sifted category

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#### Category theory
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[[!include category theory - contents]]
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#Contents#
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## Definition

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###### Definition

A [[category]] $D$ is called __sifted__ if [[colimits]] of [[diagrams]] of shape $D$ [[commutativity of limits and colimits|commute]] with [[finite products]] in [[Set]]: for every [[diagram]]

$$
  F : D \times S \to Set
  \,,
$$

where $S$ is a [[finite category|finite]] [[discrete category]] the canonical morphism

$$
  ( \underset{\underset{d \in D}{\longrightarrow}}{\lim} \prod_{s \in S} F(d,s))
   \to
  \prod_{s \in S} \underset{\underset{d \in D}{\longrightarrow}}{\lim} F(d,s)
$$

is an [[isomorphism]].

[[formal dual|Dually]], $D$ is called **cosifted** if the [[opposite category]] $D^{op}$ is sifted.

=--

A colimit over a sifted diagram is called a [[sifted colimit]].

## Properties

### Characterizations

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

An [[inhabited set|inhabited]] [[small category]] $D$ is sifted precisely if the [[diagonal]] functor

$$
  D \to D \times D
$$

is a [[final functor]].

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This is due to ([GabrielUlmer](#GabrielUlmer))

More explicitly this means that:

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

An [[inhabited set|inhabited]] [[small category]] is sifted if for every pair of objects $d_1,d_2\in D$, the category $Cospan_D(d_1,d_2)$ of [[cospans]] from $d_1$ to $d_2$ is [[connected category|connected]].

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+-- {: .num_cor}
###### Corollary

Every category with finite [[coproducts]] is sifted.

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

Since a category with finite coproducts is nonempty (it has an [[initial object]]) and each category of cospans has an initial object (the coproduct).

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We make this special case more explicit below in Example \ref{CategoriesWithFiniteProductsAreCosifted}.

## Examples

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

The [[diagram]] category for [[reflexive coequalizers]], $\{ 0 \stackrel{\overset{d_0}{\to}}{\stackrel{\overset{s_0}{\leftarrow}}{\underset{d_1}{\to}}} 1\}^{op}$ with $ s_0 \circ d_0 = s_0 \circ d_1 = id$, is sifted.

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

The presence of the degeneracy map $s_0 \colon 1 \to 0$ in example \ref{ReflexiveCoequalizerDiagram} is crucial for the statement to work: the category $\{0 \stackrel{\overset{d_0}{\to}}{\underset{d_1}{\to}} 1\}^{op}$ is _not_ sifted; there is no way to connect the cospan $(d_0,d_0)$ to the cospan $(d_1,d_1)$.

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Example \ref{ReflexiveCoequalizerDiagram} may be thought of as a truncation of:

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###### Example

The [[opposite category]] of the [[simplex category]] is sifted.

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###### Example

Every [[filtered category]] is sifted.

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

Since [[filtered colimits]] commute even with all [[finite limits]], they in particular commute with [[finite products]].

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+-- {: .num_example #CategoriesWithFiniteProductsAreCosifted}
###### Example
**([[categories with finite products are cosifted]])**

Let $\mathcal{C}$ be a [[small category]] which has [[finite products]]. Then $\mathcal{C}$ is a _[[cosifted category]]_, equivalently its [[opposite category]] $\mathcal{C}^{op}$ is a _[[sifted category]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] are _[[sifted colimits]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] _[[limits commuting with colimits|commute]] with [[finite products]]_, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to [[functors]] on the [[opposite category]] of $\mathcal{C}$ (hence two [[presheaves]] on $\mathcal{C}$) we have a [[natural isomorphism]]

$$
  \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
  \left(
    \mathbf{X} \times \mathbf{Y}
  \right)
  \;\simeq\;
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{X}
  \right)
  \times
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{Y}
  \right)
  \,.
$$

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## Related concepts

* [[distributivity of products and colimits]]

* **sifted category**, [[sifted (∞,1)-category]]

* [[filtered category]], [[filtered (∞,1)-category]]

* [[direct category]]

## References

*  [[Pierre Gabriel]], Fritz Ulmer, _Lokal präsentierbare Kategorien_ , LNM
**221**, Springer Heidelberg 1971.
{#GabrielUlmer}

*  [[Jiri Adamek]], [[Jiri Rosicky]], _On sifted colimits and generalized varieties_, TAC **8** (2001) pp.33-53. ([tac](http://www.tac.mta.ca/tac/volumes/8/n3/8-03abs.html))

*  [[Jiri Adamek]], [[Jiri Rosicky]], [[Enrico Vitale]], _What are sifted colimits?_, TAC __23__ (2010) pp. 251--260. ([tac](http://www.tac.mta.ca/tac/volumes/23/13/23-13abs.html))

* [[Jiri Adamek]], [[Jiri Rosicky]], [[Enrico Vitale]], _Algebraic Theories - a Categorical Introduction to General Algebra_ , Cambrige UP 2010. (ch. 2) ([draft](http://www.iti.cs.tu-bs.de/~adamek/algebraic_theories.pdf))

[[!redirects sifted categories]]
[[!redirects cosifted category]]
[[!redirects cosifted categories]]


sifted colimit

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###Context###
#### Limits and colimits
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#Contents#
* table of contents
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## Definition

A _sifted colimit_ is a [[colimit]] of a [[diagram]] $D \to C$ where $D$ is a [[sifted category]] (in analogy with a [[filtered colimit]], involving diagrams of shape a [[filtered category]]). Such colimits [[limits commuting with colimits|commute]] with finite [[products]] in [[Set]], by definition.

## Examples

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###### Example

A motivating example is a [[reflexive coequalizer]]. In fact, sifted colimits can "almost" be characterized as combinations of [[filtered colimits]] and reflexive coequalizers. ([Adamek-Rosicky-Vitale 10](#AdamekRosickyVitale10))

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+-- {: .num_example #CategoriesWithFiniteProductsAreCosifted}
###### Example
**([[categories with finite products are cosifted]])**

Let $\mathcal{C}$ be a [[small category]] which has [[finite products]]. Then $\mathcal{C}$ is a _[[cosifted category]]_, equivalently its [[opposite category]] $\mathcal{C}^{op}$ is a _[[sifted category]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] are _[[sifted colimits]]_, equivalently [[colimits]] over $\mathcal{C}^{op}$ with values in [[Set]] _[[limits commuting with colimits|commute]] with [[finite products]]_, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to [[functors]] on the [[opposite category]] of $\mathcal{C}$ (hence two [[presheaves]] on $\mathcal{C}$) we have a [[natural isomorphism]]

$$
  \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
  \left(
    \mathbf{X} \times \mathbf{Y}
  \right)
  \;\simeq\;
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{X}
  \right)
  \times
  \left(
    \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
    \mathbf{Y}
  \right)
  \,.
$$

=--

## Properties

See at _[[distributivity of products and colimits]]_.

## See also

* [[sifted (infinity,1)-colimit|sifted $(\infty,1)$-colimit]]

* [[finite product]]

## References

* {#GabrielUlmer71} [[Pierre Gabriel]], [[Friedrich Ulmer]], _Lokal präsentierbare Kategorien_, Springer LNM
221, Springer-Verlag 1971

*  {#AdamekRosickyVitale10} [[Jiri Adamek]], [[Jiri Rosicky]], [[Enrico Vitale]], _What are sifted colimits?_, TAC **23** (2010) pp. 251–260. ([web](http://www.tac.mta.ca/tac/volumes/23/13/23-13abs.html))

[[!redirects sifted colimit]]
[[!redirects sifted colimits]]


sigma-algebra

_The concept of a $\delta$-ring appearing in this article is distinct from that appearing in [[arithmetic differential geometry]]. For the latter, see [[Joyal delta-ring]]._

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#### Measure and probability theory
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# $\sigma$-Algebras
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## Idea

$\sigma$-algebras and their variants are collections of [[subsets]] important in classical [[measure theory]] and [[probability theory]].

Although $\sigma$-algebras are often introduced as a mere technicality in the definition of [[measurable space]] (to specify the [[measurable subsets]]), even once one has a fixed measurable space $X$, it is often useful to consider additional (typically coarser) $\sigma$-algebras of measurable subsets of $X$.

## Definitions

We assume the law of [[excluded middle]] throughout; see [[Cheng measurable space]] for the constructive theory, and compare also [[measurable locale]].

### Short version

Given a [[set]] $X$, a __$\sigma$-algebra__ is a collection of [[subsets]] of $X$ that is closed under [[complement|complementation]] and under [[unions]] and [[intersections]] of [[countable set|countable]] [[family of subsets|families]].

Notice that the [[power set]] $\mathcal{P} X$ of $X$ is a [[Boolean algebra]] under the operations of complementation and of union and intersection of [[finite set|finite]] families.  Actually, it is a [[complete lattice|complete]] Boolean algebra, since we can also take unions and intersections of all families.  A $\sigma$-algebra is an intermediate notion, since (in addition to being closed under complementation) we require that it be closed under unions and intersections of *countable* families.

### Long version

Given a [[set]] $X$ and a collection $\mathcal{M}$ of [[subsets]] $S \subseteq X$, there are really several kinds of collections that $\mathcal{M}$ could be:

*  A __ring__ on $X$ is a collection $\mathcal{M}$ which is closed under [[relative complement]]ation and under [[unions]] of [[finite set|finitary]] [[family of subsets|families]].  That is:

   1.  The [[empty set]] $\empty$ is in $\mathcal{M}$.
   2.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
   3.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.

   It follows that $\mathcal{M}$ is closed under intersections of [[inhabited set|inhabited]] finite families and under [[symmetric difference]] of finite families:

   *  If $S$ and $T$ are in $\mathcal{M}$, then so is their intersection $S \cap T = T \setminus (T \setminus S)$.
   *  If $S$ and $T$ are in $\mathcal{M}$, then so is their symmetric difference $S \uplus T = (T \setminus S) \cup (S \setminus T)$.

   We can actually use the latter as an alternative to (2), since $S \cup T = (S \uplus T) \uplus (S \cap T)$.  Or we can use the pair as an alternative to (2,3), since $T \setminus S = (S \cap T) \uplus T$.  For that matter, we can weaken (1) to simply say that *some* set $S$ is in $\mathcal{M}$; then $\empty = S \setminus S$.

   {#EmptyFamilies} While the union and symmetric difference of an [[empty family]] belong to $\mathcal{M}$ (both being the empty set), the intersection of an empty family (which is $X$ itself) might not.  The term 'ring' dates from the days when a [[ring]] in algebra was not assumed to be unital; so a ring on $X$ is simply a subring (in this sense) of the [[Boolean ring]] $\mathcal{P} X$.

*  A __$\delta$-ring__ on $X$ is a ring (as above) $\mathcal{M}$ which is closed under [[intersections]] of countably infinite families.  That is:
   1.  The [[empty set]] $\empty$ is in $\mathcal{M}$.
   2.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
   3.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
   4.  If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[intersection]] $\bigcap_i S_i$.

   Of course, every $\delta$-ring is a ring, but not conversely.  Actually, if you want to define the concept of $\delta$-ring directly, it\'s quicker if you use the symmetric difference; then (2,3) follow by the reasoning above and the [[idempotent|idempotence]] of intersection (so that $S \cap T = S \cap T \cap T \cap T \cap \cdots$).

   The symbol '$\delta$' here is from German 'Durchschnitt', meaning intersection; it may be used in many contexts to refer to intersections of countable families.

*  A __$\sigma$-ring__ on $X$ is a ring (as above) $\mathcal{M}$ which is closed under unions of [[countably infinite set|countably infinite]] families.  That is:
   1.  The [[empty set]] $\empty$ is in $\mathcal{M}$.
   2.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
   3.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
   4.  If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[union]] $\bigcup_i S_i$.

   Now (2) is simply redundant; $S \cup T = S \cup T \cup T \cup T \cup \cdots$.  A $\sigma$-ring is obviously a ring, but in fact it is also a $\delta$-ring; $\bigcap_i S_i = (\bigcup_i S_i) \setminus \bigcup_j (\bigcup_i S_i \setminus S_j)$.

