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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Higher geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Traditional [[differential geometry]] is the field of [[mathematics]] that studies the [[geometry]] of [[smooth spaces]] which are equipped with a notion of [[differentiation]]. There is a refinement of this traditional theory to the general context of "[[higher geometry]]" -- this is accordingly called _[[higher differential geometry]]_.

In higher differential geometry, [[geometry]] is paired with [[homotopy theory]]. Accordingly, [[smooth spaces]] are refined to "smooth [[homotopy types]]" -- often called _[[smooth infinity-groupoids]]_ or _[[infinity-stacks]]_ -- and [[function algebras]] of [[smooth functions]] are refined to [[strong homotopy algebras]], often called _[[infinity-algebra over an (infinity,1)-operad|infinity-algebras]]_.

Here we give an informal overview of what motivates this generalization and what it is good for.

First we list some examples of how tools from [[higher differential geometry]] help to understand plain traditional [[differential geometry]]:

* _[I) Applications of higher DG to plain DG](#ApplicationsOfHigherDGToPlainDG)_

Besides illuminating traditional DG where it exists, higher DG naturally extends plain DG to regions where it breaks down due to [[singularities]]:

* _[II) Resolution of singularities in plain DG](#ResolutionOfSingularities)_

However, much of the genuine motivation for developing [[higher differential geometry]] came and increasingly comes from structures in fundamental [[physics]], in [[quantum field theory]] and [[string theory]], which are intrinsically of [[higher geometry|higher geometric]] nature, hence for which no adequate description in plain [[differential geometry]] even exists (this is in fact already true for traditional local gauge theory, see [below](#FieldBundles)).
This we discuss in:

* [III) Motivation of higher DG from physics](#MotivationOfHigherDGFromPhysics)

## **I)** Applications of higher DG to plain DG
{#ApplicationsOfHigherDGToPlainDG}

This section lists examples of how [[higher differential geometry]] helps with understanding plain [[differential geometry]].

* [Foliations and PDEs](#FoliationsAndPDEs)

* [Poisson and symplectic geometry](#PoissonGeometry)

* [Deformation theory](#DeformationTheory)

* [Orbifolds](#Orbifolds)

* [Field bundles for local gauge theory](#FieldBundles)

* [Lie integration](#LieIntegration)

* [Variational calculus and symplectic reduction](#VariationalCalculusAndSymplecticReduction)

### Foliations and PDEs
 {#FoliationsAndPDEs}

At the roots of differential geometry, prominent in the original work of [[Eli Cartan]], is [[foliation]] theory, arising in the integration of [[partial differential equations]].

   In the context of [[Lie groupoid]] theory, foliations have a classification -- a [[moduli stack]]: the [[Haefliger groupoid]]. The textbook ([Moerdijk-Mrcun](#MoerdijkMrcun)) discusses foliation theory from this perspective. The _[[Haefliger theorem]]_ which classifies foliations on open manifolds is proven by way of the [[Haefliger groupoid]].

   Cartan had originally encoded the [[partial differential equation|PDEs]] whose integration leads to foliation by "[[exterior differential systems]]". These are equivalently ([[Chevalley-Eilenberg algebras]]) of [[L-∞ algebroids]]. The entire theory of integration is thereby part of higher [[Lie theory]] -- [[infinity-Lie theory]].

### Poisson and symplectic geometry
 {#PoissonGeometry}

A subtopic of this is [[Poisson geometry]], where the foliation is by [[symplectic leaves]]. A fundamental problem in Poisson geometry was the [[deformation quantization]] of [[Poisson manifolds]].

   The [[formal deformation quantization]] problem was solved by [[Kontsevich]], which earned him a Fields medal. [[Cattaneo]] and [[Felder]] then showed that his formula for the [[star product]] is naturally understood as a [[3-point function]] of the [[Poisson sigma model]], a [[sigma-model]] [[QFT]] whose [[target space]] is the [[Poisson Lie algebroid]] of the given Poisson manifold.

   The strengthening of this to the [[strict deformation quantization]] problem for a large class of cases was then established by forming the [[convolution algebra]] of sections of the [[prequantum bundle|prequantization]] of the [[symplectic groupoid]] with [[Lie integration|Lie integrates]] the [[Poisson Lie algebroid]].

   (In [[QFT]] the shift into [[higher geometry]] witnessed in both these proofs is a special case of what is known as the [[holographic principle]]: the 0-categorical structure of Poisson manifolds is quantized via the 1-categorical strucure of [[Poisson Lie algebroids]] and [[symplectic groupoids]].)

The [[symplectic groupoid]] that induces the [[strict deformation quantization]] of a Poisson manifold also solves its classical desingularization problem, its _[[symplectic realization]]_.
For any [[symplectic groupoid]] $\Sigma$ with base a [[Poisson manifold]] $P$ the target map is a symplectic realization of $P$ and  the source map is a symplectic realization  of the opposite structure. Thus $\Sigma$ with its symplectic structure may be regarded as a desingularization of  $P$ with its Poisson structure. Since the [[symplectic groupoid]] is the [[Lie integration]] of the [[Poisson Lie algebroid]] of the Poisson manifold, symplectic realization is reduced to a problem in [[Lie theory]].

### Deformation theory
 {#DeformationTheory}

Apart from these Poisson structures and [[symplectic structures]], differential geometry studies many other kinds of smooth structures on [[smooth manifolds]], such as [[complex structures]] in [[complex geometry]]. The classification of these structures in each case is [[infinitesimal space|infinitesimally]] given by [[deformation theory]]. Since seminal work referenced at _[[model structure for L-∞ algebras]]_ it is known that deformation theory is essentially equivalently the study of [[L-∞ algebra]] in [[∞-Lie theory]]. Classical theorems in complex differential geometry such as the discussion of [[period maps]] find their natural formulation in this context. This is amplified in  ([Fiorenza-Martinengo2012](#FiorenzaMartinengo)).

### Orbifolds
 {#Orbifolds}

Almost as basic to traditional differential geometry as [[manifolds]] are [[orbifold]], which incorporate non-free [[actions]] into the theory. Notably the [[moduli spaces]] arising in differential geometry tend to be orbifolds instead of manifolds.

But orbifolds are equivalently a [[proper Lie groupoid|proper]] [[étale Lie groupoid]].  (This relation is for some reason more famous in algebraic geometry, where the étale Lie groupoids are called [[Deligne-Mumford stacks]].)

### Field bundles for local gauge theory
 {#FieldBundles}

Differential geometry shares much of its origin and history with [[physics]]: [[tensor fields]] and [[Riemannian geometry]] grew out of the theory of [[gravity]] and [[Chern-Weil theory]] and [[differential cohomology]] out of the study of [[gauge theory]]. [[Maxwell's equations]] influenced the study of [[de Rham cohomology]].

Out of this tradition has grown the formalization of [[physical fields]] as [[sections]] of a [[fiber bundle]] over [[spacetime]] called the _[[field bundle]]_. But this notion is insufficient: there cannot be a field bundle for _both_ [[gauge field theory]] and [[local quantum field theory]]. This problem is typically not felt in [[perturbation theory]], but it affects all attempts to go beyond.

The reason is that a [[gauge field]], including its underlying [[instanton sector]]/[[charge]] sector, is not a section of a fiber bundle but of a [[fiber infinity-bundle|fiber 2-bundle]], the anolog in higher differential geometry. Detailed discussion of this is at _[[field (physics)]]_ and in the corresponding section at _[[geometry of physics]]_.

### Lie integration
 {#LieIntegration}

[[Lie integration]] of anything more general than a finite-dimensional [[Lie algebra]] in general goes out of the real of [[Lie groups]]. Local brackets = [[Lie algebroids]] integrate to [[Lie groupoids]] and
infinite-dimensional Lie algebras in general to [[Lie 2-groups]].

(...)

### Variational calculus and symplectic reduction
 {#VariationalCalculusAndSymplecticReduction}

[[derived differential geometry]] allows to resolve many otherwise singular limit/intersection constructions. Modern [[variational calculus]] in terms of [[jet bundle]] ("[[D-geometry]]") is formulated this way, as is [[symplectic reduction]] and the unification of both in the [[BV-BRST formalism]].

## **II)** Resolution of singularities in plain DG
 {#ResolutionOfSingularities}

[[singularity|Singularities]] arise in [[differential geometry]] if the two kinds of [[universal constructions]] fail to exist as [[smooth manifolds]]:

1. if an _intersection_ (in [[category theory]]: a [[limit]]) fails to exist this is resolved by passing from [[critical loci]] to [[derived critical loci]]:

   [Resolution of singular intersection](#ResolutionOfSingularIntersections)

1. if a _[[quotient]]_ (in [[category theory]]: a [[colimit]]) fails to exist, this is resolved by passing to [[homotopy quotients]]:

   [Resolution of singular quotients: stacks quotients](#ResolutionOfSingularQuotients)

### Resolution of singular qotients: stacky quotients
 {#ResolutionOfSingularQuotients}

... [[stack]] ... [[homotopy quotient]] ... [[quotient stack]] ... [[differentiable stack]] ...

### Resolution of singular intersections: derived intersections
 {#ResolutionOfSingularIntersections}

The [[category]] of [[smooth manifolds]] does not have many [[limits]]. Accordingly many constructions that one would like to perform in traditional differential geometry simply fail to exist. For instance the [[critical locus]] of a [[smooth function]] -- of central relevance in [[variational calculus]] -- rarely is again a [[smooth manifold]] and hence cannot be treated with tools of differential geometry, in general.

In the "[[derived differential geometry]]"-flavor of [[higher differential geometry]] (technically: the not-[[n-localic (infinity,1)-topos|1-localic]]) this problem is resolved by replacig [[function algebras]] of [[smooth functions]] by [[simplicial algebras]], which essentially amounts to refining [[smooth manifolds]] by [[dg-manifolds]]. This refinement resolves previously singular (hence: non-existing in DG) by the method of [[resolution]] familiar from [[homological algebra]]: a singular intersection is realized as the [[homology group]] of a [[chain complex]] of perfectly smooth manifolds (or rather their function algebras).

Applied the concept of [[critical locus]] this leads to the notion of _[[derived critical locus]]_. This is actually a construction secretly with a long tradition in differential geometry, known as the _[[BV-BRST formalism]]_. Higher and [[derived differential geometry]] provide a theoretical framework for handling [[BV-BRST formalism]] systematically.

## **III)** Motivation of higher DG from physics
 {#MotivationOfHigherDGFromPhysics}

To a large extent, [[differential geometry]] had been co-evolving with the description of [[physics]] in terms of _[[field (physics)|fields]]_. We have already seen above in _[Field bundles for local gauge theory](#FieldBundles)_ that an accurate formulation even of traditional notions of such fields requires [[higher differential geometry]] to degree 1. But modern developments in [[quantum field theory]] require notions of [[physical fields]] and [[quantum states]] that probe much more deeply into the realm of geometric [[homotopy theory]]. We indicate here some aspects. For comprehensive introductory lecture notes on this topic see at

* _[[geometry of physics]]_.

A more technical survey of is in [FSS 13](#FSS).

Below we first discuss how combining the notion of [[local quantum field theory]] in its modern incarnation as [[extended quantum field theory]] with [[prequantum field theory|Lagrangian]] quantum field theory necessitates higher differential geometric structures:

* [Extended prequantum field theory](#ExtendedPrequantumFieldTheory)

Then we turn to modern proposals for [[theory (physics)|theories]] that go beyond the [[standard model of particle physics]] and which inherently contain higher geometric fields already in their non-localized formalization:

* [Higher gauge fields](#HigherGaugeFields)

(...)

### The higher orbit method -- Extended prequantum field theory
 {#ExtendedPrequantumFieldTheory}

A high point of traditional [[differential geometry]] is its impact on [[representation theory]] via the _[[orbit method]]_. This is actually a construction deeply rooted in [[quantum mechanics]]:  given a [[symplectic manifold]] equipped with [[Hamiltonian action]] by a [[Lie group]], regarding this as a [[mechanical system]] and then applying the differential-geometric process of [[geometric quantization]] to its yields a [[space of states]] on which the group is represented. A large class of [[representations]] arises this way and the [[orbit method]] sheds light on the corresponding [[representation theory]].

