renormalization - contents

**[[renormalization]]**

**[[effective quantum field theory]]**

* [[renormalization scheme]]

* [[counterterm]]

* [[renormalization group flow]]


Renormalization and Effective Field Theory

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#### Physics
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This entry provides hyperlinks for the book

* [[Kevin Costello]],

  _Renormalization and effective field theory_

  Mathematical Surveys and Monographs, Volume 170

  American Mathematical Society (2011)

  ([AMS website](http://www.ams.org/publications/authors/books/postpub/surv-170), [book pdf](https://web.archive.org/web/20160304134950/http://math.northwestern.edu/~costello/renormalization_book.pdf), [talk slides](http://www.math.northwestern.edu/~costello/slides.pdf))

on [[renormalization]] and [[BV-quantization]] of [[effective quantum field theory]] of those [[quantum field theories]] whose [[action functional]] can be brought into "[BV-CS form](renormalization#OfTheoriesInBVForm)" $S(\phi) = \langle \phi, Q \phi\rangle + I(\phi)$.

Much of the content is based on the article

* [[Kevin Costello]], _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv:0706.1533](http://arxiv.org/abs/0706.1533))

which focuses on [[Chern-Simons theory]], as well as

* [[Kevin Costello]], _Topological conformal field theories and gauge theories_ ([arXiv:math/0605647](http://arxiv.org/abs/math/0605647))

which focuses on [[Yang-Mills theory]].

The methods of the book are designed to allow to construct [[quantum field theories]] in the form of [[factorization algebras of observables]] (see there for more details) as discussed in

* [[Kevin Costello]], [[Owen Gwilliam]], _[[Factorization algebras in perturbative quantum field theory]]_

#Contents#
* table of contents
{:toc}

## 1) Introduction

(...)

## 2) Theories, Lagrangians and counterterms

* [[theory (physics)]]

* [[prequantum field theory]]

* [[Lagrangian]]

## 3) Field theories on $\mathbb{R}^n$

* [[free field theory]]

## 4) Renormalizability

* [[renormalization]]

## 5) Gauge theory and the Batalin-Vilkovisky formalism

* [[gauge theory]]

* [[BV-BRST formalism]]

## 6) Renormalizability of Yang-Mills theory

* [[Yang-Mills theory]]

## Appendix: Asymptotics of graph integrals

* [[Feynman diagram]]

* [[path integral]]

category: reference


renormalization condition

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#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]], [[renormalization|("re"-)normalization]] is an [[extension of distributions|extension]] of [[time-ordered products]], uniquely defined away from coinciding spacetime arguments, to the locus of coinciding spacetime arguments. Before the extension these [[time-ordered products]] may generally enjoy further properties beyond the [[causal factorization]] that characterizes them as time-ordered products. Not all these properties necessarily need to be preserved by a choice of [[extension of distributions]] to coinciding spacetime arguments. A _renormalization condition_ is the condition that some such properties ought to be preserved by the choice of [[renormalization|("re-")normalization]].

The [[main theorem of perturbative renormalization]] holds true if at least the renormalization conditions "field independence" and "translation equivariance" are imposed. Depending on how many other renormalization conditions are imposed, the resulting [[Stückelberg-Petermann renormalization group]] becomes smaller. ([Duetsch 18, remark 3.102](#Duetsch18))

Besides "field indepence" and "translation invariance" a key renormalization condition for [[gauge theories]] is the _[[quantum master equation]]_ or quivalently _[[master Ward identity]]_. If this renormalization condition cannot be satisfied, one says that the corresponding [[perturbative QFT]] does not exist due to a "[[gauge anomaly]]". ([Dütsch 18, section 4.2](#Duetsch18))

## Definition

For the moment see at _[[S-matrix]]_ [this definition](S-matrix#RenormalizationConditions).

## Examples

For the moment see at _[[S-matrix]]_ [this prop.](S-matrix#BasicConditionsRenormalization)

## Related concepts

* [[protection from quantum corrections]]

## References

* {#Duetsch18} [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects renormalization conditions]]


renormalization group

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#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theories]] various concepts of "renormalization groups" describe the choices of [[renormalization|("re"-)normalization]] and their behaviour under [[scaling transformations]] or choices of cutoffs.

There are at least three different concepts referred to as "the renormalization group", only the first is in general really a [[group]]:

1. the [[Stückelberg-Petermann renormalization group]] ([Stückelberg-Petermann 53](#StueckelbergPetermann53), historically the origin of the concept)

   this is literally the [[group]] of _re-normalizations_, whose elements relate any two given [[renormalization scheme|normalization schemes]] $\mathcal{S}$ and $\mathcal{S}'$ by [[precomposition]] with a transformation $\mathcal{Z}$ of the space of [[local observable|local]] [[interaction]] [[action functionals]];

1. [[renormalization group flow]], say along [[scaling transformations]] yielding the [[Gell-Mann-Low renormalization cocycle]]

1. the [[Wilsonian RG]] of [[effective quantum field theories]] defined with a [[UV cutoff]].

(e.g. [Brunetti-Dütsch-Fredenhagen 09, p. 10](#BrunettiDuetschFredenhagen09))

In more detail:

Let

$$
  vac
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}_{kin}, \Delta_H )
$$

be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to [this def.](S-matrix#VacuumFree)) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]].

Then a [[perturbative S-matrix]] scheme/[[renormalization scheme|("re"-)normalization scheme]] around this vacuum ([this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering)) is a function

$$
  \array{
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j \rangle
      & \overset{\mathcal{S}_{vac}}{\longrightarrow} &
    PolyObs(E_{\text{BV-BRST}})_{mc}( ( \hbar ) )[ [ g, j ] ]
    \\
    g S_{int} + j A
     &\mapsto&
    \mathcal{S}_{vac}(g S_{int} + j A)
  }
$$

from [[local observables]], regared as [[adiabatic switching|adiatically switched]] [[interaction]] [[action functionals]]  to [[Wick algebra]]-elements $\mathcal{S}( g S_{int} + j A)$, encoding [[scattering amplitudes]] in the given vacuum $\mathbf{L}'$ for the given interaction $g S_{int} + j A$, with formal parameters adjoined as indicated.

The _[[Stückelberg-Petermann renormalization group]]_ is a group of transformations

$$
  \array{
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g, j \rangle
      &\overset{Z}{\longrightarrow}&
    LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g, j \rangle
    \\
    g S_{int} + J A
      &\mapsto&
    \mathcal{Z}(g S_{int} + j A)
  }
$$

such that for $\mathcal{S}$ and $\mathcal{S}'$ two [[renormalization schemes|normalization schemes]]/[[S-matrix]] schemes, there is a unique $\mathcal{Z}$ relating them by [[precomposition]], in that

$$
  \label{SMatricesRelatedBySPRenormalizationGroupElement}
  \mathcal{S}(g S_{int} + j A)
    \;=\;
  \mathcal{S}'\left( \mathcal{Z}(g S_{int} + j A) \right)
$$

for all $g S_{int} + j A$. This is the _[[main theorem of perturbative renormalization]]_. Hence this says that any two ways of choosing [[interactions]] at coincident interaction points are related by a re-definition of the original [[interaction]] $g S_{int} + j A$.

Now it may happen that

1. the [[free field]] [[vacuum]] $vac = vac(m)$ depends on a [[mass]] parameter, and with it the choice $\mathcal{S}_{vac(m)}$ of [[renormalization scheme|normalization scheme]],

1. under [[scaling transformations]] on [[local observables]] $\sigma_\rho$ ([Dütsch 18, def. 3.19](#Duetsch18))  we have that with $\mathcal{S}_{vac(m)}$ a [[perturbative S-matrix]] scheme perturbing around $vac(m)$ also

   $$
     \sigma_\rho \circ \left(\mathcal{S}_{vac(m/\rho)}\right) \circ \sigma_\rho^{-1}
   $$

   is a perturbative S-matrix around $L_{kin}(m)$.

In this case the above statement of the [[main theorem of perturbative renormalization]] implies with (eq:SMatricesRelatedBySPRenormalizationGroupElement) that there exists a unique transformation $\mathcal{Z}^m_\rho$ of the space of [[local observable|local]] [[interaction]] [[action functionals]] such that

$$
  \begin{aligned}
    &
    \sigma_\rho
      \circ
    \mathcal{S}_{vac(m/\rho)}
      \circ
    \sigma_\rho^{-1}( g S_{int} + j A )
    \\
    & =
    \mathcal{S}_{vac(m)}(\mathcal{Z}^m_\rho(g S_{int} + j A))
  \end{aligned}
$$

for all $g S_{int} + j A$.

These $\mathcal{Z}^m_\rho$ are the _[[Gell-Mann-Low cocycle]]_ elements. They do not actually form a [[group]], unless $m = 0$, but satisfy the relation

$$
  \mathcal{Z}^m_{\rho_1 \rho_2}
  \;=\;
  \mathcal{Z}^m_{\rho_1}
    \circ
  \left(
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_2}
  \right)
$$

([Brunetti-Dütsch-Fredenhagen 09 (69)](#BrunettiDuetschFredenhagen09), [Dütsch 18 (3.325)](#Duetsch18))

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

From the definition we have

$$
  \begin{aligned}
    \mathcal{S}_{vac(m)}
      \circ
    \mathcal{Z}^m_{\rho_1 \rho_2}
    & =
    \sigma_{\rho_1}
      \circ
    \underset{
      \mathcal{S}_{vac(m/\rho_1)}
        \circ
      \mathcal{Z}^{m/\rho_1}_{\rho_2}
    }{
    \underbrace{
      \sigma_{\rho_2}
        \circ
      \mathcal{S}_{vac(m/\rho_1\rho_2)}
        \circ
      \sigma_{\rho_2}^{-1}
    }}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \underset{
      =
      \mathcal{S}_{vac(m)}
        \circ
      \mathcal{Z}^m_{\rho_1}
        \circ
      \sigma_{\rho_1}
    }{
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{S}_{vac(m/\rho_1)}
        \circ
      \overset{ = id }{
        \overbrace{
          \sigma_{\rho_1}^{-1}
            \circ
          \sigma_{\rho_1}
        }
    }
    }}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \mathcal{S}_{vac(m)}
      \circ
    \mathcal{Z}^m_{\rho_1}
      \circ
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}^{m/\rho_1}_{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
  \end{aligned}
$$

To conclude, it is now sufficient to see that the perturbative S-matrix $S_{vac(m)}$, as a function form [[interaction]] [[Lagrangian densities]] to [[Wick algebra]]-elements, is an [[injective function]]. (...)

=--

## Related concepts

* [[renormalization]]

* [[renormalization group flow]]

* [[asymptotic safety]]

* [[cosmic Galois group]]

## References

The [[Stückelberg-Petermann renormalization group]] is due to

* {#StueckelbergPetermann53} [[Ernst Stückelberg]], [[André Petermann]], _La normalisation des constantes dans la theorie des quanta_, Helv. Phys. Acta 26 (1953), 499–520

The relation of the [[Stückelberg-Petermann renormalization group]] to [[renormalization group flow]] ([[Gell-Mann-Low renormalization cocycles]])

* {#GellMannLow54} [[Murray Gell-Mann]] and F. E. Low, _Quantum Electrodynamics at Small Distances_, Phys. Rev. 95 (5) (1954), 1300–1312 ([pdf](http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf))

as well as to [[Wilsonian RG]] of [[effective quantum field theories]] is due to

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

* {#DuetschFredenhagenKellerRejzner14} [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy.
55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], section 3.5.1 of _[[From classical field theory to perturbative quantum field theory]]_, 2018

See also

* Kiyoshi Higashijima, Kazuhiko Nishijima, _Renormalization Groups of Gell-Mann and  Low and of Callan  and Symanzik_, Progress in theoretical physics, vol. 64, no. 6, December 1980 ([pdf](https://academic.oup.com/ptp/article-pdf/64/6/2179/5275368/64-6-2179.pdf))

* {#Shomer07} Assaf Shomer, _A pedagogical explanation for the non-renormalizability of gravity_ ([arXiv:0709.3555](https://arxiv.org/abs/0709.3555))

* Alessandro Giuliani, Vieri Mastropietro, Slava Rychkov, _Gentle introduction to rigorous Renormalization Group: a worked fermionic example_ ([arXiv:2008.04361](https://arxiv.org/abs/2008.04361))

* Wikipedia _[Renormalization group](http://en.wikipedia.org/wiki/Renormalization_group)_

[[!redirects renormalization groups]]

[[!redirects renormalisation group]]
[[!redirects renormalisation groups]]

[[!redirects exact renormalization group]]
[[!redirects exact renormalization groups]]

[[!redirects exact renormalisation group]]
[[!redirects exact renormalisation groups]]


renormalization group flow

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#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] the construction of the [[scattering matrix]] $\mathcal{S}$, hence of the [[interacting field algebra of observables]] for a given [[interaction]] $g S_{int}$ [[perturbation theory|perturbing]] around a given [[free field theory|free field]] [[vacuum]], involves choices of _normalization_ of [[time-ordered products]]/[[Feynman diagrams]] (traditionally called _[[renormalization|"re"-normalizations]]_) encoding new [[interactions]] that appear where several of the original interaction vertices defined by $g S_{int}$ coincide.