   The symbol '$\sigma$' here is from German 'Summe', meaning union; it may be used in many contexts to refer to unions of countable families.

*  An __algebra__ or __field__ on $X$ is a ring (as above) $\mathcal{M}$ to which $X$ itself belongs.  That is:
   1.  The [[empty set]] $\empty$ is in $\mathcal{M}$.
   2.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
   3.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
   4.  The [[improper subset]] $X$ is in $\mathcal{M}$.

   Actually, (2) is now redundant again; $S \cup T = X \setminus ((X \setminus T) \setminus S)$.  But perhaps more importantly, $\mathcal{M}$ is closed under *absolute* [[complement|complementation]] (that is, complementation relative to the entire ambient set $X$); that is:

   *  If $S$ is in $\mathcal{M}$, then so is its [[complement]] $\neg{S}$.

   In light of this, the most common definition of algebra is probably to use this fact together with (1,2); then (3) follows because $T \setminus S = \neg(S \cup \neg{T})$ and (4) follows because $X = \neg\empty$.  On the other hand, one could equally well use intersection instead of union; absolute complements allow the full use of [[de Morgan duality]].

   The term 'field' here is even more archaic than the term 'ring' above; indeed the only field in this sense which is a [[field]] (in the usual sense) under symmetric difference and intersection is the field $\{\empty, X\}$ (for an [[inhabited set]] $X$).

*  Finally, a __$\sigma$-algebra__ or __$\sigma$-field__ on $X$ is a ring $\mathcal{M}$ that is both an algebra (as above) and a $\sigma$-ring (as above).  That is:
   1.  The [[empty set]] $\empty$ is in $\mathcal{M}$.
   2.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[union]] $S \cup T$.
   3.  If $S$ and $T$ are in $\mathcal{M}$, then so is their [[relative complement]] $T \setminus S$.
   4.  The [[improper subset]] $X$ is in $\mathcal{M}$.
   5.  If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[union]] $\bigcup_i S_i$.

   As with $\sigma$-rings, (2) is redundant; as with algebras, it\'s probably most common to use the absolute complement in place of (3,4).  Thus the usual definition of a $\sigma$-algebra states:
   1.  The [[empty set]] $\empty$ is in $\mathcal{M}$.
   2.  If $S$ is in $\mathcal{M}$, then so is its [[complement]] $\neg{S}$.
   3.  If $S_1, S_2, S_3, \ldots$ are in $\mathcal{M}$, then so is their [[union]] $\bigcup_i S_i$.

   And again we could again just as easily use [[intersection]] as union, even in the infinitary axiom.  That is, a $\delta$-algebra is automatically a $\sigma$-algebra, because $\bigcup_i S_i = \neg\bigcap_i \neg{S_i}$.

Any and all of the above notions have been used by various authors in the definition of measurable space; for example, Kolmogorov used algebras (at least at first), and Halmos used $\sigma$-rings.  Of course, the finitary notions (ring and algebra) aren\'t strong enough to describe the interesting features of [[Lebesgue measure]]; they are usually used to study very different examples ([[finitely additive measure]]s).  On the other hand, $\delta$‑ or $\sigma$-rings may be more convenient than $\sigma$-algebras for some purposes; for example, [[topological vector space|vector]]-valued measures on $\delta$-rings make good sense even when the absolute measure of the whole space is infinite.

Note that the collection of measurable sets with finite measure (in a given [[measure space]]) is a $\delta$-ring, while the collection of measurable sets with $\sigma$-finite measure is a $\sigma$-ring.

### Measurable sets

A __[[measurable space]]__ is usually defined to be a [[set]] $X$ with a $\sigma$-algebra $\mathcal{M}$ on $X$; sometimes one of the more general variations above is used.

In any case, an __$\mathcal{M}$-[[measurable subset]] of $X$__, or just a __measurable set__, is any [[subset]] of $X$ that belongs to $\mathcal{M}$.  If $\mathcal{M}$ is one of the more general variations, then we also want some subsidiary notions: $S$ is __[[relatively measurable subset|relatively measurable]]__ if $S \cap T$ belongs to $\mathcal{M}$ whenever $T$ does, and $S$ is __$\sigma$-[[sigma-measurable subset|measurable]]__ if it is a countable union of elements of $\mathcal{M}$.  Notice that every relatively measurable set is measurable iff $S$ is at least an algebra; in any case, the relatively measurable sets form a ($\sigma$)-algebra if $\mathcal{M}$ is at least a ($\delta$)-ring.  Similary, every $\sigma$-measurable set is measurable iff $S$ is at least a $\sigma$-ring; in any case, the $\sigma$-measurable sets form a $\sigma$-ring if $\mathcal{M}$ is at least a $\delta$-ring.

### Generating $\sigma$-algebras

As a $\sigma$-algebra is a collection of subsets, we might hope to develop a theory of [[base|bases]] and [[subbase|subbases]] of $\sigma$-algebras, such as is done for [[topological space|topologies]] and [[uniform space|uniformities]].  However, things do not work out as nicely.  (It *is* quite easy to generate rings or algebras, but generating $\delta$-rings and $\sigma$-rings is just as tricky as generating $\sigma$-algebras.)

We do get something by [[general abstract nonsense]], of course.  It\'s easy to see that the [[intersection]] of any collection of $\sigma$-algebras is itself a $\sigma$-algebra; that is, we have a [[Moore closure]].  So given any collection $\mathcal{B}$ of sets whatsoever, the intersection of all $\sigma$-algebras containing $\mathcal{B}$ is a $\sigma$-algebra, the $\sigma$-algebra __generated__ by $\mathcal{B}$.  (We can similarly define the $\delta$-ring generated by $\mathcal{B}$ and similar concepts for all of the other notions defined above.)

What is missing is a simple description of the $\sigma$-algebra generated by $\mathcal{B}$.  (For a mere algebra, this is easy; any $\mathcal{B}$ can be taken as a _[[subbase]]_ of an algebra, the symmetric unions of finite families of elements of $\mathcal{B}$ form a _[[base]]_ of the algebra, and the intersections of finite families of elements of the base form an algebra.  For a ring, the only difference is to use intersections only of inhabited families.  But for anything from a $\delta$-ring to a $\sigma$-algebra, nothing this simple will work.)

In fact, the question of how to generate a $\sigma$-algebra is the beginning of an entire field of mathematics, [[descriptive set theory]].  For our purposes, we need this much:

*  Start with a collection $\Sigma_0$ (our collection $\mathcal{B}$ above), and let $\Pi_0$ be the collection of the complements of the members of $\Sigma_0$.
*  Let $\Sigma_1$ be the collection of unions of countably infinite families of sets in $\Pi_0$, and let $\Pi_1$ be the collection of their complements (the intersections of countably infinite families of sets in $\Sigma_0$); even $\Sigma_1 \cup \Pi_1$ is not in general a $\sigma$-algebra.
*  Continue by [[recursion|recursively]], defining $\Sigma_n$ for all [[natural numbers]] $n$.
*  Let $\Sigma_\omega$ be the union of the various $\Sigma_n$; although this is closed under complement, it is still not in general a $\sigma$-algebra.
*  Continue by [[transfinite induction|transfinite]] recursion, defining $\Sigma_\alpha$ for all countable [[ordinal numbers]] $\alpha$.
*  Let $\Sigma_{\omega_1}$ be the union of the various $\Sigma_\alpha$; *this* is finally a $\sigma$-algebra.

So we need an $\aleph_1$ steps, not just $2$.

(This is only the beginning of descriptive set theory; our $\Sigma_\alpha$ are their $\Sigma^0_\alpha$ ---except that for some reason they start with $\Sigma^0_1$ instead of $\Sigma^0_0$---, and the subject continues to higher values of the superscript.)

Note that [[countable choice]] is essential here and elsewhere in measure theory, to show that a countable union of a countable union is a countable union.  But the full [[axiom of choice]] is not; in fact, much of descriptive set theory (although this is irrelevant to the small portion above) works better with the [[axiom of determinacy]] instead.

## Examples

*  Of course, the [[power set]] of $X$ is closed under all operations, so it is a $\sigma$-algebra.

*  If $X$ is a [[topological space]], the $\sigma$-algebra generated by the open sets (or equivalently, by the closed sets) in $X$ is the __Borel $\sigma$-algebra__; its elements are called the __[[Borel sets]]__ of $X$. In particular, the Borel sets of [[real numbers]] are the Borel sets in the [[real line]] with its usual topology.

*  In the application of [[statistical physics]] to [[thermodynamics]], there is both a microcanonical [[phase space]] $P$ (typically something like $\mathbb{R}^N$ for $N$ on the order of [[Avogadro's number]]) describing every last detail of a [[physical system]] and a macrocanonical phase space $p$ (typically $\mathbb{R}^2$ or at least $\mathbb{R}^n$ for $n \lt 10$) describing those features of the system that can be measured practically, with a projection $P \to p$.  Then the [[preimage]] under this projection of the Borel $\sigma$-algebra of $p$ is a $\sigma$-algebra on $P$, and the thermodynamic [[entropy]] of the system is (theoretically) its information-theoretic entropy with respect to this $\sigma$-algebra.

*  If a measurable space $(X,\mathcal{M})$ is equipped with a (positive) measure $\mu$, making it into a [[measure space]], then the sets of measure zero form a $\sigma$-[[sigma-ideal|ideal]] of $\mathcal{M}$ (that is an [[ideal]] that is also a sub-$\sigma$-ring).  Let a __[[null set]]__ be *any* set (measurable or not) contained in a set of measure zero; then the null sets form a $\sigma$-ideal in the [[power set]] of $X$.  Call a set __$\mu$-measurable__ if it is the union of a measurable set and a null set; then the $\mu$-measurable sets form a $\sigma$-algebra called the __completion__ of $\mathcal{M}$ under $\mu$.  (Even if $\mathcal{M}$ is only a $\delta$-ring, still the null sets will form a $\sigma$-ring; in any case, we get as completion the same kind of structure as we began with.)  Note that we can also do this by starting with any $\sigma$-ideal $\mathcal{N}$ and simply declaring by fiat that these are the null sets, as with a [[localisable measurable space]]; then we speak of the __completion__ of $\mathcal{M}$ with respect to $\mathcal{N}$ (or sometimes with respect to the $\delta$-[[delta-filter|filter]] $\mathcal{F}$ of [[full sets]]).

*  In particular, the __Lebesgue-measurable__ sets in the real line are the elements of the completion of the Borel $\sigma$-algebra under [[Lebesgue measure]].

## Alternatives

We are now learning ways to understand [[measure theory]] and [[probability]] away from the traditional reliance on sets required with $\sigma$-algebras; see [[measurable space]] for a summary of other ways to define this concept.  We still need to know what happens to all of the *other* $\sigma$-algebras of measurable sets in a measurable space.  One solution may be to use [[quotient space|quotient]] measurable spaces in place of sub-$\sigma$-algebras; for example, see explicit quotient in the example of macroscopic entropy above.

## Related concepts

* [[sigma-topological space|$\sigma$-topological space]]

category: probability

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sigma-compact topological space

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#Contents#
* table of contents
{:toc}

## Definition

A [[topological space]] is called _sigma-compact_ if it is the [[union]] of a [[countable set]] of [[compact topological space|compact]] [[subspaces]].

