1)-categories]].

Therefore to amplify that one is looking at general [[cohomology]] without restricting to [[abelian sheaf cohomology|abelian cohomology]] one sometimes speaks of  **nonabelian cohomology**.

## History

It was originally apparently John Roberts who understood (remarkably: while thinking about [[quantum field theory]] in the guise of [[AQFT]]) that general cohomology is about coloring simplices in $\infty$-categories.

* John E. Roberts, _Mathematical Aspects of Local Cohomology_
talk at Colloqium on Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, 1977 .

This is recounted for instance by Ross Street in

* [[Ross Street]], _Categorical and combinatorial aspects of descent theory_ ([pdf](http://arxiv.org/ftp/math/papers/0303/0303175.pdf))

and

* [[Ross Street]], _An australian conspectus on higher categories_ ([pdf](http://www.math.mq.edu.au/~street/Minneapolis.pdf)).

Parallel to this development of the notion of [[descent|descent and codescent]] there was the development of [[homotopical cohomology theory]] as described in

* Kenneth S. Brown, _Abstract Homotopy Theory and Generalized Sheaf Cohomology_, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 ([pdf](http://www.math.uni-hamburg.de/home/schreiber/Abstract%20homotopy%20theory%20and%20generalized%20sheaf%20cohomology.pdf))

The two approaches are different, but closely related. Their relation is via the notion of [[descent|codescent]].

There is a chain of inclusions

$$
  AbelianGroups \hookrightarrow
  ChainComplexesOfAbelianGroups
  \hookrightarrow
  CrossedComplexes
  \hookrightarrow
  \omega Groupoids
  \hookrightarrow
  \omega Categories
$$

along which one can generalize the coefficient objects of ordinary cohomology. (See [[strict omega-groupoid]], [[strict omega-category]]). Since doing so in particular generalizes abelian groups to nonabelian groups (but goes much further!) this is generally addressed as leading to _nonabelian cohomology_.

Depending on the [[model category|models]] chosen, there are different concrete realizations of nonabelian cohomology.

For instance nonabelian [[Čech cohomology]] played a special role in the motivation of the notion of [[gerbe]]s (see in particular [[gerbe (in nonabelian cohomology)]]), concretely thought of in terms of [[pseudofunctor]]s at least in the context of nonabelian [[group cohomology]], while more abstract (and less explicit) [[homotopy theory]] methods dominate the discussion of [[infinity-stack]]s.

Either way, one obtains a notion of _cohomology on $\infty$-categories with coefficients in $\infty$-catgories_. This is, most generally, the setup of "[[nonabelian cohomology]]".

This is conceptually best understood today in terms of [[Higher Topos Theory|higher topos theory]], using [[(infinity,1)-category of (infinity,1)-sheaves|(infinity,1)-categories of (infinity,1)-sheaves]].

This perspective on nonabelian cohomology is discussed for instance in [Toen 02](#Toen02)

## Properties

### Postnikov decomposition and Whitehead principle

In an [[(∞,1)-topos]] every object has a [[Postnikov tower in an (∞,1)-category]]. This means that in some sense general nonabelian cohomology can be decomposed into nonabelian cohomology in degree 1 and abelian cohomology in higher degrees, twisted by this nonabelian cohomology. This has been called ([Toën](#Toen02)) the [[Whitehead principle of nonabelian cohomology]].

## Special cases

### Nonabelian group cohomology

Sometimes the term _nonabelian cohomology_ is used in a more restrictive sense. Often people mean
[[nonabelian group cohomology]] when they say nonabelian cohomology, hence restricting to the domains to [[group|groups]], which are [[groupoid|groupoids]] with a single object.

This kind of nonabelian cohomology is discussed for instance in

* [[John Baez]], [[Mike Shulman]], [[Lectures on n-Categories and Cohomology]] ([arXiv](http://arxiv.org/abs/math.CT/0608420)).

That and how ordinary [[group cohomology]] is reproduced from the [[homotopical cohomology theory]] of [[strict omega-groupoid|strict omega-groupoids]] is discussed in detail in chapter 12 of

* [[Ronnie Brown]], P. Higgins, R. Sivera, [[nonabelian algebraic topology|Nonabelian algebraic topology]].