Whenever a [[group]] $RG$ [[action|acts]] on the space of [[observables]] of the theory such that [[conjugation]] by this action takes [[renormalization scheme|("re"-)normalization schemes]] into each other, then these  choices of [[renormalization|("re"-)normalization]]  are parameterized by -- or "flow with" -- the elements of $RG$.  This is called _renormalization group flow_  (prop.
\ref{FlowRenormalizationGroup} below); often called _RG flow_, for short.

The archetypical example here is the [[group]] $RG$ of [[scaling transformations]] on [[Minkowski spacetime]] (def. \ref{ScalingTransformations} below), which induces a [[renormalization group flow]] (prop. \ref{RGFlowScalingTransformations} below) due to the particular nature of the [[Wightman propagator]] resp. [[Feynman propagator]] on [[Minkowski spacetime]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below). In this case the choice of [[renormalization|("re"-)normalization]] hence "flows with scale".

Now the _[[main theorem of perturbative renormalization]] states that (if only the basic [[renormalization condition]] called "field independence" is satisfied) any two choices of [[renormalization scheme|("re"-)normalization schemes]] $\mathcal{S}$ and $\mathcal{S}'$ are related by a unique [[interaction vertex redefinition]] $\mathcal{Z}$, as

$$
  \mathcal{S}' = \mathcal{S} \circ \mathcal{Z}
  \,.
$$

Applied to a parameterization/flow of renormalization choices by a group $RG$ this hence induces an [[interaction vertex redefinition]] as a function of $RG$. One may think of the shape of the interaction vertices as fixed and only their ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] as changing under such an [[interaction vertex redefinition]], and hence then one has [[coupling constants]] $g_j$ that are parameterized by elements $\rho$ of $RG$:

$$
  \mathcal{Z}_{\rho_{vac}}^\rho
  \;\colon\;
  \{g_j\}
    \mapsto
  \{g_j(\rho)\}
$$

This dependendence is called _running of the coupling constants_ under the renormalization group flow (def. \ref{CouplingRunning} below).

One example of [[renormalization group flow]] is that induced by [[scaling transformations]] (prop. \ref{RGFlowScalingTransformations} below). This is the original and main example of the concept ([Gell-Mann & Low 54](#GellMannLow54))

In this case the [[running of the coupling constants]] may be understood as expressing how "more" [[interactions]] (at higher energy/shorter [[wavelength]]) become visible (say to [[experiment]]) as the scale resolution is increased. In this case the dependence of the coupling $g_j(\rho)$ on the parameter $\rho$ happens to be [[differentiable function|differentiable]]; its [[logarithm|logarithmic]] [[derivative]] (denoted "$\psi$" in [Gell-Mann & Low 54](#GellMannLow54)) is known as the _[[beta function]]_ ([Callan 70](#Callan70), [Symanzik 70](#Symanzik70)):

$$
  \beta(g) \coloneqq \rho \frac{\partial g_j}{\partial \rho}
  \,.
$$

Notice that this is related to, but conceptually different from, _[[Polchinski's flow equation]]_ in the context of [[Wilsonian RG]].

The [[running of the coupling constants]] is not quite a [[representation]] of the [[renormalization group flow]],
but it is a "twisted" representation, namely a [[group cocycle|group 1-cocycle]] (prop. \ref{CocycleRunningCoupling} below). For the case of [[scaling transformations]] this may be called the _[[Gell-Mann-Low renormalization cocycle]]_ ([Brunetti-Dütsch-Fredenhagen 09](#BrunettiDuetschFredenhagen09)).

For more see at

* [[geometry of physics -- A first idea of quantum field theory]] the section _[Renormalization](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#Renormalization)_

## Definition

+-- {: .num_prop #FlowRenormalizationGroup}
###### Proposition
**([[renormalization group flow]])**

Let

$$
  vac
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
$$

be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to [this def.](S-matrix#VacuumFree)) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]].

Consider a [[group]] $RG$ equipped with an [[action]] on the [[Wick algebra]] of [[off-shell]] [[microcausal polynomial observables]] with formal parameters adjoined (as in [this def.](S-matrix#FormalParameters))

$$
  rg_{(-)}
  \;\colon\;
  RG \times PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ g, j ] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{mc}((\hbar))[ [ \hbar, g, j ] ]
  \,,
$$

hence for each $\rho \in RG$ a [[continuous linear map]] $rg_\rho$ which has an [[inverse]] $rg_\rho^{-1} \in RG$ and is a [[homomorphism]] of the [[Wick algebra]]-product (the [[star product]] $\star_H$ induced by the [[Wightman propagator]] of the given vauum $vac$)

$$
  rg_\rho( A_1 \star_H A_2 )
  \;=\;
  rg_\rho(A_1) \star_H rg_\rho(A_2)
$$

such that the following conditions hold:

1. the action preserves the subspace of [[off-shell]] polynomial [[local observables]], hence it [[restriction|restricts]] as

   $$
     rg_{(-)}
     \;\colon\;
     RG \times LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
     \longrightarrow
     LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j ] ]\langle g,j\rangle
   $$

1. the action respects the [[causal order]] of the spacetime support ([this def.](A+first+idea+of+quantum+field+theory#SpacetimeSupport)) of local observables, in that for $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]$ we have

   $$
     \left(
       supp(O_1)
       \,{\vee\!\!\!\wedge}\,
       supp(O_2)
     \right)
     \phantom{A}
       \Rightarrow
     \phantom{A}
     \left(
       supp(rg_\rho(O_1))
       \,{\vee\!\!\!\wedge}\,
       supp(rg_\rho(O_2))
     \right)
   $$

   for all $\rho \in RG$.

Then:

The operation of [[conjugation]] by this action on [[observables]] induces an [[action]] on the [[set]] of [[S-matrix]] [[renormalization schemes]] ([this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering), [this remark](S-matrix#calSFunctionIsRenormalizationScheme)),
in that for

$$
  \mathcal{S}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
$$

a perturbative [[S-matrix scheme]] around the given [[free field theory|free field]] [[vacuum]] $vac$, also the [[composition|composite]]

$$
  \mathcal{S}^\rho
  \;\coloneqq\;
  rg_\rho \circ \mathcal{S} \circ rg_{\rho}^{-1}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar , g, j] ]
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})( (\hbar) )[ [ g, j] ]
$$

is an [[S-matrix]] scheme, for all $\rho \in RG$.

More generally, let

$$
  vac_\rho
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}'_\rho, \Delta_{H,\rho} )
$$

be a collection of [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]] parameterized by elements $\rho \in RG$,
all with the same underlying [[field bundle]]; and
consider $rg_\rho$ as above, except that it is not an [[automorphism]] of any [[Wick algebra]],
but an [[isomorphism]] between the [[Wick algebra]]-structures on various vacua, in that

$$
  \label{IntertwiningWickProductsActionRG}
  rg_{\rho}( A_1 \star_{H, \rho^{-1} \rho_{vac}} A_2 )
  \;=\;
  rg_{\rho}(A_1) \star_{H, \rho_{vac}} rg_{\rho}(A_2)
$$

for all $\rho, \rho_{vac} \in RG$

Then if

$$
  \{ \mathcal{S}_{\rho} \}_{\rho \in RG}
$$

is a collection of [[S-matrix schemes]], one around each of the [[gauge fixing|gauge fixed]] [[free field theory|free field]] [[vacua]]
$vac_\rho$, it follows that for all pairs of group elements $\rho_{vac}, \rho \in RG$ the [[composition|composite]]

$$
  \label{RGConjugateSmatrix}
  \mathcal{S}_{\rho_{vac}}^\rho
  \;\coloneqq\;
  rg_\rho \circ \mathcal{S}_{\rho^{-1}\rho_{vac}} \circ rg_\rho^{-1}
$$

is an [[S-matrix scheme]] around the vacuum labeled by $\rho_{vac}$.

Since therefore each element $\rho \in RG$ in the [[group]] $RG$ picks a different choice of [[renormalization|normalization]]
of the [[S-matrix]] scheme around a given vacuum at $\rho_{vac}$, we call the assignment $\rho \mapsto \mathcal{S}_{\rho_{vac}}^{\rho}$
a _[[renormalization group flow|re-normalization group flow]]_.

=--

([Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1](#BrunettiDuetschFredenhagen09), [Dütsch 18, section 3.5.3](#Duetsch18))

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

It is clear from the definition that each $\mathcal{S}^{\rho}_{\rho_{vac}}$ satisfies the axiom "perturbation" (in [this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering)).

In order to verify the axiom "[[causal additivity]]",
observe, for convenience, that by [this prop.](S-matrix#CausalFactorizationAlreadyImpliesSMatrix) it is sufficient to check [[causal factorization]].

So consider $O_1, O_2 \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]\langle g,j\rangle$
two local observables whose spacetime support is in [[causal order]].

$$
  supp(O_1)
  \;{\vee\!\!\!\wedge}\;
  supp(O_2)
  \,.
$$

We need to show that the

$$
  \mathcal{S}_{\rho_{vac}}^{\rho}(O_1 + O_2)
  =
  \mathcal{S}_{\rho_{vac}}^\rho(O_1)
    \star_{H,\rho_{vac}}
  \mathcal{S}_{vac_e}^\rho(O_2)
$$

for all $\rho, \rho_{vac} \in RG$.

Using the defining properties of $rg_{(-)}$ and the [[causal factorization]] of $\mathcal{S}_{\rho^{-1}\rho_{vac}}$ we directly compute as follows:

$$
  \begin{aligned}
    \mathcal{S}_{\rho_{vac}}^\rho(O_1 + O_2)
    & =
    rg_\rho
      \circ
    \mathcal{S}_{\rho^{-1} \rho_{vac}}
      \circ
    rg_\rho^{-1}( O_1 + O_2 )
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1}\rho_{vac}}
      \left(
        rg_\rho^{-1}(O_1) + rg_\rho^{-1}(O_2)
      \right)
      {\, \atop \,}
    \right)
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \left(
        \mathcal{S}_{\rho^{-1}\rho_{vac}}\left(rg_\rho^{-1}(O_1)\right)
      \right)
        \star_{H, \rho^{-1} \rho_{vac}}
      \left(
        \mathcal{S}_{ \rho^{-1} \rho_{vac} }\left(rg_\rho^{-1}(O_2)\right)
      \right)
      {\, \atop \,}
    \right)
    \\
    & =
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1} \rho_{vac}}\left(rg_{\rho^{-1}}(O_1)\right)
      {\, \atop \,}
    \right)
    \star_{H, \rho_{vac}}
    rg_\rho
    \left(
      {\, \atop \,}
      \mathcal{S}_{\rho^{-1} \rho_{vac}}\left( rg_\rho^{-1}(O_2)\right)
      {\, \atop \,}
    \right)
    \\
    & =
    \mathcal{S}^\rho_{\rho_{vac}}( O_1 )
      \,
      \star_{H, \rho_{vac}}
      \,
     \mathcal{S}_{\rho_{vac}}^\rho(O_2)
    \,.
  \end{aligned}
$$

=--

+-- {: .num_defn #CouplingRunning}
###### Definition
**([[running coupling constants]])**

Let

$$
  vac
  \coloneqq
  vac_e
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )
$$

be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] (according to [this def.](S-matrix#VacuumFree)) around which we consider [[interacting field theory|interacting]] [[perturbative QFT]], let $\mathcal{S}$ be an [[S-matrix]] scheme
around this vacuum and let $rg_{(-)}$ be a [[renormalization group flow]] according to prop. \ref{FlowRenormalizationGroup}, such that each re-normalized [[S-matrix scheme]] $\mathcal{S}_{vac}^\rho$ satisfies the [[renormalization condition]] "field independence".