## Properties

* [[locally compact and sigma-compact spaces are paracompact]]

## Examples

* [[locally compact and second-countable spaces are sigma-compact]]

## related concepts

* [[countable cover]]

## References

* Wikipedia, _[σ-compact space](https://en.wikipedia.org/wiki/%CE%A3-compact_space)_

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sigma-complete lattice

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## Definition ##

A __$\sigma$-complete lattice__ is a [[lattice]] $(L, \leq, \bot, \vee, \top, \wedge)$ with a function

$$\Vee_{n:\mathbb{N}} (-)(n): (\mathbb{N} \to L) \to L$$

such that

* for all natural numbers $n \in \mathbb{N}$ and sequences $s: \mathbb{N} \to L$
$$ s(n) \leq \Vee_{n:\mathbb{N}} s(n)$$

* for all elements $x \in L$ and sequences $s: \mathbb{N} \to L$ if $s(n) \leq x)$ for all natural numbers $n \in \mathbb{N}$, then
$$\Vee_{n:\mathbb{N}} s(n) \leq x$$

## See also ##

* [[lattice]]

* [[sigma-frame]]

* [[sigma-continuous valuation]]

* [[sigma-continuous probability valuation]]

## References ##

* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))


sigma-continuous probability valuation

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## Definition ##

In measure theory, a __$\sigma$-continuous probability valuation__ on a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[probability valuation]] $\mu:L \to [0, 1]$ such that the $\sigma$-continuity condition is satisfied: for all sequences $s:\mathbb{N} \to L$, if $s(n) \leq s(n + 1)$ for all natural numbers $n \in\mathbb{N}$, then
$$\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n))$$

## See also ##

* [[unit interval]]

* [[sigma-continuous valuation]]

* [[probability valuation]]

* [[probability measure]]

## References ##

* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))


sigma-continuous valuation

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## Definition ##
### In set theory ###
In measure theory, a __$\sigma$-continuous valuation__ on a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[valuation (measure theory)|valuation]] $\mu:L \to [0, \infty]$ such that the $\sigma$-continuity condition is satisfied: for all sequences $s:\mathbb{N} \to L$, if $s(n) \leq s(n + 1)$ for all natural numbers $n \in\mathbb{N}$, then
$$\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n))$$

## See also ##

* [[valuation (measure theory)]]

* [[sigma-continuous probability valuation]]

* [[content (measure theory)]]

* [[measure]]

## References ##

* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))


sigma-frame

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## Definition ##

A __$\sigma$-frame__ is a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ such that for all elements $a \in L$ and sequences $s:\mathbb{N} \to L$,
$$a \wedge \Vee_{n:\mathbb{N}} s(n) = \Vee_{n:\mathbb{N}} a \wedge s(n)$$

## Examples ##

* [[Sierpinski space]], denoted as $\Sigma$ or $1_\bot$, is the initial $\sigma$-frame.

* [[Rosolini dominance]]

* The [[intrinsic topology]] on a set $A$, which is the [[function set]] $\Sigma^A$

* Malcev topologies for set $A$, which are a $\sigma$-[[subobject|subframe]] of the intrinsic topology $\Sigma^A$.

* Any [[frame]] is a $\sigma$-frame. In particular, the [[poset of truth values]] is a $\sigma$-frame.

## See also ##

* [[sigma-complete lattice]]

* [[distributive lattice]]

* [[measure]]

* [[probability measure]]

* [[Dedekind cut]]

* [[Dedekind real numbers]]

* [[sigma-pretopos]]

* [[streak]]

* [[sigma-topological space]]

* [[sigma-locale]]

* [[first countable space]]

* [[limited principle of omniscience]]

## References ##

* Francesco Ciraulo, *$\sigma$-locales in Formal Topology*, Logical Methods in Computer Science, Volume 18, Issue 1 (January 12, 2022) ([doi:10.46298/lmcs-18%281%3A7%292022](https://doi.org/10.46298/lmcs-18%281%3A7%292022), [arXiv:1801.09644](https://arxiv.org/abs/1801.09644))

* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))

* [[Andrej Bauer]], *Spreen spaces and the synthetic Kreisel-Lacombe-Shoenfield-Tseitin theorem* ([arXiv:2307.07830](https://arxiv.org/abs/2307.07830))

* Martin E. Bidlingmaier, Florian Faissole, [[Bas Spitters]], *Synthetic topology in Homotopy Type Theory for probabilistic programming*. Mathematical Structures in Computer Science, 2021;31(10):1301-1329. [[doi:10.1017/S0960129521000165](https://doi.org/10.1017/S0960129521000165), [arXiv:1912.07339](https://browse.arxiv.org/abs/1912.07339)]

* Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)


sigma-ideal

# $\sigma$-Ideals
* table of contents
{: toc}

## Idea

A $\sigma$-ideal is a collection of sets (either [[subsets]] of an ambient set or [[pure sets]]) that are considered 'small' in some fashion.  Unlike the notion of 'small' in [[small category]], this is not expected to be closed under most infinitary operations, but it *is* expected to be closed under [[countable set|countably]] infinitary operations, in particular under countable [[union]].

If we use 'large' sets instead, then we have a $\delta$-filter.

## Definitions

Let $X$ be a [[set]].  Then a $\sigma$-ideal on $X$ is a collection $\mathcal{I}$ of [[subsets]] of $X$ such that:
1.  If $A \subset B$ and $B \in \mathcal{I}$, then $A \in \mathcal{I}$;
2.  If $A_1, A_2, \ldots \in \mathcal{I}$, then there exists $B$ such that $B \in \mathcal{I}$ and $\bigcup_i A_i \subseteq B$; in light of (1), $B$ may be assumed to be the [[union]] $\bigcup_i A_i$ itself.
3.  Some set belongs to $\mathcal{I}$; in light of (1), the [[empty set]] $\empty \in \mathcal{I}$.

A __[[base]]__ of a $\sigma$-ideal is any collection satisfying (2,3); a base is precisely what generates a $\sigma$-ideal by closing under subsets.  A __[[subbase]]__ of a $\sigma$-ideal is any collection at all; a subbase generates a base by closing under countable unions.

Instead of a set, $X$ may easily be a [[proper class]]; then the elements of $\mathcal{I}$ may be restricted to subclasses that are actually sets.  One may take $X$ to be the class of all [[pure sets]]; from the perspective of [[material set theory]], this actually includes the general case above.

As defined above, a $\sigma$-ideal $\mathcal{I}$ is a subset of the [[power set]] (or power class) $\mathcal{P}X$; we can just as easily make $\mathcal{I}$ a subset of any [[complete lattice]] $\mathcal{L}$.  Actually, it works just as well if $\mathcal{L}$ if replaced by a sup-[[semilattice]] with all countablary [[supremum|suprema]], or probably even more generally (see [[ideal]] for some idea of how to do that).  Note the grammar: a $\sigma$-ideal *in* $L$ but *on* $X$ (which is the same as *in* $\mathcal{P}X$).

Dually, a __$\delta$-[[filter]]__ on $X$ is a collection $\mathcal{F}$ of [[subsets]] of $X$ such that:
1.  If $A \subset B$ and $A \in \mathcal{F}$, then $B \in \mathcal{F}$;
2.  If $A_1, A_2, \ldots \in \mathcal{F}$, then there exists $B$ such that $B \in \mathcal{F}$ and $B \subseteq \bigcap_i A_i$; in light of (1), $B$ may be assumed to be the [[intersection]] $\bigcap_i A_i$ itself.
3.  Some set belongs to $\mathcal{F}$; in light of (1), the [[improper subset]] $X \in \mathcal{F}$.

Using [[de Morgan duality]], $\delta$-filters and $\sigma$-ideals are essentially the same; we have
$$ \mathcal{F} = \{ A \subseteq X \;|\; \neg{A} \in \mathcal{I} \} $$
and vice versa.  In [[constructive mathematics]], however, they are not equivalent.  Also, the two notions are not equivalent when $X$ is a [[proper class]].

## Examples

*  The power set of $X$ is both a $\sigma$-ideal and a $\delta$-filter on $X$; it is the _improper $\sigma$-ideal_ on $X$.  (Compare [[proper ideal]].)
*  The [[null sets]] in a [[measure space]] $X$ form a $\sigma$-ideal on $X$, while the [[full sets]] form the corresponding $\delta$-filter.
*  The [[meager subspace|meagre sets]] in a [[topological space]] $X$ form a $\sigma$-ideal on $X$, while the comeagre sets form the corresponding $\delta$-filter.
*  The [[pure sets]] form a $\sigma$-ideal on (or, equivalently in this case, in) the class of all pure sets; actually, this collection is closed under arbitrary unions rather than merely countablary unions, so we may call it a _complete ideal_.

## Properties

Of course, any $\sigma$-ideal in an [[ideal]].  A $\sigma$-ideal is a $\sigma$-[[sigma-ring|ring]] in its own right.  In fact, a $\sigma$-ideal on $X$ is precisely simultaneously an ideal in and a sub-$\sigma$-ring of $\mathcal{P}X$.  Dual results hold for $\delta$-filters.

[[!redirects sigma-ideal]]
[[!redirects sigma-ideals]]
[[!redirects ∞-ideal]]
[[!redirects ∞-ideals]]

[[!redirects delta-filter]]
[[!redirects delta-filters]]
[[!redirects ∞-filter]]
[[!redirects ∞-filters]]


sigma-locale

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## Idea

...

## Definition

A **$\sigma$-locale** consists of a set $X$ with a [[sigma-frame]] $\mathcal{O}(X)$.

A **[[continuous function]]** $f:X \to Y$ between $\sigma$-locales $X$ and $Y$ is a sigma-frame [[homomorphism]] $f^*:\mathcal{O}(Y) \to \mathcal{O}(X)$, a [[monotonic function]] which preserves [[finite set|finite]] [[meets]] and [[countable set|countable]] [[joins]].

## Related concepts

* [[sigma-frame]]

* [[sigma-topological space]]

* [[locale]]

## References

* Francesco Ciraulo, *$\sigma$-locales in Formal Topology*, Logical Methods in Computer Science, Volume 18, Issue 1 (January 12, 2022) ([doi:10.46298/lmcs-18%281%3A7%292022](https://doi.org/10.46298/lmcs-18%281%3A7%292022), [arXiv:1801.09644](https://arxiv.org/abs/1801.09644))

* [[Alex Simpson]], *Measure, randomness and sublocales*, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. ([doi:10.1016/j.apal.2011.12.014](https://doi.org/10.1016/j.apal.2011.12.014))


sigma-model

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# Contents
* table of contents
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## Idea
 {#Idea}

A _$\sigma$-model_ is a particular kind of [[physical theory]] of certain [[field (physics)|fields]].  The basic data describing a specific $\sigma$-model is some kind of "[[space]]" $X$, in a [[category]] of "[[spaces]]" which includes [[smooth manifolds]].  We call $X$ the [[target space]], and we define the "[[configuration space]] of [[field (physics)|fields]]" $Conf_\Sigma$  over a manifold $\Sigma$ to be the [[mapping space]]/[[mapping stack]] $Map(\Sigma, X)$.  That is, a "configuration of fields" over a manifold $\Sigma$ is like  an $X$-valued [[function]] on $\Sigma$.

We assign a [[dimension]] $n \in \mathbb{N}$ to our $\sigma$-model, take $dim \Sigma \leq n$ and assume that [[target space]] $X$ is equipped with a "[[circle n-bundle with connection]]".

For $n = 1$ this is an ordinary [[circle bundle]] [[connection on a bundle|with connection]] and models a configuration of the [[electromagnetic field]] on $X$. To distinguish this "field" on $X$ from the fields on $\Sigma$ we speak of a _[[background gauge field]]_. (This remains fixed background data unless and until we pass to _[[second quantization]]_.) A field configuration $\Sigma \to X$ on $\Sigma$ models a trajectory of a [[charge]]d particle subject to the [[force]]s exerted by this background field.

For $n = 2$, a circle $n$-bundle with connection is a [[circle n-group|circle 2-group]] [[principal 2-bundle]] or equivalently a [[bundle gerbe]] with connection. This models a "higher electromagnetic field", called a [[Kalb-Ramond field]]. Now $\Sigma$ is taken to be 2-dimensional and a map $\Sigma \to X$ models the trajectory of a [[string]] on $X$, subject to forces exerted on it by this higher order field.

This pattern continues. In the next dimension a [[membrane]] with 3-dimensional [[worldvolume]] is charged under a [[circle n-bundle with connection|circle 3-bundle]] with connection, for instance something called the [[supergravity C-field]].

