relevance monoidal category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Relevance monoidal categories

* table of contents
{:toc}

## Idea

A *relevance monoidal category* is a [[symmetric monoidal category]] which has [[monoidal category with diagonals|diagonal maps]] $A\to A\otimes A$, but not projection maps $A \to I$.  (If it had both, it would be a [[cartesian monoidal category]]; while if it had projection maps but not diagonals it would be a [[semicartesian monoidal category]].)  The name comes from the connection with [[relevance logic]], and seems to have been introduced in ([Petrić 2002](#Petric02)).

## Definition

Given a [[symmetric monoidal category]] $C$, let $CCSG(C)$ denote the category of commutative co-[[semigroups]] [[internalization|in]] $C$, i.e. objects $A$ equipped with a "comultiplication" $A\to A\otimes A$ that is coassociative and cocommutative.  There is an obvious forgetful functor $CCSG(C) \to C$.  Moreover, $CCSG(C)$ has a symmetric monoidal structure making the forgetful functor strict symmetric monoidal: if $A$ and $B$ are cosemigroups then $A\otimes B \to (A\otimes A)\otimes (B\otimes B) \cong (A\otimes B) \otimes (A\otimes B)$ makes $A\otimes B$ into a commutative cosemigroup as well, and the unit is the unit object $I$ of $C$ with its a canonical commutative cosemigroup structure given by the coherence isomorphism $I\cong I\otimes I$.

We say that $C$ is a **relevance monoidal category** if this functor $CCSG(C) \to C$ is equipped with a strict section that is also a strict symmetric monoidal functor.  That is, we have a functor assigning to every object of $C$ a commutative cosemigroup structure on that object, in such a way that every morphism becomes a cosemigroup map, the structure on $A\otimes B$ is induced from those on $A$ and $B$ as above, and the structure on $I$ is the canonical one.  This amounts to a natural assignment of "diagonal maps" $A\to A\otimes A$ satisfying some straightforward axioms.

(If we replaced cosemigroups with comonoids, then the analogous property would characterize [[cartesian monoidal categories]], while if instead we used copointed objects --- that is, the slice $C/I$ --- it would characterize [[semicartesian monoidal categories]].  Interestingly, unlike in those two cases, the relevance case doesn't seem to imply any universal property for the monoidal product or the unit.)

One can of course additionally ask that a relevance monoidal category be [[closed monoidal category|closed]], or that it have finite [[products]] or [[coproducts]].  One might also ask it to be [[star-autonomous]], although in that case there might need to be some compatibility between the star-autonomy and the relevance.

## Examples

* The category of [[pointed sets]], which is equivalent to the category of sets and [[partial functions]], is a relevance monoidal category with its pointed [[smash product]] ([Došen-Petrić 2007](#DosenPetric07)).  The unit for this tensor product is not the terminal object, which is the 1-element pointed set, but instead the 2-element pointed set, so this relevance monoidal category is not cartesian.  The category of pointed sets can also be given a cartesian monoidal structure.

* Any [[Church monoid]] is a relevance monoidal category whose underlying category is a [[poset]].

## Related concepts

* [[monoidal category with diagonals]]

* [[semicartesian monoidal category]]

* [[monoidal category]]

* [[relevance logic]]

## References

* {#Jacobs94} B. Jacobs, _Semantics of Weakening and Contraction_, Annals of Pure and Applied Logic, volume 69, Issue 1, (1994) pp.73-106, doi:[10.1016/0168-0072(94)90020-5](https://doi.org/10.1016/0168-0072%2894%2990020-5)

* {#Petric02} Z. Petrić, _Coherence in Substructural Categories_, Studia Logica volume 70, (2002) pp.271–296, doi:[10.1023/A:1015186718090](https://doi.org/10.1023/A:1015186718090), arXiv:[math/0006061](https://arxiv.org/abs/math/0006061)

* {#DosenPetric07} K. Došen and Z. Petrić, _Relevant Categories and Partial Functions_, Publications de l'Institut Mathématique, Nouvelle Série, Vol. 82(96), pp. 17–23 (2007) ([doi:10.2298/PIM0796017D](https://doi.org/10.2298/PIM0796017D), [arxiv:math/0504133](http://arxiv.org/abs/math/0504133))

[[!redirects relevance monoidal categories]]


relevant monad

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

#Contents#
* table of contents
{:toc}

## Idea

A relevant monad is a special kind of monad which is related to [[relevance logic]].