Then by the [[main theorem of perturbative renormalization]] ([this prop.](Stückelberg-Petermann+renormalization+group#AnyTwoSMatrixRenormalizationSchemesDifferByAUniqueVertexRedefinition)) there is for every [[pair]] $\rho_1, \rho_2 \in RG$ a unique [[interaction vertex redefinition]]

$$
  \mathcal{Z}_{\rho_{vac}}^{\rho}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
    \longrightarrow
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g, j] ]
$$

which relates the corresponding two [[S-matrix]] schemes via

$$
  \label{SMatrixScemesRelatedByRunningFunction}
  \mathcal{S}_{\rho_{vac}}^{\rho}
  \;=\;
  \mathcal{S}_{\rho_{vac}}
    \circ
  \mathcal{Z}_{\rho_{vac}}^\rho
  \,.
$$

If one thinks of an [[interaction]] vertex, hence a [[local observable]] $g S_{int}+ j A$, as specified by the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j \in C^\infty_{cp}(\Sigma)\langle g \rangle$ multiplying the corresponding [[interaction]] [[Lagrangian densities]] $\mathbf{L}_{int,j} \in \Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})$ as

$$
  g S_{int}
  \;=\;
  \underset{j}{\sum} \tau_\Sigma \left( g_j \mathbf{L}_{int,j} \right)
$$

(where $\tau_\Sigma$ denotes [[transgression of variational differential forms]]) then $\mathcal{Z}_{\rho_1}^{\rho_2}$ exhibits a dependency of the ([[adiabatic switching|adiabatically switched]]) [[coupling constants]] $g_j$ of the [[renormalization group flow]] parameterized by $\rho$. The corresponding functions

$$
  \mathcal{Z}_{\rho_{vac}}^{\rho}(g S_{int})
    \;\colon\;
   (g_j)
     \mapsto
   (g_j(\rho))
$$

are then called _[[running coupling constants]]_.

=--

([Brunetti-Dütsch-Fredenhagen 09, sections 4.2, 5.1](#BrunettiDuetschFredenhagen09), [Dütsch 18, section 3.5.3](#Duetsch18))

## Properties

+-- {: .num_prop #CocycleRunningCoupling}
###### Proposition
**([[running coupling constants]] are  [[group cocycle]] over [[renormalization group flow]])**

Consider [[running coupling constants]]

$$
   \mathcal{Z}_{\rho_{vac}}^{\rho}
   \;\colon\;
   (g_j)
     \mapsto
   (g_j(\rho))
$$

as in def. \ref{CouplingRunning}. Then for all $\rho_{vac}, \rho_1, \rho_2 \in RG$ the following equality is satisfied
by the "running functions" (eq:SMatrixScemesRelatedByRunningFunction):

$$
  \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
  \;=\;
  \mathcal{Z}_{\rho_{vac}}^{\rho_1}
    \circ
  \left(
    \sigma_{\rho_1}
      \circ
    \mathcal{Z}_{\rho^{-1} \rho_{vac}}^{\rho_2}
      \circ
    \sigma_{\rho_1}^{-1}
  \right)
  \,.
$$

=--

([Brunetti-Dütsch-Fredenhagen 09 (69)](#BrunettiDuetschFredenhagen09), [Dütsch 18, (3.325)](#Duetsch18))

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

Directly using the definitions, we compute as follows:

$$
  \begin{aligned}
    \mathcal{S}_{\rho_{vac}}
      \circ
    \mathcal{Z}_{\rho_{vac}}^{\rho_1 \rho_2}
    & =
    \mathcal{S}_{\rho_{vac}}^{\rho_1 \rho_2 }
    \\
    & =
    \sigma_{\rho_1}
      \circ
    \underset{
      =
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
        =
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}
        \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_vac}^{\rho_2}
    }{
    \underbrace{
      \sigma_{\rho_2}
        \circ
      \mathcal{S}_{\rho_2^{-1}\rho_1^{-1}\rho_{vac}}
        \circ
      \sigma_{\rho_2}^{-1}
    }}
      \circ
    \sigma_{\rho_1}^{-1}
    \\
    & =
    \underset{
      =
      \mathcal{S}_{\rho_{vac}}
        \circ
      \mathcal{Z}_{\rho_{vac}}^{\rho_1}
        \circ
      \sigma_{\rho_1}
    }{
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{S}_{\rho_1^{-1} \rho_{vac}}
        \circ
      \overset{ = id }{
        \overbrace{
          \sigma_{\rho_1}^{-1}
            \circ
          \sigma_{\rho_1}
        }
    }
    }}
      \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
      \circ
      \sigma_{\rho_1}^{-1}
    \\
    & =
    \mathcal{S}_{\rho_{vac}}
      \circ
    \mathcal{Z}_{\rho_{vac}}^{\rho_1}
      \circ
    \underbrace{
      \sigma_{\rho_1}
        \circ
      \mathcal{Z}_{\rho_1^{-1} \rho_{vac}}^{\rho_2}
        \circ
      \sigma_{\rho_1}^{-1}
    }
  \end{aligned}
$$

This demonstrates the equation between vertex redefinitions to be shown after [[composition]] with an S-matrix scheme. But by the uniqueness-clause in the [[main theorem of perturbative renormalization]] the composition operation $\mathcal{S}_{\rho_{vac}} \circ (-)$ as a function from [[vertex redefinitions]] to S-matrix schemes is [[injective function|injective]]. This implies the equation itself.

=--

## Examples

### Scaling transformations
 {#ScalingTransformationsRGFlow}

We discuss (prop. \ref{RGFlowScalingTransformations} below) that, if the field species involved have well-defined [[mass dimension]] (example \ref{ScalarFieldMassDimensionOnMinkowskiSpacetime} below) then [[scaling transformations]] on [[Minkowski spacetime]] (example \ref{ScalingTransformations} below) induce a [[renormalization group flow]] (def. \ref{FlowRenormalizationGroup}). This is the original and main example of [[renormalization group flows]] ([Gell-Mann& Low 54](#GellMannLow54)).

+-- {: .num_example #ScalingTransformations}
###### Example
**([[scaling transformations]] and [[mass dimension]])**

Let

$$
  E \overset{fb}{\longrightarrow} \Sigma
$$

be a [[field bundle]] which is a [[trivial vector bundle]] over [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$.

For $\rho \in (0,\infty) \subset \mathbb{R}$ a [[positive real number]],
write

$$
  \array{
    \Sigma
      &\overset{\rho}{\longrightarrow}&
    \Sigma
    \\
    x &\mapsto& \rho x
  }
$$

for the operation of multiplication by $\rho$ using the [[real vector space]]-[[structure]] of the [[Cartesian space]] $\mathbb{R}^{p+1}$ underlying [[Minkowski spacetime]].

By [[pullback of differential forms|pullback]] this acts on [[field histories]] ([[sections]] of the [[field bundle]]) via

$$
  \array{
    \Gamma_\Sigma(E)
      &\overset{\rho^\ast}{\longrightarrow}&
    \Gamma_\Sigma(E)
    \\
    \Phi &\mapsto& \Phi(\rho(-))
  }
  \,.
$$

Let then

$$
  \rho
    \mapsto
  vac_\rho
    \;\coloneqq\;
  (E_{\text{BV-BRST}}, \mathbf{L}'_{\rho}, \Delta_{H,\rho} )
$$

be a 1-parameter collection of [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to [this def.](S-matrix#VacuumFree), and consider a decomposition into a set $Spec$ of field species ([this def.](S-matrix#VerticesAndFieldSpecies)) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] $\Delta_{F,\rho,sp}$ for that species _scales homogeneously_ in that there exists

$$
  dim(sp) \in \mathbb{R}
$$

such that for all $\rho$ we have (using [[generalized functions]]-notation)

$$
  \label{FeynmanPropagatorScalingBehaviour}
  \rho^{ 2 dim(sp) }  \Delta_{F, 1/\rho, sp}( \rho x )
  \;=\;
  \Delta_{F,sp, \rho = 1}(x)
  \,.
$$

Typically $\rho$ rescales a [[mass]] parameter, in which case $dim(sp)$ is also called the _[[mass dimension]]_ of the field species $sp$.

Let finally

$$
  \array{
    PolyObs(E)
    &
      \overset{
        \sigma_\rho
      }{\longrightarrow}
    &
    PolyObs(E)
    \\
    \mathbf{\Phi}_{sp}^a(x)
      &\mapsto&
    \rho^{- dim(sp)} \mathbf{\Phi}^a( \rho^{-1} x )
  }
$$

be the [[function]] on [[off-shell]] [[polynomial observables]] given on [[field observables]] $\mathbf{Phi}^a(x)$ by [[pullback of differential forms|pullback]] along $\rho^{-1}$ followed by multiplication by $\rho$ taken to the negative power of the [[mass dimension]], and extended from there to all [[polynomial observables]] as an [[associative algebra|algebra]] [[homomorphism]].

This constitutes an [[action]] of the [[group]]

$$
  RG
    \coloneqq
  \left(
    \mathbb{R}_+, \cdot
  \right)
$$

of [[positive real numbers]] (under [[multiplication]]) on [[polynomial observables]], called the group of _[[scaling transformations]]_ for the given choice of field species and [[mass]] parameters.

=--

([Dütsch 18, def. 3.19](#Duetsch18))

+-- {: .num_example #ScalarFieldMassDimensionOnMinkowskiSpacetime}
###### Example
**([[mass dimension]] of [[scalar field]])**

Consider the [[Feynman propagator]] $\Delta_{F,m}$ of the [[free field theory|free]] [[real scalar field]] on [[Minkowski spacetime]] $\Sigma = \mathbb{R}^{p,1}$  for [[mass]] parameter $m \in (0,\infty)$; a [[Green function]] for the [[Klein-Gordon equation]].

Let the group $RG \coloneqq (\mathbb{R}_+, \cdots)$ of [[scaling transformations]] $\rho \in \mathbb{R}_+$ on [[Minkowski spacetime]] (def. \ref{ScalingTransformations}) act on the mass parameter by inverse multiplication

$$
  (\rho , \Delta_{F,m})
    \mapsto
  \Delta_{F,\rho^{-1}m}(\rho (-))
  \,.
$$

Then we have

$$
  \Delta_{F,\rho^{-1}m}(\rho (-))
  \;=\;
  \rho^{-(p+1) + 2}
  \Delta_{F,1}(x)
$$

and hence the corresponding [[mass dimension]] (def. \ref{ScalingTransformations}) of the [[real scalar field]] on $\mathbb{R}^{p,1}$ is

$$
  dim(\text{scalar field})
  =
  (p+1)/2 - 1
  \,.
$$

=--

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

By  ([this prop.](Feynman+propagator#FeynmanPropagatorAsACauchyPrincipalvalue)) the [[Feynman propagator]] in question is given by the [[Cauchy principal value]]-formula (in [[generalized function]]-notation)

$$
  \begin{aligned}
  \Delta_{F,m}(x)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu - \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \,.
  \end{aligned}
$$

By applying [[change of integration variables]] $k \mapsto \rho^{-1} k$ in the [[Fourier transform of distributions|Fourier transform]] this becomes

$$
  \begin{aligned}
  \Delta_{F,\rho^{-1}m}(\rho x)
  & =
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu \rho x^\mu}
  }{
    - k_\mu k^\mu - \left( \rho^{-1} \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1)}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - \rho^{-2} k_\mu k^\mu - \rho^{-2} \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1)+2}
  \underset{ {\epsilon \in (0,\infty)} \atop {\epsilon \to 0} }{\lim}
  \frac{+i}{(2\pi)^{p+1}}
  \int
  \int_{-\infty}^\infty
  \frac{
     e^{i k_\mu x^\mu}
  }{
    - k_\mu k^\mu -  \left( \tfrac{m c}{\hbar} \right)^2 \pm i \epsilon
  }
  \, d k_0 \, d^p \vec k
  \\
  & =
  \rho^{-(p+1) + 2}
  \Delta_{F,m}(x)
  \end{aligned}
$$

=--

+-- {: .num_prop #RGFlowScalingTransformations}
###### Proposition
**([[scaling transformations]] are [[renormalization group flow]])**

Let

$$
  vac \coloneqq vac_m
  \coloneqq (E_{\text{BV-BRST}}, \mathbf{L}', \Delta_{H,m})
$$

be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacua]] on that field bundle, according to [this def.](S-matrix#VacuumFree) equipped with a decomposition into a set $Spec$ of field species ([this def.](S-matrix#VerticesAndFieldSpecies)) such that for each $sp \in Spec$ the collection of [[Feynman propagators]] the corresponding field species has a well-defined [[mass dimension]] $dim(sp)$ (def. \ref{ScalingTransformations})

Then the [[action]] of the [[group]] $RG \coloneqq (\mathbb{R}_+, \cdot)$ of [[scaling transformations]] (def. \ref{ScalingTransformations}) is a [[renormalization group flow]] in the sense of [this prop.](renormalization+group+flow#FlowRenormalizationGroup).