While one can speak of [[principal ∞-bundles|higher bundles]] in full generality and full analogy to ordinary [[principal bundle]]s, it is useful to observe that any circle $n$-bundle is characterized by a classifying map $\alpha : X \to \mathbf{B}^n U(1)$ in our category of [[space]]s, so we can just think about classifying maps instead.  Here $U(1)$ is the [[circle group]], and $\mathbf{B}^n$ denotes its $n$th _[[delooping]]_ ; thus such a map is also a sort of [[cocycle]] in "[[smooth infinity-groupoid|smooth]] $n$th [[cohomology]] of $X$ with coefficients in $U(1)$". The additional data of a _connection_ refines this to a cocycle in the _[[ordinary differential cohomology|differential cohomology]]_ of $X$.

Such connection data $\nabla$ on a circle $n$-bundle defines -- and is defined by -- a notion of [[higher parallel transport]] over $n$-dimensional trajectories: for closed $n$-dimensional $\Sigma$ it defines a map $hol : (\gamma : \Sigma \to X) \mapsto \exp(i \int_\Sigma \gamma^*\nabla) \in U(1)$ that sends trajectories to elements in $U(1)$: the [[holonomy]] of $\nabla$ over $\Sigma$, given by [[integration]] of local data over $\Sigma$. The local data being integrated is called the _[[Lagrangian]]_ of the $\sigma$-model. Its integral is called the [[action functional]].

In the _[[QFT|quantum]]_ $\sigma$-model one considers in turn the integral of the [[action functional]] over all of [[configuration space]]: the "[[path integral]]". In the _classical $\sigma$-model_ one considers only the [[critical locus]] of the action functional (where the rough idea is that the path integral to some approximation localizes around the critical locus). Points in this critical locus are said to be configurations that satisfy the "[[Euler-Lagrange equations]] of motion". These are supposed to be the physically realized trajectories among all of them, in the classical approximation.

Finally, just like an ordinary [[circle group]]-[[principal bundle]] has an [[associated bundle|associated]] [[vector bundle]] once we fix a [[representation]] of $U(1)$ to be the [[fiber]]s, any "[[circle n-bundle]]" has an associated "[[n-vector bundle]]" once we fix a "[[∞-representation]]" $\rho : \mathbf{B}^n U(1) \to n Vect$ on "[[n-vector spaces]]".  Just as for the ordinary $U(1)$, here we usually pick the canonical 1-dimensional such "representation".  Finally, we define bundles $V_\Sigma : Conf_\Sigma \to \mathcal{C}$ of "internal states" by [[transgression]] of these associated bundles.

The passage from [[principal ∞-bundle]]s to [[associated ∞-bundle]]s is necessary for the description of the quantum $\sigma$-model: it assigns in positive codimension spaces of [[section]]s of these associated bundles. For a 1-categorical description of the resulting [[QFT]] ordinary vector bundles (assigned in codimension 1) would suffice, but the $\sigma$-model should determine much more: an [[extended quantum field theory]]. This requires sections of higher vector bundles. For instance for $n = 2$ some boundary conditions of the $\sigma$-model are given by sections of the background [[n-vector bundle|2-vector bundle]]: these are the [[twisted bundle|twisted vector bundles]] known as the [[Chan-Paton bundles]] on the [[boundary]]-[[D-brane]]s of the [[string]]. (...)

We now try to fill this with life by spelling out some standard examples. Further [below](#Exposition) we look at precise formalizations of the situation.

## Terminology and history
 {#TerminologyAndHistory}

In physics one tends to speak of a _model_ if one specifies a particular [[quantum field theory]] for describing a particular situation, for instance by specifying a [[Lagrangian]] or [[local action functional]] on some [[configuration space]]. This is traditionally not meant in the mathematical sense of _[[model]]_ of some [[theory]]. But in light of progress of mathematically formalizing [[quantum field theory]] (see [[FQFT]] and [[AQFT]]), it can with hindsight be interpreted in this way:

a $\sigma$-model is supposed to be a type of [[model]] for the [[theory]] called [[quantum field theory]]. This sounds like a tautology, but much effort in mathematical physics is devoted to eventually making this a precise statement. In special cases and toy examples this has been achieved, but for the examples that seem to be directly relevant for the phenomenological description of the observed world, lots of clues still seem to be missing.

As to the "$\sigma$" in "$\sigma$-model": back in the 1960s people were interested in a hypothetical particle called the
$\sigma$-particle.  [[Murray Gell-Mann]] came up with a theory of them.  It was called 'the $\sigma$-model'.  It was an old-fashioned field theory where the field took values in a vector space.   Then someone came up with a modified version of the σ-model where the field took values in some other [[manifold]] and this was called 'the nonlinear σ-model'.

While the parameter space (the domain space of the fields) of the original $\sigma$-models was supposed to be our [[spacetime]] and the target space was some abstract space, with the advent of [[string theory]] the nonlinear $\sigma$-models gained importance as quantum field theories whose _target space_ is [[spacetime]] $X$ and whose parameter space is some low dimensional space, usually denoted $\Sigma$. A field configuration $\Sigma \to X$ is then interpreted as being the trajectory of an extended fundamental particle -- a fundamental [[brane]] -- in $X$, and the $\sigma$-model describes the quantum mechanics of that brane propagating in $X$.

In particular the [[quantum mechanics]] of a [[relativistic particle]] propagating on $X$ is described by a $\sigma$-model on the [[real line]] $\Sigma = \mathbb{R}$ -- the _worldline_ of the particle.

In [[string theory]] one considers 2-[[dimension]]al $\Sigma$ and thinks of maps $\Sigma \to X$ as being the _worldsheets_ of the trajectory of a string propagating in spacetime.

In the context of [[11-dimensional supergravity]] there is a $\sigma$-model with 3-dimensional $\Sigma$, describing the propagation of a [[membrane]] in [[spacetime]].

## Exposition of classical sigma-models
 {#ExpositionClassical}

We survey, starting from the very basics, [[classical field theory]] aspects of $\sigma$-models that describe dynamics of [[particles]], [[strings]] and [[branes]] on geometric [[target spaces]].

The content of this section is at

* [[sigma-model -- exposition of classical sigma-models]]

## Exposition of higher gauge theories as $\sigma$-models
 {#GaugeTheoryAsSigmaModel}

We discuss how [[gauge theories]] and their higher analogs are
naturally regarded as $\sigma$-models.

The content of this section is at

* [[sigma-model -- exposition of higher gauge theories as sigma-models]].

## Exposition of quantum $\sigma$-models
 {#ExpositionQuantumSigmaModel}
 {#StringTopology}

Above we have discussed some standard [classical sigma-models](#ExpositionClassical) and [higher gauge theories as sigma-models](#GaugeTheoryAsSigmaModel), also mostly classically. Here we talk about the [[quantization]] of these models (or some of them) to [[QFT]]s: _quantum $\sigma$-models_ .

The content of this section is at

* [[sigma-model -- exposition of quantum sigma-models]].

See there for discussion of [[string topology]], [[Gromov-Witten theory]], [[Chern-Simons theory]].

## Exposition of a general abstract formulation
 {#Exposition}

We give a leisurely exposition of a general abstract formulation $\sigma$-models, aimed at readers with a background in [[category theory]] but trying to assume no other prerequisites.

What is called an _$n$-dimensional $\sigma$-model_ is first of all an instance of an $n$-dimensional [[quantum field theory]] (to be explained). The distinctive feature of those quantum field theories that are $\sigma$-models is that

1. these arise from a simpler kind of field theory -- called a [[classical field theory]] -- by a process called [[quantization]]

1. moreover, this simpler kind of field theory encoded by [[geometry|geometric data]] in a nice way: it describes physical [[configuration space]]s that are [[mapping space]]s into a [[geometry|geometric]] [[space]] equipped with some [[synthetic differential geometry|differential geometric]] [[structure]].

We give expositions of these items step-by-step:

1. [Quantum field theory](#ExpositionQuantumFieldTheory)

1. [Classical field theory](#ExpositionClassicalFieldTheory)

1. [Quantization](#ExpositionQuantization)

1. [Classical sigma-models](#ExpositionClassicalSigmaModels)

1. [Quantum sigma-models](#ExpositionQuantumSigmaModels)

We draw from ([FHLT, section 3](#FHLT)).

The content of this section is at

* [[sigma-model -- exposition of a general abstract formulation]].

## Exposition of second quantization of $\sigma$-models
 {#SecondQuantization}

We discuss [[second quantization]] in the context of $\sigma$-models.

The content of this section is at

* [[sigma-model -- exposition of second quantization of sigma-models]]

## Examples

### Non-topological $\sigma$-models

* The canonical textbook example of a quantum mechanical system is of this form for $n=1$: A line [[vector bundle|bundle]] with [[connection on a bundle|connection]] $E \to X$ on a ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian manifold]] $X$ induces the 1-dimensional [[quantum field theory]] which is the quantum mechanics of a point particle which propagates on $X$, subject to the forces of [[gravity]] (given by the [[pseudo-Riemannian metric]] on $X$) and [[electromagnetism]] (given by the line bundle with connection). The [[Hamilton operator]] encoding this quantum dynamics in this case is the Laplace-operator of $T X$ twisted by the line bundle $E$.

  For $X$ a [[spacetime]] this is called the [[relativistic particle]].

  For $\Sigma$ or $X$ a [[supermanifold]] this is the [[superparticle]].

* Generalizing in the above example the line bundle $E$ by an abelian [[bundle gerbe]] with a connection yields a background for a 2-dimensional $\sigma$-model which mayb be thought of as describing the propgation of a [[string theory|string]].  The best-studied version of this is the case where $X = G$ is a [[Lie group]], in which case this $\sigma$-model is known as the _[[WZW model|Wess--Zumino--Witten model]]_.

### Topological $\sigma$-models

* [[Dijkgraaf-Witten theory]] is the (2+1)-dimensional $\sigma$-model induced from an abelian 2-gerbe on $\mathbf{B} G$, for $G$ a finite group.

* [[Chern-Simons theory]] is supposed to be analogously the $\sigma$-model induced from an abelian 2-gerbe with connection on $\mathbf{B}G$, but now for $G$ a Lie group.

* [[topological membrane]]

* the [[Poisson sigma-model]] is a model whose target is a [[Poisson Lie algebroid]].

* in [[AKSZ theory]] this is generalized to a large class of sigma models with [[symplectic Lie n-algebroid]]s as target.

* Rozansky--Witten theory is essentially the $\sigma$-model for $X$ a smooth projective variety.

* generally [[schreiber:∞-Chern-Simons theory]] is a $\sigma$-model with a [[smooth ∞-groupoid]] of [[connection on an ∞-bundle|∞-connections]].

## Related concepts

* **sigma model**

  * [[worldvolume]]

  * [[target space]]

  * [[background gauge field]]

  * [[genus]]

* [[gauged linear sigma-model]]

* [[supersymmetric sigma-model]]

  * [[2d (2,0)-supersymmetric sigma-model]]

* [[CP^N sigma-model]]

* [[brane]]

  * [[particle]], [[superparticle]]

  * [[string]]

  * [[membrane]]

* [[homological mirror symmetry]]

  * [[A-model]], [[B-model]]

  * [[Landau-Ginzburg model]]

## References

The concept of [[sigma-models]] originates with the introduction of the [[σ-meson]] to [[chiral perturbation theory]] in

* [[Murray Gell-Mann]], M. Lévy, _The axial vector current in beta decay_, Il Nuovo Cimento (1955-1965) volume 16, pages 705–726 (1960) ([doi:10.1007/BF02859738](https://doi.org/10.1007/BF02859738))

A standard reference on 2-dimensional [[string]] $\sigma$-models is

* [[Pierre Deligne]], [[Dan Freed]], _Classical field theory_ , chapter 5, page 211

  and

  [[Krzysztof Gawedzki]], _Lectures on conformal field theory_ , part 3, lecture 3

  in

  [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey,
[[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds.  _[[Quantum Fields and Strings]], A course for mathematicians_, 2 vols. Amer. Math. Soc. Providence 1999. ([web version](http://www.math.ias.edu/qft))

General exposition of $n$-dimensional $\sigma$-models, including the fact that these are [[renormalization|non-renormalizable]] for $n\geq 3$:

* [[Christopher Hull]], *Lectures on Non-Linear Sigma-Models and Strings*, in *Super field theories*, eds. H. C. Lee, V. Elias, G. Kunstatter, R. B. Mann, K. S. Viswanathan, NATO Science Series **160**, Springer (1987) 77--168 [[doi:10.1007/978-1-4613-0913-0_4](https://doi.org/10.1007/978-1-4613-0913-0_4)]

Further (and original) discussion of [[string]] sigma-models and their [[Ricci flow]] [[renormalization group flow]]  is for instance in

* [[Daniel Friedan]]. *Nonlinear models in $2+\epsilon$ dimensions*. Annals of Physics, 163(2):318–419, 1980.