For more see

* [[nonabelian group cohomology]].

### Nonabelian sheaf cohomology with constant coefficients {#NonabelianSheafCohomology}

For $X$ a [[topological space]] and $A$ an [[∞-groupoid]], the standard way to define the [[nonabelian cohomology]] of $X$ with coefficients in $A$ is to define it as the intrinsic cohomology as seen in [[∞Grpd]] $\simeq$ [[Top]]:

$$
  H(X,A) := \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A)
  \,,
$$

where $|A|$ is the [[geometric realization]] of $A$ and $Sing X$ the [[fundamental ∞-groupoid]] of $X$.

But both $X$ and $A$ here naturally can be regarded, in several ways, as objects of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf (∞,1)-topos]]es $\mathbf{H} = Sh_{(\infty,1)}(C)$ over nontrivial [[(∞,1)-site]]s $C$. The intrinsic cohomology of such $\mathbf{H}$ is a [[nonabelian cohomology|nonabelian sheaf cohomology]]. The following discusses two such choices for $\mathbf{H}$ such that the corresponding nonabelian sheaf cohomology coincides with $H(X,A)$ (for [[paracompact space|paracompact]] $X$).

#### Petit $(\infty,1)$-sheaf $(\infty,1)$-topos

For $X$ a [[topological space]] and $Op(X)$ its [[category of open subsets]] equipped with the canonical structure of an [[(∞,1)-site]], let

$$
  \mathbf{H}
  :=
  Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))
$$

be the [[(∞,1)-category of (∞,1)-sheaves]] on $X$. The space $X$ itself is naturally identified with the [[terminal object]] $X = * \in Sh_{(\infty,1)}(X)$. This is the [[petit topos]] incarnation of $X$.

Write

$$
  (LConst \dashv \Gamma) :  Sh_{(\infty,1)}(X)
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
$$

be the [[global section]]s terminal [[geometric morphism]].

Under the [[constant (∞,1)-sheaf]] functor $LConst$ an an [[∞-groupoid]] $A \in \infty Grpd$ is regarded as an object $LConst A \in Sh_{(\infty,1)}(X)$.

There is therefore the _intrinsic_ [[cohomology]] of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ with coefficients in the [[constant ∞-stack|constant (∞,1)-sheaf]] on $A$

$$
  H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A)
  \,.
$$

This is [[cohomology with constant coefficients]].

Notice that since $X$ is in fact the [[terminal object]] of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that [[global section]]s functor, this is equivalently

$$
  \cdots \simeq \pi_0 \Gamma LConst A
  \,.
$$

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

If $X$ is a [[paracompact space]], then these two definitins of [[nonabelian cohomology]] of $X$ with [[constant ∞-stack|constant coefficients]] $A \in \infty Grpd$ agree:

$$
  H(X,A) := \pi_0 \infty Grpd(Sing X,A)  \simeq Sh_{(\infty,1)}(X)(X,LConst A)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, theorem 7.1.0.1]]. See also [[(∞,1)-category of (∞,1)-sheaves]] for more.

#### Gros $(\infty,1)$-sheaf $(\infty,1)$-topos

Another alternative is to regard the space $X$ as an object in the [[cohesive (∞,1)-topos]] [[ETop∞Grpd]].

$$
  (\Pi \dashv LConst \dashv \Gamma)
  :
  ETop\infty Grpd
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd
  \,,
$$

with the further [[left adjoint]] $\Pi$ to $LConst$ being the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] functor.  The intrinsic [[nonabelian cohomology]] in there also coincides with nonabelian cohomology in [[Top]]; even the full [[cocycle]] [[∞-groupoid]]s are equivalent:

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

For [[paracompact space|paracompact]] $X$ we have an equivalence of [[cocycle]] [[∞-groupoid]]s

$$
  ETop\infty Grpd(X, LConst A)
  \simeq
  Top(X, |A|)
$$

and hence in particular an isomorphism on cohomology

$$
  H(X,A) \simeq \pi_0   ETop\infty Grpd(X, LConst A)
$$

=--

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

See [[ETop∞Grpd]].