But speaking of [[physics]] in the first place, certainly the [[quantum mechanics]] governing the [[orbit method]] is just the simplest fragment of the foundational theory of [[physics]], which is [[quantum field theory]] in arbitrary [[dimension]]. Quantum mechanics can be thought of as quantum field theory in dimension 1 ("on the [[worldline]]"). Passing in [[geometric quantization]]/[[orbit method]] from dimension 1 to higher dimensional quantum field theory corresponds precisely, as we discuss now, to passing to [[higher differential geometry]]: "[[extended prequantum field theory|extended prequantum geometry]]".

The notion of _[[quantum field theory]]_ exists without reference to any predefined notion of _[[configuration space]]_ of _[[quantum fields]]_, _[[action functional]]_, _[[phase space]]_ etc.:

A [[quantum field theory]] in [[FQFT]]-[[axiomatization]] is simply a consistent assignment of [[spaces of quantum states]], whereas in [[AQFT]]-[[axiomatization]] it is a consistent assignment of [[algebras of quantum observables]], and that's it.

However, most (or maybe all?) quantum field theories of interest in actual [[physics]] (as opposed to as devices of pure [[mathematics]]) are not random models of these axioms, but do arise under a process called [[quantization]] from a ([[local Lagrangian|local]]/[[extended Lagrangian|extended]]) [[Lagrangian]], hence from an [[action functional]], defined on a [[configuration space]] of [[quantum fields]], or else arise as [[holographic duals]] of quantum field theories that arise by quantization. Moreover, the extra information provided by the Lagrangian is commonly used (and is maybe strictly necessary) to interpret the mathematical structure of the axiomatic QFT in actual [[physics]] (though notably in [[AQFT]] there are results that re-extract at least parts of this data from the axiomatic QFT, for instance the [[Doplicher-Roberts reconstruction theorem]] which extract the [[global gauge group]] from the [[local net of quantum observables]]).

There are in turn two formalizations of the notion of [[quantization]]: _algebraic [[deformation quantization]]_ and _[[geometric quantization]]_. In the latter one speaks of _[[prequantization]]_ when referring to a precursor step to the actual quantization step, in which the [[symplectic form]] on [[phase space]] is lifted from to [[differential cohomology]], hence to a [[prequantum bundle]]. But in the context of [[higher geometry]] and [[higher geometric quantization]] this prequantization step is already part of the data of the [[Lagrangian]] itself: an [[extended Lagrangian]] already encodes not just the [[action functional]] but also the [[prequantum bundle]] and all the [[prequantum n-bundle|prequantum (n-k)-bundles]] in each [[dimension]] $k$. The action functional itself is the _[[prequantum 0-bundle]]_  in this context.

Therefore, in the refined picture of [[higher geometry]]/[[extended quantum field theory]] it makes good sense to refer in a unified way to **prequantum field theory** for all of the data related to [[Lagrangians]] that is not yet the final [[quantum field theory]].

In particular, an **extended prequantum field theory** of [[dimension]] $n$ is a rule that assigns

* to every (suitably [[orientation|oriented]]) [[closed manifold]] of [[dimension]] $k$ a [[prequantum n-bundle|prequantum (n-k)-bundle]]

* to every (suitably [[orientation|oriented]]) [[compact topological space|compact]] [[manifold with boundary]] $\Sigma_k$ a [[section]] of the prequantum $(n-k+1)$-bundle assigned to the [[boundary]] $\partial \Sigma_k$ and pulled back to the space of fields over $\Sigma_k$

such that this data is related suitably under [[transgression]].

The actual [[extended quantum field theory]] would be obtained from such a data by passing from the assignment of a given prequantum $(n-k)$-bundle to that of the [[n-vector space|(n-k)-vector space]] of [[polarization|polarized]] [[sections]] of a suitable [[associated infinity-bundle|associated]] [[fiber infinity-bundle|fiber bundle]].

This is summarized in the following table:

[[!include extended prequantum field theory - table]]

### Higher gauge fields
 {#HigherGaugeFields}

Beyond the [[prequantum n-bundles]] involved in any [[extended prequantum field theory]], there are various [[theory (physics)|theories]]in physics that involve [[field (physics)|fields]] which themselves are already given by higher geometric structure, notably [[higher gauge fields]] given by [[connection on an infinity-bundle|higher connections]] on [[principal infinity-bundle|higher principal bundles]]. These arise notably in higher dimensional [[supergravity]] [[theory (physics)|theories]] and their [[UV-completion]]: [[string theory]].

The first example along these lines is the [[Kalb-Ramond field]] or [[B-field]], which is a higher order version of the [[electromagnetic field]]. In order to fully capture the nature of these fields, one needs to describe them in higher geometry. Notably [[quantum anomaly]] cancellation conditions can lead to fairly intricate structures of the [[moduli infinity-stacks]] of such fields. For instance anomaly-free [[heterotic string theory|heterotic]]  [[background gauge fields]] are given by moduli 2-stacks of [[twisted differential string structures]].

[[!include table of branes]]

## Related pages

* [[geometry of physics]]

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[twisted smooth cohomology in string theory]]

* [[motives in physics]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

* [[motivation for cohesion]]

## References

The basic relation between [[foliation]] theory and [[Lie groupoid]]-theory is discussed in

* [[Ieke Moerdijk]], [[Janez Mrčun]], _Introduction to foliations and Lie groupoids_, Cambridge Studies in Advanced Mathematics __91__, 2003. x+173 pp. ISBN: 0-521-83197-0
 {#MoerdijkMrcun}

The identification of [[orbifolds]] as (proper) [[etale Lie groupoids]] is due to

* [[Ieke Moerdijk]], [[Dorette Pronk]], _Orbifolds, sheaves and groupoids_, K-theory 12 3-21 (1997) ([pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/pronk.pdf))
  {#MoerdijkPronk}

A discussion of how [[∞-Lie theory|∞-Lie theoretic]] [[deformation theory]] neatly captures various traditional theorems is in

* [[Domenico Fiorenza]], [[Elena Martinengo]], _A short note on ∞-groupoids and the period map for projective manifolds_, Publications of the nLab [[publications:FiorenzaMartinengo2012|vol. 2 no. 1]] (2012)

For a survey of how higher differential geometry serves to capture subtle aspects of [[prequantum field theory]] see

* _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_
 {#FSS}

(...)

[[!redirects higher differential geometry applied to plain differential geometry]]


motivation for sheaves > history

< [[motivation for sheaves]]


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### Context
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

The following is (supposed to be) a pedagogical motivation of the concepts [[sheaf]], [[stack]], [[infinity-stack]] and [[Higher Topos Theory|higher topos theory]]. It assumes only that the reader has a working knowledge of [[topological space]]s and aims to provide from that an intuitive but useful idea of the relevance of the circle of ideas of [[Categories and Sheaves|categories and sheaves]], [[cohomology]], [[sheaf cohomology]] and a bit of [[Higher Topos Theory|higher topos theory]].

Actually, the reader need not even know anything about topological spaces. The only assumption we need is that a space is something that has _probes_ by test spaces (points, say). The reader can jump right to _[The basic idea of sheaves](#TheBasicIdeaOfSheaves)_.

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

## Motivation: maps between topological spaces
 {#MapsBetweenTopologicalSpaces}

Classes of maps between [[topological space]]s encode interesting information.

To pick just one example: there is a [[topological space]]  that goes by the name $K(\mathbb{Z},2)$ or $\mathcal{B}U(1)$ and is called an [[Eilenberg-Mac Lane space]].  It has the  peculiar property that for any other space $X$, the [[homotopy]] classes $[X, \mathcal{B}U(1)]$ of continuous maps from $X$ to $\mathcal{B}U(1)$ characterize the following:

* they correspond to equivalence classes of [[circle bundle]]s over $X$: that is, to spaces $P$ equipped with a map to $X$ such that over
a contractible open subset $U \subset X$ the space $P$ looks like
the [[product]] of $U$ with the circle, $P|_U \simeq U \times S^1$ (and such that the identification respects the action of the [[group]] $U(1)$ on the circle).

  * Thus, one says that $\mathcal{B}U(1)$ is the [[classifying space]] for circle [[bundle]]s ($[X, \mathcal{B}U(1)] \simeq \{ $circle bundles over $X \}/{\sim}$) or that it [[representable functor|represents]] the assignment of circle bundles to spaces -- to some extent the topic of sheaves, and cohomology is all about the notion of such [[representable functor|representability]]

  * A famous such circle bundle you may have seen is the [[Hopf fibration]], which is a way to map the 3-dimensional sphere $S^3$ to the familiar 2-dimensional sphere $S^2$, such that over small parts $U \subset S^2$ of the 2-sphere the 3-sphere looks like the [[product]] space $U \times S^1$.

  * This particular circle bundle corresponds to a very special class of maps $X \to \mathcal{B}U(1)$: namely the collection of maps $[S^2, \mathcal{B}U(1)]$ happens to have the structure of the abelian group $\mathbb{Z}$ of integers under addition, and the [[Hopf fibration]] corresponds to the generating element $1$ of this group;

  * generally, this abelian group formed by $[X, \mathcal{B}U(1)]$
  is famous under another name: it is called the second integral [[cohomology]] group of $X$, denoted $H^2(X, \mathbb{Z})$.

* There is a similar such cohomology group $H^{n+1}(X,\mathbb{Z})$
of $X$ for every natural number $n$: the $(n+1)$st _integral cohomology group_ of $X$ -- and all these groups are naturally
realized in terms of homotopy classes of maps from $X$ into
some topological space -- called the [[Eilenberg-MacLane space]]s $K(\mathbb{Z}, n+1)$ or $\mathcal{B}^{n} U(1)$ -- and again each element in these groups can be identified with a certain geometric structure living over $X$, called a circle $n$-bundle or $(n-1)$-[[gerbe]]: $[X, \mathcal{B}^{n} U(1)] = \{ $circle $n$-bundles on $X \}/{\sim}$.

So these cohomology groups encode a lot of interesting information about topological spaces; but there is some useful information about a space, which is not
encoded in maps from it to another topological space:

* in particular, if the space $X$ in question
is not just a [[topological space]] but carries extra [[stuff, structure, property|structure]], such as being a _[[smooth manifold]]_, then one is interested  in _smooth_ circle bundles over $X$ which are equipped with a [[connection]] $\nabla$: a good rule which assigns to each smooth path  $\gamma : x \to y$ in $X$ an identification between the two circles over its endpoints;
$$
  \array{
    x && \stackrel{\nabla}{\mapsto} & S^1_x
    \\
    \downarrow^\gamma
    &&&
    \downarrow^{\nabla(\gamma)}
    \\
    y && \stackrel{\nabla}{\mapsto} & S^1_y
  }
$$

  * this is the point where these considerations become of interest for [[physics]]: there [[bundles]] with [[connection on a bundle|connection]], and then [[principal infinity-bundle|n-bundles]], or $(n-1)$-[[gerbe]]s, with connection encode _[[gauge fields]]_ such as the [[electromagnetic field]]. See at _[[geometry of physics]]_ for more on this.

  * you might guess that there is a smooth manifold classifying smooth circle bundles with connection -- but there is not, not unless one stretches the meaning of _manifold_ and of _space_ in general a bit

  * so maybe we just need a slightly more flexible notion of what "space" should mean.

* the generalization of the notion of [[space and quantity|space]] which does accomplish this: the notion of a [[sheaf]]

  * (there is a very general abstract nonsense way to understand [[sheaf|sheaves]] as generalized spaces in the context of a very general abstract [[duality]] between the notions of  [[space and quantity]]. The following is an informal way to understand this).

## The basic idea of sheaves
 {#TheBasicIdeaOfSheaves}

To understand what a [[sheaf]] is, recall which aspect of the notion of [[topological space]] was relevant in the above examples $\mathcal{B}^{n} U(1)$: these spaces were entirely characterized by how one can _map_ other spaces _into_ them:
$$ X \mapsto [X,\mathcal{B}^{n} U(1)] .$$

This is a general strategy that one can adopt: suppose I dream up a space but don't tell you which one it is, but I give you hints: for each other space $U$ that you can dream up, I tell you how you can _probe_ my space by mapping your space $U$ into it.