* [[Daniel Friedan]]. *Nonlinear models in $2+\epsilon$ dimensions*. Physical Review Letters, 45(13):1057, 1980.

* {#Tseytlin89} [[Arkady Tseytlin]], _Sigma model approach to string theory_, Int. J. Mod. Phys. A 4, 1257 (1989).

* C. Callan, L. Thorlacius, _Sigma models and string theory_, Particles, Strings and Supernovae, Volumes I and II. Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, held June 6 - July 2, 1988, at Brown University, Providence, Rhode Island. Edited by A. Jevicki and C.-I. Tan. Published by World Scientific, New York, 1989, p.795 ([pdf](http://www.damtp.cam.ac.uk/user/tong/string/sigma.pdf))

* [[Arkady Tseytlin]], _Sigma model approach to string theory effective actions with tachyons_, J. Math.Phys.42:2854-2871 (2001) ([arXiv:hep-th/0011033](http://arxiv.org/abs/hep-th/0011033))

* {#Tseytlin06} [[Arkady Tseytlin]], _On sigma model RG flow, "central charge" action and Perelman's entropy_, Phys.Rev.D75:064024,2007 ([arXiv:hep-th/0612296](http://arxiv.org/abs/hep-th/0612296))

First indications on how to formalize $\sigma$-models in a [[higher category theory|higher categorical]] context were given in

* [[Dan Freed]], _Higher algebraic structures and quantization_, ([arXiv](http://arxiv.org/abs/hep-th/9212115))

A grand picture developing this approach further is sketched in

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_,  ([arXiv](http://arxiv.org/abs/0905.0731))
 {#FHLT}

A discussion of 2- or (2+1)-dimensional $\Sigma$-models whose target is an [[derived stack]]/[[infinity-stack]] is in

* [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], _Integral transforms and Drinfeld centers in derived geometry_, ([arXiv](http://arxiv.org/abs/0805.0157)) .

More discussion of the latter is at [[geometric infinity-function theory]].

A discussion of $\sigma$-models of higher gauge theory type is at

* [[Urs Schreiber]], _[[schreiber:infinity-Chern-Simons theory]]_ .

Concrete applications of $\sigma$-models with target [[stacks]] (typically smooth ones, hence [[smooth groupoids]], and typically smooth [[gerbes]] among these) in [[string theory]] and [[supergravity]] are discussed in

* [[Tony Pantev]], [[Eric Sharpe]], _String compactifications on Calabi-Yau stacks_, Nucl.Phys. B733 (2006) 233-296, ([arXiv:hep-th/0502044](http://arxiv.org/abs/hep-th/0502044))

* {#PantevSharpe} [[Tony Pantev]], [[Eric Sharpe]], _Gauged linear sigma-models for gerbes (and other toric stacks)_, Adv.Theor.Math.Phys. 10 (2006) 77-121 ([arXiv:hep-th/0502053](http://arxiv.org/abs/hep-th/0502053))

* {#HellermanSharpe} S. Hellerman, [[Eric Sharpe]], _Sums over topological sectors and quantization of Fayet-Iliopoulos parameters_, ([arXiv:1012.5999](http://arxiv.org/abs/1012.5999))

with review in

* [[Eric Sharpe]], _Sigma models on gerbes_, talk at _[Higher Structures in M-Theory, Durham, August 2018](http://www.maths.dur.ac.uk/lms/109/index.html)_ ([pdf slides](http://www.maths.dur.ac.uk/lms/109/talks/1864sharpe.pdf), [video recording](http://www.maths.dur.ac.uk/lms/109/movies/1864shar.mp4))

Discussion of [[geometric Langlands duality]] in terms of 2d sigma-models on stacks ([[moduli stacks]] of [[Higgs bundles]] over a given [[algebraic curve]]) is in

* {#Witten08} [[Edward Witten]], section 6 of _Mirror Symmetry, Hitchin's Equations, And Langlands Duality_, in Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon, _The many faces of geoemtry: A tribute to Nigel Hitchin_, Oxford Scholarship Online ([arXiv:0802.0999](http://arxiv.org/abs/0802.0999))

See also:

* Rafał R. Suszek. *Towards higher super-$\sigma$-model categories*. (2023). ([arXiv:2311.01768](https://arxiv.org/abs/2311.01768)).

[[!redirects sigma-models]]
[[!redirects sigma model]]
[[!redirects sigma models]]
[[!redirects σ-model]]
[[!redirects σ-models]]


sigma-model -- exposition of a general abstract formulation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Phyiscs
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
=--
=--

> This is a sub-entry of [[sigma-model]]. See there for further background and context.

***

#Contents#
* table of contents
{:toc}

## Exposition of a general abstract formulation

We give a leisurely exposition of a general abstract formulation $\sigma$-models, aimed at readers with a background in [[category theory]] but trying to assume no other prerequisites.

What is called an _$n$-dimensional $\sigma$-model_ is first of all an instance of an $n$-dimensional [[quantum field theory]] (to be explained). The distinctive feature of those quantum field theories that are $\sigma$-models is that

1. these arise from a simpler kind of field theory -- called a [[classical field theory]] -- by a process called [[quantization]]

1. moreover, this simpler kind of field theory encoded by[[geometry|geometric data]] in a nice way: it describes physical [[configuration space]]s that are [[mapping space]]s into a [[geometry|geometric]] [[space]] equipped with some [[synthetic differential geometry|differential geometric]] [[structure]].

We give expositions of these items step-by-step:

1. [Quantum field theory](#ExpositionQuantumFieldTheory)

1. [Classical field theory](#ExpositionClassicalFieldTheory)

1. [Quantization](#ExpositionQuantization)

1. [Classical sigma-models](#ExpositionClassicalSigmaModels)

1. [Quantum sigma-models](#ExpositionQuantumSigmaModels)

We draw from ([FHLT, section 3](#FHLT)).

### Quantum field theory
 {#ExpositionQuantumFieldTheory}

#### Definition

For our purposes here, a [[quantum field theory]] of [[dimension]] $n$ is a [[symmetric monoidal functor]]

$$
  Z : Bord_n^S \to \mathcal{C}
  \,,
$$

where

* $Bord_n^S$ is a [[cobordism category|category of n-dimensional cobordisms]] that are equipped with some [[structure]] $S$;

  * its [[object]]s are $(n-1)$-[[dimension]]al [[topological manifold]]s;

  * its [[morphism]]s are [[isomorphism class]]es of $n$-[[dimension]]al [[cobordism]]s between such manifolds

  * and all manifolds are equipped with _$S$-[[structure]]_ . For instance $S$ could be _[[Riemannian structure]]_ . Then we would call $Z$ a [[Euclidean quantum field theory]] (confusingly). If $S$ is "no structure" then we call $Z$ a [[topological quantum field theory]].

  * the [[monoidal category]]-structure is given by [[disjoint union]] of manifolds

    $$
      \Sigma_1 \otimes \Sigma_2 := \Sigma_1 \coprod \Sigma_2
      \,;
    $$

* $\mathcal{C}$ is some [[symmetric monoidal category]].

We think of data as follows:

* $Bord_n^S$ is a model for _being_ and _becoming_ in [[physics]] (following [[Bill Lawvere]]'s terminology): the objects of $Bord_n^S$ are archetypes of _physical spaces that are_ and the morphisms are _physical spaces that evolve_ ;

* the [[object]] $Z(\Sigma)$ that $Z$ assigns to any $(n-1)$-manifold $\Sigma$ is to be thought of as the _[[space]] of all possible [[state]]s_ over the space $\Sigma$ of a the physical system to be modeled;

* so $\mathcal{C}$ is the category of [[n-vector space]]s among which the _spaces of [[state]]s_ of the quantum theory can be picked;

* the [[morphism]] $Z(\hat \Sigma) : Z(\Sigma_{in}) \to Z(\Sigma_{out})$ that $\Sigma$ assigns to any [[cobordism]] $\hat \Sigma$ with incoming [[boundary]] $\Sigma_{in}$ and outgoing boundary $\Sigma_{out}$ is the [[propagator]] along $\hat \Sigma$: it maps every state $\psi \in Z(\Sigma_{in})$ of the system over $\Sigma_{in}$ to the state $Z(\Psi) \in \Sigma_{out}$ that is the result of the evolution of $\psi$ along $\hat \Sigma$ by the _dynamics_ of the system. Or conversely: the action of $Z$ encodes what this dynamics is supposed to be.

Notice that since $Z$ is required to be a symmetric monoidal functor it sends disjoint unions of manifolds to [[tensor product]]s

$$
  F(\Sigma_1 \coprod \Sigma_2) \simeq Z(\Sigma_1) \otimes Z(\Sigma_2)
  \,.
$$

Moreover, for $\hat \Sigma$ a closed [[cobordism]], hence a morphism $\emptyset \stackrel{\hat \Sigma}{\to} \emptyset$ from the [[empty set|empty]] manifold to itself, we have that

* $Z(\emptyset) = \mathbf{1}$ is the tensor unit of $\mathcal{C}$;

* $Z(\hat \Sigma)  \in End(\mathbb{1})$ is an [[endomorphism]] of this tensor unit, a _number_ as seen internal to $\mathcal{C}$ -- this is the _invariant_ associated to $\hat \Sigma$ by $Z$, called the [[partition function]] of $Z$ over $\hat \Sigma$. We can think of $Z$ as being a rule for computing such invariants by building them up from smaller pieces. This is the _locaity_ of quantum field theory.

#### Examples
 {#ExpositionQFTExamples}

* A simple but archetypical example is this: let $S := Riem$ be [[Riemannian manifold|Riemannian structure]]. Then the category $Bord_1^{Riem}$ of 1-dimensional cobordisms equipped with Riemannian structure is generated (as a [[symmetric monoidal category]]) from [[interval]]s

  $$
    \bullet \stackrel{t}{\to} \bullet
  $$

  equipped with a [[length]] $t \in \mathbb{R}_+$. Composition is given by addition of lengths

  $$
    (\bullet \stackrel{t_1}{\to} \bullet \stackrel{t_2}{\to})
    =
    (\bullet \stackrel{t_1 + t_2}{\to} \bullet)
    \,.
  $$

  Therefore a 1-dimensional [[Euclidean quantum field theory]]

  $$
    Z : Bord_1^{Riem} \to Vect
  $$

  is specified by

  * a [[vector space]] $\mathcal{H}$ ("of [[state]]s") assigned to the point;

  * for each $t \in \mathbb{R}_+$ a linear endomorphism

    $$
      U(t) : \mathcal{H} \to \mathcal{H}
    $$

    such that

    $$
      U(t_1 + t_2) = U(t_2) \circ U(t_1)
      \,.
    $$

  This is just a system of [[quantum mechanics]]. If we demand that $Z$ respects the [[smooth structure]] on the space of morphisms in $Bord_1^{Riem}$ then there will be a linear map $i H : \mathcal{H} \to \mathcal{H}$ such that

   $$
     U(t) = \exp(i H t)
     \,.
   $$

   This $H$ is called the [[Hamilton operator]] of the system.

