restriction category

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### Context
#### Category theory
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# Restriction category
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## Idea

The concept of **restriction category** is an [[element]]-free formalisation of the idea of a [[category]] of [[partial maps]], where the [[domain]] of a map $f\colon X\to Y$ is encoded in a specified [[idempotent]] [[endomorphism]] $\overline{f}$ of $X$.

## Definition

Given a [[category]] $C$, a _restriction structure_ consists of the assignment to each [[morphism]] $f\colon X\to Y$ a morphism $\overline{f}\colon X\to X$ satisfying the following conditions:

* $f\circ \overline{f} = f$,
* $\overline{f}\circ \overline{g} = \overline{g}\circ\overline{f}$ whenever $s(f)=s(g)$,
* $\overline{g\circ \overline{f}} = \overline{g}\circ \overline{f}$ whenever $s(f)=s(g)$, and
* $\overline{g}\circ f = f\circ \overline{g\circ f}$ whenever $s(g) = t(f)$

where $s(f) = \text{dom}(f)$ is the domain of $f$. A _restriction category_ is a category with a restriction structure.

Note that a restriction structure is a [[structure]] in the technical sense of the word, not a [[property]]. Note that it follows from the above definition that $\overline{f}$ is idempotent for composition: $\overline{f}\circ \overline{f} = \overline{f}$, and that the operation $f \mapsto \overline{f}$ is also idempotent: $\overline{\overline{f}} = \overline{f}$ (among other properties).

## Examples

Every category admits the _trivial_ restriction structure, with $\overline{f} = id_{s(f)}$.

Conversely the [[wide subcategory]] consisting off all the [[objects]] together with the morphisms $f$ satsfying $\overline{f}=id_{s(f)}$—the _total_ morphisms—has a trivial induced restriction structure.

The category $PF$ of [[sets]] and [[partial functions]] is the prototypical example, where for a partial function $X \supseteq U \stackrel{f}{\to} Y$, the partial endomorphism $\overline{f}$ is the partially-defined identity function $X \supseteq U \hookrightarrow X$. Many other examples are listed in ([Cockett–Lack 2002](#CockettLack02))

## Related concepts

* [[partial function]]
* [[cartesian restriction category]]
* [[Turing category]]

## References

* {#CockettLack02} [[Robin Cockett]] and [[Steve Lack]], _Restriction categories I: categories of partial maps_, Theoretical Computer Science **270** 1-2 (2002) 223-259 [doi:10.1016/S0304-3975(00)00382-0]

* {#CockettLack03} [[Robin Cockett]] and [[Steve Lack]], _Restriction categories II: partial map classification_, Theoretical Computer Science **294** 1-2 (2003) 61-102 [doi:10.1016/S0304-3975(01)00245-6]

* {#CockettLack07} [[Robin Cockett]] and [[Steve Lack]], _Restriction categories III: colimits, partial limits and extensivity_, Mathematical Structures in Computer Science **17** 4 (2007)  775-817 [[arXiv:math/0610500](https://arxiv.org/abs/math/0610500), [doi:10.1017/S0960129507006056](https://doi.org/10.1017/S0960129507006056)]

* {#CockettGarner14} [[Robin Cockett]] and [[Richard Garner]], _Restriction categories as enriched categories_, Theoretical Computer Science **523** (2014) 37-55 [[arXiv:1211.6170](https://arxiv.org/abs/1211.6170), [doi:10.1016/j.tcs.2013.12.018](https://doi.org/10.1016/j.tcs.2013.12.018)]

A [[double category|double categorical]] approach to restriction categories is proposed in:

* [[Robert Paré]], _A double category take on restriction categories_, ([slides](https://www.mscs.dal.ca/~pare/Azores.pdf))

On showing that restriction categories are [[categories enriched in a double category]]:

* [[Robin Cockett]] and [[Richard Garner]], _Restriction categories as enriched categories_, Theoretical Computer Science 523 (2014): 37-55.

On [[free cocompletions]] of restriction categories:

* [[Richard Garner]] and Daniel Lin, _Cocompletion of restriction categories_, Theory and Applications of Categories 35.22 (2020): 809-844.

On using restriction categories to model [[essentially algebraic theories]]:

* [[Ivan Di Liberti]], [[Fosco Loregian]], [[Chad Nester]], [[Paweł Sobociński]]. _Functorial semantics for partial theories_, Proceedings of the ACM on Programming Languages 5.POPL (2021): 1-28.

On *range restriction categories*:

* Xiuzhan Guo, *Ranges, restrictions, partial maps, and fibrations* (2004), Master's thesis ([pdf](https://www.collectionscanada.gc.ca/obj/s4/f2/dsk4/etd/MQ93373.PDF?is_thesis=1&oclc_number=60669889)).