=--

([Dütsch 18, exercise 3.20](#Duetsch18))

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

It is clear that rescaling preserves [[causal order]] and the [[renormalization condition]] of "field indepencen".

The condition we need to check is that for $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g, j] ]$ two [[microcausal polynomial observables]]  we have for any $\rho, \rho_{vac} \in \mathbb{R}_+$ that

$$
  \sigma_\rho
  \left(
    A_1 \star_{H, \rho^{-1} \rho_{vac} c} A_2
  \right)
  \;=\;
  \sigma_\rho(A_1)
    \star_{H,\rho_{vac}}
  \sigma_\rho(A_2)
  \,.
$$

By the assumption of decomposition into free field species $sp \in Spec$, it is sufficient to check this for each species $\Delta_{H,sp}$.
Moreover, by the nature of the [[star product]] on [[polynomial observables]], which is given by iterated contractions with the [[Wightman propagator]], it is sufficient to check this for one such contraction.

Observe that the scaling behaviour of the [[Wightman propagator]] $\Delta_{H,m}$ is the same as the behaviour (eq:FeynmanPropagatorScalingBehaviour) of the correspponding [[Feynman propagator]]. With this we directly compute as follows:

$$
  \begin{aligned}
    \sigma_\rho (\mathbf{\Phi}(x))
     \star_{F, \rho_{vac} m}
    \sigma_\rho (\mathbf{\Phi}(y)
    & =
    \rho^{-2 dim }
    \mathbf{\Phi}(\rho^{-1} x)
      \star_{F, \rho_{vac} m}
    \mathbf{\Phi}(\rho^{-1} y)
    \\
    & =
    \rho^{-2 dim }
    \Delta_{F, \rho_{vac} m}(\rho^{-1}(x-y))
    \\
    & =
    \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
    \\
    & =
    rg_{\rho}\left(
     \Delta_{F, \rho^{-1}\rho_{vac}m }(x,y) \mathbf{1}
    \right)
    \\
    & =
    rg_{\rho}
    \left(
      \mathbf{\Phi}(x)
        \star_{F, \rho^{-1} \rho_{vac} m}
      \mathbf{\Phi}(y)
    \right)
  \end{aligned}
  \,.
$$

=--

## Related concepts

* [[gauge coupling unification]]

* [[threshold correction]]

* [[Wilsonian RG]]

## References

The original informal discussion for RG-flow along [[scaling transformations]] is due to

* {#GellMannLow54} [[Murray Gell-Mann]] and F. E. Low, _Quantum Electrodynamics at Small Distances_, Phys. Rev. 95 (5) (1954), 1300–1312 ([pdf](http://www.fafnir.phyast.pitt.edu/py3765/GellManLow.pdf))

* {#Callan70} [[Curtis Callan]], _Broken Scale Invariance in Scalar Field Theory_, Phys. Rev. D 2, 1541, 1970 ([doi:10.1103/PhysRevD.2.1541](https://doi.org/10.1103/PhysRevD.2.1541))

* {#Symanzik70} [[Kurt Symanzik]],  _Small distance behaviour in field theory and power counting_, Communications in Mathematical Physics. 18 (3): 227–246 ([doi:10.1007/BF01649434](https://doi.org/10.1007/BF01649434))

Formulation in the rigorous context of [[causal perturbation theory]]/[[pAQFT]], via the [[main theorem of perturbative renormalization]], is due to

* {#BrunettiDuetschFredenhagen09} [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](https://arxiv.org/abs/0901.2038))

reviewed in

* {#Duetsch18} [[Michael Dütsch]], section 3.5.3 of _[[From classical field theory
to perturbative quantum field theory]]_, 2018

In the context of [[factorization algebras]], this is given by the book [[Renormalization and Effective Field Theory]]

[[!redirects renormalization group flows]]

[[!redirects rg-flow]]
[[!redirects rg-flows]]

[[!redirects rg flow]]
[[!redirects rg flows]]

[[!redirects RG-flow]]
[[!redirects RG-flows]]

[[!redirects RG flow]]
[[!redirects RG flows]]

[[!redirects Gell-Mann-Low renormalization cocycle]]
[[!redirects Gell-Mann-Low renormalization cocycles]]

[[!redirects Gell-Mann-Low cocycle]]
[[!redirects Gell-Mann-Low cocycles]]

[[!redirects Gell-Mann-Low renormalization group]]
[[!redirects Gell-Mann-Low renormalization groups]]

[[!redirects mass dimension]]
[[!redirects mass dimensions]]

[[!redirects running coupling constant]]
[[!redirects running coupling constants]]

[[!redirects running of the coupling constant]]
[[!redirects running of the coupling constants]]


renormalization scheme

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Effective field theory and renormalization
+--{: .hide}
[[!include renormalization - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]], for a suitable [[free field theory]] to perturb around, given an  [[interaction]] [[Lagrangian density]] then the [[renormalization]] of the corresponding [[Lagrangian field theory]] is in each order $\omega \in \mathbb{Z}$ a choice from an [[affine space]] of renormalization parameters. A _renormalization scheme_ is a way to pick for all interaction Lagrangians at once an origin in these affine spaces, making them [[finite dimensional vector spaces]] relative to this choice.

This means that, given a [[free field theory]] $\mathbf{L}_{kin}$, then a renormalization scheme makes the construction of the [[perturbative S-matrix]] become a [[function]]

$$
  S
    \;\colon\;
  LocObs(E)[ [\hbar] ]
    \longrightarrow
  PolyObs(E,\mathbf{L}_{kin})_{mc}
$$

which assigns to each [[local observable]] $\mathbf{L}_{int} \in LocObs(E)$, regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[Lagrangian density]], the corresponding perturbative [[scattering matrix]] $S(\mathbf{L}_{int}) \in PolyObs(E,\mathbf{L}_{kin})$ (in the form of an element of the [[Wick algebra]] $(PolyObs(E,\mathbf{L})_{mc}, \star_{H})$) satisfying some conditions (notably [[causal additivity]]).

Concretely, in [[causal perturbation theory]] [[renormalization]] is understood as the choice of [[extension of distributions]] of the [[Feynman amplitudes]] (which are [[products of distributions]], along [[edges]] of [[Feynman diagrams]], of [[Feynman propagators]] with the [[interaction]] [[Lagrangian densities]]) to the [[diagonal]] where the interaction points coincide. For translation-invariant theories this amounts, locally, to [[extension of distributions]] from the [[complement]] $\mathbb{R}^n \setminus \{0\}$ of the origin in some (high-dimensional) [[Cartesian space]] to the full space. For given [[scaling degree of a distribution]] $\omega$, the space of such extensions is an [[affine space]] modeled on the [[finite dimensional vector space]] of [[point-supported distributions]] at the origin of degree $\leq \omega$.

A choice of such extensions for all distributions at once is naturally fixed by a choice of [[projection]] from the space of all [[bump functions]] on $\mathbb{R}^n$ to the subspace of those all whose [[partial derivatives]] to order $\omega$ vanish at the origin ([Brunetti-Fredenhagen 00, p. 24-25](#BrunettiFredenhagen00), following [Epstein-Glaser 73, p. 27-28 (236-237)](#EpsteinGlaser73)).

Such a choice hence amounts to a _renormalization scheme_.

The [[main theorem of perturbative renormalization]] states that any two choices of renormalization schemes differ by a unique re-definition of the [[interaction]] [[Lagrangian density]] by an element in the [[Stückelberg-Petermann renormalization group]].

For more see at _[[geometry of physics -- A first idea of quantum field theory]]_.

## References

Discussion in [[relativistic field theory]] via [[causal perturbation theory]] is due to

* {#EpsteinGlaser73} [[Henri Epstein]] and [[Vladimir Glaser]], section 5 of _[[The role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 5.2 of _Microlocal analysis and interacting quantum field theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

Review is in

* [[Frédéric Paugam]], section 20.6 of _Towards the mathematics of quantum field theory_, Springer 2014 ([pdf](https://webusers.imj-prg.fr/~frederic.paugam/documents/enseignement/master-mathematical-physics.pdf))

Discussion in [[Euclidean field theory]] is in

* [[Kevin Costello]], def. 1.4.1, def. 8.0.2 _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv:0706.1533](http://arxiv.org/abs/0706.1533))

Further discussion of dependence of quantities on choice of renormalization scheme includes

* _Renormalization scheme dependence_, lecture notes ([pdf](http://bolvan.ph.utexas.edu/~vadim/classes/2012f/ms.pdf))

* Matin Mojaza, Stanley J. Brodsky, Xing-Gang Wu, _A Systematic All-Orders Method to Eliminate Renormalization-Scale and Scheme Ambiguities in PQCD_ ([arXiv:1212.0049](http://arxiv.org/abs/1212.0049))

[[!redirects renormalization schemes]]


renormalon

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[non-perturbative effect]]

* [[instanton]]

## References

* M. Beneke, _Renormalons_, Phys. Rept. 317 (1999) 1-142 ([arXiv:hep-ph/9807443](http://arxiv.org/abs/hep-ph/9807443), doi:10.1016/S0370-1573(98)00130-6)

See also:

* Wikipedia, _[Renormalon](https://en.wikipedia.org/wiki/Renormalon)_

A detailed toy example in [[quantum mechanics]]:

* Cihan Pazarbasi, Dieter Van den Bleeken, _Renormalons in quantum mechanics_, J. High Energ. Phys. 2019, 96 (2019) ([arXiv:1906.07198](https://arxiv.org/abs/1906.07198))

[[!redirects renormalons]]


Renyi entropies -- table

| order | | $\phantom{\to} 0$ | | $\to 1$ | |  $\phantom{\to}2$ | | $\to \infty$ |
|--|--|--|--|--|--|--|--|--|
| **[[Rényi entropy]]** | | [[Hartley entropy]] | $\geq$ |  [[Shannon entropy]] | $\geq$ | [[collision entropy]] | $\geq$ |  [[min-entropy]] |


Renzo A. Piccinini

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=8614)

## Selected writings

On [[homotopy theory]]:

* [[Renzo A. Piccinini]], *Lectures on Homotopy Theory*, Mathematics Studies **171**, North Holland 1992 ([ISBN:978-0-444-89238-6](https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C))

On [[cell complexes]] ([[CW complexes]]) in [[topology]] ([[algebraic topology]]):

* {#FritschPiccinini90} [[Rudolf Fritsch]], [[Renzo Piccinini]], _Cellular structures in topology_, Cambridge University Press (1990) ([doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948), [pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf))

## Related nLab entries

* [[simplicial approximation theorem]]

category: people

[[!redirects Renzo Piccinini]]


René Descartes

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### Context
#### Philosophy
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#### Mathematics
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[[!include mathematicscontents]]
=--
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* [Wikipedia entry](http://en.wikipedia.org/wiki/René_Descartes)

Latin name: Renatus Cartesius

Hence the adjective “Cartesian” or “cartesian”, including: [[cartesian space]],
[[cartesian product]],
[[cartesian monad]], [[cartesian logic]], [[cartesian functor]], [[cartesian morphism]], [[cartesian monoidal category]],
[[cartesian closed category]],
[[cartesian fibration]], and many others.