   (We are glossing here over some technical fine print in the definition of $Bord_1^{Riem}$. Done right we have that $\mathcal{H}$ may indeed be an infinite-dimensional vector space. See [[(1,1)-dimensional Euclidean field theories and K-theory]])

### Classical field theory
 {#ExpositionClassicalFieldTheory}

A special class of examples of $n$-dimensional quantum field theories, as discussed [above](#ExpositionQuantumFieldTheory), arise as [[deformation quantization|deformations]] or _[[path integral|averages]]_ of similar, but simpler structure: _[[classical field theories]]_ . The process that constructs a quantum field theory out of a classical field theory is called _[[quantization]]_ . This is discussed [below](#ExpositionQuantization). Here we describe what a classical field theory is. We shall inevitably oversimplify the situation such as to still count as a leisurely exposition. The kind of examples that the following discussion applies to strictly are field theories of Dijkgraaf-Witten type. But despite its simplicity, this case accurately reflects most of the general abstract properties of the general theory.

For our purposes here, a [[classical field theory]] of dimension $n$ is

* a [[symmetric monoidal functor]]

  $$
    \exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C})
    \,,
  $$

  where

  * $Bord_n^S$ is the same [[category of cobordisms]] as before;

  * $Span(Grpd, \mathcal{C})$ is the [[category of spans]] of [[groupoid]]s over $\mathcal{C}$:

    * [[object]]s are [[groupoid]]s $K$ equipped with [[functor]]s $\phi : K \to \mathcal{C}$;

    * [[morphism]]s $(K_1, \phi_1) \to (K_2, \phi_2)$ are [[diagram]]s

      $$
        \array{
           && \hat K
           \\
           & \swarrow && \searrow
           \\
           K_1 &&\swArrow&& K_2
           \\
           & \searrow && \swarrow
           \\
           && \mathcal{C}
        }
        \,,
      $$

      where in the middle we have a [[natural transformation]];

    * [[composition]] of morphism is by forming [[2-pullback]]s:

      $$
        (\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2
        \,.
      $$

Let $\hat \Sigma : \Sigma_1 \to \Sigma_2$ be a [[cobordism]] and

$$
  \exp(i S(-))_{\Sigma} =
  \left(
        \array{
           && Conf_{\hat \Sigma}
           \\
           & {}^{\mathllap{(-)|_{in}}}\swarrow && \searrow^{\mathrlap{(-)|_{out}}}
           \\
           Conf_{\Sigma_1} &&\swArrow_{\exp(i S(-)_{\hat \Sigma})}&& Conf_{\Sigma_2}
           \\
           & {}_{V_{\Sigma_1}}\searrow && \swarrow_{\mathrlap{V_{\Sigma_2}}}
           \\
           && \mathcal{C}
        }
  \right)
$$

the value of a classical field theory on $\hat \Sigma$.
We interpret this data as follows:

* $Conf_{\Sigma_1}$ is the [[configuration space]] of a [[classical field theory]] over $\Sigma_1$: [[object]]s are "field configurations" on $\Sigma_1$ and [[morphism]]s are [[gauge transformation]]s between these. Similarly for $Conf_{\Sigma_2}$.

  Here a "physical field" can be something like the [[electromagnetic field]]. But it can also be something very different. For the special case of $\sigma$-models that we are eventually getting at, a "field configuration" here will instead be a way of an particle of shape $\Sigma_1$ sitting in some [[target space]].

* $Conf_{\hat \Sigma}$ is similarly the groupoid of field configurations on the whole cobordism, $\hat \Sigma$. If we think of an object in $Conf_{\hat \Sigma}$ of a way of a [[brane]] of shape $\Sigma_1$ sitting in some [[target space]], then an object in $Conf_{\hat Sigma}$ is a _trajectory_ of that brane in that target space, along which it evolves from shape $\Sigma_1$ to shape $\Sigma_2$.

* $V_{\Sigma_i} : Conf_{\Sigma_i} \to \mathcal{C}$ is the [[classifying space|classifying map]] of a kind of [[vector bundle]] over configuration space: a [[state]] $\psi \in Z(\Sigma_1)$ of the [[quantum field theory]] that will be associated to this [[classical field theory]] by [[quantization]] will be a [[section]] of this vector bundle. Such a section is to be thought of as a generalization of a
[[probability distribution]] on the space of classical field configurations. The [[generalized element]]s of a [[fiber]] $V_{c}$ of $V_{\Sigma_1}$ over a configuration $c \in Conf_{\Sigma_1}$ may be thought of as an _internal state_ of the brane of shape $\Sigma_1$ sitting in target space.

* $\exp(i S(-))_{\hat \Sigma}$ is the [[action functional]] that defines the classical field theory: the component

  $$
   \exp(i S(\gamma))_{\hat \Sigma}
    :
    V_{\gamma|_{in}}
    \to
    V_{\gamma|_{out}}
  $$

 of this [[natural transformation]] on a trajectory $\gamma \in Conf_{\hat \Sigma}$ going from a configuration $\gamma|_{in}$ to a configuration $\gamma|_{out}$ is a morphism in $\mathcal{C}$ that maps the internal states of the ingoing configuration $\gamma|_{\Sigma_1}$ to the internal states of the outgoing configuration $\gamma|_{\Sigma_2}$. This evolution of internal states encodes the _classical dynamics_ of the system.

Notice that this way a classical field theory is taken to be a special case of a [quantum field theory](#ExpositionQuantumFieldTheory), where the codomain of the symmetric monoidal functor is of the special form $Span(Grpd, \mathcal{C})$. For more on this see [[classical field theory as quantum field theory]].

### Quantization
 {#ExpositionQuantization}

We assume now that $\mathcal{C}$ has [[colimit]]s and in fact [[biproduct]]s.

Then for every [[functor]] $\phi : K \to \mathcal{C}$ the [[colimit]]

$$
  \int^{K} \phi \in \mathcal{C}
$$

exists, and (using the existence of [[biproduct]]s) this construction extends to a [[functor]]

$$
  \int : Span(Grpd, \mathcal{C}) \to \mathcal{C}
  \,.
$$

We call this the _[[path integral]]_ functor.

For

$$
  \exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C})
$$

a [classical field theory](#ExpositionClassicalFieldTheory), we get this way a [quantum field theory](#ExpositionQuantumFieldTheory) by forming the composite functor

$$
  Z
    :=
  \int \circ \exp(i S(-))
   :
  Bord_n^S
    \stackrel{\exp(i S(-))}{\to}
  Span(Grpd, \mathcal{C})
    \stackrel{\int}{\to}
  \mathcal{C}
  \,.
$$

This $Z$ we call the _[[quantization]]_ of $\exp(i S(-))$.

It acts

* on [[object]]s by sending

  $$
    \begin{aligned}
      \Sigma_{in} & \mapsto (V_{\Sigma_{in}} : Conf_{\Sigma_{in}} \to \mathcal{C})
      \\
      & \mapsto \mathcal{H}_{\Sigma_{in}} := \int^K V_{\Sigma_{in}}
    \end{aligned}
  $$

  the vector bundle on the [[configuration space]] over some boundary $\Sigma_{in}$ of [[worldvolume]] to its space $\mathcal{H}_{\Sigma_{in}}$ of [[gauge transformation|gauge invariant]] [[section]]s. In typical situations this $\mathcal{H}_{\Sigma_{in}}$ is the famous [[Hilbert space]] of [[state]]s in [[quantum mechanics]], only that here it is allowed to be any object in $\mathcal{C}$;

* on [[morphism]]s by sending a [[natural transformation]]

  $$
    \begin{aligned}
      \hat \Sigma & \mapsto
       (\exp(i S(-))_{\hat \Sigma} : \gamma \mapsto
       V_{\gamma|_{in}} \to V_{\gamma|_{out}})
       \\
       & \mapsto
       (\int^K \exp(i S(-))_{\hat \Sigma}
        :
        \mathcal{H}_{\Sigma_1} \to \mathcal{H}_{\Sigma_2}
       )
    \end{aligned}
  $$

  to the [[integral transform]] that it defines, weighted by the [[groupoid cardinality]] of $Conf_{\hat \Sigma}$ : the _[[path integral]]_ .

### Classical $\sigma$-models
 {#ExpositionClassicalSigmaModels}

A _classical $\sigma$-model_ is a [classical field theory](#ExpositionClassicalFieldTheory) such that

* the [[configuration space]]s $Conf_{\Sigma}$ are [[mapping spaces]] $\mathbf{H}(\Sigma,X)$ in some suitable category -- some [[cohesive (infinity,1)-topos|higher topos]] in fact -- , for $X$ some fixed [[object]] of that category called _[[target space]]_ ;

* the bundles $V_{\Sigma} : Conf_{\Sigma} \to \mathcal{C}$ "of internal states" over these mapping spaces are

  * the [[transgression]] to these [[mapping space]]s...

  * ...of an [[associated infinity-bundle|associated higher bundle]]...

  * ...asscociated to a [[circle n-bundle with connection]] on [[target space]]...

  * ...encoded by a classifying morphism

    $$
      \alpha : X \to \mathbf{B}^{n} U(1)
    $$

    into the [[circle n-group]] in $\mathbf{H}$; equipped with an [[connection on an infinity-bundle|n-connection]] $\nabla$...

  * ...where the [[associated bundle|association]] is via a [[representation]]

    $$
      \rho : \mathbf{B}^{n+1} U(1) \to (n+1) Vect
    $$

    on [[n-vector spaces]], which is usually taken to be the canonical 1-dimensional one.

  One calls $(\alpha,\nabla)$ the [[background gauge field]] of the $\sigma$-model.

* The [[action functional]]s $\exp(i S(-))_{\hat \Sigma}$ are given by the [[higher parallel transport]] of $\nabla$ over $\hat \Sigma$.

So an $n$-dimensional $\sigma$-model is a classical field theory that is [[representable functor|represented]], in a sense, by a [[circle n-bundle with connection]] on some [[target space]].

More specifically and more simply, in cases where $X$ is just a [[discrete ∞-groupoid]]  -- the case of  sigma-models of Dijkgraaf-Witten type, every [[principal ∞-bundle]] on $X$ is necessarily flat, hence the [[background gauge field]] is given just by the morphism

$$
  \alpha : X \to \mathbf{B}^{n} U(1)
  \,.
$$

Then for $\hat \Sigma$ a closed $n$-dimensional manifold, the [[action functional]] of the sigma-model on $\Sigma$ on a field configuration $\gamma : \hat \Sigma \to X$ has the value

$$
  \exp(i S(\gamma))_{\hat \Sigma}
  =
  \int_{\hat \Sigma} [\alpha]
$$

being the evaluation of $[\alpha]$ regarded as a class in [[ordinary cohomology]] $H^n(\hat \Sigma, U(1))$ evaluated on the [[fundamental class]] of $X$.

One says that $[\alpha]$ is the [[Lagrangian]] of the theory.

### Quantum $\sigma$-models
 {#ExpositionQuantumSigmaModels}

(...)

## References

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_, A celebration of the mathematical legacy of Raoul Bott, 367--403, Amer. Math. Soc. 2010 ([arXiv:0905.0731](https://arxiv.org/abs/0905.0731))
 {#FHLT}


sigma-model -- exposition of classical sigma-models

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Phyiscs
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
=--
=--

> This is a subentry of [[sigma-model]]. See there for background and context.

***

#Contents#
* table of contents
{:toc}

## Exposition of classical $\sigma$-models

We survey, starting from the very basics, [[classical field theory]] aspects of $\sigma$-models that describe dynamics of [[particles]], [[strings]] and [[branes]] on geometric [[target spaces]].