## Definition

Let $\mathcal{C}$ be a [[category]] with [[cartesian products|Cartesian products]]. Let $(T, \mu,\eta)$ be a [[monad]] on $\mathcal{C}$.

Recall that $T$ is said to be **right-strong** with respect to the Cartesian product (see [[strong monad]]) if it is equipped with a natural transformation $\alpha_{X,Y} : T(X)\times Y\to T(X\times Y)$ satisfying some coherence conditions which can be found on the linked page.

In what follows, fix a monad $(T, \mu,\eta)$ together with a choice of [[strength|right strength]] $\alpha$ for the Cartesian product.

Using the [[symmetric monoidal category|symmetry isomorphism]] of the Cartesian product it is possible to give a [[strength|left strength]] $\alpha'_{A,B} : A\times T(B)\to T(A\times B)$ by
\begin{equation}
A\times T(B) \cong T(B)\times A \xrightarrow{\alpha_{B,A}}T(B\times A)\cong T(A\times B)
\end{equation}

\begin{definition}
Let $T$ be a strong monad. The **right double-strength** on $T$ is the [[natural transformation]]
\begin{equation}
dst_{X,Y} : T(X)\times T(Y)\xrightarrow{\alpha_{X,T(Y)}}T(X\times T(Y))\xrightarrow{T(\alpha^{'}_{X,Y})}T^2(X\times Y)\xrightarrow{\mu}T(X\times Y)
\end{equation}
\end{definition}

\begin{definition}
If $(T,\mu,\eta)$ is a right-strong monad with respect to the Cartesian product, then $T$ is said to be
**relevant** if the following diagrammatic law holds:

\begin{centre}
\begin{tikzcd}
      T(X\times Y)\arrow[dr,equal]
      \arrow[d,"(T(\pi_{X}){,}T(\pi_{Y}))"']\\
      T(X)\times T(Y)\arrow[r,"dst_{X,Y}"] &T(X\times Y)
\end{tikzcd}
\end{centre}
\end{definition}

\section{References}

Concept was first introduced here, see Prop 2.2:

- [[Anders Kock]], _Bilinearity and Cartesian-Closed Monads_ In: Mathematica Scandinavica 29.2 (1971), pp. 161–74. [JSTOR: 24491025](https://www.jstor.org/stable/24491025), [pdf](https://users-math.au.dk/kock/BCCM.pdf)

Jacobs is probably the first one to introduce the name 'relevant monad':

- [[Bart Jacobs]], _Semantics of Weakening and Contraction_ In: Annals of
Pure and Applied Logic 69.1 (Sept. 1994), pp. 73–106. issn: 01680072.
[doi: 10.1016/0168-0072(94)90020-5]( https://linkinghub.elsevier.com/retrieve/pii/0168007294900205)

See also:

- Louis Parlant et al., _Preservation of Equations by Monoidal Monads_, July 7, 2020, [arXiv: 2001.06348 [cs, math]](http://arxiv.org/abs/2001.06348)


Remy Tuyeras

Remy is a PhD alumnus and past member of the Australian school of higher category theory (Macquarie University).

His PhD research was about the [[homotopy hypothesis|Homotopy Hypothesis]] for Grothendieck's [[infinity-groupoid|$\infty$-groupoids]].

Remy did his postdoctoral training at MIT -- first in the Math department and then in [CSAIL](https://www.csail.mit.edu/) -- regarding the application of [[category theory]] to [[biology]]. More about his work can be found on his [personal webpage](http://www.normalesup.org/~tuyeras/)


Renata Kallosh

* [Wikipedia entry](http://en.wikipedia.org/wiki/Renata_Kallosh)

## Selected writings

Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:

* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]

On [[moduli stabilization]] in [[M-theory on 8-manifolds]] for the [[product manifold]] of [[K3]] with itself:

* [[Paul Aspinwall]], [[Renata Kallosh]], _Fixing All Moduli for M-Theory on $K3 \times K3$_, JHEP 0510:001, 2005 ([arXiv:hep-th/0506014](https://arxiv.org/abs/hep-th/0506014))

## Related entries

* [[landscape of string theory vacua]]

category: people


Renato Betti

* [institute page](https://matematica.unibocconi.it/autore/renato-betti)