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Descartes](Lectures+on+the+History+of+Philosophy#Descartes)_

> René Descartes ist in der Tat der wahrhafte Anfänger der modernen Philosophie, insofern sie das Denken zum Prinzip macht. ([LecthHistPhil, Descartes](#Lectures+on+the+History+of+Philosophy#DescartesDerAnfaenger))

#Contents#
* table of contents
{:toc}

## On space, matter and mechanics
 {#OnSpaceMatterAndMechanics}

> Extension and motion are the fundamental conceptions in mechanical physics; they represent the truth of the corporeal world. It is thus that ideality comes before the mind of Descartes, and he is far elevated above the reality of the sensuous qualities, although he does not reach so far as to the separation of this ideality. He thus remains at the point of view of mechanism pure and simple. Give me matter (extension) and motion and I will build worlds for you, is what Descartes virtually says.(33) Space and time were hence to him the only determinations of the material universe. In this, then, lies the mechanical fashion of viewing nature, or the natural philosophy of Descartes is seen to be purely mechanical. Hence changes in matter are due merely to motion, so that Descartes traces every relationship to the rest and movement of particles, and all material diversity such as colour, and taste — in short, all bodily qualities and animal phenomena — to mechanism. In living beings processes such as that of digestion are mechanical effects which have as principles, rest and movement. We here see the ground and origin of the mechanical philosophy; but further on we find that this is unsatisfactory, for matter and motion do not suffice to explain life. Yet the great matter in all this is that thought goes forward in its determinations, and that it constitutes from these thought-determinations the truth of nature. ([Hegel, LectHistPhil](Lectures+on+the+History+of+Philosophy#DescartesMechanism))

> {#SEPDescartesOnSpace} For Descartes argued in his 1644 _Principles of Philosophy_ (see Book II) that the essence of matter was extension (i.e., size and shape) because any other attribute of bodies could be imagined away without imagining away matter itself. But he also held that extension constitutes the nature of space, hence he concluded that space and matter were one and the same thing. An immediate consequence of the identification is the impossibility of the vacuum; if every region of space is a region of matter, then there can be no space without matter. Thus Descartes' universe is 'hydrodynamical' — completely full of mobile matter of different sized pieces in motion, rather like a bucket full of water and lumps of ice of different sizes, which has been stirred around. Since fundamentally the pieces of matter are nothing but extension, the universe is in fact nothing but a system of geometric bodies in motion without any gaps. (from [Stanf. Enc. Phil, Theories of Space and Motion](#SEPTheoriesOfSpaceAndMotion))

> Also like the Scholastics, Descartes rejects any form of atomism, which is the view that there exists a smallest indivisible particle of matter. Rather, he holds that since any given spatially extended length is divisible in thought, thus God has the power to actually divide it (Pr II 20). The material entities that interact in Descartes' physics come in distinct units or corpuscles (see Section 7), which explains the "corpuscularian" title often attributed to his mechanical system, but these corpuscles are not indivisible. ([from Stanf. Enc. Phil., Descartes -- Space, Body, and Motion](http://plato.stanford.edu/entries/descartes-physics/#3))

> {#OnDusekOnDescartes} In places, Dusek seems to suggest (quite correctly, in my opinion) that Descartes' physics is closer in spirit to later−day [[continuum mechanics]], which is supposedly holistic, than to the sort of atomistic, reductionist physics of the Newtonians ([Dusek 1999, 23, 204](#Dusek99)) ([pdf source](http://www.philsci.org/archives/psa2000/descartes-science.pdf))

>  In addition, Descartes rejects any explanation of the solidity of a body that employs a bond among its particles (since the bond itself would be either a substance or property, and thus the solidity of the bond would presumably need to be explained; Pr II 55). A macroscopic material body is, essentially, held together just by the relative rest of its constituent material parts. This raises the obvious difficulty that the impact of such bodies should result in their dispersion or destruction (for there is nothing to hold them together). These sorts of complications prompted many later natural philosophers, who were generally sympathetic to Descartes' mechanical philosophy, to search for an internal property of matter that could serve as a type of individuating and constitutive principle for bodies; e.g., Leibniz' utilization of "force". ([from Stanf. Enc. Phil., Descartes -- Space, Body, and Motion](http://plato.stanford.edu/entries/descartes-physics/#3))

## related $n$Lab entries

* [[Cartesian space]]

## References

* {#SEPTheoriesOfSpaceAndMotion} Stanford Encyclopedia of Philosophy, _[Absolute and Relational Theories of Space and Motion](http://plato.stanford.edu/entries/spacetime-theories/#3)_

* {#Dusek99} V. Dusek, _The Holistic Inspirations of Physics_ NewBrunswick: Rutgers University Press (1999) ([review](http://www.jstor.org/stable/188634?seq=1))

[[!redirects Descartes]]
[[!redirects Rene Descartes]]

category: people


René Guitart

**René Guitart** (born 1947) is a French category theorist. A former student of [[Charles Ehresmann]] his work revolves around the theory of [[sketch|sketches]] and develops a broad philosophical perspective on mathematics with influences from semiotics and Lacanian psychonalysis and on the dialectics between logic and algebra on one side and topology and geometry on the other side.

* [homepage](https://web.archive.org/web/20231003032438/http://rene.guitart.pagesperso-orange.fr/)

* [ResearchGate page](https://www.researchgate.net/profile/Rene-Guitart)

## Selected writings

* [Guitart's articles on Numdam](http://numdam.mathdoc.fr/numdam-bin/recherche?format=short&h=aur&f=1&n=30&bibitems.au_op=and&bibitems.text_op=and&aur=Guitart,+Ren%E9&ti_op=and&bibitems.ti_op=and&au_op=and)

On [[free strict cocompletion]]:

* {#BPG2021} Erwan Beurier, Dominique Pastor, [[René Guitart]] _Presentations of clusters and strict free-cocompletions_, [[Theory and Applications of Categories]] **36** 17 (2021) 492-513 [[tac:36-17](http://www.tac.mta.ca/tac/volumes/36/17/36-17abs.html)]

## Related entries

* [[relative pseudomonad]]

[[!redirects Rene Guitart]]
[[!redirects Guitart]]

category:people


René Lavendhomme

* *Biographie de René Lavendhomme*, [[Cahiers]] de Topologie et Géométrie Différentielle Catégoriques **44** 4 (2003) 242-246 [[numdam:](http://www.numdam.org/item/?id=CTGDC_2003__44_4_242_0)]

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=87399)

## Selected writings

On [[synthetic differential geometry]]:

* [[René Lavendhomme]], *Basic concepts of synthetic differential geometry*, Kluwer Texts in the Mathematical Sciences **13**,  Springer (1996) [[doi:10.1007/978-1-4757-4588-7](https://doi.org/10.1007/978-1-4757-4588-7)]

On [[non-abelian cohomology]] of [[algebra|algebraic]] [[structures]]:

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

category: people

[[!redirects Rene Lavendhomme]]


René Thom

* [Wikipedia entry](http://en.wikipedia.org/wiki/Ren%C3%A9_Thom)

## Selected writings

Introducing [[bordism theory]] ([[Thom collapse map]], [[Cohomotopy charge]], [[Thom's theorem]] and [[cobordism cohomology theory]]):

* {#Thom54} [[René Thom]], _[[Quelques propriétés globales des variétés différentiables]]_, Comment. Math. Helv. 28, (1954). 17-86 ([doi:10.1007/BF02566923](https://doi.org/10.1007/BF02566923), [dml:139072](https://eudml.org/doc/139072), [digiz:GDZPPN002056259](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002056259), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/thomcob.pdf))

Also:

* R. Thom, _Problèmes rencontrés dans mon parcours mathématique : un bilan_ , Publications Mathématiques de l'IHÉS **70** (1989) pp.199-214. ([pdf](http://archive.numdam.org/article/PMIHES_1989__70__199_0.pdf))

## Quotes
 {#Quotes}

> Ce qui limite le vrai, ce n’est pas le faux, c’est l’insignifiant.

i.e.

> The true is bounded not by the false but rather by the insignificant.

from R. Thom _Prédire n’est pas expliquer_, Eshel 1991

See also:

* Alain Chenciner, _Le vrai, le faux, l’insignifiant_ ([pdf](https://perso.imcce.fr/alain-chenciner/Vrai_faux_insignifiant.pdf))

## Related entries

* [[Thom space]]

* [[Thom spectrum]]

* [[Thom's theorem]]

* [[Thom's transversality theorem]]

* [[Pontrjagin-Thom collapse map]]

* [[Dold-Thom theorem]]

[[!redirects Rene Thom]]

category: people


repeat-until-success computing

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### Context
#### Computation
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#### Quantum systems
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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum computing]] (but the idea applies generally to [[nondeterministic computation]]), *repeat-until-success* [[Shah & Oi (2013)](#ShahOi13), [Lim, Beige & Kwek (2005)](#LimBeigeKwek05)] is a computation scheme where a given non-deterministic [[logic gate]] is re-applied to its given input data until a failure syndrome vanishes to indicate that the desired kind of gate operation has been performed.

Concretely, in [[quantum computation]] this means that a [[quantum logic gate]] is executed followed by [[quantum measurement]] of parts of its output data, and depending on this measurement either the remaining quantum state output is accepted and forwarded to the next [[logic gate]] in the [[quantum circuit]], or else the result is [[uncomputation|uncomputed]] and the procedure repeated.

While it has been shown [[Lim, Beige & Kwek (2005)](#LimBeigeKwek05)] that such repeat-until-success gates may be efficient (as they can avoid use of gate elements which yield guaranteed outcomes but at a high resource cost), they fall out of the plain [[quantum logic circuit]] scheme in that the nature and even the number of [[quantum gates]] being executed depends on [[classically controlled quantum gate|classical control]] measurement results known only at runtime  (i.e. on "[[dynamic lifting]]" of measurement results back into the classical context).

Phrased differently, repeat-until-success algorithms do have a generalized [[quantum circuit]]-description, but involving an infinite number of "wires"  (since it may take arbitrarily long until the failure syndrome vanishes).

## Related concepts

* [[classically controlled quantum gate]]

* [[dynamic lifting]]

* [[non-deterministic computation]]

## References

* {#ShahOi13} Kerem Halil Shah, Daniel Kuan Li Oi, *Ancilla Driven Quantum Computation with arbitrary entangling strength*, Theory of Quantum Computation **22** (2013) [[arXiv:1303.2066](https://arxiv.org/abs/1303.2066), [doi:10.4230/LIPIcs.TQC.2013.1](https://doi.org/10.4230/LIPIcs.TQC.2013.1)]

* {#LimBeigeKwek05} Yuan Liang Lim, Almut Beige, Leong Chuan Kwek, *Repeat-Until-Success Quantum Computing*, Phys. Rev. Lett. **95** (2005)  030505 [[doi:10.1103/PhysRevLett.95.030505](https://doi.org/10.1103/PhysRevLett.95.030505), [arXiv:quant-ph/0408043](https://arxiv.org/abs/quant-ph/0408043)]

* Yuan Liang Lim, Sean D. Barrett, Almut Beige, Pieter Kok, Leong Chuan Kwek, *Repeat-Until-Success quantum computing using stationary and flying qubits*, Phys. Rev. A **73** (2006) 012304 [[arXiv:quant-ph/0508218](https://arxiv.org/abs/quant-ph/0508218), [doi:10.1103/PhysRevA.73.012304](https://doi.org/10.1103/PhysRevA.73.012304)]

* Adam Paetznick, Krysta M. Svore, *Repeat-Until-Success: Non-deterministic decomposition of single-qubit unitaries*, Quantum Information & Computation **14** 15-16 (2014) 1277-1301 [[arXiv:1311.1074](https://arxiv.org/abs/1311.1074), [doi:10.5555/2685179.2685181](https://dl.acm.org/doi/10.5555/2685179.2685181)]

Discussion of the setup where "fail" (non-success) is guaranteed to act as the [[identity]] on the input [[quantum states]] ("success or draw")