=--

## Examples

Examples of *generalized* nonabelian cohomology (with general classifying spaces):

* [[unstable topological K-theory]]

* [[Cohomotopy]]

## Objects classified by nonabelian cohomology

For $g : X \to A$ a [[cocycle]] in nonabelian cohomology, we say the [[fibration sequence|homotopy fibers]] of $g$ is the object _classified_ by $g$.

For examples and discussion of this see

* [[principal bundle]]

* [[principal 2-bundle]], [[gerbe]]

* [[principal 3-bundle]], [[2-gerbe]]

* [[principal ∞-bundle]], [[∞-gerbe]]

* [[twisted form|twisted forms]] .

## Related concepts

* [[nonabelian de Rham cohomology]]

* [[differential non-abelian cohomology]]

* [[generalized cohomology]]

* [[Chern-Weil theory]]

## References
 {#References}

### Classical theory -- bundles and groups

The classical notion of non-abelian ([[Čech cohomology|Čech]]-)cohomology in degree 1 and its relation to [[fiber bundles]]/[[principal bundles]]:

* [[Alexander Grothendieck]], Chapter V of: *A General Theory of Fibre Spaces With Structure Sheaf*, University of Kansas, Report No. 4 (1955, 1958) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf), [[Grothendieck-FibreSpaces.pdf:file]]]

* [[Jean Frenkel]], *Cohomologie à valeurs dans un faisceau non abélien*, C. R. Acad. Se., t. 240 (1955) 2368-2370

* [[Jean Frenkel]], *Cohomologie non abélienne et espaces fibrés*, Bulletin de la Société Mathématique de France, **85** (1957) 135-220 [[numdam:BSMF_1957__85__135_0](http://www.numdam.org/item/BSMF_1957__85__135_0)]

Review:

* [[Torsten Wedhorn]], *Torsors and Non-abelian Čech Cohomology*, chapter 7 in: *Manifolds, Sheaves, and Cohomology*, Springer (2016) [[doi:10.1007/978-3-658-10633-1](https://doi.org/10.1007/978-3-658-10633-1)]

Review in topological spaces (via [[classifying spaces]]):

* Nicolas Addington, _Fiber bundles and nonabelian cohomology_ (2007) [[pdf](http://pages.uoregon.edu/adding/notes/gstc2007.pdf)]

* {#Mitchell11} [[Stephen Mitchell]], around Theorem 7.4 in: _Notes on principal bundles and classifying spaces_, Lecture Notes. University of Washington (2011) [[pdf](https://sites.math.washington.edu/~mitchell/Notes/prin.pdf), [[MitchellPrincipalBundles.pdf:file]]]

* {#RudolfSchmidt17} [[Gerd Rudolph]], [[Matthias Schmidt]], Thm. 3.5.1 of: *Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer (2017) [[doi:10.1007/978-94-024-0959-8](https://link.springer.com/book/10.1007/978-94-024-0959-8)]

See also:

*  Jinpeng An. Zhengdong Wang, _Nonabelian cohomology with coefficients in Lie groups_, Trans. Amer. Math. Soc. **360** (2008) 3019-3040 [[doi:10.1090/S0002-9947-08-04278-5](https://doi.org/10.1090/S0002-9947-08-04278-5)]

The case of [[nonabelian group cohomology]]:

* {#Milne17} [[James Milne]], Section 27.a (of the [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf), not present in the published version) of: *Algebraic Groups*, Cambridge University Press 2017  ([doi:10.1017/9781316711736](https://doi.org/10.1017/9781316711736), [webpage](http://www.jmilne.org/math/Books/iag.html), [pdf](https://www.jmilne.org/math/CourseNotes/iAG200.pdf))

### Categorified theory -- 2-bundles/gerbes
 {#CategorifiedTheory}

Early discussion, of higher non-abelian cohomology with [[coefficients]] in certain [[2-groups]] (implicitly):

* [[Paul Dedecker]], *Cohomologie à coefficients non abéliens et espaces fibrés*,  Bulletins de l'Académie Royale de Belgique **41** (1955) 1132-1146 [[persee:barb_0001-4141_1955_num_41_1_69497](https://www.persee.fr/doc/barb_0001-4141_1955_num_41_1_69497)]