* Let's now call the space which I dreamed up $X$, the generic symbol for spaces. For every space $U$ that you come up with, I do some secret computation and then present you with the result: I hand you a [[set]], let's call it $X(U)$, and tell you  that this is the set of ways that $U$ can be mapped into $X$:

  : $X(U) = \{ $ways to map $U$ into some hypothetical $X \}$
  :   =: {probes of $X$ by $U$}.

* Will you be able, in general, to guess my space $X$ from this information $U \mapsto X(U)$? No. So, to be fair, I should provide a bit more information. To actually get a feeling for what my space $X$ is like, you need to know something about how different ways of probing my space relate to each other.

* If you give me a map $U \stackrel{f}{\to} V$ from one test space to another, then for any element $V \stackrel{p}{\to} X$ of $X(V)$, I can produce an element $U \stackrel{f}{\to} V \stackrel{p}{\to} X$ of $X(U)$. In other words, your map $U \to V$ induces a transformation $X(U) \leftarrow X(V)$. To be fair, I should tell you at least what this transformation is!

* Well, fine: I'll do it. If you give me a map $U \stackrel{f}{\to} V$ from one test space to another, I'll give you a map of sets, denoted $X(U) \stackrel{X(f)}{\leftarrow} X(V)$, that supposedly indicates how $V$-probes of $X$ turn into $U$-probes when you precompose them with $f$.

Do you need still more information to guess my space $X$? No, you have enough! This somewhat remarkable fact is closely related to a fundamental statement of [[category theory]]: the [[Yoneda lemma]] and its implication on [[representable functor]]s.  These terms will be discussed in a moment...

First, there is a bit more to the game than mentioned so far: you need to be sure that even if I  won't reveal my space $X$ to you directly, the little information about it which I do provide, that rule $U \mapsto X(U)$,  I should provide honestly and consistently. Some consistency checks to assure that I am not just making things up
but am giving you consistent information about my secret space $X$ are the following:

* it must be true that when you hand me the identity map $Id_U : U \to U$ on a test space, that then I return you the identity map $Id_{X(U)} : X(U) \to X(U)$ on the set of probes of $X$ by $U$;

* also, it must be true that when you first hand me two consecutive maps of test spaces, $f : U \to V$ and $g : V \to W$, and then hand me their composite $g \circ f : U \to W$, that my reply $X(g \circ f) : X(W) \to X(U)$ to the latter is the result of composing my two replies about the former: $X(g \circ f) = X(W) \stackrel{X(g)}{\to} X(V) \stackrel{X(f)}{\to} X(U)$.

Such an assignment $U \mapsto X(U)$ of sets to spaces which satisfies the above two properties is what is called a [[presheaf]]: this, in turn, is nothing but what is called a [[functor]] on a [[category]]:

* the [[category]] in question here is the collection of test [[topological space]]s $U$, called [[object]]s, equipped with continuous maps $f : U \to V$ between them, called [[morphism]]s, which may be composed when their domains and targets match. This composition is associative in the obvious way and on every object $U$ there is an [[identity morphism|identity]] map $Id_U : U \to U$,  composition with which does nothing. This category is known by the name [[Top]]; $Top = \{ $topological spaces and continuous maps between them$ \}$

* the other [[category]] in the game, namely the [[category]] called [[Set]] whose [[object]]s are [[set]]s and whose [[morphism]]s are functions between sets: $Set = \{ $sets and functions between them$ \}$

* the probe-game which we are playing can hence be understood as being an assignment of [[object]]s in [[Set]] to [[object]]s in [[Top]] and of [[morphism]]s in [[Set]] to [[morphism]]s in [[Top]] which satisfies the above consistency conditions: this is called a [[contravariant functor|contravariant]] [[Set]]-valued [[functor]], or a [[presheaf]], on [[Top]] and it is denoted $X(-) : Top^{op} \to Set$

We say [[presheaf]] instead of just [[functor]], even though taken at face value a presheaf is just a functor, to indicate that there is one further, more nontrivial consistency check on the information about probes of my secret space $X$ which you should do: it must be true that you can piece together the information which I give for small test spaces to deduce information about probes by bigger test spaces.

* More precisely, if I tell you the set $X(U)$ of probes of my secret space $X$ by the test space $U$, and if you then chop up $U$ into two pieces $V_1$ and $V_2$  sitting inside $U$ by inclusion maps $p_i : V_i \hookrightarrow U$, with a bit of overlap $V_1 \cap V_2$, then it ought to be true that all the probes by $U$ in $X(U)$ can be entirely and exactly reconstructed from taking probes by $V_1$ in $X(V_1)$ and by $V_2$ in $X(V_2)$ and see if they match over the overlap $V_1 \cap V_2$ of the two small probes:

* in symbols, the collection of pairs in $\{ $matching maps from $V_1$ and $V_2$ to $X \} \hookrightarrow X(V_1) \times X(V_2)$ whose elements coincide when restricted to the overlap

  $$
    X(V_1) \times X(V_2)
      \underoverset
       {\underset{restrict\;second\;element}{\longrightarrow}}
       {\overset{restrict\;first\;element}{\longrightarrow}}
       {\phantom{AAA}}
    X(V_1 \cap V_2)
  $$

  should be exactly all the possible maps in $X(U)$: $\{ $matching probes of $X$ on $V_1$ and $V_2 \} \simeq X(U)$.

* one says that the set $\{ $matching probes of $X$ on $V_1$ and $V_2 \}$ is the [[equalizer]] of the two [[parallel morphisms]]

  $$
    X(V_1) \times X(V_2)
     \underoverset
       {\underset{restrict\;second\;element}{\longrightarrow}}
       {\overset{restrict\;first\;element}{\longrightarrow}}
       {\phantom{AAA}}
    X(V_1 \cap V_2)
  $$

  the optimal solution to mapping into $X(V_1) \times X(V_2)$ such that these two maps become equal;

* this condition is called a [[sheaf]] or [[descent]] condition: one thinks of the sheaf of _descending_ from the cover to the base along the map $(V_1 \sqcup V_2) \stackrel{p_1 \sqcup p_2}{\to} U$;

This now is a sensible game to play: I don't tell you directly which space $X$ I am thinking of, but I do give you all the information $U \mapsto X(U)$ about its probes $X(U)$  by test spaces $U$, subject to the conditions that

* $X(-)$ is functorial and hence a [[presheaf]];

* $X(-)$ satisfies [[descent]] and is therefore a [[sheaf]]

## Sheaves more general than spaces

So far we have been talking about using topological spaces as probe spaces, hence about presheaves on the category [[Top]]. One can use other systems $S$ of test spaces as long as $S$ forms a [[category]] and such that there is an agreement about how some of its [[object]]s $U$, can be [[cover]]ed by other [[object]]s, such as the $V_1$ and $V_2$ above. Such an agreement of what counts as a [[cover]] in a [[category]] is called a [[coverage]] or a [[Grothendieck topology]], and a [[category]] equipped with such an information is called a [[site]];

Here are some important examples:

* A popular choice is to fix one particular topological space $Z$ and probe all generalized spaces using only open subsets $U \subset Z$ of $Z$. These open subsets form the [[category of open subsets]] $S = Op(Z)$ and the choice of [[coverage]] is the obvious one. [[sheaf|Sheaves]] on $Op(Z)$ are generalized spaces _over_ $Z$.

* Another relevant choice is to use instead of all topological test spaces in the category [[Top]] all smooth manifolds as test spaces, in the category $S =$ [[Diff]].

* Or the combination of these: the test spaces in the category $Op(Z)$ for $Z$ a smooth manifold.

We had motivated this language of sheaves as a reformulation of the familar notion of spaces in a somewhat more indirect form: a rule for how to probe a space. But now something strange happens: it turns out that there are collections of probe information $X(-) : S^{op} \to Set$ which do obey the rules, i.e. do form a sheaf, but for which there is nevertheless _no_ test space in $S$ that it describes probes into. And in fact, this is the generic case.

* On the other hand, maybe this just means that our notion of "ordinary space" was a bit too restrictive: after all, we noticed that all that really matters usually are the
ways we can probe a given space by other spaces; and
that the information about such probes is useful if only
it obeys the rules of being a sheaf.

* So we have every justification then to regard sheaves in general as perfectly valid _generalized spaces_; or more precisely: we regard sheaves as rules for how to  probe generalized spaces, and we take these generalized spaces to be entirely specified by their [[sheaf|sheaves]] of probes.

Examples:

* The notion of "generalized" here depends on the perspective of our probes. For the probe category of the form $S = Op(Z)$ with $Z$ a topological space, most sheaves are "generalized spaces" in that they do not describe maps into a space in $S$, i.e. an open subset of $Z$. But every sheaf on $Op(Z)$ can be shown to correspond to maps from $Z$ to an ordinary topological space $E$ which is equipped with a continuous map $p : E \to Z$ down to $Z$, such that $p$ is locally a homeomorphism. Such $E$ are called _[[etale space|étale spaces]]_ over $Z$.

* A sheaf on $S = $ [[Diff]] is a [[generalized smooth space]]. These may be very different from ordinary manifolds and ordinary topological spaces. There are generalized smooth spaces with a single point and still many curves in them. In fact, the generalized classifying space for smooth circle bundles with connection that we are still after is of this form.

## Maps between generalized spaces

Since we are thinking of sheaves $F_X : S^{op} \to Set$ and $F_Y : S^{op} \to Set$ as characterizing 'generalized spaces' $X$ and $Y$, we better have a good notion of maps between sheaves $F_X \to F_Y$ that corresponds to a sensible notion of maps $X \to Y$ between generalized spaces.

That's easy to figure out: if $f : X \to Y$ is supposed to be a map between generalized spaces, whatever these are, then in particular it turns every probe $p \in X(U)$, of $X$ which we are thinking of as a probe map $p : U \to X$, to a probe $f \circ p \in Y(U)$ of $Y$, which we should think of as the composite map $f \circ p : U \stackrel{p}{\to} X \stackrel{f}{\to} Y$ that simply maps the image of $U$ in $X$ further to an image of $U$ in $Y$ using the map $f : X \to Y$.

* nota bene: for the time being there is no space $X$ such that thinking of probes $p \in X(U)$ as maps $p : U \to X$ of some sort would literally make sense. However, once we impose some consistency conditions on the probe sets $X(U)$ precisely this will make sense. And to motivate these consistency conditions in the first place, it is best to already think of probes $p \in X(U)$ as being maps $p : U \to X$.

* so in terms of their collections (sheaves) of probes a map $f : X \to Y$ of generalized spaces should be given for each probe space $U$ by a map of sets of probes $f_U : X(U) \to Y(U)$, to be thought of as encoding the post-composition map $f_U : (U \stackrel{p}{\to} X) \mapsto (U \stackrel{p}{\to} X \stackrel{f}{\to} Y)$

* but we can also pre-compose: given a map $\phi : U \to V$ of test spaces, we also have the probe transformation map from above, $X(\phi) : X(V) \to X(U)$ which is to be thought of as acting by $X(\phi) : (V \stackrel{p}{\to} X) \mapsto (U \stackrel{\phi}{\to} V \stackrel{p}{\to} X)$

* and we can combine both pre- and postcomposition and send $(V \stackrel{p}{\to} X) \mapsto (U \stackrel{\phi}{\to} V \stackrel{p}{\to} X \stackrel{f}{\to} Y)$

* in this notation it is manifestly clear that it must not matter whether we first pre- and then post-compose, or the other way around. But recall that so far writing $p : V \to X$ for a probe $p \in X(V)$ is just notation. So we need to enforce the obvious here on our formalism explicitly, lest the interpretation of generalized spaces breaks down.