* Qingxiuxiong Dong, Marco Túlio Quintino, Akihito Soeda, Mio Murao, *Success-or-Draw: A Strategy Allowing Repeat-Until-Success in Quantum Computation*, Phys. Rev. Lett. **126** (2021) 150504 [[arXiv:2011.01055](https://arxiv.org/abs/2011.01055), [doi:10.1103/PhysRevLett.126.150504](https://doi.org/10.1103/PhysRevLett.126.150504)]

On repeat-until-success algorithms for creating "quantum imaginary time evolution":

* {#AolitaEtAl21} Thais de Lima Silva, Márcio M. Taddei, Stefano Carrazza, [[Leandro Aolita]], *Fragmented imaginary-time evolution for early-stage quantum signal processors* [[arXiv:2110.13180](https://arxiv.org/abs/2110.13180), [spire:1952417](https://inspirehep.net/literature/1952417)]

Review:

* [[Leandro Aolita]], *Imaginary-time evolution algorithms for intermediate-scale quantum signal processors*, talk at *[Atomtronics@AbuDhabi2021](https://qrcseminar.tii.ae)* (2021) [[video](https://youtu.be/G2eigbPiMKU)]

[[!redirects repeat-until-success]]

[[!redirects repeat-until-success quantum gate]]
[[!redirects repeat-until-success quantum gates]]

[[!redirects repeat-until-success gate]]
[[!redirects repeat-until-success gates]]


replete subcategory

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Notions of subcategory
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#### Higher category theory
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[[!include higher category theory - contents]]
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# Replete subcategories
* table of contents
{: toc}

## Idea

A [[subcategory]] $D$ of a [[strict category]] $C$ is called __replete__ if it respects [[isomorphism]] of morphisms in the [[arrow category]] of $C$.  It is a subcategory for which the property of (strictly) belonging to it respects the [[principle of equivalence]] of categories.

## Definition

A [[subcategory]] $D$ of $C$ is __replete__ if for any [[object]] $x$ in $D$ and any [[isomorphism]] $f\colon x\cong y$ in $C$, both $y$ and $f$ are also in $D$.  Equivalent ways to state this include:

* If $f \in D$ and $f \cong g$ in the [[arrow category]] $Arr(C)$, then $g \in D$.

* The inclusion $D\hookrightarrow C$ is an [[isofibration]].

## Repletion and replete images
 {#RepletionAndRepleteImages}

Since repleteness is a "closure condition," the [[intersection]] of any collection of replete subcategories is again replete.  Therefore, any subcategory is contained in a smallest replete subcategory, called its **repletion**.  We can construct the repletion $repl(D)$ of $D\subset C$ explicitly as follows:

* its objects are those objects of $C$ which admit an isomorphism to some object of $D$, and
* its morphisms are those morphisms of $C$ which can be written as a composite of morphisms in $D$ and isomorphisms in $C$.

Repleteness and repletions are most often applied to [[full subcategories]], in which case the repletion is simply the full subcategory of $C$ determined by those objects which are isomorphic to some object of $D$.  In particular, in this case, the repletion is [[equivalence of categories|equivalent]] to $D$.  More generally, we can say:

+-- {: .un_prop}
###### Proposition
The inclusion $D\hookrightarrow repl(D)$ is an equivalence if and only if the inclusion $D\hookrightarrow C$ is [[pseudomonic functor|pseudomonic]].
=--
+-- {: .proof}
###### Proof
The inclusion of a replete subcategory is always pseudomonic, so if $D\hookrightarrow repl(D)$ is an equivalence, and in particular full and faithful, then $D\hookrightarrow C$ must also be pseudomonic.

Conversely, if $D\hookrightarrow C$ is pseudomonic, then every morphism of $repl(D)$ can be written as $g f h^{-1}$ for some morphism $f$ in $D$ and isomorphisms $g$ and $h$ in $C$.  If the domain and codomain are in $D$, then since $D\hookrightarrow C$ is pseudomonic, both $g$ and $h$ must also be in $D$, hence such a morphism is itself necessarily in $D$.  Thus the inclusion $D\hookrightarrow repl(D)$ is full.  It is clearly faithful and essentially surjective, so it is an equivalence.
=--

The **replete image** of a functor is the repletion of its image.  The **replete full image** of a functor is the repletion of its [[full image]], i.e. the full subcategory of its target determined by those objects isomorphic to some object in its image.  See also [[essential image]].

## Higher-categorical versions

More generally, for $n\geq 0$, a subcategory $D$ of an $n$-[[n-category|category]] $C$ is replete if all [[equivalence]]s in $C$ whose either source or target is in $D$ are themselves in $D$. In particular, all $k$-cells equivalent in $C$ to some $k$-cell in $D$ are themselves in $D$ (because they are sources of some equivalence in $D$).  Then replete subcategories of $1$-[[1-category|categories]] are those which are replete in this sense.

Here we are using the weakest notion of equivalence; one could also talk about $k$-replete $n$-subcategories if every $k$-equivalences in $C$ belongs to $D$ if either its source or target does.  Then using the usual definition of 'category' as a model for $1$-categories, replete subcategories are already those which are $0$-replete (indeed, $0$-equivalence is isomorphism in this model); but using a model of $1$-categories in which [[hom-sets]] are [[setoid]]s (as is common, for example, in [[type theory|type-theoretic]] [[foundations]]), we must insist on $1$-repleteness.

Note that a subcategory $D$ of an $n$-category $C$ is $(k-1)$-replete if and only if:

*  for every $(n-k)$-cell in $D$ all $(n-k)$-cells $(k-1)$-equivalent to an $(n-k)$-cell in $D$ are themselves in $C$, and
*  for any two $(n-k)$-cells $x,y$ the hom $k$-category $hom(x,y)$ is $(k-2)$-replete.

[[!redirects replete subcategory]]
[[!redirects replete subcategories]]
[[!redirects replete image]]
[[!redirects replete images]]
[[!redirects full replete image]]
[[!redirects full replete images]]
[[!redirects repletion]]


representability determines functoriality

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### Context
#### Category theory
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[[!include category theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

If a [[functor]] represents a given [[profunctor]], then the action of the functor on morphisms is determined by the action of the profunctor and the representation isomorphism.

This theorem can make it easier to define certain [[adjoint functors]], and is also implicit in [[type theory]] where type connectives are presentation of functors, but the action on morphisms is derivable rather than primitive.

## Statement

Let $R : C $⇸$ D$ be a profunctor, i.e., a functor from $D^{op}\times C \to Set$ and $F : C\to D$ a functor that [[representable functor|represents]] it by a natural isomorphism $\alpha : R \Rightarrow D(-,F=)$.
Then the action of $F$ on morphisms is determined by $R,\alpha$ by the formula:

$$F(f : c \to c') = \alpha \Big( R(id_{F c}, f)\big(\alpha^{-1}(id_{F c})\big) \Big)$$
or the following composite acting on $id_{F c}$:
$$ D(F c, F c) \xrightarrow{\alpha^{-1}} R (Fc, c) \xrightarrow{R (id_{F c}, f)} R (F c, c') \xrightarrow{\alpha} D(F c, F c') $$

## Representation Presentation of a Functor

Since the action of the functor on morphisms is determined by the data of a representation of a profunctor, we can even *define* the functor by giving that data. Crucially though, what does it mean for $\alpha$ to be natural if we have not yet defined the action of $F$? It turns out that it is sufficient to require naturality in $D$ only. Furthermore, $\alpha^{-1}$ can be recovered from its action on the identity by the [[Yoneda lemma]].

A *right functor presentation* from $C$ to $D$ consists of

1. A profunctor $R : C $⇸$ D$.
2. An object function $F_0 : C_0 \to D_0$.
3. An "introduction rule" $\iota_{c,d} : R(d,c) \to D(d,F_0c)$.
4. A "universal morphism" $\epsilon_c : R(F_0c,c)$.

satisfying

1. *left-naturality of $\iota$*, meaning $\iota \circ R(f,id) = D(f,id) \circ \iota$
2. *isomorphism*, $R(-,id)(\epsilon)$ is an inverse to $\iota$.

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

Given a profunctor $R : C $⇸$ D$, right functor presentations representing $R$ are in bijection with functors $F : C \to D$ with a natural isomorphism $\alpha_{c,d} : R(c,d) \equiv D(c,Fd)$.

=--

Given a right functor presentation $(R, F_0, \iota, \epsilon)$, we can define a functor that represents $R$ by extending $F$ to act on morphisms by $F_1(f) = \iota(R(id, f)(c))$. Then the functoriality of $F$ and naturality of $\alpha$ and $\alpha^{-1}$ follow from the above equalities.

Given a functor representing $R$, the universal morphism is defined as in the [[Yoneda lemma]]: $\epsilon_c = \alpha^{-1}(id)$.

## In Type Theory

A right functor presentation consists of exactly the data that define a [[negative type]] in type theory. The introduction rule of the presentation, unsurprisingly corresponds to the introduction rule in the type theory. The universal morphism corresponds to the (0 or more) elimination rules that are collected into $R(c,F_0 c)$. Left-naturality corresponds to the definition of [[substitution]] for the introduction rule. The isomorphism property corresponds to the $\beta$ and $\eta$ equalities.

This theorem is usually not explicitly invoked in initiality theorems for type theories, but gives an explanation for why the "dogma" of introduction and elimination rules is so effective in practice.

## Examples

1. Decategorifying to the poset case, we get that if a function $f : A \to B$ represents a "relational profunctor" $R$ , i.e. a subset $R \subseteq A \times B$ that is downward closed in $A$ and upper closed in $B$, then $f$ is monotone.
2. Consider the [[cartesian product]] functor, which can be defined by saying that it represents the profunctor $R_{\times} : C^2 $⇸$ C$ defined by
\[ R_{\times}\big(a,(b_1,b_2)\big) = C(a,b_1) \times C(a,b_2) \]

whose action is defined as $R_{\times}(f,(g,h))(k,l) = (g \circ k \circ f, h \circ l \circ f)$.
Then the universal property of the product functor $\times : C^2 \to C$ is encapsulated in the natural isomorphism $\alpha_{a,b_1,b_2} : R_{\times}(a,(b_1,b_2)) \Rightarrow C(a,b_1\times b_2)$.
By the [[Yoneda lemma]], the inverse of $\alpha$ can be reconstructed from its action on the identity, which in this case gives us the two projections: $(\pi_1,\pi_2) = \alpha^{-1}_{b_1\times b_2, b_1,b_2}(id)$.

Then, by the above theorem, the action of $\times$ on functions is equal to:

$$f \times g = \alpha(f \circ \pi_1, g \circ \pi_2)$$

## References

A similar theorem, but for presenting [[adjoint functors]] without defining $R$ first is given (and generalized to the internal setting) in

* [[Ross Street]], [The core of adjoint functors](https://arxiv.org/abs/1112.0094)


Representability Theorems

[[!redirects Representability theorems]]

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#### Higher geometry
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[[!include higher geometry - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
=--
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This page collects some links related to the article

* [[Jacob Lurie]],

  _Representability theorems_

  ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XIV.pdf))

of [[representable|representability]] by [[Deligne-Mumford stacks]] in [[E-∞ geometry]] ("[[geometric ∞-stacks]]").

This provides the main technical ingredient, Proposition 4.1, in _[[A Survey of Elliptic Cohomology]]_, on the [[Goerss-Hopkins-Miller-Lurie theorem]].

More concrete implementations of the main theorem appear in

* [[Jonathan Pridham]], _Representability of derived stacks_ ([arXiv:1011.2189](http://arxiv.org/abs/1011.2189))

#Contents#
* table of contents
{:toc}

## 1. Cotangent complex

* [[cotangent complex]]

## 2. Properties of moduli functors

### 2.1 Nilcomplete, Cohesive, and Integrable Functors

* [[cohesive ∞-presheaf on E-∞ rings]]

### 2.2 Relativized properties of functors

### 2.3 Finiteness conditions on functors

### 2.4 Moduli of spectral Deligne-Mumford stacks

* [[spectral Deligne-Mumford stack]]

## 3. Representability theorems

## 4. Tangent complexes and dualizing modules

### 4.1 The tangent complex

### 4.2 Dualizing modules

* [[dualizing object in a closed category]]

* [[dualizing module]] ([[Verdier duality]])

### 4.3 Existence of dualizing modules

### 4.4 A linear representability criterion

### 4.5 Existence of cotangent complexes

category: reference


representable 2-category

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### Context
#### $\infty$-Limits
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[[!include infinity-limits - contents]]
=--
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[2-category]] is **representable** when it admits [[pullbacks]] and [[cotensors]] with the [[interval category]]. A 2-category has [[2-limit|finite limits]] when it is representable and has a [[terminal object]].