* [[Paul Dedecker]], *Cohomologie de dimension 2 à coefficients non abéliens*, C. R. Acad. Sci. Paris **247** (1958) 1160-1163 [[BnF](https://gallica.bnf.fr/ark:/12148/bpt6k7258/f192.item.r=dedecker)]

* [[Paul Dedecker]], *Sur La Cohomologie Non Abelienne I (Dimension Deux)*, Canadian Journal of Mathematics **12** (1960) 231-251 [[doi:10.4153/CJM-1960-019-7](https://doi.org/10.4153/CJM-1960-019-7)]

and with coefficients in certain [[3-groups]] presented by [[crossed squares]]:

* [[Paul Dedecker]], A. Frei, _Les relations d'équivalence des morphismes de la suite exacte de cohomologie non abêlienne_, C. R. Acad. Sci. Paris **262** (1966) 1298-1301

* [[Paul Dedecker]], _Three dimensional non-abelian cohomology for groups_, Category theory, homology theory and their applications, II (Battelle Institute Conf.) 1969 ([MathSciNet](http://www.ams.org/mathscinet/pdf/263894.pdf?pg1=IID&s1=55880&vfpref=html&r=15))

The correct definition using [[crossed modules]] of sheaves then appeared in

* {#Debremaeker76} Raymond Debremaeker, _Cohomologie met waarden in een gekruiste groepenschoof op een situs_, PhD thesis, 1976 (Katholieke Universiteit te Leuven). English translation:
_Cohomology with values in a sheaf of crossed groups over a site_, arXiv:[1702.02128](https://arxiv.org/abs/1702.02128)

Of [[algebra|algebraic]] [[structures]]:

*  [[René Lavendhomme]], J. R. Roisin, _Cohomologie non abélienne de structures algèbriques_, Journal of Algebra **67** 2 (1980) 385-414 [doi:10.1016/0021-8693(80)90168-4]

Discussion in terms of [[gerbes]]:

* [[Jean Giraud]], _Cohomologie non abélienne_ , Springer  (1971) [[doi:10.1007/978-3-662-62103-5](https://link.springer.com/book/10.1007/978-3-662-62103-5)]

  > (aspects of classification of $G$-[[gerbes]] by cohomology with coefficients in the [[automorphism 2-group]] $AUT(G)$, but imposes extra constraints)

* [[Larry Breen]], _Bitorseurs et cohomologie non-Abélienne_, The Grothendieck Festschrift: a collection of articles written in honour of the 60th birthday of Alexander Grothendieck, Vol. I, edited P.Cartier, et al., Birkhäuser, Boston, Basel, Berlin, 401-476, (1990) ([doi:10.1007/978-0-8176-4574-8_10](https://doi.org/10.1007/978-0-8176-4574-8_10))

* [[John Duskin]], _Non-abelian cohomology in a topos_, reprinted as: Reprints in Theory and Applications of Categories **23** (2013) 1-165 [[tac:tr23](http://www.tac.mta.ca/tac/reprints/articles/23/tr23abs.html)]

* [[Ieke Moerdijk]], _Lie Groupoids, Gerbes, and Non-Abelian Cohomology _ ([journal](http://www.springerlink.com/content/ul554x3077444545/))

* [[Amnon Yekutieli]], Combinatorial descent data for gerbes, Journal of Noncommutative Geometry Volume 8, Issue 4, 2014, pp. 1083–1099, arXiv:1109.1919 ([webpage](https://www.math.bgu.ac.il/~amyekut/publications/comb-descent/comb-descent.html))

* [[Alexander Campbell]], _A higher categorical approach to Giraud's non-abelian cohomology_, PhD thesis, Macquarie University (2016) [[hdl:1959.14/1261186](http://hdl.handle.net/1959.14/1261186)]

Existence of [[classifying spaces]] for [[principal 2-bundles]]/[[nonabelian gerbes]]:

* [[John Baez]], [[Danny Stevenson]], _The Classifying Space of a Topological 2-Group_,  In: Baas N., Friedlander E., Jahren B., Østvær P. (eds.) *Algebraic Topology*, Abel Symposia, vol 4. Springer 2009 ([arXiv:0801.3843](https://arxiv.org/abs/0801.3843), [doi:10.1007/978-3-642-01200-6_1](https://doi.org/10.1007/978-3-642-01200-6_1))

### General theory -- $\infty$-bundles/$\infty$-gerbes

Discussion of the general theory via [[principal ∞-bundles]] and/or [[∞-gerbes]] and/or [[∞-stacks]]:

* [[Paul Glenn]], _Realization of cohomology classes in arbitrary exact categories_, J. Pure Appl. Algebra 25, 1982, no. 1, 33- 105 (doi:10.1016/0022-4049(82)90094-9)

In a context of [[nonabelian Hodge theory]]:

* [[Carlos Simpson]], _The Hodge filtration on nonabelian cohomology_, in: János Kollár, Robert Lazarsfeld, [[David Morrison]] (eds.) _Algebraic Geometry Santa Cruz 1995, Part 2_, Proceedings of Symposia in Pure Mathematics Volume 62.2, AMS 1997 ([arXiv:alg-geom/9604005](http://arxiv.org/abs/alg-geom/9604005), [doi:10.1090/pspum/062.2](https://doi.org/10.1090/pspum/062.2))

* [[Carlos Simpson]], _Secondary Kodaira-Spencer classes and nonabelian Dolbeault cohomology_ ([arXiv:alg-geom/9712020](http://arxiv.org/abs/alg-geom/9712020))

* [[Carlos Simpson]], _Algebraic aspects of higher nonabelian Hodge theory_,  in: Fedor Bogomolov, Ludmil Katzarkov (eds.), _Motives, polylogarithms and Hodge theory, Part II_ (Irvine, CA, 1998), Int. Press Lect. Ser., 3, II, Int. Press, 2002, 2016, 417-604.  ([arXiv:math/9902067](http://arxiv.org/abs/math/9902067), [ISBN:9781571462909](https://www.intlpress.com/site/pub/pages/books/items/00000089/index.php))

* [[Carlos Simpson]], [[Tony Pantev]], [[Ludmil Katzarkov]], _Nonabelian mixed Hodge structures_ ([arXiv:math/0006213](http://arxiv.org/abs/math/0006213))

Generally:

* {#Toen02} [[Bertrand Toën]], _Stacks and Non-abelian cohomology_, lecture at _[Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory](https://www.msri.org/realvideo/index04.html)_, MSRI 2002 ([slides](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/index.html), [ps](http://www.msri.org/publications/ln/msri/2002/introstacks/toen/1/meta/aux/toen.ps), [pdf](https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.pdf), [[ToenStacksAndNonabelianCohomology.pdf:file]])

* [[J. F. Jardine]], Z. Luo, _Higher principal bundles_, Mathematical Proceedings of the Cambridge Philosophical Society,  Volume 140, Issue 2 March 2006 , pp. 221-243 ([pdf](http://www.math.uiuc.edu/K-theory/0681/cocycles6.pdf), [doi:10.1017/S0305004105008911](https://doi.org/10.1017/S0305004105008911))

* [[J. F. Jardine]], _Cocycle categories_, In: [[Nils Baas]], [[Eric Friedlander]], B. Jahren , [[Arne Østvær]] (eds.) _Algebraic Topology_, Abel Symposia, vol 4. Springer 2009 ([arXiv:math/0605198](https://arxiv.org/abs/math/0605198), [doi:10.1007/978-3-642-01200-6_8]( https://doi.org/10.1007/978-3-642-01200-6_8))

* [[Jacob Lurie]], Thm. 7.1.0.1 in: _[[Higher Topos Theory]]_, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

* {#Wendt} [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ , Journal of  Homotopy and Related Structures 6(1), 2011, pp. 1--38.  ([arXiv:1009.2930](http://arxiv.org/abs/1009.2930)) ([published version](http://tcms.org.ge/Journals/JHRS/volumes/2011/volume6-1.htm))

* {#RobertsStevenson16} [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parametrized spaces_, New York Journal of Mathematics Volume 22 (2016) 405-440 ([arXiv:1203.2460](https://arxiv.org/abs/1203.2460), [nyjm:22-19](http://nyjm.albany.edu/j/2016/22-19.html))