[[!redirects restriction categories]]

[[!redirects cartesian restriction category]]
[[!redirects cartesian restriction categories]]


restriction of scalars

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#### Algebra
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#Contents#
* table of contents
{:toc}

## Definition

Let $f \colon R \to S$ be a [[homomorphism]] of [[algebra|algebraic]] objects such as [[rings]]. Let $\cdot_S$ be an [[action]] of $S$ on a [[module]] $M$, then

$$
  r\cdot_R m \coloneqq f( r )\cdot_S m
$$

defines an action of $R$ on $M$. This construction extends to a  [[functor]]
$$
  f^\ast \colon SMod \longrightarrow RMod
$$
between [[categories of modules]], sending $\cdot_S$ to $\cdot_R$. This is called *restriction of scalars* (along $f$).

This functor has a [[left adjoint functor]]

$$
  f_! \; \coloneqq\;
  S \otimes_R (-)
  \;\colon\;
  RMod
   \longrightarrow
  SMod
$$

called *[[extension of scalars]]*, since for an $R$-module $M$ and an $R$-module $S$ we have that $M\otimes_R S$ is a well defined [[tensor product of modules|tensor product]] of $R$ modules which becomes an $S$ module by the operation of $S$ on itself in the second factor of the tensor. We have an [[adjoint functors|adjunction]] $f_! \dashv f^*)$.

Not only is restriction of scalars a right adjoint, it is also a [[monadic functor]].  This can be shown using the [[monadicity theorem]] or by direct computation.

Furthermore, not only is restriction scalars a right adjoint, it is also a left adjoint.  That is, it has a right adjoint of its own, called [[coextension of scalars]]:

$$
  f_* \; \coloneqq\;
  RMod(S, -)
  \;\colon\;
  RMod
   \longrightarrow
  SMod
$$

## Related concepts

* [[extension of scalars]] $\dashv$ **restriction of scalars** $\dashv$ [[coextension of scalars]]


Restriction Theory

[[Kakeya Conjecture]]


resultant

## Idea

The __resultant__ of two monic polynomials is an expression which is zero if the polynomials have a common multiple: it may be defined as a product of all possible difference between the pairs of roots of the two polynomials. They are an analogue of a discriminant of a single quadratic equation: the discriminant is zero if the equation has a double [[root]]. Resultant for $n$ homogeneous polynomials of $n$ variables defines a condition of solvability for a system of $n$ homogeneous polynomials, as the determinant does for a system of linear equations.

When the number of polynomials and variables is bigger than few smallest integers, it is very little known. There are remarkable determinant expressions for resultants of 2 or 3 polynomials. In what is the first article on [[homological algebra]], Cayley has shown how to descibe a resultant as a determinant of a complex which is a certain Koszul resolution.

While syzygies show linear relations, linear relations among linear relations and so on and thus belong to linear homological algebra, resultants seem to point to a nonlinear homological algebra for polynomial relations among polynomials and so on.

## References

* I. M. Gelfand, [[Mikhail Kapranov|M. M. Kapranov]], A. Zelevinsky, _Discriminants, [[resultants]], and multidimensional determinants_, Birkhäuser 1994, 523 pp.

* A. Anokhina, A. Morozov, Sh. Shakirov, _Resultant as determinant of Koszul complex_, ITEP/TH-70/08. Theor.Math.Phys. 160:3 (2009) 1203-1228, [arXiv:0812.5013](http://arXiv.org/abs/0812.5013)

* A.Morozov, Sh.Shakirov, _New and old results in resultant theory_, [arxiv/0911.5278](http://arxiv.org/abs/0911.5278)

* H. B. Griffiths, _Cayley's version of the resultant of two polynomials_, Amer. Math. Monthly __88__, No. 5 (May, 1981), pp. 328-338, [jstor](http://www.jstor.org/stable/2320108)

* A. Morozov, Sh. Shakirov, _Resultants as contour integrals_, [arXiv:0807.4539](http://arxiv.org/abs/0807.4539)

* [wikipedia](http://en.wikipedia.org/wiki/Resultant)

[[!redirects resultants]]


resummation

#Contents#
* table of contents
{:toc}

## Idea

A __resummation__ is an operation which applies to some class of [[series]] and which makes some divergent [[series]] become [[convergence|convergent]].
Normally one requires that the operation makes sense also to the series which are convergent to start with and then the resummation gives a series with the same sum as the original one.

* [[Borel resummation]]

## Related concepts

* [[perturbation theory]]

* [[resurgence]]

## References

The main historical reference is the monograph of Hardy, still very readable (and of course, accurate).

* [[G. H. Hardy]], _Divergent series_, Clarendon Press, Oxford, 1949.

Some resummation methods can be made sense also from the point of view of [[nonstandard analysis]] are discussed in

* Vladimir Kanovei, Michael Reeken, _Summation of divergent series from the nonstandard piont of view_, Real Anal. Exchange __21__:2 (1995) 473-497 [euclid](http://projecteuclid.org/euclid.rae/1339694079)

More recent developments include

* {#CostinDunne17} Ovidiu Costin, Gerald V. Dunne, _Convergence from Divergence_ ([arXiv:1705.09687](https://arxiv.org/abs/1705.09687))

[[!redirects resummation method]]
[[!redirects resummation methods]]


resurgence theory

#Contents#
* table of contents
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## Idea

What is called _resurgence theory_ is a mathematical formalization of what in [[quantum field theory]] is known as reading off [[non-perturbative effects]] from the [[Feynman perturbation series]] ([[S-matrix]]) in [[perturbative quantum field theory]]. It tries to regard this non-converging [[formal power series]] in the [[coupling constant]] $g$ as an [[asymptotic series]] of a function involving [[analytic function|non-analytic]] contributions such as $e^{-1/g^2}$.