### The Newtonian particle
 {#ExpositionNewtonianParticle}

With hindsight, the earliest $\sigma$-model ever considered was also the very origin of the science of [[physics]]:

In order to describe the motion of matter [[particles]] in [[space]], [[Isaac Newton]] wrote down a [[differential equation]] with the famous symbols

$$
  \vec F = m \vec a
  \,.
$$

More in detail, this is meant to describe the following situation:

* write $X := \mathbb{R}^3$ for the [[Cartesian space]] of [[dimension]] 3; think of this as a model for physics space;

* write $\Sigma := \mathbb{R}$ for the [[Cartesian space]] of [[dimension]] 1; think of this as the abstract trajectory of a point particle;

* write $\gamma : \Sigma \to X$ for a [[smooth function]]; think of this as an actual trajectory of a point particle in $X$;

  write furthermore

  * $\vec v := \dot \gamma \in Hom(T \Sigma, T X)$ for the [[derivative]] of $\gamma$; think of this as the [[velocity]] of the particle; and

  * $\vec a := \ddot \gamma$ for the second [[derivative]], the [[acceleration]] of the particle (strictly speaking this is the [[covariant derivative]] with respect to the trivial [[connection on a bundle|connection]] on the (canonically trivialized) [[tangent bundle]] on $\mathbb{R}^3$, see [below](#RelativisticParticle) for the fully fledged discussion).

We call then the collection of all [[smooth function]]s

$$
  Conf := C^\infty(\Sigma, X)
$$

the [[configuration space]] of a physical model of a point particle propagating on $X$.

In order to define the model -- the model of some physical situation --

* pick a [[vector field]] $\vec F \in \Gamma(T X)$ on $X$. Think of this as expressing at each point a [[force]] acting on the particle with trajectory.

  For instance $\vec F := q \vec E$ could be an [[electromagnetic field|electric field]] $\vec E$ influencing the propagation of an electrically charged particle of [[charge]] $q$.

In modern language we may say:

* $\Sigma$ is the [[worldline]] of the particle;

* $X$ is the [[target space]] (expressing the fact that it is the [[codomain]] of a function $\gamma : \Sigma \to X$)

* $\vec F$ is the [[background gauge field]];

and the collection $(\Sigma, X, \vec F)$ of all three is a _$\sigma$-model_ .

Given this data, the space of solutions to the original [[differential equation]]

$$
  P := \{ \gamma \in Conf | \forall t \in \mathbb{R} : \vec F_{\gamma(t)} = m \vec a(t) \}
$$

is called the [[covariant phase space]] of the model. The configurations in $P \subset Conf$ have the interpretation of being those potential configurations, that describe actual trajectories of [[particle]]s observed in nature.

Notice that in the case of vanishing [[force]] field $\vec F = 0$, the [[equations of motion]] of the [Newtonian particle](#ExpositionNewtonianParticle)

$$
  \vec a = 0
$$

may be read as characterizing precisely the [[geodesic]]s in $\mathbb{R}^3$ regarded as a [[Riemannian manifold]] using the canonical [[metric]]. This is a special and limiting case of the [relativistic particle](#RelativisticParticle) discussed below.

A cautionary note is in order. While the Newtonian particle may serve as an introductory example for motivating the concept of $\sigma$-models, it in general lacks some of the nice properties that later on we shall take to be characteristic of $\sigma$-models. Mainly this is due to the fact that the Newtonian particle is but a limiting approximation to the [[relativistic particle]] to which we turn next.

### The relativistic particle
 {#RelativisticParticle}

The Newtonian particle propagating on $\mathbb{R}^3$, discussed [above](#ExpositionNewtonianParticle), is a special and limiting case of a particle propagating on a 4-dimensional [[pseudo-Riemannian manifold]]: [[spacetime]]. For historical reasons (the same that led to the theory of [[gravity]] being called a theory of _[[general relativity|relativity]]_) this is called the _[[relativistic particle]]_.

The $\sigma$-model describing the relativistic particle is the following.

* [[target space|Target space]] is a  [[pseudo-Riemannian manifold]] $(X,g)$, thought of as [[spacetime]].

* Parameter space $\Sigma = \mathbb{R}$ is the [[real line]], thought of as the abstract [[worldline]] of the particle.

* The [[background gauge field]] is given by a [[circle n-bundle with connection|circle bundle with connection]], which for the moment we shall assume to be topologically trivial and hence be equivalently given by a smooth [[differential form|1-form]] $A \in \Omega^1(X)$. Its [[curvature]] [[exterior derivative]] $F := d A$ is the [[field strength]] of an [[electromagnetic field]] on $X$.

* The [[configuration space]] is the [[quotient]] (or rather [[action groupoid]])

  $$
    Conf = C^\infty(\mathbb{R}, X)//Diff(\mathbb{R})
    \,,
  $$

  of [[smooth function]]s $\Sigma \to X$ by [[diffeomorphism]]s $\Sigma \stackrel{\simeq}{\to} \Sigma$. Each [[object]] in [[configuration space]] is a trajectory $\gamma : \Sigma \to X$ of a particle in spacetime, each morphism/equivalence $\gamma_1 \stackrel{\simeq}{\to} \gamma_2$ : a [[gauge transformation]].

* The [[covariant phase space]] -- the subspace of the configuration space of those configurations that satisfy the [[equations of motion]] -- is defined to be

  $$
    P =
    \{ [\gamma] \in Conf
     |\;\;
    m g(\nabla_{\dot \gamma} \dot \gamma/{\vert \dot \gamma}\vert , - ) =  q F(\dot \gamma, -)
    \;\; \}
    \subset Conf
    \,,
  $$

  where

  * $\nabla$ denotes the [[covariant derivative]] of the [[Levi-Civita connection]] of the background metric $g$ on the [[tangent bundle]] $T X$ of $X$;

  * the equation is taken to hold in each [[cotangent space]] $T^*_{\gamma(\tau)} X$ for each $\tau \in \mathbb{R}$.

To see what this means, consider some special cases. First regard the case that the background field strength vanishes, $F = 0$. Then the equations of motion reduce to

$$
  \nabla_{\dot \gamma} \dot \gamma = 0
  \,.
$$

This says that the trajectory $\gamma$ exhibits [[parallel transport]] of its tangent vectors with respect to the [[Levi-Civita connection]] of the background metric. These curves are precisely the _[[geodesics]]_ of the background geometry. This models motion under the [[force]] exerted by the field of [[gravity]] on our particle.

In the even more special case that $X$ is [[Minkowski spacetime]], where we may find a global [[coordinate chart]] $(\mathbb{R}^4, \eta) \simeq (X,g)$, these are exactly the straight lines in $\mathbb{R}^4$. Given any such, there is precisely one representative in the diffeomorphism class for which $\mathbb{R} \stackrel{\gamma}{\to} \mathbb{R}^4 \stackrel{x^0}{\to} \mathbb{R}$ is the identity, hence for which the [[worldline]] parameter coincides precisely with the chosen global time coordinate $t := x^0$ on $\mathbb{R}^4$. For these the equations of motions are again those of the free Newtonian particle $\vec a = 0$.

Remaining in the case that $X$ is [[Minkowski space]] but allowing now a nontrivial background field, notice that we may write the 2-form $F$ always as

$$
  F = \vec E_i \cdot d x^i \wedge d t + \vec B^i \epsilon_{i j k} d x^j \wedge d x^k
  \,,
$$

where $\vec E \in \mathbb{R}^3$ is the electric field strength vector and $\vec B \in \mathbb{R}^3$ the magnetic field strength vector. The spatial part of the above equations of motion are in this case again as for a Newtonian particle

$$
  m \vec a = q \vec E + q \vec v \times \vec B
  \,,
$$

where in the second term we have the [[cross product]] of vectors in $\mathbb{R}^3$. The [[force]] on the right is the _[[Lorentz force]]_ exerted by an electromagnetic field on a charged particle.

Notice that the equations of motion imply, generally, that the [[norm]] of $\dot \gamma$ is constant along the trajectory

$$
  \begin{aligned}
    \frac{d}{d \tau} g(\dot \gamma, \dot \gamma)
    &= 2 g(\nabla_{\dot \gamma} \dot \gamma, \dot \gamma)
    \\
    & =
    2 g(g^{-1}(F(\dot \gamma,-), -), \dot \gamma)
    \\
    & \propto F(\dot \gamma, \dot \gamma)
    \\
    & =
    0
  \end{aligned}
   \,.
$$

Therefore a trajectory that solves the equations of motion and whose tangent vector is [[timelike]] or [[spacelike]] or [[lightlike]], respectively, at any instant is so throughout. In particular, no choice of gravitational and electromagnetic background field strength can accelerate a physical particle from being timelike to being light-like.

Experiments around the second half of the 19th and the beginning of the 20th century established that this covariant phase space correctly describes the dynamics of gravitationally and electromagnetically charged relativistic particles. But also formally this phase space is not a randomly chosen space; instead, it is the [[critical locus]] of a (mathematically) natural [[action functional]].

The points in the [[covariant phase space]]

$$
  P := \{ [\gamma] \in Conf |
   m g(\nabla_{\dot \gamma} \dot \gamma / {\vert \dot \gamma}\vert,-) =  q\iota_{\dot \gamma} F
    \}
  \subset Conf
$$

happen to be the local critical points of the [[nonlinear functional|functional]]

$$
  S : Conf \to \mathbb{R}
$$

given by

\[
  \label{ParticleActionForGlobalA}
  \begin{aligned}
    S([\gamma])
     & :=
     S_{kin}([\gamma])
     +
     S_{gauge}([\gamma])
     \\
     &
     :=
     m \int_\Sigma dvol(\gamma^*g)  + q \int_\Sigma \gamma^* A
  \end{aligned}
  \,,
\]

where on the left we have the integral of the [[volume form]] of the pullback $\gamma^* g \in Sym^2 T^* \Sigma$ of the metric on target space to the [[worldline]].

This is called the _[[action functional]]_ of the relativistic particle $\sigma$-model. The first summand is called the _kinetic action_, the second is called the _gauge coupling_  action.

Typically one characterizes $\sigma$-models in terms of such action functionals, so that the [[covariant phase space]] is then given as their [[critical locus]]. This usually yields a simpler and deeper description of the model.

Notably the above action functional has an evident generalization to the case where the background [[electromagnetic field]] is not given by a globally defined 1-form, but more generally by a [[circle n-bundle with connection|circle bundle with connection]] $\nabla$: if we pass to the exponentiated action functional

$$
  \exp(i S(-)) : Conf \to U(1)
$$

$$
  \exp(i S(\gamma)) =
  \exp(S_{kin}(\gamma))
   \;\;
  \exp(i q \int_\Sigma \gamma^* A)
$$

the second factor is precisely the [[holonomy]] of $\nabla$ over the worldline. Hence for general electromagnetic background gauge fields the action functional is (assuming for simplicity now closed curves with $\Sigma = S^1$)

$$
  \exp(i S(\gamma))
  =
  \exp(S_{kin}(\gamma))
   \;\;
  hol(\nabla, \gamma)
  \,.
$$

This is the beginning of an important pattern: most $\sigma$-models are determined by a kind of higher [[gauge field]] $\nabla$ on [[target space]] (a cocycle in the [[differential cohomology]] of [[target space]]) and their dynamics is determined by an [[action functional]] that is the [[higher parallel transport|higher holonomy]] functional of this gauge field.

At the same time the kinetic action functional factor is usually to be understood as part of the [[measure]] on [[configuration space]] $Conf$. For the particle this has been made precise: the [[path integral]]

$$
 \int_{\gamma \in Conf} hol(\nabla,\gamma) \;\;\exp(i S_{kin}(\gamma)) [d \gamma]
  :=
 \int_{\gamma \in Conf} hol(\nabla,\gamma) d \mu_{Wien}
$$

can be interpreted as the [[integral]] with respect to the [[Wiener measure]] on path space (after [[Wick rotation]], at least). The kinetic part of the action functional is then absorbed into the Wiener measure $d \mu_{Wien}$

$$
  \exp(i S_{kin}(\gamma)) [d\gamma] := d \mu_{Wien}
  \,;
$$

(at least after replacing the kinetic [[Nambu-Goto action]] by the classically equivalent [[Polyakov action]])
and the path integral is just the "expectation value" (after Wick rotation) of the holonomy, taken over all trajectories.