* [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](https://arxiv.org/abs/1203.2461))

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801 ([doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

* [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 ([doi:10.1007/s40062-014-0077-4](http://link.springer.com/article/10.1007/s40062-014-0077-4), [arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

On [[cohomology operations]] on components of [[Whitehead-generalized cohomology theories]] seen in non-abelian cohomology:

* [[John Michael Boardman]], [[David Copeland Johnson]], [[W. Stephen Wilson]], _Unstable Operations in Generalized Cohomology_ ([pdf](https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf)), in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

On the non-abelian [[Chern-Dold character]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

[[!redirects non-abelian cohomology]]

[[!redirects nonabelian cohomology theory]]
[[!redirects nonabelian cohomology theories]]

[[!redirects non-abelian cohomology theory]]
[[!redirects non-abelian cohomology theories]]


nonabelian cosheaf homotopy

_Nonabelian cosheaf homotopy_ is a notion in the context of [[nonabelian cohomology]]:

Given an [[higher category theory|infinity-category]]-valued (pseudo)copresheaf $\mathbf{B} : Spaces \to \infty-Cat$, its _homotopy_ $\pi(-,\mathbf{B})$ is the $\infty$-category valued (pseudo)copresheaf which assigns to each space the limit over all [[descent|codescent data]]:
$$
  \pi(X, \mathbf{B}) := lim_{Y^\bullet \to X}
  Codesc(Y^\bullet, \mathbf{B})
  \,.
$$
Here the limit is over all [[hypercover|hypercovers]] of $X$.

#Remarks#

* [[duality|Dual]] to nonabelian cosheaf homotopy is [[nonabelian sheaf cohomology]].

* A co-presheaf whose value on each space is [[equivalence|equivalent]] to its codescent $\infty$-category for any cover of that space is an [[infinity-stack|infinity-co-stack]].


nonabelian differential cohomology

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#### Differential cohomology
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#Contents#
* table of contents
{:toc}

## Idea

What may be called *nonabelian differential cohomology* -- in combination of *[[nonabelian cohomology]]* and *[[differential cohomology]]* -- is a notion of [[connections on infinity-bundles|connections on higher bundles]] with [[higher group|higher]] [[gauge group]] being a possibly non-[[abelian infinity-group|abelian]] [[n-group]].

Early motivation was the observation ([SSS12](#SatiSchreiberStasheff12)) that with the [[Green-Schwarz mechanism]] imposed, the [[B-field]] in [[heterotic string theory]] combines with the ordinary non-abelian [[gauge field]] into a non-abelian higher connection (a [[twisted differential string structure]]), and analogously so after lifting the situation to the [[C-field]] in [[Hořava-Witten theory]] ([FSS15](#FSS15)).

Later, the desire to more accurately model more of the expected subtle topological properties of the [[C-field]] (such as the [[shifted C-field flux quantization]] combined with [[Page charge]]-quantization) led to the hypothesis ("[[schreiber:Hypothesis H]]") that it ought to be [[flux quantization|flux quantized]] in unstable [[differential Cohomotopy|differential]] 4-[[Cohomotopy]], hence with [[homotopy type]] of its higher gauge group being the [[loop group|loop]] [[infinity-group|$\infty$-group]] of the 4-sphere. This [[schreiber:Hypothesis H]] turns out to subtly reproduce the [[Green-Schwarz mechanism]] in its lift to [[Hořava-Witten theory]] ([FSS22](#FSS22)) and also that on [[M5-branes]] ([SS20](#SS20), [FSS21](string+structure#FSS21)).

Generally, one may understand [[differential cohomology|differential]] [[nonabelian cohomology]] as modelling [[flux quantization|flux quantized]] [[higher gauge theory|higher gauge fields]] ([SS23](#SS23)).