* namely this means that the map $X(V) \stackrel{f_V}{\to} Y(V) \stackrel{\phi^*}{\to} Y(U)$ which first turns a $V$-probe of $X$ into a $V$-probe of $Y$ and then turns the result into a $U$-probe of $Y$, must coincide with the map $X(V) \stackrel{\phi^*}{\to} X(U) \stackrel{f_U}{\to} Y(U)$ which first turns a $V$-probe of $X$ into a $U$-probe of $X$ and then turns the result into a $U$-probe of $Y$.

* Such an equality between two different composite maps is conveniently depicted as a square
$$
  \array{
     X(U)
     &\stackrel{f_U}{\to}&
     Y(U)
     \\
     \uparrow^{X(\phi)}
     &&
     \uparrow^{Y(\phi)}
     \\
     X(V)
     &\stackrel{f_V}{\to}&
     Y(V)
  }
$$
The fact that the two ways of going from the bottom left to the top right of the square are supposed to be equal is called. "the square commutes", or "the square is commutative".

So that then is our definition of maps $f : X \to Y$ between generalized spaces $X$ and $Y$: if $U \mapsto X(U)$ and $V \mapsto X(V)$ are the probe-assignments that define $X$ and $Y$, then the map $f$ is defined by probe-transformations $f_U : X(U) \to Y(U)$ such that for all maps $\phi : U \to V$ the above squares commute.

* Such a collection of data is called a [[natural transformation]] between the probe-assigning [[functor]]s $X(-)$ and $Y(-)$. Clearly, it's a transformation, and it being "natural" just means that it satisfies the obvious consistency condition we discussed.

With this understanding of maps between generalized spaces in hand, we should go back and compare it to the notion of _probes_ of generalized spaces by ordinary spaces.

* we had started the discussion with observing that every ordinary topological space $X$ defines a [[sheaf]] of probes by the assignment $U \mapsto X(U)$ := {maps from $U$ to $X$} =: $Maps(U,X)$.

* this way every ordinary space is a generalized space, of course;

* but that should make us nervous: because it means that now there are two _different_ definitions of what a map from a test topological space $U$ to a topological space should be:

  * on the one hand, it should be an element in $X(U) = Top(U,X)$, where $Top$ indicates a set of maps of topological spaces;

  * on the other hand, we can regard both $U$ and $X$ as generalized spaces and then look at the set $Sh(U,X)$ of maps of generalized spaces between them, as just discussed above;

  * if  these two notions of maps don't coincide, then the whole picture of generalized spaces so far runs into inconsistencies!

  * luckily it doesn't, as one can check. Both notions of maps, while defined differently, happen to be perfectly equivalent. This statement is a simple exercise to prove, and still is the fundamental fact which makes the important idea of generalized spaces work -- this fact is called the [[Yoneda lemma]]: it says that
  $$
  \begin{aligned}
    Top(U,X) &=: X(U)
    \\ & = Sh(U,X)
  \end{aligned}
  $$
  or more formally
  $$
  Hom_{Sheaves}(U,X) \simeq X(U)
  .$$

* as a slogan: **The Yoneda lemma says that the idea of generalized spaces determined by their collections of probes is consistent.**

This point is so important that it is worthwhile to build it into our very notation, so that we can't help but think of the situation in this way.

* We agree for any two generalized spaces $X$ and $Y$ to write $X(Y)$ for the collection of maps from $Y$ to $X$: $ X(Y) := Sh(Y,X)$, the set of maps of generalized spaces from $Y$ to $X$.

* If $Y = U$ happens to be an ordinary space, then $X(U)$ consistently and equivalently denotes both: the collection of probes of $X$ by the ordinary space $U$, and the collection of maps of $U$ into $X$, with $U$ regarded as a generalized space itself.

* (This notation is for instance used in the book [[Categories and Sheaves]], see page 25.)

## And more: a topos of generalized spaces

It turns out that we can keep going this way. There is not just a natural notion of maps between generalized spaces given by  [[sheaf|sheaves]], but every important type of operation on spaces has its analogs as an operation on sheaves: one says that  sheaves form a [[topos]]: a place where we can go to study generalized [[homotopy theory]], more general than the place of [[topological space]]s, but where still all the crucial constructions familiar from topological spaces make sense.

## Higher sheaves: $\infty$-stacks

Once we are at this point, we should go even a bit further: we have been talking about _sets_ of probes $X(U)$. But of course
for $X$ a true [[topological space]] and $U$ a [[topological space]], there
is not just a set, but a [[topological space]] of maps $U \to X$ (using the [[compact-open topology]] on spaces of continuous maps).  So we should be looking at sheaves with values in topological spaces: functors $X(-) : S^{op} \to Top$ satisfying some suitable condition.

There is only one thing we need to take care of, then: since a
topological space is a bit more flexible than a plain set,
we should be a bit careful about when two sheaves $U \mapsto X(U)$
with values in topological spaces are to be regarded as
equal for all practical purposes: recall that $X(U)$ is
to be thought of as the space of all ways of mapping a probe
space $U$ into a space $X$. It's not the single points (maps) in this
space which matter so much, but they way they hang together.

* All "ways of hanging together" in a topological space $X(U)$
  are measured by a collection of groups called the
  [[homotopy group]]s $\pi_n(X(U))$, one for each natural number $n$.

* So if we are being careful, we should recognize that two sheaves  $X : S^{op} \to Top$ and $Y : S^{op} \to Top$ must characterize essentially the same generalized space if there is a rule $f : X \to Y$ for turning for each test space $U$, probes of $X$ into probes of $Y$, $f_U : X(U) \to Y(U)$, as before, such that for _sufficiently small_ $U$ this comparison map of
  probe spaces does not change the homotopy groups, i.e. so that
  all induced homomorphism $\pi_n(f_U) : \pi_n(X(U), x) \to \pi_n(Y(U), f_U(x))$ of [[homotopy group]]s are isomorphisms of groups. One says that such $f_U$ is a [[weak homotopy equivalence]].

* If a comparison map $f : X \to Y$ exists with these properties (for small enough $U$, recall), then one says that our generalized spaces $X$ and $Y$ defined by these sheaves are _weakly equivalent_ or [[quasi-isomorphism|quasi-isomorphic]].

There are several ways to deal with this complication that
sheaves with values in topological spaces are very flexible, so flexible
that it may require a bit of effort to determine if two of them
determine the same generalized space, for all practical purposes.

One nice way to deal with it is to again go back to our attitude that everything  should be determined by how we map other things into it: if $f : X \to Y$ is a weak equivalence of sheaves, as above, then we
would expect that for $Z$ any other sheaf, the induced map of spaces of probes $Z(f) : Z(Y) \to Z(X)$
of $Z$ by $X$ and $Y$ is a [[weak homotopy equivalence]].

* Recall from the discussion in the above section "maps of generalized spaces" that $Z(Y)$ is just another way saying "space of maps of generalized spaces from $Y$ to $Z$", and similarly for $Z(X)$.

Now, that may fail to be true, simply because we are using
an awkward realization of the sheaf $Z$. It may be that we have to _straighten out_ $Z$ a bit, without really changing it, for the above to be true. Indeed, if things are set up correctly, there should always be a sheaf of topological spaces $Z'$ which is equivalent to $Z$, but for which the above consistency condition is true: the map $Z'(f) : Z'(Y) \to Z'(X)$ between spaces of probes of $Z'$ by $X$ and by $Y$ is a homotopy equivalence.

The sheaves with this niceness property are called [[infinity-stack]]s or [[(infinity,1)-sheaf|(infinity,1)-sheaves]]. Recall that despite the fancy terminology, these things are nothing but consistent rules for something that may be probed with test spaces $U$, $V$, ...

* in applications this means that before we compute the space of maps $[X,A]$ between generalized spaces, we first have to ensure that at least one of them is sufficiently _straightened out_. One says one picks a _resolution_. More on that below.

## Generalized cohomology

We come back to our original motivation. Recall that we were looking
at cohomology groups of spaces $X$ by mapping $X$ into other spaces.
We noticed that there is in general interesting information about
$X$ which cannot be gained by mapping it into an ordinary topological
space. Starting from this we have now arrived at a more general notion
of space, called a higher sheaf or an $\infty$-stack.

* So let's state for the record that we succeeded: we will say now for
$X$ and $A$ generalized spaces, $\infty$-stacks, that the
collection of maps
$$
  H(X,A) := [X,A]
$$
from $X$ to $A$ is the [[cohomology]] of $X$ with coefficients in $A$.

* Indeed, one finds that despite their enlarged generality, these $\infty$-stacks
behave in all relevant aspects just like ordinary sheaves and ordinary
topological spaces do. So one says that $\infty$-stacks form an [[infinity-topos]]
(more precisely: an [[(infinity,1)-topos]]): a place where we can go in order
to do the familiar [[homotopy theory]] of topological spaces in a more
general context.

## Complexes of sheaves: abelian $\infty$-stacks

Another term for this [[cohomology|generalized cohomology]] theory thus obtained is
[[nonabelian cohomology]]. The reason for that terms is a historic one: before arriving at the full picture of higher topos theory as described here, people had a pretty good guess about some aspects of this story, and this aspect they called
[[abelian sheaf cohomology|sheaf cohomology]]. It turns out that sheaf cohomology is precisely the _abelian_ part of general cohomology. So now general cohomology is sometimes called [[nonabelian cohomology]] to distinguish it from the more traditional cohomology theory.

Here is what "abelian part of general cohomology" means:

* For many sheaves that appear in practice, the topological space $X(U)$ assigned to any test space $U$ is not just a topological space, but happens to have the structure of an abelian group, too.

* In this case a cascade of simplifications kicks in: first of all,
  such topological spaces, by a special case of something called the
  [[homotopy hypothesis]] (which is now really a precise theorem)
  can be identified with simplicial abelian groups.

* Second, by another theorem called the [[Dold-Kan correspondence]],
  simplicial abelian groups are equivalently encoded more simply
  in the collection of data given by
  (non-positively graded) chain complexes in abelian groups.

* \{abelian chain complexes} $\simeq$ {simplicial abelian groups} $\subset$ {Kan complexes} $\simeq$ {nice topological spaces}

By an [[abelian sheaf]] one means a sheaf $U \mapsto X(U)$ which
takes values in [[chain complex]]es of abelian groups
(or,  more generally, in [[complex]]es in another [[abelian category]]).
Under the above correspondences, we understand this as a special case
of a sheaf with values in topological spaces.

* By [[abelian sheaf cohomology]] one means nothing but the above
general [[Higher Topos Theory|higher topos theoretic]] notion of
cohomology given by the simple idea of collections of maps from
one generalized space to another. Only that in the special
case of abelian sheaves a wealth of particular computational
techniques are used to compute these mapping spaces
(such as [[derived category|derived categories]] and [[derived functor|derived functors]]) which are not available in the fully general case. This wealth of computational machinery
employed, useful as it is, sometimes hides the simple conceptual
reasoning underlying everything.

There is a collection of tools available for _computing_ explicitly the cohomologies  $H(X,A) := [X,A]$ for the case that the coefficient space can be thought of as a complex of abelian groups, or a complex of modules, these tools go by names like

* [[model category]]

  * [[derived category]]

  * [[calculus of fractions]] / [[factorization system]]

  * [[localization]]

* [[derived functor]]

  * [[injective object|injective resolution]]

  * [[bar and cobar construction]]

and so forth; historically many concepts in abelian sheaf theory were thought to be _defined_ in terms of these constructions. From a more modern perspective all these are just tools for unraveling an independent reality. This also explains why there are so many different tools which yield the same results. Most notably in the context of [[abelian sheaf cohomology]] there are two general ways for computing this cohomology called

* right derived global section functor

* Čech cohomology;

these two methods differ in whether one computes using "resolutions" of either the coefficient object $A$ (this happens in the right derived global section functor approach) or in the domain object $X$ (this happens in Čech cohomology).

## Examples

Recall that we started the search for a generalized notion of space
motivated by the fact that a circle bundle on a space $X$ may be
encoded by a map from $X$ to some topological space $\mathcal{B}U(1)$, but
that the same was not true for circle bundles _with connection_.
We had wanted to obtain a generalized notion of space such that
this is remedied.