## References

* [[John Gray]], _The meeting of the Midwest Category Seminar in Zurich August 24–30, 1970_,  Lecture Notes in Mathematics, vol 195. Springer 1971, pp. 254–255 ([doi:10.1007%2FBFb0072315](https://link.springer.com/chapter/10.1007%2FBFb0072315))


representable > history

< [[representable]]
[[!redirects representable -- history]]


representable fibered category

Given an [[object]] $X$ in a [[category]] $B$ the [[domain cofibration|domain functor]] $(Y\stackrel{f}\to X)\mapsto Y$ from the [[slice category]] $B/X$ to $B$ is a [[fibered category]] (i.e. Grothendieck fibration).

Any fibered category isomorphic to the $dom:B/X\to B$ is said to be __representable__. This is because under the [[Grothendieck construction]] representable fibered categories correspond precisely to [[representable functor]]s $B^{op} \to Set \hookrightarrow Cat$: the category $B/X$ is the [[category of elements]] of the representable functor $B(-,X)$.

[[!redirects representable Grothendieck fibration]]

[[!redirects representable fibered categories]]


representable functor

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## Idea

For a [[locally small category]] $C$, a [[presheaf]] on $C$ or equivalently a [[functor]]

$$F: C^{op} \to Set$$

on the [[opposite category]] of $C$ with values in [[Set]] is **representable** if it is [[natural isomorphism|naturally isomorphic]] to a [[hom-functor]] $h_X := \hom_C(-, X): C^{op} \to Set$, which sends an object $U \in C$ to the [[hom-set]] $Hom_C(U,X)$ in $C$

and which sends a [[morphism]] $\alpha : U' \to U$ in $C$ to the [[function]] which sends each morphism $U \to X$ to the composite $U' \stackrel{\alpha}{\to} U \to X$

If we picture $Hom_C(U,X)$ as strands of morphisms as above, then the morphism $\alpha:U'\to U$ serves to "comb" the strands back from $Hom_C(U,X)$ to $Hom_C(U',X)$, i.e.

$$h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).$$

The object $X$ is determined uniquely up to [[isomorphism]] in $C$, and is called a **representing object** for $F$.

Representability is one of the most fundamental concepts of [[category theory]], with close ties to the notion of [[adjoint functor]] and to the [[Yoneda lemma]]. It is the crucial concept underlying the idea of [[universal property]]; thus for example crucial concepts such as "[[limit]]", "[[colimit]]", "[[exponential object]]", "[[Kan extension]]" and so on are naturally expressed in terms of representing objects. The concept permeates much of algebraic geometry and algebraic topology.

## Definition

### In ordinary category theory

For a functor $F: C^{op} \to Set$ (also called a [[presheaf]] on $C$), a **representation** of $F$ is a specified [[natural isomorphism]]

$$\theta: \hom_C(-, c) \stackrel{\sim}{\to} F$$

By the [[Yoneda lemma]], any such transformation $\theta$ (isomorphism or not) is uniquely determined by an element $\xi \in F(c)$. As above, the object $c$ is called a **representing object** (or often, **universal object**) for $F$, and the element $\xi$ is called a **[[universal element]]** for $F$. Again, it follows from the [[Yoneda lemma]] that the pair $(c, \xi)$ is determined uniquely up to unique isomorphism.

Following the proof of the Yoneda lemma, representability means precisely this: given any object $x$ of $C$ and any element $\alpha \in F(x)$, there exists a unique morphism $f: x \to c$ such that the function $F(f)$ carries the universal element $\xi \in F(c)$ to $\alpha \in F(x)$. Such a dry formulation fails to convey the remarkable power of this concept, which can really only be appreciated through the myriad examples which illustrate it.

Abstractly, a presheaf $F$ is representable if and only if it admits a [[relative adjunction|left adjoint relative to]] the functor $1 \colon \mathbf 1 \to Set$ from the [[terminal category]], picking out the [[terminal object]] in [[Set]]. The representing object is given by evaluating the left relative adjoint at the unique object of $\mathbf 1$.

### In enriched category theory

The above definition generalizes straightforwardly
to [[enriched category theory]].

Let $V$ be a [[closed monoidal category]] and $C$ a $V$-[[enriched category]].

Then for every object $c \in C$ there is a
$V$-[[enriched functor]]

$$
  C(c,-) : C \to V
$$

from $C$ to $V$ regarded canonically as a $V$-[[enriched category]].

This is defined

* on objects by $C(c,-) : d \mapsto C(c,d) \in V$

* on morphisms between $d$ and $d'$ by

$$
  C(c,-)_{d,d'} : C(d,d') \to [C(c,d), C(c,d')]
  \,,
$$

being the [[adjunct]] of the composition morphism

$$
  \circ_{c,d,d'} : C(d,d') \otimes C(c,d) \to C(c,d')
  \,.
$$

A $V$-enriched functor $F : C \to V$ is **representable** if there is $c \in C$ and a $V$-[[enriched natural transformation]] $\eta : F \to C(c,-)$.

If $V$ is [[symmetric monoidal category|symmetric monoidal]] one can form the [[opposite category]] $C^{op}$ and have the analogous definition for representable functors $F : C^{op} \to V$.

### In higher category theory

The notion of representable functors has its straightforward analogs also in [[higher category theory]].

* For [[2-category theory]] see ... .

* For [[(∞,1)-category theory]] theory see [[(∞,1)-presheaf]]

### In homotopy type theory
 {#InHomotopyTypeTheory}

Discussion for [[internal category in homotopy type theory|category theory in]] [[homotopy type theory]]:

\begin{proposition}
\label{RepresentabilityIsAMereProposition}
**(representability is a mere proposition)**
\linebreak
If $A$ is a [[univalent category]], then the type "F is representable" is a [[mere proposition]].
\end{proposition}
This is Theorem 9.5.9 in the [[HoTT book]].
\begin{proof}
By definition "F is representable" is just the fiber of $\mathbf{y}_0$ over $F$. Since $\mathbf{y}_0$ is an embedding by Corollary 9.5.7 in the HoTT book (see [[Yoneda lemma]]), this fiber is a mere proposition.
\end{proof}

## Examples

The central point about examples of representable functors is:

_Representable functors are ubiquitous_ .

To a fair extent, [[category theory]] is all about representable functors and the other [[universal constructions]]: [[Kan extensions]], [[adjoint functors]], [[limits]], which are all special cases of representable functors -- and representable functors are special cases of these.

Listing examples of representable functors in [[category theory]] is much like listing examples of [[integral]]s in [[analysis]]: one can and does fill books with these. (In fact, that analogy has more to it than meets the casual eye: see [[coend]] for more).

Keeping that in mind, we do list some special cases and special classes of examples that are useful to know. But any list is necessarily wildly incomplete.

### Limits
  {#ExampleLimits}

If $F:J\to C$ is a [[diagram]] in $C$, we obtain a diagram $\hom_C\big(-,F(-)\big) \,\colon\, J\to Set^{C^{op}}$ in the [[functor category]] ([[presheaf category]]) $Set^{C^{op}}$ as the composite of $F$ with the curried [[hom-functor]] $C\to\Set^{C^{op}}$ (the [[Yoneda embedding]]).  The object-wise [[limit]] of this diagram in [[Set]], that is, the functor $C^{op}\to\Set$ sending an object $c \in C$ to the set which is the limit of the diagram $\hom_C\big(c,F(-)\big) \,\colon\, J \to\Set$, is representable iff the diagram $F$ has a limit in $C$; in fact, a representing object for that limit functor is exactly $\lim F$, and we obtain a [[natural isomorphism]]

$$
  \underset{
    \underset{
      j \in J
    }{\longleftarrow}
  }{\lim}
    \; \hom_C\big(-,F(j)\big)
  \;\cong\;
  \hom_C\big(
    -
    ,\,
    \underset{
      \underset{
        j \in J
      }{\longleftarrow}
    }{\lim} F(j)
  \big)
$$

(see at *[[hom-functor preserves limits]]*).

#### Products

For an example in the case of the [[product]], let $c, d$ be objects of $C$, and consider the presheaf given by a product of [[hom-functor|hom-functors]]

$$\hom_C(-, c) \times \hom_C(-, d): C^{op} \to Set;$$

that is, the functor which takes an object $x$ of $C$ to the set $\hom_C(x, c) \times \hom_C(x, d)$. A product $c \times d$ is precisely a representing or universal object for this presheaf, where the universal element is precisely the pair of projection maps

$$(\pi_c, \pi_d) \in \hom(c \times d, c) \times \hom(c \times d, d)$$

We leave it to the reader to check that the representability here means precisely that given a pair
of maps

$$(f, g) \in \hom(x, c) \times \hom(x, d)$$

there exists a unique element in $\hom(x, c \times d)$, denoted $\langle f, g \rangle$, such that

$$\pi_c \langle f, g \rangle = f \qquad \pi_d \langle f, g \rangle = g.$$

#### Weighted limits

The above example has an important straightforward generalization.

Noticing that the limit over the functor $H : J \to Set$ is just the collection of [[cone]]s over $H$ whose tip is the point
$$
  lim H = [J,Set](\Delta pt, H)
$$
the above expression
$\lim\hom_C(-,F)$ can be rewritten equivalently as
$[J,Set](\Delta pt, C(-,F(-)))$. Replacing in this expression the constant terminal functor $\Delta pt : J \to Set$ by any other functor leads to the notion of [[weighted limit]], as described there.

### Exponential objects

Suppose $C$ is a category which admits finite products; given objects $c, d$, consider the presheaf

$$\hom_C(- \times c, d): C^{op} \to Set.$$

A representing or universal object for this presheaf is an [[exponential object]] $d^c$; the universal element

$$e \in \hom_C(d^c \times c, d)$$

is a morphism called the **evaluation** map $eval: d^c \times c \to d$.

### Classifying bundles

Consider a category $Top$ of 'nice' spaces (just to fix the discussion, let's say paracompact spaces, although this is a technical point), and a topological group $G$ therein, i.e., a group internal to $Top$. There is a presheaf

$$G\Bund: Top^{op} \to Set$$

which assigns to each space $X$ the set of isomorphism classes of $G$-bundles over $X$, and assigns to each continuous map $f: X \to Y$ the function

$$G\Bund(f): G\Bund(Y) \to G\Bund(X)$$

which carries a (class of a) $G$-bundle $E \to Y$ to the (class of the) pullback bundle $f^*E \to X$. It is well-known that the pullback construction is invariant with respect to homotopic deformations; that is, this presheaf descends to a functor on the [[homotopy category]],

$$G\Bund: Ho_{Top}^{op} \to Set.$$

A **[[classifying space]]** $\mathcal{B}G$ is precisely a representing object for this functor; the universal element is the (isomorphism class of the) classifying $G$-bundle $[\pi: \mathcal{E}G \to \mathcal{B}G]$.

These general considerations are quite commonplace in algebraic topology, where they crop up for example in the connection between generalized cohomology theories and spectra; cf. Brown's representability theorem.

## Related concepts

* [[representable functor theorem]]

* [[Yoneda lemma]], [[Yoneda embedding]]

* [[classifying space]], [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]]

* [[classifying morphism]]

* [[representable morphism of stacks]]

* [[Artin representability theorem]], [[Artin-Lurie representability theorem]]

* [[multirepresentable functor]]

* [[relative adjunction]]

## References

Early accounts:

* [[Alexander Grothendieck]], Section A.1 of: *Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p.  ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/?id=SB_1958-1960__5__369_0))

A discussion of representable functors in the context of [[enriched category theory]] is in section 1.6 and section 1.10 of

* [[Max Kelly]], _Basic concepts of enriched category theory_ ([pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

A query discussion on differences between representable functor and representation of a functor is archived [here](https://nforum.ncatlab.org/discussion/4497/representable-functor/).