For example, also the "Hypothesis K" of [[D-brane charge quantization in topological K-theory]] is, at face value, a [[flux quantization]] (of the [[RR-field]] combined with the [[B-field]]) in a non-abelian differential cohomology (see the overview [here](/schreiber/show/Introduction+to+Hypothesis+H#OverviewLogicOfFluxQuantization)) which however may be and commonly is regarded as *twisted abelian* namely as [[twisted differential K-theory]], by regarding the [[B-field]] as a "[[background field]]" and considering the [[RR-field]] in its dependence.

## Related concepts

* [[nonabelian cohomology]]

* [[differential cohomology]]

* [[higher gauge theory]]

## References

### Via higher Cartan connections from (adjusted) Weil algebras

Early explorative notes:

* [[Urs Schreiber]], _On nonabelian differential cohomology_, March 17, 2008, [Slides](https://www.math.uni-hamburg.de/home/schreiber/diffcohslides.pdf).

The local structure of [[L-infinity algebra valued differential forms|$L_\infty$-algebra valued differential forms]], via [[dg-algebra]] [[homomorphisms]] out of ("[[adjusted Weil algebra|adjusted]]") [[Weil algebras]] in the [[de Rham complex]] of the base manifold:

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], example 5 in section 6.5.1, p. 54 of _[[schreiber:L-infinity algebra connections|$L_\infty$ algebra connections and applications to String- and Chern-Simons n-transport]]_, in: *Quantum Field Theory*, Birkhäuser (2009) 303-424 \[doi:0801.3480, [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)\]

The [[Lie integration]] of this local structure to globally possibly nontrivial actual [[connections on infinity-bundles|connections on higher bundles]]:

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _Čech cocycles for differential characteristic classes_, Advances in Theoretical and Mathematical Physics 16:1 (2012), 149–250, [arXiv:1011.4735](http://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916).

For the application of this construction to modelling the [[Green-Schwarz mechanism]] see [below](#HigherGaugeTheoryOfTheGreenSchwarzMechanismReferences).

### Via the nonabelian character map

A more axiomatic formulation of [[differential cohomology|differential]] [[non-abelian cohomology]] (not yet proven to subsume the above construction) in non-abelian generalization of the original [Hopkins-Singer construction](differential}cohomology#HopSin) of abelian (namely [[Whitehead-generalized cohomology]]) [[differential cohomology]], now using a nonabelian generalization of the [[Chern-Dold character map]]:

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], Def. 9.3 ([p. 161](/schreiber/files/NonabelianCharacterWS231212.pdf#page=171)) in: *[[schreiber:The Character Map in Non-Abelian Cohomology]]*, World Scientific (2023) [[doi:10.1142/13422](https://doi.org/10.1142/13422)]

On how this generally serves to reflect [[flux quantization]] in [[higher gauge theories]]:

* {#SS23} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Flux Quantization on Phase Space]]* [[arXiv:2312.12517](https://arxiv.org/abs/2312.12517)]

reviewed in:

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Introduction to Hypothesis H]]* ([here](/schreiber/show/Introduction+to+Hypothesis+H#OverviewFluxQuantizationViaNonabDiffCohomology))

### Differential cohomotopy for the 11d SuGra C-field

The example of unstable (and as such non-abelian) [[differential Cohomotopy]] (in the context of modeling the [[supergravity C-field]] via [[schreiber:Hypothesis H]]):

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §4 in: *[[schreiber:The WZW term of the M5-brane and differential cohomotopy]]*, J. Math. Phys. **56** 102301 (2015) [[arXiv:1506.07557](https://arxiv.org/abs/1506.07557), [doi:10.1063/1.4932618](https://doi.org/10.1063/1.4932618)]

* [[Daniel Grady]], [[Hisham Sati]], Def. 3.2 in: *Differential cohomotopy versus differential cohomology for M-theory and differential lifts of Postnikov towers*, Journal of Geometry and Physics **165** (2021) 104203 [[arXiv:2001.07640](https://arxiv.org/abs/2001.07640), [doi:10.1016/j.geomphys.2021.104203](https://doi.org/10.1016/j.geomphys.2021.104203)]

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Ex. 9.3 in: _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

[[!include higher gauge theory of the Green-Schwarz mechanism -- references]]

[[!redirects non-abelian differential cohomology]]
[[!redirects differential nonabelian cohomology]]
[[!redirects differential non-abelian cohomology]]