* Indeed, this is now the case. There is a famous
  abelian sheaf, called the second [[Deligne cohomology|Deligne complex]] which we may denote by
  $[P_1(-), \mathcal{B}U(1)]$ or $\bar \mathcal{B}_1 U(1)$ (as explained at [[Deligne cohomology]] and at [[groupoid of Lie-algebra valued forms]]), which has precisely the property that maps from a manifold $X$ into it classify circle bundles with connection on $X$. Moreover, by varying the two parameters here we get the following cases

  * $A = [P_0(-), \mathcal{B}U(1)] = \mathcal{B}U(1)$;  $H(X,A) = H^2(X,A)$ is second integral cohomology -- classifies line bundles;

  * $A = [P_1(-), \mathcal{B}U(1)] = \bar \mathcal{B}_1 U(1)$; $H(X,A) = \bar H^2(X,\mathbb{Z})$ is second differential integral cohomology -- classifies line bundles with [[connection]];

  * $A = [P_2(-), \mathcal{B}U(1)] = \bar \mathcal{B}_2 U(1)$; $ H(X,A) = \bar H_{flat}^2(X,\mathbb{Z})$ is second flat differential integral cohomology -- classifies smooth line bundles with flat [[connection]];

  * $A = [P_0(-), \mathcal{B}^2U(1)] = \mathcal{B}^2U(1)$;  $H(X,A) = H^3(X,A)$ is third integral cohomology -- classifies line [[bundle gerbe]]s;

  * $A = [P_1(-), \mathcal{B}^2U(1)] = \bar \mathcal{B}^2_1 U(1)$ -- classifies line [[bundle gerbe]]s with connective structure;

  * $A = [P_2(-), \mathcal{B}^2U(1)] = \bar \mathcal{B}^2_2 U(1)$; $ H(X,A) = \bar H_{flat}^3(X,\mathbb{Z})$ is third differential integral cohomology -- classifies smooth line [[bundle gerbe]]s with connection;

  * $A = [P_3(-), \mathcal{B}^2U(1)] = \bar \mathcal{B}^2_3 U(1)$; $ H(X,A) = \bar H_{flat}^3(X,\mathbb{Z})$ is third flat differential integral cohomology -- classifies line [[bundle gerbe]]s with flat connection;

Notice that all these objects $A$ do indeed behave, and can be treated like, generalized spaces. For instance one can look at paths in these spaces and for instance form their [[fundamental groupoid]]s. One finds results such that for instance

* $\Pi_1(\bar \mathcal{B}_2 U(1)) \simeq \mathcal{B}R$

* more generally: for $G$ any Lie group, we have
  $\Pi_1(\bar \mathcal{B}_2 G) \simeq \mathcal{B}\hat G$, where $\hat G$ is the unique simply connected Lie group covering $G$.

## Related entries

* [[geometry of physics]]

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[twisted smooth cohomology in string theory]]

* [[motives in physics]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

* [[motivation for cohesion]]

category: motivation

[[!redirects motivation for sheaves, cohomology and higher stacks]]
[[!redirects Motivation for sheaves, cohomology and higher stacks]]
[[!redirects heuristic introduction to sheaves, cohomology and higher stacks]]
[[!redirects heuristic introduction to sheaves, cohomology and higher sta]]
[[!redirects introduction to sheaves, cohomology and higher stacks]]
[[!redirects motivation for sheaves]]


motive

+-- {: .rightHandSide}
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#### Motivic cohomology
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[[!include motivic cohomology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
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=--

[[!redirects motives]]

# Contents
* table of contents
{: toc}

## Idea

> In order to express this kinship of these different cohomological theories, I formulated the notion of "motive" associated to an algebraic variety. By this term I want to suggest that it is the "common motive" (or "common reason") behind this multitude of cohomological invariants attached to an algebraic variety, or indeed, behind all cohomological invariants that are a priori possible. ([[Grothendieck]], [[Récoltes et Semailles]])

The similarity of the behaviour of various [[cohomologies]] of [[algebraic variety|varieties]] over a [[field]] suggests that there is a universal one among them with values in an intermediate [[abelian category]], called the _[[category]] of motives_. The idea is that to every variety $X$ is associated a motive $M(X)$, such that every good [[cohomology theory]] factors through the functor $M$. (Here not every motive is supposed to be the image of a single variety.)

One distinguishes a theory of [[pure motives]] for smooth projective varieties from a more general theory of [[mixed motives]] for arbitrary smooth varieties.
So far, pure motives and mixed motives have only been defined conditionally. However there are several equivalent definitions of a [[triangulated category|triangulated]] [[tensor category]] which has all conjectured structural properties of the derived category of mixed motives (except for the t-structure which would make it a derived category).

[[Grothendieck]]'s original realization of this idea is the [[category of Chow motives]], which is a certain abelianization and completion of a [[category of spans]] of [[smooth variety|smooth]] [[projective varieties]]. Later a more [[homotopy theory|homotopy-theoretic]] version was given, the [[Voevodsky motives]] or _derived motives_, see [below](#DerivedMotives), which subsume the Chow motives faithfully. More generally still, there are definitions for _[[noncommutative motives]]_ obtained by passing to [[noncommutative algebraic geometry]]. Finally, the construction principle of motives can also be adapted to other flavors of [[geometry]]. For instance in [[noncommutative topology]] the role of the category of [[noncommutative motives]] is played by [[KK-theory]].

Constructions of motives often depend on whether we work in prime [[characteristic]]s or in characteristic zero. Part of the formalism involves more general [[schemes]] than varieties.

Another crucial idea leading to motives is that the various cohomology theories lead to the same pieces of information; therefore there is a symmetry related to this, which is of [[Galois theory]] nature. For example, over the [[complex numbers]] one can compare the [[Betti cohomology]] and [[de Rham cohomology]] "realizations". Thus one has a  [[motivic Galois group]], and as usually with representations one has a [[tensor category]] structure which is also [[rigid monoidal category|rigid]]. Thus one has in fact an [[abelian category|abelian]] [[tensor category]] of motives. [[Tannakian reconstruction]] plays a major role; for pure motives we have neutral [[Tannakian category|Tannakian categories]], and for mixed motives we have mixed Tannakian categories. Functions on the [[torsor]] of the isomorphism between "realizations" correspond to the matrices of [[period]]s in [[Hodge theory]].

$L$-functions (and $\zeta$-functions in particular) of varieties are also invariants of their motives. The [[Langlands program]] indirectly involves motives; in particular its essential part can be expressed as a general modularity conjecture relating $L$-functions to automorphic functions. Most of the deep properties of [[elliptic curve]]s are of motivic nature, and in particular a major step of the proof of [[Fermat's last theorem]] by Wiles and Taylor can be interpreted as a proof of a special case of the modularity conjecture (for elliptic curves).

## Constructions of the abelian category of mixed motives {#AbelianMotives}

There is no generally accepted construction of a $\mathbb{Q}$-linear [[abelian category]] of mixed motives, and its existence remains conjectural. However, there exist candidate and conditional constructions which are useful in practice.

Note that "the" abelian category of mixed motives depends on choosing a base [[scheme]] $S$, and one speaks of motives (or _motivic sheaves_) over $S$. Traditionally, $S$ is the spectrum of a [[field]], often of characteristic zero.

### Nori motives

[[Madhav Nori]] has an approach to the theory of motives based on a peculiar kind of Tannakian reconstruction, the so called _[[Nori's Tannakian theorem]]_. Nori's construction unconditionally produces a $\mathbb{Q}$-[[Tannakian category]] of mixed motives over any sub[[field]] of $\mathbb{C}$.

### Deligne motives

[[Pierre Deligne]] gave a definition of a category of mixed motives over [[number fields]] as compatible systems of realizations, essentially bundling together all the structure that mixed motives should give rise to. This approach automatically yields a $\mathbb{Q}$-[[Tannakian category]] of mixed motives with all the desired realization functors (Betti, $\ell$-adic, de Rham, and crystalline). See [Deligne](#DeligneP1) for details.

### As the heart of a t-structure on the derived category of mixed motives

[[Pierre Deligne|Deligne]] first suggested that it might be easier to define the [[derived category]] $DM(S,\mathbb{Q})$ of the hypothetical abelian category of mixed motives. Once this is done, one can in principle recover the abelian category as the heart of a [[t-structure]] on $DM(S,\mathbb{Q})$. It is now well-understood what the [[triangulated category]] $DM(S,\mathbb{Q})$ is over any base [[scheme]] ([see below](#DerivedMotives)). The hypothetical t-structure on $DM(S,\mathbb{Q})$ whose heart is the abelian category of mixed motives over $S$ is called the **motivic t-structure**.

[[Alexander Beilinson|Beilinson]] proved that, over fields of characteristic zero, the existence of the motivic t-structure implies the [[standard conjectures on algebraic cycles]] (see [Beilinson](#Beilinson10)), and [[Mikhail Bondarko|Bondarko]] proved that it implies the existence of motivic t-structures for more general schemes (see [Bondarko](#Bondarko13)).

While the derived category of mixed motives can also be defined with integral rather than rational coefficients, [[Vladimir Voevodsky|Voevodksy]] observed that the derived category of integral motives cannot have a motivic t-structure ([Voevodsky, Prop. 4.3.8](#TriCatMixMot)). Thus, the _abelian_ category of motives always refers to motives with rational coefficients.

### References

* [[Alexander Beilinson]], _Remarks on Grothendieck's standard conjectures_, 2010, ([arXiv](http://arxiv.org/abs/1006.1116))
{#Beilinson10}

* [[Mikhail Bondarko]], _Mixed motivic sheaves (and weights for them) exist if 'ordinary' mixed motives do_, 2013, ([arXiv](http://arxiv.org/abs/1105.0420))
{#Bondarko13}

* [[Pierre Deligne]], _Le groupe fondamental de la droite projective moins trois points_, ([pdf](http://publications.ias.edu/sites/default/files/61_LeGroupeFondamentalDroite.pdf))
{#DeligneP1}

* [[Vladimir Voevodsky]], _Triangulated categories of motives over a field_, ([K-theory](http://www.math.uiuc.edu/K-theory/0074/))
{#TriCatMixMot}

## Constructions of the derived category of mixed motives {#DerivedMotives}

The [[derived category]] of the hypothetical abelian category of mixed motives has been unconditionally defined over any [[Noetherian scheme]]. The first definition was proposed by [[Vladimir Voevodsky|Voevodksy]] in the mid 1990s. Since then, several other definitions were formulated: one by [[Fabien Morel|Morel]], one by [[Joseph Ayoub|Ayoub]], and one by [[Denis-Charles Cisinski|Cisinski]] and [[Frédéric Déglise|Déglise]]. The latter three are equivalent and support a full-fledged formalism of [[six operations]]. However, they are only known to be equivalent to Voevodsky's definition over [[excellent scheme|excellent]] and [[geometrically unibranch scheme|geometrically unibranch]] schemes.

On the other hand, Voevodsky's definition is the only one among these four which also makes sense with integral coefficients rather than rational coefficients. Recently, [[Markus+Spitzweck|Spitzweck]] proposed a definition of the category of integral motives over general base schemes which also supports a formalism of [[six operations]]. It is known to agree with Voevodsky's definition for [[fields]] of characteristic zero. Rationally, however, it agrees with the Morel/Ayoub/Cisinski-Déglise definition over any base scheme.

### As homotopy invariant Nisnevich sheaves with transfers (Voevodsky motives)
 {#VoevodskyMotives}

Associated to a [[Noetherian scheme]] $S$ there is an [[additive category]] $SmCor_S$ of "finite" [[correspondences]] of [[schemes]], whose

* [[objects]] are smooth schemes of finite type over $S$;

* [[morphism]]s $SmCor_S(X,Y)$ are the abelian group of cycles on the [[fiber product]] $X \times_S Y$ that are "universally integral relative to $X$" and each of whose components are finite and and surjective over $X$.

See at _[[pure motive]]_ for more (see also [MaVoWe, Appendix 1A](#MaVoWe)). Associating to a morphism of schemes its graph defines a [[faithful functor]] $Sm/S\hookrightarrow SmCor_S$.