Since there is a good general abstract theory of higher gauge fields and their higher holonomies (see [[differential cohomology]] and [[cohesive (infinity,1)-topos|differential cohomology in a cohesive topos]]), this suggests that there should be a general abstract theory of $\sigma$-models. Aspects of this are discussed [below](Exposition).

### The relativistic $(n-1)$-brane
 {#RelativisticNBrane}

It is hard not to consider the following generalization of the relativistic particle $\sigma$-model, that we discussed [above](#RelativisticParticle):

notice that nothing in the structure of the relativistic particle's [[action functional]] (eq:ParticleActionForGlobalA) relies on the [[dimension]] of $\Sigma$ being $1$. Instead, it is just the degree-1 case of the following family of types of classical $\sigma$-models, that make sense for all $n \in \mathbb{N}$:

* let $\Sigma$ be of [[dimension]] $n$;

* let [[target space]] be a [[pseudo-Riemannian manifold]] $(X,g)$ as before;

* let the [[background gauge field]] be given by a smooth [[differential form|differential n-form]] $A \in \Omega^n(X)$.

* let [[configuration space]] be the [[action groupoid|weak quotient]]

  $$
    Conf_\Sigma := C^\infty(\Sigma, X)//Diff(\Sigma)
    \,.
  $$

* let then finally the [[action functional]] be given by

  $$
    S([\gamma]) = \int_\Sigma dvol(\gamma^* g) + \int_\Sigma \gamma^* A
    \,.
  $$

This is the same formula as for the relativistic particle as before, only that now the differential forms are taken to be of degree $n$ and integrals to be over $n$-dimensional spaces.

Moreover, for each $n \in \mathbb{N}$ there is an analog of the generalization

$$
  \{1-forms\} \hookrightarrow \{circle bundles with connection\}
$$

to the generalization

$$
  \{n-forms\} \hookrightarrow \{circle n-bundles with connection\}
  \,.
$$

Given any [[circle n-bundle with connection]] $\nabla$ and closed $\Sigma$ of dimension $n$, there is a [[higher parallel transport|higher]] [[holonomy]] [[nonlinear functional|functional]]

$$
  hol(\nabla, -) : (\Sigma \stackrel{\gamma}{\to} X) \mapsto
  hol(\nabla, \gamma) \in U(1)
$$

that extends the functional $A \mapsto \exp(i \int_\Sigma \gamma^* A)$.

Therefore, generally, we may take for $n \in \mathbb{N}$

* the [[background gauge field]] on $X$ to be a [[circle n-bundle with connection]] $\nabla$;

* the exponentiated [[action functional]] to be

  $$
    [\gamma] \mapsto \exp(i \int_\Sigma dvol(\gamma^* )) \;\;\; hol(\nabla,\gamma)
   \,.
  $$

For $n = 3$ such a $\sigma$-model describes an analog of a relativistic particle which is not point-like, but 2-dimensional (with 3-dimensional trajectory) hence which reminds one of a [[membrane]]. Inspired by this term, the general case has come to be known as the _relativistic $(n-1)$-[[brane]]_.

The case $n = 2$ is called the relativistic [[string]], which we consider in more detail [below](#RelativisticString). This has received a lot of attention (in _[[string theory]]_) not just because it is the next simplest in an infinite hierarchy of cases, but also because its quantum theory turns out to have various interesting features that seem to make it special. Moreover, many of the $(n-1)$-branes for other $n$ re-appear in one way or other in the study of the string (as its boundary [[D-brane]]s in all dimensions $0 \leq n \leq 10$, as its "strongly coupled" version: the [[M-theory membrane]], or as its [[electric-magnetic duality|electric-magnetic dual]]: the [[NS5-brane]]). If nothing else, the seemingly innocent step from $n = 1$ to $n = 2$ in the $\sigma$-model shows that there is a rich pattern of higher dimensional ($\sigma$-model) quantum field theories that are all interrelated in intricate ways.

Another important special case for the general discussion of $\sigma$-models is the case of the [[membrane]], $n = 3$, for which the [[background gauge field]] is a _[[Chern-Simons circle 3-bundle]]_ for some $G$-[[principal bundle]] on $X$, for $G$ some suitable [[Lie group]]. In this case the gauge-coupling [[Lagrangian]] of the $\sigma$-model is, locally, the [[Chern-Simons form]] $CS(\nabla_\mathfrak{g})$ of a $G$-[[connection on a bundle|connection]] $\nabla_{\mathfrak{g}}$, hence the [[action functional]] is (locally) the [[Chern-Simons theory|Chern-Simons functional]]

$$
  (\Sigma \stackrel{\gamma}{\to} X) \mapsto \int_\Sigma CS(\gamma^* \nabla_\mathfrak{g})
  \,.
$$

Below we will see that when $\sigma$-models are considered [[internalization|internal to]] a suitable [[cohesive (∞,1)-topos]], then there are _universal_ $\sigma$-models of this Chern-Simons type, whose [[target space]] is no longer a [[smooth manifold]], but a [[smooth ∞-groupoid]] incarnation of a [[classifying space]] $B G$.

### The relativistic string
 {#RelativisticString}

The important case $n = 2$ of the general [(n-1)-brane sigma-model](RelativisticNBrane) that we considered above is called the [[string]]-$\sigma$-model. Even though this is just the first step after the relativistic particle, the theory of this $\sigma$-model is already considerably richer classically and all the more so after quantization. For the purposes of this exposition here we only briefly indicate the physical interpretation of the $\sigma$-model and then consider some qualitatively new higher [[gauge theory]] aspects, that appear in this dimension.

First notice that by the general reasoning of relativistic $(n-1)$-branes, the [[background gauge field]] is now given (if we assume for the moment a topological trivial class) by a 2-form, which is traditionally denoted $B \in \Omega^2(X)$ and called the _[[B-field]]_. Its 3-form [[curvature]] [[field strength]] is traditionally denoted $H := d B$.

The [[action functional]] of the [[string]]'s $\sigma$-model for a [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[target space]] $(X,g)$ with background gauge field $B$ is

$$
  [\gamma] \mapsto \int_\Sigma dvol(\gamma^* g) + \int_\Sigma \gamma^* B
  \,.
$$

To gain insight into the physical meaning of this, consider the simple case that [[target space]] $(X,g)$ is [[Minkowski spacetime]] and that the [[worldsheet]] $\Sigma = \mathbb{R} \times S^1$ is the [[cylinder]]. With $(\tau,\sigma)$ the two canonical [[coordinate]]s on $\Sigma$, we still write

$$
  \dot \gamma := \partial_\tau \gamma
$$

for the derivative "along the trajectory" (along the $\mathbb{R}$-factor), but now we also have the derivative $\partial_\sigma \gamma$ which we may think of as being _tangential to the string_ at any instant of its trajectory. A field configuration $\gamma : \Sigma \to X$ may be thought of as the trajectory of a [[circle]] propagating in $X$.

The critical trajectories $\gamma : \Sigma \to X$ are found to be those that satisfy the 2-dimensional [[wave equation]]

$$
  g(\ddot \gamma,-)  - g(\partial_\sigma^2 \gamma,-)
  =
  H(\dot \gamma, \partial_\sigma \gamma,-)
$$

on the [[worldsheet]]. Comparison with the equation of motion of the  [relativistic particle](#RelativisticParticle) shows that $H(\partial_\sigma \gamma, -,-)$ plays the role of an electromagnetic [[field strength]] 2-form.  Hence the string behaves as if electric [[charge]] is spread out evenly along it.

For _point particle limit configurations_  $\gamma$, where the string has vanishing extension in that $\partial_\sigma \gamma = 0$, the above equation reduces again to free motion

$$
  \ddot \gamma = \vec a = 0
$$

and for general $(X,g)$ to the corresponding [[geodesic]] motion.

Therefore close to these point particle configurations the string looks like a little oscillating loop whose dynamics is that of its "center of mass" point, but slightly modified by the energy in the oscillations and the way these interact with the background fields. After quantization of the $\sigma$-model, these oscillations have a discrete ( quantized!) set of possible frequencies, and indeed each of the oscillation modes makes the string appear in the point particle limit as one species or other of a relativistic particle. (For more on this see _[[string theory]]_.)

Next we have a look at aspects of higher [[gauge theory]] that appears in $n = 2$.

The above 2-form $B$ is in general just the local connection form of a _[[circle n-bundle with connection|circle 2-bundle with connection]]_ $\nabla$ on $X$, given (as its [[homotopy fiber]]) by a morphism of [[smooth ∞-groupoid]]s $\alpha : X \to \mathbf{B}^2 U(1)$. Equivalently this is a $U(1)$-[[bundle gerbe]] with connection.

There is a canonical 1-dimensional [[infinity-representation|2-representation]] of the [[circle n-group|circle 2-group]] $\mathbf{B} U(1)$ on [[2-vector space]]s:

$$
  \rho : \mathbf{B} \mathbf{B}U(1) \to 2 Vect
  \,.
$$

Hence the corresponding [[associated infinity-bundle|associated 2-bundle]] is classified by a morphism

$$
  \rho(g) : X \stackrel{g}{\to} \mathbf{B}^2 U(1)
   \stackrel{\rho}{\to}
  2 Vect
  \,.
$$

One can consider the string $\sigma$-model for [[worldsheet]]s with [[boundary]]. A careful analysis then shows that the consistent Dirichlet-type boundary conditions that can be added correspond, roughly,  to certain subspaces of [[target space]] -- called [[D-brane]]s -- that are equipped with a section $V : \mathbf{1} \to \rho(g)|_{D-brane}$ of the background gauge field
[[n-vector bundle|2-vector bundle]] restricted to the $D$-brane. Such a section is precisely a [[twisted bundle|twisted vector bundle]] on the brane, where the twist is the class in [[integral cohomology]] $H^3(X, \mathbb{Z})$ of the background gauge field. More generally, these twisted bundles are cocycles in [[twisted K-theory]] and [[differential K-theory]]. Hence more [[differential cohomology]] appears on the target space for the string in the presence of string boundaries.

More generally, the structure [[2-group]] of the background [[principal 2-bundle]] need not be $\mathbf{B}U(1)$, which is given by the [[crossed module]] $[U(1) \to 1]$. Instead, it can be the [[automorphism 2-group]] $AUT(U(1))$, which is given by the crossed module $(U(1) \to Aut(U(1)) \simeq \mathbb{Z}_2)$. An $AUT(U(1))$-[[principal 2-bundle]] on $X$ is equivalently a double cover of $X$, equipped with a circle 2-bundle that has a twisted [[equivariant cohomology|equivariance]] under the $\mathbb{Z}_2$-[[action]].  Such a background gauge field structure is called a _string [[orientifold]]_ background. This is a kind of [[higher category theory|higher structure]] that the relativistic particle alone cannot see.

More such higher structure appears as one passes to the [[supergeometry]] analogs of the $\sigma$-models that we have considered so far: the [[superstring]]. The presence of the additional [[fermion]] fields that this brings with it (both on [[target space]] as well as on the [[worldsheet]]) influences all the structures that we have considered so far. For instance, a phenonemon called a fermionic [[quantum anomaly]] forces the above background [[circle n-bundle|circle 2-bundle]] to become a [[twisted cohomology|twisted 2-bundle]], where the twist is given by a [[fivebrane]] [[charge]] [[Chern-Simons circle 3-bundle]]. This is discussed in detail at _[[differential string structure]]_

These are the first examples of a general phenomenon: as $n$ increases, a background gauge $n$-bundle with connection may constitute considerably more structure then one might naively expect from a generalization of the ordinary notion of a connection. More examples of this phenomenon arise when we allow our [[target space]]s to be general [[smooth ∞-groupoid]]s, below.

## Related concepts

* [[harmonic map]]