## Selected writings

On [[closed bicategories]] and [[locally internal categories]]:

* {#BettiWalters1985} [[Renato Betti]], [[Robert F. C. Walters]], *Closed bicategories and variable category theory*, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, **26** (2020) 1-27 $[$[tac:tr26](http://www.tac.mta.ca/tac/reprints/articles/26/tr26abs.html)$]$

category: people


Renato P dos Santos

* [GoogleScholar page](https://scholar.google.com/citations?user=tCPkqLYAAAAJ&hl=en)

## Selected writings

On [[quantum computation]], [[blockchains]] and [[machine learning]]:

* [[Melanie Swan]], [[Renato P dos Santos]], [[Frank Witte]], Between Science and Economics, Volume 2: *Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks*, World Scientific 2020 ([doi:10.1142/q0243](https://doi.org/10.1142/q0243))

  * On [[holographic quantum error correction]]:

    Part 5 (Ch. 13, 14): *The AdS/CFT Correspondence and Holographic Codes* ([doi:10.1142/9781786348210_0013](https://doi.org/10.1142/9781786348210_0013), [doi:10.1142/9781786348210_0014](https://www.worldscientific.com/doi/abs/10.1142/9781786348210_0014))

category: people


Renato Renner

* [institute page](https://itp.phys.ethz.ch/people/person-detail.html?persid=59275)

* [Wikipedia entry](https://de.wikipedia.org/wiki/Renato_Renner)

## Selected writings

On [[min-entropy]]:

* Robert Koenig, [[Renato Renner]], Christian Schaffner, *The operational meaning of min- and max-entropy*, IEEE Trans. Inf. Th., vol. 55, no. 9 (2009) [[arXiv:0807.1338](https://arxiv.org/abs/0807.1338), [doi:10.1109/TIT.2009.2025545](https://doi.org/10.1109/TIT.2009.2025545)]

On [[quantum state tomography]]:

* [[Matthias Christandl]], [[Renato Renner]], *Reliable Quantum State Tomography*, Physical Review Letters **109** 120403 (2012) [[arXiv:1108.5329](https://arxiv.org/abs/1108.5329), [doi:10.1103/PhysRevLett.109.120403](https://doi.org/10.1103/PhysRevLett.109.120403)]

On [[quantum cryptography]]:

* [[Renato Renner]], Ramona Wolf, *Quantum Advantage in Cryptography*, AIAA Journal  (2023) 1-16 [[arXiv:2206.04078](https://arxiv.org/abs/2206.04078), [doi:10.2514/1.J062267](https://doi.org/10.2514/1.J062267)]

category: people


Renato Vasconcellos Vieira

* [personal page](http://lattes.cnpq.br/9012433941779564)

* [GoogleScholar page](https://scholar.google.com/citations?user=67_tDYsAAAAJ&hl=en)

## Selected writings

On [[recognition of relative loop spaces]]:

* {#VVieira2020} [[Renato Vasconcellos Vieira]], _Relative recognition principle_, Algebr. Geom. Topol. 20(3): 1431-1486 (2020) [[arXiv:1802.01530](https://arxiv.org/abs/1802.01530), [doi:10.2140/agt.2020.20.1431](https://doi.org/10.2140/agt.2020.20.1431)]

category: people


Rene Thom > history

< [[Rene Thom]]


renormalizable interaction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In the construction of [[interacting field theory|interacting]] [[perturbative quantum field theory]] the specification of

1. the [[gauge fixing|gauge-fixed]] [[free field theory|free field]] [[vacuum]] $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H)$ that the [[perturbation theory|perturbation]] is about ([this def.](S-matrix#VacuumFree));

1. the [[local observable|local]] [[interaction]] [[action functional]] $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g\rangle $ which prescribes the nature of the perturbation ([this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering))

still leaves, in general, a [[countable set]] of [[finite dimensional vector space|finite dimensional]] [[affine spaces]] of choices $\{c^\alpha_k \in \mathbb{C}\}_{k \in \mathbb{n}}$ to fully specify the corresponding [[interacting field theory|interacting]] [[perturbative QFT]] ([this prop.](renormalization#RenormalizationChoices)). This is called the choice of _[[renormalization|("re"-)normalization]]_ constants of the [[perturbative QFT]].