[[!redirects representable functor]]
[[!redirects representable functors]]
[[!redirects represented functor]]
[[!redirects represented functors]]
[[!redirects representable]]
[[!redirects representables]]
[[!redirects representable presheaf]]
[[!redirects representable presheaves]]
[[!redirects representing object]]
[[!redirects representing objects]]


representable functor theorem

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# The representable functor theorem
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## Statement

The __representable functor theorem__ states that:

- A [[presheaf]] $C^{op} \to Set$ on a [[cocomplete category]] $C$ is [[representable functor|representable]] (i.e. of the form $C({-}, c)$ for some object $c \in C$) if and only if it sends [[colimits]] in $C$ to [[limits]] in $Set$ and has a [[solution set condition|solution set]].

Dually, the __corepresentable functor theorem__ states that:

- A [[copresheaf]] $C \to Set$ on a [[complete category]] $C$ is [[corepresentable functor|corepresentable]] (i.e. of the form $C(c, {-})$ for some object $c \in C$) if and only if it is [[continuous functor|continuous]] and has a [[solution set condition|solution set]].

## Related facts

As representable functors are closely connected to [[adjoint functors]], this theorem is essentially equivalent to the [[adjoint functor theorem]] and to theorems guaranteeing the existence of [[limits]].

Specifically, assuming that $C$ is [[copowered]] over [[Set]] (in particular, this is true if $C$ is [[cocomplete]]), a functor $C\to Set$
is a right adjoint functor if and only if it is representable, in which case the left adjoint functor $Set\to C$ sends the [[singleton]] set to the representing object, and more generally a set $X$ to the [[copower]] of $X$ with the representing object.

## Related concepts

* [[adjoint functor theorem]]

## References
 {#References}

* {#Pareigis70} [[Bodo Pareigis]], Thm. 1 on p. 109 in: *Categories and Functors*, Pure and Applied Mathematics **39**, Academic Press (1970) [[doi:10.5282/ubm/epub.7244](https://doi.org/10.5282/ubm/epub.7244), [pdf](https://epub.ub.uni-muenchen.de/7244/1/7244.pdf)]

* {#MacLane97} [[Saunders MacLane]], p. 118 (2nd ed: 130) in: _[[Categories for the Working Mathematician]]_, Graduate texts in mathematics **5**, Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects corepresentable functor theorem]]
[[!redirects representability theorem]]
[[!redirects corepresentability theorem]]


representable morphism

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## Idea

In the philosophy of the [[Grothendieck]] school, one starts with some [[category]] $C$ of "local models" of [[spaces]], equips it with a [[subcanonical topology|subcanonical]] [[Grothendieck topology]], $\tau$, and enlarges $C$ to some category of [[sheaf|sheaves of sets]] on the [[site]] $(C,\tau)$ playing the role of *spaces*. There are further generalizations to [[stacks]] and so on.

When doing this, we often find that properties of "local model spaces" $X\in C$ have to be extended to properties of arbitrary spaces (i.e. sheaves on $(C,\tau)$).  In fact it is most natural to do this in a *relative* situation, i.e. to talk about properties of _morphisms_ rather than properties of _objects_, with an object $X$ regarded as the morphism $X\to 1$.  Thus, one of the main steps in the construction of the theory is to extend good classes of morphisms of local models to the category of spaces. Grothendieck axiomatizes the situation, actually for general presheaves.

Representable morphisms are also important in [[algebraic set theory]] and appear implicitly in the notion of [[category with families]].

## Definition

Let $\mathcal{P}$ be a class of [[morphisms]] in a [[category]] $C$ which is closed under isomorphisms, i.e. it is [[replete subcategory|replete]] when regarded as a [[full subcategory]] of the [[arrow category]] of $C$.

+-- {: .un_defn}
###### Definition
A morphism $\alpha : F\to G$ of [[presheaves]] of sets on $C$ is said to be __representable by a morphism in__ $\mathcal{P}$ if for every morphism from a representable presheaf $h_X\to G$, the projection from the pullback $F\times_G h_X\to h_X$ is (the image under the [[Yoneda embedding]] of) a morphism in $\mathcal{P}$.

When $\mathcal{P}$ is the class of all morphisms in $C$, we simply say that $\alpha$ is __representable__.
=--

In geometrical contexts, we usually assume that $\mathcal{P}$ is itself closed under pullbacks in $C$, i.e. if $f: X\to Y$ is in $\mathcal{P}$ and $g : V\to Y$ a morphism in $C$, then the [[pullback]] $X\times_Y V$ exists and the projection $X\times_Y V\to V$ is in $\mathcal{P}$.  If $C$ has all pullbacks, then the class of *all* morphisms in $C$ satisfies this property.

If $\mathcal{P}$ is closed under pullback, then a morphism $h_X\to h_Y$ between representable presheaves is representable by a morphism in $\mathcal{P}$ if and only if it is itself (the image under the Yoneda embedding of) a morphism in $\mathcal{P}$.  In this way, the class $\mathcal{P}$ of morphisms in $C$ is extended to a class $\hat{\mathcal{P}}$ of morphisms in the category of presheaves of sets $\hat{C} = Set^{C^{op}}$.

## Examples

* [[representable morphism of stacks]] (extends this notion from presheaves to stacks)

[[!redirects representable morphisms]]
[[!redirects representable morphism of presheaves]]
[[!redirects representable morphisms of presheaves]]


representable morphism of stacks

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## Definition

A [[morphism]] $f : X \to Y$ of [[stack]]s over a [[site]] $C$ is called **representable** if for all [[representable functor|representable objects]] $U \in C \stackrel{Y}{\hookrightarrow} Stacks(C)$ and all [[morphism]]s $U \to Y$ the [[homotopy pullback]] $X \times_Y U$ in

$$
  \array{
    X \times_Y U &\to& X
    \\
    \downarrow &{}^{\simeq}\swArrow& \downarrow^f
    \\
    U &\to& Y
  }
$$

is again representable.

## Properties

### Push-forward in generalized cohomology

Along representable morphisms $f$ of stacks over [[smooth manifolds]] ([[smooth infinity-groupoids]]) is induced a [[push-forward in generalized cohomology]] operation.

## Related concepts

* [[representable morphism]].

## References

The general definition appears for instance as def. 38.5 in

* [[Aise Johan de Jong]],  _[[The Stacks Project]] -- Categories_ ([pdf](http://stacks.math.columbia.edu/download/categories.pdf))

(there with stacks perceived equivalently and dually under the [[Grothendieck construction]] as [[fibered categories]]).

Applications of [[push-forward in generalized cohomology]] along representable morphisms appear for instance in

* [[Johannes Ebert]], [[Jeffrey Giansiracusa]], _Pontrjagin–Thom maps and the homology of the moduli stack of stable curves_, Math. Ann. (2011) 349:543–575 ([arXiv:0712.0702](http://arxiv.org/abs/0712.0702))

[[!redirects representable morphisms of stacks]]


representable multicategory

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# Representable multicategories #
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## Idea

The _representable multicategory_ underlying a [[monoidal category]] $C$ is a [[multicategory]] whose $n$-ary morphisms are the [[morphisms]] out of $n$-fold [[tensor products]] in $C$, i.e.

$$
  Rep(C)_n(c_1, \cdots, c_n,c)
  :=
  Hom_C(c_1\otimes \cdots \otimes c_n, c)
  \,.
$$

## Definition ##

There is one flavor of representable multicategory for every flavor of [[generalized multicategory]].  Here we focus on the two best-known: one for ordinary [[monoidal categories]], giving an ordinary [[multicategory]], and one for [[symmetric monoidal categories]], giving a [[symmetric multicategory]].  Mostly the discussion of both cases proceeds in parallel.

We first give the simple pedestrian definition in terms of explicit components, and then a more abstract definition, which is useful for studying general properties.

### In terms of components

For $(C,\otimes, I)$ a ([[symmetric monoidal category|symmetric]]) [[monoidal category]], the **representable multicategory** $Rep(C)$ is the [[symmetric multicategory|(symmetric)]] [[multicategory]] whose objects are the objects of $C$, and whose objects of $n$-ary operations are the [[hom objects]]

$$
  Rep(C)(c_1, \cdots, c_n ; c) :=
C(c_1 \otimes \cdots \otimes c_n,\; c)
  \,,
$$

This comes with the obvious composition operation induced from the composition in $C$. Moreover, in the symmetric case there is a canonical action of the symmetric group induced.

The full sub-multicategory of $Rep(C)$ on an object $c\in C$, being a one-object multicategory, is a ([[symmetric operad|symmetric]]) [[operad]], called the [[endomorphism operad]] of $c\in C$.  Note that some authors use the term "endomorphism (colored) operad" for the whole multicategory $Rep(C)$.

### In terms of Cartesian monads

Let $T : Set \to Set$ be the [[free monoid]] [[monad]]. Notice, from the discussion at _[[multicategory]]_, that a [[multicategory]] $P$ over [[Set]] with object set $C$ is equivalently a monad in the [[bicategory]] of $T$-[[spans]]

$$
  \array{
     && P
     \\
     & \swarrow && \searrow
     \\
     T C && && C
  }
  \,.
$$

In this language, for $C$ a (strict) [[monoidal category]], the corresponding representable multicategory is given by the $T$-span

$$
  \array{
      && & & T Obj(C) \times_{Obj(C)} Mor(C)
      \\
      && & \swarrow && \searrow
      \\
      && T Obj(C) && && Mor(C)
     \\
     & {}^{\mathllap{id}}\swarrow && \searrow^{\mathrlap{\otimes}}
    && {}^{\mathllap{s}}\swarrow
    && \searrow^{\mathrlap{t}}
     \\
     T Obj(C) &&&& Obj(C) &&&& Obj(C)
  }
  \,,
$$

where $\otimes : T Obj(C) \to C$ denotes the iterated tensor product in $C$, and
where the top square is defined to be the [[pullback]], as indicated.

## Properties

### Adjunction
 {#UniversalCharacterization}

The functor $Rep$ is a [[right adjoint]]; its left adjoint constructs the free monoidal category on a multicategory, which is also known as the [[prop]] associated to the multicategory.  See ([Hermida, theorem 7.2](#Hermida)) for a precise statement in the context of [[non-symmetric operads]] and strict monoidal categories.  In the case of semicartesian multicategories, this free monoidal category is the [[category of operators]] associated to the multicategory.

## Related concepts

* [[endomorphism operad]]

## References

The basic definition of representable symmetric multicategories (there called "endomorphism operads") is for instance in section 1 of

* [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_ Contemp. Math. 431 (2007) 31-58 ([arXiv:0512576](http://arxiv.org/abs/math/0512576))

A general account of the definition of representable multicategories is in section 3.3 of

* [[Tom Leinster]], _Higher Operads, Higher Categories_ ([arXiv:math/0305049](http://arxiv.org/abs/math/0305049))

The notion of _representable multicategory_ is due to

* [[Claudio Hermida]], _Representable multicategories_, Adv. Math. 151 (2000), no. 2, 164-225 ([pdf](http://wslc.math.ist.utl.pt/s84.www/cs/claudio/articles/rep-mult.pdf))
  {#Hermida}

Discussion of the [[2-adjunction]] with the left adjoint [[prop]]-construction is around theorem 7.2 there. Characterization of representable multicategories by [[fibrations of multicategories]] is in

* [[Claudio Hermida]], _Fibrations for abstract multicategories_, Field Institute Communications, Volume 43 (2004) ([pdf](http://sqig.math.ist.utl.pt/pub/HermidaC/fib-mul.pdf))

and in section 9 of

* [[Claudio Hermida]], _From Coherent Structures to Universal Properties_ ([arXiv:0006161](http://arxiv.org/abs/math/0006161))

Discussion in the context of [[generalized multicategories]] is in section 9 of

* G. Cruttwell, [[Mike Shulman]], _A unified framework for generalized multicategories_ Theory and Applications of Categories, Vol. 24, 2010, No. 21, pp 580-655. ([TAC](http://www.tac.mta.ca/tac/volumes/24/21/24-21abs.html))

[[!redirects representable multicategory]]
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