An **(∞,1)-[[presheaf with transfers]]** on the category $Sm/S$ of smooth schemes of finite type is an [[(∞,1)-presheaf]] on $SmCor_S$ which transforms finite sums into finite (∞,1)-products (and hence take values in connective [[chain complexes]]).

The (∞,1)-category $DM^{eff}_{\geq 0}(S)$ is a certain reflexive [[localization]] of the (∞,1)-category of presheaves with transfers: it consists of those presheaves with transfers whose underlying presheaves on $Sm/S$ are [[(∞,1)-sheaves]] for the [[Nisnevich topology]] and are [[A1-homotopy theory|A1-homotopy invariant]].

The **Tate motive** $\mathbb{Z}(1)[2]$ is the image of the pointed scheme $(\mathbb{P}^1,\infty)$ in $DM^{eff}_{\geq 0}(S)$.

+-- {: .num_defn}
###### Definition
The stable (∞,1)-category of (integral) motives $DM(S)$ is the [[stabilization]] of $DM^{eff}_{\geq 0}(S)$ at the Tate motive $\mathbb{Z}(1)[2]$.
=--

The (∞,1)-category $DM(S)$ is a [[stable (∞,1)-category|stable]], [[symmetric monoidal (∞,1)-category|symmetric monoidal]], and [[locally presentable (∞,1)-category|locally presentable]] (∞,1)-category which is [[enriched category|enriched]] in [[chain complexes]]. It is slightly larger than the category $DM^-(S)$ defined in [MaVoWe, p. 110](#MaVoWe) which is not cocomplete.

_Voevodsky's cancellation theorem_ states that the canonical functor $DM^{eff}_{\geq 0}(S)\to DM(S)$ is [[fully faithful]] if $S$ is a [[perfect field]].

### As rational stable motivic homotopy types with trivial action of the Hopf element (Morel motives)

Let $\epsilon : \mathbb{G}_m\wedge \mathbb{G}_m\to \mathbb{G}_m\wedge \mathbb{G}_m$ be the transposition. In the [[stable motivic homotopy category]] $SH(S)$ this becomes an endomorphism of the motivic sphere spectrum $S^0$ such that $\epsilon^2=1$. Rationally (or even away from 2), we obtain a pair of idempotent elements

$$ \frac{1+\epsilon}{2}, \quad \frac{1-\epsilon}{2} $$

which induce a splitting $SH(S)_{\mathbb{Q}}\simeq SH(S)_{\mathbb{Q}_+}\times SH(S)_{\mathbb{Q}_-}$.

+-- {: .num_defn}
###### Definition
$SH(S)_{\mathbb{Q}_+}$ is the stable $(\infty,1)$-category of _Morel motives_.
=--

In other words, a Morel motive is a rational stable motivic homotopy type on which $\epsilon$ acts as $-1$.

The *Hopf element* $\eta\in \pi_{1,1}(S^0)$ is the stabilization of the algebraic [[Hopf fibration]] $\mathbb{A}^2-0\to\mathbb{P}^1$ over $S$. Morel motives can also be characterized as those rational stable motivic homotopy types that are acted on trivially by the Hopf element.

We have $\epsilon=-1$ if and only if $-1$ is a sum of squares in all the residue fields of $S$, in which case $SH(S)_{\mathbb{Q}}= SH(S)_{\mathbb{Q}_+}$. Thus, the other summand $SH(S)_{\mathbb{Q}_-}$ only appears over [[formally real field]]s. It is called the category of _Witt motives_.

### As homotopy invariant étale sheaves without transfers (Ayoub motives)

According to Ayoub, the stable $(\infty,1)$-category of motives over a scheme $S$ can be constructed in the same way as the stable motivic homotopy category $SH(S)$, with two variations:

* Instead of the [[Nisnevich topology]] on the category of smooth $S$-schemes, use the [[étale topology]]
* Instead of [[(∞,1)-sheaves]] of [[∞-groupoids]], use (∞,1)-sheaves of connective [[chain complexes]] of $\mathbb{Q}$-[[vector spaces]].

The resulting (∞,1)-category is denoted $DA^{\mathrm{et}}(S,\mathbb{Q})$. Its objects are thus $\mathbb{P}^1$-spectra of $\mathbb{A}^1$-invariant étale [[(∞,1)-sheaves]] with values in connective rational [[chain complexes]].

### As modules over a summand of rational homotopy invariant algebraic K-theory (Beilinson motives)

This definition is due to Cisinski and Déglise. The rationalization of the homotopy invariant [[algebraic K-theory]] spectrum $KGL\in SH(S)$ splits as a direct sum

$$ KGL_{\mathbb{Q}} = \bigoplus_{n\in\mathbb{Z}} \Sigma_T^n H_B $$

for some $E_\infty$ rational motivic ring spectrum $H_B\in SH(S)$.

+-- {: .num_defn}
###### Definition
The stable $(\infty,1)$-category of _Beilinson motives_ is the $(\infty,1)$-category of [[module]]s over $H_B$. Equivalently, it is the full subcategory of $SH(S)_{\mathbb{Q}}$ consisting of $H_B$-local objects.
=--

Cisinski and Déglise have shown that $H_B$ is exactly the $+$-summand $S^0_{\mathbb{Q}_+}$ of the rational motivic sphere spectrum, and hence that a Beilinson motive is the same thing as a Morel motive. They have also shown that Beilinson/Morel motives are equivalent to Ayoub motives. Finally, they have shown that Beilinson motives are equivalent to rational Voevodsky motives $DM(S,\mathbb{Q})$ when $S$ is [[excellent scheme|excellent]] and [[geometrically unibranch scheme|geometrically unibranch]]. Over such schemes, all four definitions of the derived category of mixed motives are therefore equivalent.

### As modules over a spectrum representing motivic cohomology over $Spec \mathbb{Z}$ (Spitzweck motives)

One idea to define a category of integral motives with a formalism of [[six operations]] is to first define an $E_\infty$ motivic ring spectrum $M_{\mathbb{Z}}\in SH(Spec \mathbb{Z})$. If $f: S\to Spec \mathbb{Z}$ is any scheme, we obtain an $E_\infty$-algebra $M_S = f^\ast(M_{\mathbb{Z}})$ in $SH(S)$. The categories of modules over $M_S$ for varying $S$ then inherit a complete formalism of six operations from $SH$.

[[Markus Spitzweck|Spitzweck]] defined such an $E_\infty$-algebra $M_{\mathbb{Z}}$ such that

* if $S$ is smooth over a [[Dedekind domain]], $M_S$ represents [[motivic cohomology|Bloch-Levine motivic cohomology]],
* if $S$ is smooth over a field, $M_S$ is equivalent to Voevodsky's motivic Eilenberg-Mac Lane spectrum $H\mathbb{Z}$,
* $M_{\mathbb{Z}}\otimes\mathbb{Q}$ is the Beilinson motive $H_B$.

The stable $(\infty,1)$-category of $M_S$-modules is thus a well-behaved candidate for a derived category of integral motives, but it is only known to agree with Voevodsky's definition when $S$ is a field of characteristic zero (by [Rondigs-Ostvaer, Theorem 5.5](#RO08)).

## Variations and extensions

Correspondences are interesting in [[noncommutative geometry]] of the [[operator algebra]] flavour. For example, KK-groups are in fact themselves sort of correspondences; [[Alain Connes|Connes]] and [[Georges Skandalis|Skandalis]] had an early reference very much paralleling some ideas from the algebraic world. More recently, motives in the operator algebraic setup have been approached by [[Alain Connes|Connes]], [[Matilde Marcolli|Marcolli]] and others.

{#Noncommutative} In derived [[noncommutative algebraic geometry]] based on $A_\infty$-categories, [[Maxim Kontsevich|Kontsevich]] proposed a theory of [[noncommutative motive]]s. There is now a more general setup (than Kontsevich's) due to [[Denis-Charles Cisinski|Cisinski]] and [[Gonçalo Tabuada|Tabuada]], see the [references](noncommutative+motive#References) at *[[noncommutative motive]]*.

In [[birational geometry]], [[Bruno Kahn]] defined the appropriate version. In [[rigid analytic geometry]], $A^1$-homotopy theory is replaced by $B^1$-homotopy theory and the appropriate analogue of the Voevodsky's category of mixed motives has been constructed; the construction follows the same basic pattern.

## Relation to other fields

### Relation to physics

Motivic structures show up in [[quantum field theory]], for instance

* [[motivic multiple zeta values]] in [[scattering amplitudes]]

* the [[motivic Galois group|motivic]] [[cosmic Galois group]] in [[renormalization]].

The [[path integral as a pull-push transform|pull-push quantization]] in [[Gromov-Witten theory]] is naturally understood as a "[[motivic quantization]]" in terms of [[Chow motives]] of [[Deligne-Mumford stacks]] ([Behrend-Manin 95](#BehrendManin95)).

### Relation to KK-theory

See at _[[KK-theory]]_ in the section _[As an analog of motives in noncommutative topology](KK-theory#AsAnAnalogOfMotives)_.

## Related concepts

* [[pure motive]], [[mixed motive]], [[Voevodsky motive]], [[Nori motive]], [[Chow motive]]

* [[K-motive]], [[noncommutative motive]]

* [[presheaf with transfer]]

* [[motivic homotopy theory]]

* [[motivic cohomology]]

* [[zeta function]], [[motivic multiple zeta values]]

* [[Hodge theory]]

* [[motivic function]]

* [[motivic Donaldson-Thomas invariant]]

* [[Fourier-Mukai transform]]

* See also at _[KK-theory -- Relation to motives](KK-theory#AsAnAnalogOfMotives)_.

* [[motives in physics]]

## References

### General

A brief exposition is in

* [[Barry Mazur]], _What is... a Motive?_, Notices of the AMS, volume 51, Number 10, 2004 ([pdf](http://www.ams.org/notices/200410/what-is.pdf))

A review is also in chapter I of

* [[Alain Connes]], [[Matilde Marcolli]], _[[Noncommutative Geometry, Quantum Fields and Motives]]_

Lectures include

* {#Levine06} [[Marc Levine]], _Six lectures on motives_, lectures at _ICTP Workshop on K-theory and Motives_ 2006 ([[LevineMotiveLecture.pdf:file]])

The definition of Voevodsky motives can be found in

* [[Carlo Mazza]], [[Vladimir Voevodsky]], [[Charles Weibel]], _Lectures in motivic cohomology_ ([web](http://math.rutgers.edu/~weibel/motiviclectures.html) [pdf](http://www.claymath.org/library/monographs/cmim02.pdf))
{#MaVoWe}

Voevodsky's formalization of motives was sketched in

* [[Pierre Deligne]], _Voevodsky lectures on cross functors,
Fall 2001. ([pdf](http://mat.uab.cat/~kock/tmp/delnotes01.pdf))
 {#Deligne}

and worked out in detail in

* [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I)_, Astérisque, vol. 314, Soc. Math. France, 2007.