One says that the interaction $g S_{int}$ is _renormalizable_ if this potentially countably infinite dimensional space of choices happens to be a finite dimensional space, hence if there is $n \in \mathbb{N}$ such that the only available choice is $c^\alpha_k = 0$ for all $k \geq n$. Otherwise one says that $g S_{int}$ is _non-renromalizable_.

(If only a [[finite set]] of [[Feynman diagrams]] contributes to the ("re"-)normalization choices to be made for a renormaliable interaction, then it is called _super-renormalizable_).

## Interpretation

Beware that this terminology may be misleading: The [[main theorem of perturbative renormalization]] says, among other things, that for _all_  [[perturbative QFTs]] there do exist constructions of [[time-ordered products]]/[[Feynman amplitudes]] on the diagonal, hence of [[renormalization|("re"-)normalizations]], hence of choices of these $c_k^\alpha$, namely by [[Epstein-Glaser renormalization]] ([Epstein-Glaser 73](causal+perturbation+theory#EpsteinGlaser73)); even if there are (countably) infinitely many. In [Gomis-Weinberg 96](#GomisWeinberg96) this is called "renormalizable in the modern sense" (at least if [[renormalization conditions]] such as [[Ward identities]] may be met).

Therefore the issue expressed with the term "non-renormalizable" at this point is not a mathematical [[obstruction]], but a philosophical sentiment: The need to specify an infinite set of further physical constants to specify a [[theory (physics)|physical theory]] is (or has been) felt to defeat the purpose or meaning of having specified a [[theory (physics)|physical theory]].

But in practice this is less of a problem than it might seem: By the [[renormalization group flow|Gell-Mann-Low renormalization group flow]]/[[Polchinski's flow equation|Wilson-Polchinski RG flow]] the higher the order of the renormalization constants, the higher the [[energy]]/the smaller the [[wavelength]] at which their effect is visible in [[experiment]]. Hence in _practice_ only a finite number of renormalization constants is observable anyway. As the energy/resolution of an experiment is increased by a finite amount, a finite number of further renormalization constants may become visible.

For famous "non-renormalizable" theories such as [[perturbative quantum gravity]] the first such corrections would become visible to [[experiment]] only at energies way beyond current reach, so that for all practical purpose the construction of "non-renormalizable" [[perturbative QFTs]] such as [[perturbative quantum gravity]] via [[renormalization|("re"-)normalization]] does exist ([Scharf 01, chapter 5](#Scharf01)). (A wide-open problem is, instead, the construction of [[interacting field theory|interacting]] [[non-perturbative quantum field theories]] for [[spacetime]] [[dimension]] $\geq 3+1$.)

It has become popular to think of such "non-renormalizable" [[perturbative QFTs]] as "[[effective field theories]]". See there for more.

## Examples

* [[Yang-Mills theory]] (...condition on gauge group...) hence [[QCD]] is renormalizable (t'Hooft, ...)

* [[Einstein gravity]] is non-renormalizable ([Goroff-Sagnotti 85](quantum+gravity#GoroffSagnotti85), [van de Ven 92](#vandeVen92))

## References

The renormalization of _any_ [[interaction]] in [[perturbative QFT]] (by a possibly countably infinite number of renormalization constants) was established by [[Epstein-Glaser renormalization]] in

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[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 ))

This point was later amplified as "renormalization in the modern sense" in

* {#GomisWeinberg96} [[Joaquim Gomis]], [[Steven Weinberg]], _Are Nonrenormalizable Gauge Theories Renormalizable?_, Nucl. Phys. B469 : 473-487, 1996 ([arXiv:hep-th/9510087](https://arxiv.org/abs/hep-th/9510087))

Discussion in the rigorous context of [[causal perturbation theory]]/[[pAQFT]] is in

* {#Scharf95} [[Günter Scharf]], p. 271 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

See also

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

[[!redirects renormalizable interactions]]

[[!redirects renormalizable perturbative QFT]]
[[!redirects renormalizable perturbative QFTs]]

[[!redirects renormalizable QFT]]
[[!redirects renormalizable QFTs]]

[[!redirects renormalizability]]

[[!redirects non-renormalizable pertrubative QFT]]
[[!redirects non-renormalizable perturbative QFTs]]

[[!redirects non-renormalizable QFT]]
[[!redirects non-renormalizable QFTs]]