  _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II)_, Astérisque, vol. 315, Soc. Math. France, 2007.
 {#Ayoub}

the definition of Ayoub motives in

* [[Joseph Ayoub]], _La réalisation étale et les opérations de Grothendieck_ ([pdf](http://user.math.uzh.ch/ayoub/PDF-Files/Realisation-Etale.pdf))

For the definition of Beilinson and Morel motives, the equivalences of the various definitions, and the formalism of six operations, see

* [[Denis-Charles Cisinski]], [[Frédéric Déglise]], _Triangulated categories of mixed motives_ ([arXiv](http://arxiv.org/abs/0912.2110))
{#CD12}

The fact that $DM(k)$ is equivalent to the category of $H(\mathbb{Z})$-modules if $\mathrm{char}(k)=0$ is proved in

* [[Oliver Röndigs]], [[Paul Arne Østvær]], _Modules over motivic cohomology_ ([pdf](http://www.math.uni-bielefeld.de/~oroendig/MZfinal.pdf))
{#RO08}

Spitzweck's definition of a motivic cohomology spectrum over $Spec \mathbb{Z}$ is in

* [[Markus Spitzweck]], _A commutative P^1-spectrum representing motivic cohomology over Dedekind domains I_ ([arXiv](http://arxiv.org/abs/1207.4078))

A summary of the axioms and of the main theorems is in the introduction of

* [[Denis-Charles Cisinski]], [[Frédéric Déglise]], _Triangulated categories of mixed motives_, [arxiv/0912.2110](http://arxiv.org/abs/0912.2110)
 {#CisinskiDeglise}

An outline of the big picture can be found in the introduction to

* [[Dmitri Kaledin]], _Motivic structures in non-commutative geometry_, [arxiv/1003.3210](http://arxiv.org/abs/1003.3210)

Some pretty useful comments on motives are at this MathOverflow thead:

* [What's the "Yoga of Motives?"](http://mathoverflow.net/questions/2146/whats-the-yoga-of-motives/2230#2230)

See also the blog post

* [[David Speyer]], _Motive-ating the Weil conjecture proof_ ([blog](http://sbseminar.wordpress.com/2010/06/10/motive-ating-the-weil-conjecture-proof/))

A formal discussion of motives can be found in [lecture 14](http://www.math.rutgers.edu/~weibel/MVWnotes/prova-hyperlink.pdf#page=121) of

* [[Carlo Mazza]], [[Vladimir Voevodsky]] and [[Charles Weibel]], _Lectures in motivic cohomology_ ([web](http://math.rutgers.edu/~weibel/motiviclectures.html) [pdf](http://www.math.rutgers.edu/~weibel/MVWnotes/prova-hyperlink.pdf))

There is also

* [[James Milne|James S. Milne]], _[Motives -- Grothendieck's Dream](http://www.jmilne.org/math/xnotes/MOT.pdf)_

* [[Minhyong Kim]], _[Classical Motives: Motivic $L$-functions](http://www.ucl.ac.uk/~ucahmki/ihes3.pdf)_

* [[Bruno Kahn]], [pdf slides](http://www.aimath.org/WWN/motivesdessins/PaloAlto1.pdf) on pure motives

* Florence Lecomte, Nathalie Wach, _Réalisations des complexes motiviques de Voevodsky_, [arxiv:0911.5611](http://de.arxiv.org/abs/0911.5611)

* [[Marc Levine]], _Smooth motives_, [arxiv:0807.2265](http://arxiv.org/abs/0807.2265)

* [[Marc Levine]], _Mixed motives_, Math. Surveys and Monographs __57__, Amer. Math. Soc.  1998, free [pdf](http://www.ams.org/online_bks/surv57/surv57.pdf)

For a noncommutative analogue to the theory of motives, see [[noncommutative motives]].

Some other aspects

* [[Mikhail Bondarko|Mikhail V. Bondarko]], _Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general)_, [arxiv/0704.4003](http://arxiv.org/abs/0704.4003)

* [[Yuri Manin]], _Motives and quantum cohomology_, talk at Colloque Grothendieck, [video](http://www.dailymotion.com/video/x8juco_colloque-grothendieck-yuri-manin_tech)

Motives from the point of view of Grothendieck topoi are studied in

* [[Olivia Caramello]], _Motivic toposes, [arxiv/1507.06271](http://arxiv.org/abs/1507.06271)

### Relation to Hodge theory

Explicit discussion of the relation to [[Hodge theory]] is in

* [[Chris Peters]], _Tata Lectures on Motivic Aspects of Hodge Theory_ ([pdf](http://www-fourier.ujf-grenoble.fr/~peters/motivic.f/Tatalects.pdf))

### Relation KK-theory and bivariant K-theory, K-motives
 {#RelationToKKTheory}

Relation of [[motivic cohomology]] to [[bivariant algebraic K-theory]] (see also at _[[KK-theory]]_) is discussed in

* [[Alain Connes]], [[Caterina Consani]], [[Matilde Marcolli]], _Noncommutative geometry and motives: the thermodynamics of endomotives_ ([arXiv:math/0512138](http://arxiv.org/abs/math/0512138))
 {#ConnesConsaniMarcolli05}

* [[Guillermo Cortiñas]], [[Andreas Thom]], _Bivariant algebraic K-theory_. J. Reine Angew. Math. 510 (2007), 71-124. ([arXiv:math/0603531](http://arxiv.org/abs/math/0603531))

* [[Grigory Garkusha]], Ivan Panin, _K-motives of algebraic varieties_ ([arXiv:1108.0375](http://arxiv.org/abs/1108.0375))

* [[Grigory Garkusha]], _Algebraic Kasparov K-theory. II_ ([arXiv:1206.0178](http://arxiv.org/abs/1206.0178))

See also at _[KK-theory -- Relation to motives](http://ncatlab.org/nlab/show/KK-theory#AsAnAnalogOfMotives)_.

For a collection of literature see also paragraph 1.5 in

* [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_ ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282))

(in the context of [[noncommutative motives]]).

### In physics

* A. Rej, [[Matilde Marcolli]], _Motives, an introductory survey for physicists_ ([pdf](http://www.its.caltech.edu/~matilde/ObiMotivesSurveyFinal.pdf))

See also at _[[motivic multiple zeta values]]_.

That the [[path integral as a pull-push transform|pull-push quantization]] in [[Gromov-Witten theory]] is naturally understood as a "[[motivic quantization]]" in terms of [[Chow motives]] of [[Deligne-Mumford stacks]] was suggested in

* [[Kai Behrend]], [[Yuri Manin]], _Stacks of Stable Maps and Gromov-Witten Invariants_ ([arXiv:alg-geom/9506023](http://arxiv.org/abs/alg-geom/9506023))
 {#BehrendManin95}

Further investigation of these stacky Chow motives then appears in

* [[Bertrand Toën]], _On motives for Deligne-Mumford stacks_, International Mathematics Research Notices 2000, 17 (2000) 909-928 ([arXiv:math/0006160](http://arxiv.org/abs/math/0006160), [web](http://hal.archives-ouvertes.fr/hal-00773027), [pdf](http://hal.archives-ouvertes.fr/docs/00/77/30/27/PDF/motdm.pdf))
 {#Toen00}

* [[Utsav Choudhury]], _Motives of Deligne-Mumford Stacks_ ([arXiv:1109.5288](http://arxiv.org/abs/1109.5288))

For more see at _[[motives in physics]]_.

category: algebraic geometry

[[!redirects motif]]
[[!redirects motive]]
[[!redirects motives]]
[[!redirects category of motives]]
[[!redirects categories of motives]]
[[!redirects derived category of motives]]
[[!redirects derived category of mixed motives]]
[[!redirects motivic]]


motives in physics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[motive|Motivic]] structures enters [[quantum physics]] in two dual guises, related, on the one hand, to [[algebraic deformation quantization]] and, on the other hand, to [[geometric quantization]].

### Via perturbative algebraic deformation quantization
 {#ViaPerturbativeAlgebraicDeformationQuantization}

In the first case one observes that [[formal deformation quantization]] of $n$-dimensional [[prequantum field theory]] amounts to choosing an inverse [[equivalence]] to the [[formality]] map from [[En-algebras]] to [[Pn-algebras]], this is explained at _[Motivic Galois group action on the space of quantizations](deformation+quantization#MotivicGaloisGroup)_.

The [[automorphism infinity-group]] of either side therefore naturally [[infinity-action|acts]] on the space of formal deformation quantization choices obtained this way and one shows (conjectured by [[Maxim Kontsevich]] ([Kontsevich 99](#Kontsevich99)), recently proven by Dolgushev) that the connected component group of this is the
[[Grothendieck-Teichmüller group]], a [[quotient]] of the [[motivic Galois group]].
Related to this in some way is [[Alain Connes]]' "[[cosmic Galois group]]" [[action|acting]] on the space of [[renormalization|renormalizations]] of [[perturbative quantum field theory]].

According to Kontsevich, this explains the role of motivic [[periods]] in [[correlation functions]] and hence in [[scattering amplitudes]] (see also at _[[amplituhedron]]_) in [[perturbative field theory]].

More concretely, [[scattering amplitudes]] typically are expressions in [[multiple zeta values]] and handling them involves intricate combinatorics. By just re-expressing these in terms of [[motivic multiple zeta values]] (see the rerferences) much of the combinatorics becomes more tractable (in terms of some [[Hopf algebra]]). This is how "motivic" structures are used by many practicioners. The actual [[motives]] do not play much of a role in these computations, but one makes use of the combinatorial simplification obtained by re-expressing multiple zeta values by their motivic version.

For more see the motive-related references at _[[Feynman diagram]]_, at _[[motivic multiple zeta values]]_ and at _[[motivic L-function]]_.

### Via non-perturbative geometric quantization

On the other hand, in full [[non-perturbative quantum field theory|non-perturbative]] [[geometric quantization]] in its modern cohomological form as [geometric quantization by push-forward](http://ncatlab.org/nlab/show/geometric+quantization+by+push-forward) one finds a "[[cohesion|cohesive]]" form of actual [[motivic cohomology]] exhibited by "[[cohesive]] [[pure motives]]". In effect, a [[local action functional]] on a [[moduli space]] of [[field (physics)|field]] [[trajectories]] is exhibited by a [[correspondence]]
with the [[local action functional]] itself exhibited by a [[twisted bivariant cohomology|twisted bivariant cocycle]] on the correspondence space, and the [[motivic quantization|motivic path integral quantization]] of this corresponds to the induced pull-push index transform. See the [references below](#ReferencesNonPerturbativeGeometric).

## References

We list references

1. [on motives in perturbative algebraic quantization](#ReferencesPerturbativeAlgebraic)

1. [on motives in non-perturbative geometric quantization](#ReferencesNonPerturbativeGeometric)

### On motivic structures in perturbative quantum field theory
 {#ReferencesPerturbativeAlgebraic}

The action of a [[motivic Galois group]] ("[[cosmic Galois group]]") on the space of choices in [[deformation quantization]] was first observed/conjectured in

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

A quick and rough survey of this and vaguely related motivic structures in physics is in the appendix of

* A. Rej, [[Matilde Marcolli]], _Motives, an introductory survey for physicists_ ([pdf](http://www.its.caltech.edu/~matilde/ObiMotivesSurveyFinal.pdf))

A detailed review of the motivic [[cosmic Galois group]] [[action]] on the space of [[renormalization]] procedures is in section 7 of

* [[Alain Connes]], [[Matilde Marcolli]], _[[Noncommutative Geometry, Quantum Fields and Motives]]_

Discussion of motivic structure in [[periods]] in [[scattering amplitudes]] is also the lecture

* [[Spencer Bloch]], _Dilogarithm motives appearing in physics_ ([video](http://blip.tv/pifagorov/spencer-bloch-dilogarithm-motives-arising-in-physics-3854827)); S. Bloch, H. Esnault, D. Kreimer, _On motives associated to graph polynomials_ [pdf](http://www.math.uchicago.edu/~bloch/graphpoly050928b.pdf); Spencer Bloch, Pierre Vanhove, _The elliptic dilogarithm for the sunset graph_, IHES preprint P-13-24, [pdf](http://preprints.ihes.fr/2013/P/P-13-24.pdf).

See also the material linked at

* Conference _Amplitudes and Periods_, Dec 3-7, 2012, IHÉS ([web](http://pagesperso.ihes.fr/~vanhove/QFT2012)) (videos of talks).

More recent developments on motivic structures in [[scattering amplitudes]] include for instance

* John Golden, Alexander B. Goncharov, Marcus Spradlin, Cristian Vergu, Anastasia Volovich, _Motivic Amplitudes and Cluster Coordinates_ ([arXiv:1305.1617](http://arxiv.org/abs/1305.1617))

### On motivic structures in non-perturbative local quantum field theory
 {#ReferencesNonPerturbativeGeometric}

The formulation of [[quantization]] of [[local prequantum field theory]] as a passage to [[cohesive]] generalized [[pure motives]], hence "[[motivic quantization]]" is formulated as such in the last section of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ ([arXiv:1310.7930](http://arxiv.org/abs/1310.7930))

based on

* [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_

The following is a list of "precursors" of aspects of the idea [[motivic quantization]] as laid out there

$\,$

The formulation of [[geometric quantization]] as an [[index]] map in [[K-theory]] is attributed to [[Raoul Bott]].

The proposal that the natural [[domain]] for geometric quantization are [[Lagrangian correspondences]] is due to

* [[Lars Hörmander]], _Fourier Integral Operators I._, Acta Math. __127__ (1971)
 {#Hoermander71}

* [[Alan Weinstein]], _Symplectic manifolds and their lagrangian submanifolds_, Advances in Math. __6__ (1971) {#Weinstein71}

In [[semiclassical quantization]] the precursor to this are works of [[Maslov]] in 1960s (which are quite related to Hoermander's and Sato's work).

With the recognition of [[supersymmetric quantum mechanics]] in the 1980s, [[index theory]] (hence [[push-forward in generalized cohomology]] to the point) was understood to be about [[partition functions]] of systems of [[supersymmetric quantum mechanics]] in

* [[Luis Alvarez-Gaumé]], _Supersymmetry and the Atiyah-Singer index theorem_, Comm. Math. Phys. __90__:2 (1983) 161-173, [euclid](http://projecteuclid.org/euclid.cmp/1103940278) {#AlvarezGaume83}

* [[Ezra Getzler]], _Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem_, Comm. Math. Phys. 92 (1983), 163-178. ([pdf](http://math.northwestern.edu/~getzler/Papers/1103940796.pdf))
 {#Getzler83}

*  [[Ezra Getzler]], _A short proof of the Atiyah-Singer index theorem_, Topology 25 (1986), 111-117 ([pdf](http://math.northwestern.edu/~getzler/Papers/local.pdf))

In higher analogy to this but much more subtly, the [[partition function]] of the [[heterotic string]], hence the [[Witten genus]], was understood to be the [[push-forward in generalized cohomology|push-forward]] to the point in [[tmf]]:

* [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))
  {#AndoHopkinsRezk}

The general perspective of the [[path integral as a pull-push transform]] was originally laid out, somewhat implicitly, in

* [[Daniel Freed]], _Quantum groups from path integrals_, [arXiv:q-alg/9501025](http://xxx.lanl.gov/abs/q-alg/9501025); _Higher algebraic structures and quantization_, [arXiv:hep-th/9212115](http://arxiv.org/abs/hep-th/9212115)

and then fully explicitly in

* [[Daniel Freed]], _Twisted K-theory and the Verlinde ring_, Andrejewski lecture ([pdf slides](http://www.ma.utexas.edu/users/dafr/Andrejewski%20Lectures.html))

Discussion along these lines of a pull-push quantization over [[KU]] of a 2-dimensional [[Chern-Simons theory]]-like [[gauge theory]] is in

* {#FreedHopkinsTeleman07} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _Consistent Orientation of Moduli Spaces_ ([arXiv:0711.1909](http://arxiv.org/abs/0711.1909)), chapter XIX, pages 395-419 in: Oscar Garcia-Prada, Jean Pierre Bourguignon, Simon Salamon (eds.) _The Many Facets of Geometry: A Tribute to Nigel Hitchin_, Oxford University Press 2010 ([doi:10.1093/acprof:oso/9780199534920.001.0001](http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199534920.001.0001/acprof-9780199534920))

More in detail a functorial quantization of suitable correspondences of smooth manifolds to [[KK-theory]] by pull-push has been given in ([Connes-Skandalis 84](#ConnesSkandalis84)) and the generalization of that to [[equivariant K-theory]] (hence to [[groupoid K-theory]] of [[action groupoids]]) is in

* [[Heath Emerson]], [[Ralf Meyer]], _Bivariant K-theory via correspondences_, Adv. Math. 225 (2010), 2883-2919 ([arXiv:0812.4949](http://arxiv.org/abs/0812.4949))
 {#EmersonMeyer08}

The point of view that the [[path integral as a pull-push transform|pull-push quantization]] of [[Gromov-Witten theory]] should be thought of as a theory of [[Chow motives]] of [[Deligne-Mumford stacks]] is expressed in

* [[Kai Behrend]], [[Yuri Manin]], _Stacks of Stable Maps and Gromov-Witten Invariants_ ([arXiv:alg-geom/9506023](http://arxiv.org/abs/alg-geom/9506023))
 {#BehrendManin95}

* [[Bertrand Toën]], _On motives for Deligne-Mumford stacks_, International Mathematics Research Notices 2000, 17 (2000) 909-928 ([arXiv:math/0006160](http://arxiv.org/abs/math/0006160), [web](http://hal.archives-ouvertes.fr/hal-00773027), [pdf](http://hal.archives-ouvertes.fr/docs/00/77/30/27/PDF/motdm.pdf))
 {#Toen00}

The proposal that the natural [[codomain]] for geometric quantization is [[KK-theory]] is due to

* [[Klaas Landsman]], _Functorial quantization and the Guillemin-Sternberg conjecture_ Proc. Bialowieza 2002 ([arXiv:math-ph/0307059](http://arxiv.org/abs/math-ph/0307059))

* [[Klaas Landsman]], _Functoriality of quantization: a KK-theoretic approach_, talk at ECOAS, ECOAS, Dartmouth College, October 2010 ([web](http://www.academia.edu/1992202/Functoriality_of_quantization_a_KK-theoretic_approach))

The point of view that pull-push along [[correspondences]] equipped with [[operator K-theory]] [[cycles]] in [[KK-theory]] is a [[K-theory]]-analog of [[motives]] was amplified in

* [[Alain Connes]], [[Georges Skandalis]], _The longitudinal index theorem for foliations_. Publ. Res. Inst. Math. Sci. 20,
no. 6, 1139–1183 (1984) ([pdf](http://www.alainconnes.org/docs/longitudinal.pdf))
 {#ConnesSkandalis84}

* [[Alain Connes]], Caterina Consani, [[Matilde Marcolli]], _Noncommutative geometry and motives: the thermodynamics of endomotives_ ([arXiv:math/0512138](http://arxiv.org/abs/math/0512138))
 {#ConnesConsaniMarcolli05}

The proof that the [[universal property]] that characterizes [[noncommutative motives]] is the analog in [[noncommutative algebraic geometry]] of the [[universal property]] that characterizes [[KK-theory]] in [[noncommutative topology]] is due to

* [[Gonçalo Tabuada]], _K-theory via universal invariants_, Duke Math. J. 145 (2008), no.1, 121–206.
 {#Tabuada08}

* [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], _Non connective K-theory via universal invariants_. Compositio
Mathematica 147 (2011), 1281–1320 ([arXiv:0903.3717](http://arxiv.org/abs/0903.3717))
 {#CisinskiTabuada11}

* [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_,  Geometry and Topology ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282))
 {#BlumbergGepnerTabuada10}

The description of [[string topology]] operations as an [[HQFT]] defined by pull-push transforms in [[ordinary homology]]/[[ordinary cohomology]] was originally realized in

* [[Ralph Cohen]], [[Veronique Godin]],  _[[A Polarized View of String Topology]]_ ([arXiv:math/0303003](http://arxiv.org/abs/math/0303003))

* Hirotaka Tamanoi, _Loop coproducts in string topology and triviality of higher genus TQFT operations_ (2007) ([arXiv:0706.1276](http://arxiv.org/abs/0706.1276))
 {#Tamanoi07}

A detailed discussion and generalization to [[open strings]] and an open-closed [[HQFT]] in the presence of a single space-filling [[brane]] is in

* [[Veronique Godin]], _Higher string topology operations_ (2007)([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))
 {#Godin}

and for arbitrary [[branes]] in

* [[Alexander Kupers]], _String topology operations_ MS thesis (2011) ([pdf](http://math.stanford.edu/~kupers/thesis7thjune2011.pdf))

That [[D-brane charge]] and [[T-duality]] is naturally understood in terms of pull-push/[[indices]] along [[correspondences]] in [[noncommutative topology]]/[[KK-theory]] was amplified in

* [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]],  _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))
 {#BMRS2}

The general analogy between such [[KK-theory]] cocycles and [[pure motives]] is noted explicitly in

* [[Alain Connes]], Caterina Consani, [[Matilde Marcolli]], _Noncommutative geometry and motives: the thermodynamics of endomotives_ ([arXiv:math/0512138](http://arxiv.org/abs/math/0512138))
 {#ConnesConsaniMarcolli05}

* [[Alain Connes]], [[Matilde Marcolli]], _[[Noncommutative Geometry, Quantum Fields and Motives]]_

This analogy is given a precise form in

*  [[Snigdhayan Mahanta]], _Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality_, ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf), [talk notes](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KKTD.pdf))
 {#Mahanta13}

where it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from [[KK-theory]] to the category of [[noncommutative motives]], which is the category of [[dg-categories]] and dg-[[profunctors]] up to homotopy between them. This is given by sending a [[C*-algebra]] to the [[dg-category]] of [[perfect complexes]] of (the unitalization of) its underlying [[associative algebra]].

Linearization of correspondences of [[discrete groupoid|geometrically discrete groupoids]] was considered in

* [[John Baez]], [[Jim Dolan]], _Groupoidification_, November 2009 ([web](http://math.ucr.edu/home/baez/groupoidification/))

and applied to a pull-push quantization of [[Dijkgraaf-Witten theory]] in

* [[Jeffrey Morton]], _2-Vector Spaces and Groupoids_ ([arXiv:0810.2361](http://arxiv.org/abs/0810.2361))

* [[Jeffrey Morton]], _Cohomological Twisting of 2-Linearization and Extended TQFT_ ([arXiv:1003.5603v4](http://arxiv.org/abs/1003.5603v4))
 {#Morton10}

following previous work by [[Daniel Freed]] and [[Frank Quinn]].

An unpublished predecessor note on quantization of correspondences of moduli stacks of fields is

* [[Urs Schreiber]], _[[schreiber:Nonabelian cocycles and their quantum symmetries]]_ (2008)
 {#Schreiber08}

Quantization of [[correspondences]] of [[perfect ∞-stacks]] by pull-push of [[stable (∞,1)-categories]] of [[quasicoherent sheaves]] is discussed in

* [[David Ben-Zvi]], [[John Francis]], [[David Nadler]],
_[[geometric function theory|Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry]]_, J. Amer. Math. Soc. __23__ (2010), no. 4, 909-966,
([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))

Linearization of [[higher correspondences]] of [[discrete ∞-groupoids]] as the quantizaton of [[∞-Dijkgraaf-Witten theories]] is indicated in section 3 and 8 of

* [[Daniel Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]], in P. Kotiuga (ed.), _A Celebration of the Mathematical Legacy of Raoul Bott,  AMS, (2010) ([arXiv:0905.0731](http://arxiv.org/abs/0905.0731))
 {#FreedHopkinsLurieTeleman09}

and in

* [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_, talk at Notre Dame Graduate Summer School on Topology and Field Theories and Harvard lecture 2012 ([pdf](http://www.math.harvard.edu/~lurie/papers/Ambidexterity.pdf))

A clear picture of [[fiber integration]] in [[twisted cohomology]] is developed in

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))
 {#ABG}

A proposal to axiomatize [[perturbative field theory|perturbative]] [[prequantum field theory]] by functors from [[cobordisms]] to a [[symplectic category]] of [[symplectic manifolds]] and [[Lagrangian correspondences]] is in

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical and quantum Lagrangian field theories with boundary_ ([arXiv:1207.0239](http://arxiv.org/abs/1207.0239), [pdf slides](http://users.uoa.gr/~iandroul/GAP%202012%20WEBPAGE/GAP2012_Cattaneo.pdf))
 {#CattaneoMnevReshetikhin12}

* [[Damien Calaque]], _Lagrangian structures on mapping stacks and semi-classical TFTs_ ([arXiv:1306.3235](http://arxiv.org/abs/1306.3235))

## Related entries

* [[geometry of physics]]

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[twisted smooth cohomology in string theory]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

* [[motivation for cohesion]]

[[!redirects motives and physics]]
[[!redirects motivic cohomology in physics]]