In [[quantum mechanics]] and [[quantum field theory]], [[renormalization|renormalized]] [[pQFT|perturbation theory]] produces expansions in powers of the [[coupling constant]] $g$. By appropriate [[resummation]] using the [[renormalization group]] one can also capture terms involving [[logarithm|logarithmic]] contributions of the form $\log (g/g_0)$. However, it is well-known that, e.g., [[instantons]] contribute terms of order $e^{-c/g}$ or $e^{-c/g^2}$, whose [[Taylor expansion]] is identically zero at $g = 0$, so that they do not contribute at all to the power series expansion. These contributions are therefore intrinsically [[non-perturbative effect|nonperturbative]].

A [[transseries]] is an expansion of a function $f(x)$ as a [[power series]] in a [[vector]] $z$ whose components are fixed functions of $x$. In [[quantum mechanics]] and [[quantum field theory]], the relevant case is $x=g$ or $x=g^2$ and $z_1=x$, $z_2=e^{-c/x}$, and in QFT usually also $z_3=\log(x/x_0)$. Clearly, transseries are more flexible than ordinary power series.

Resurgent functions are functions arising in the analysis of singular points of [[germs]] of [[analytic functions]]. Resurgent transseries are [[transseries]] that arise in methods for [[resummation]] that generalize [[Borel summation]] by looking at the [[obstructions]] for Borel summability such as [[renormalons]].

It turns out that using resurgent transseries, one can obtain under certain conditions information from the [[non-perturbative quantum field theory|nonperturbative sector]], starting from the [[Feynman perturbation series]] alone. The basic idea is to construct from the perturbative series the terms appearing in the [[renormalization group]] equation (RGE) by Callan and Symanzik, to substitute into it the leading terms of an appropriate transseries, and to determine the coefficients so that they match the RGE to the order specified.

To check that the method really works one can apply it to simple examples form quantum mechanics where other methods can be used to compute nonperturbative information. For example, for the double well potential, this was done by [Jentschura & Zinn-Justin 04](#JentschuraZinnJustin04) , and for some other cases by ([Dunne-Unsal 14](#DunneUnsal14)). For applications to [[gauge theories]] and to [[string theory]], see ([Marino 12](#Marino12)); of course, in the latter cases there is no known alternative way of checking the results. A more mathematically oriented overview is presented in ([Dorigoni 14](#Dorigoni14))

Thus it seems possible (and sometimes is claimed) that the perturbative series already contains all nonperturbative information. The future will tell.

## Related concepts

* [[resummation]]

* [[Borel resummation]]

## References

### General

Original articles include

* {#JentschuraZinnJustin04} Jentschura, [[Jean Zinn-Justin]], Phys. Lett. B 596 (2004), 138-144

* {#DunneUnsal14} [[Gerald Dunne]], Mithat Unsal, _Uniform WKB, Multi-instantons, and Resurgent Trans-Series_, Phys. Rev. D 89, 105009 (2014) ([arXiv:1401.5202](https://arxiv.org/abs/1401.5202))

A clear account is in

* M. V. Berry, C. J. Howls, _Hyperasymptotics for Integrals with Saddles_, Proceedings: Mathematical and Physical Sciences Vol. 434, No. 1892 (Sep. 9, 1991), pp. 657-675 ([jstor:51890](https://www.jstor.org/stable/51890))

See p. 9 of https://arxiv.org/abs/1206.1890

Review includes

* {#Marino12} [[Marcos Marino]], _Lectures on non-perturbative effects in large N gauge theories, matrix models and strings_ ([arXiv:1206.6272](https://arxiv.org/abs/1206.6272))

* {#Dorigoni14} Daniele Dorigoni, _An Introduction to Resurgence, Trans-Series_ ([arXiv:1411.3585](https://arxiv.org/abs/1411.3585))

* {#Costin17} Ovidiu Costin, _The Mathematical Theory of Resurgence and Resummation, and Their Applications in Mathematics and Physics_, 2017 ([web](http://online.kitp.ucsb.edu/online/resurgent_c17/costin/))

* [[Gerald Dunne]], _Introduction to Resurgence, Trans-series and Non-perturbative Physics_, 2018 ([pdf](https://www.icts.res.in/sites/default/files/NUMSTRINGS2018-2018-01-27-Gerald-Dunne.pdf))

Part of the above text is taken from

* {#Neumaier16} [[Arnold Neumaier]], [PO comment](https://physicsoverflow.org/36857/what-are-resurgent-transseries?show=36859#a36859), 2016

See also

* Marco Serone, Gabriele Spada, Giovanni Villadoro, _Instantons from Perturbation Theory_ ([arXiv:1612.04376](https://arxiv.org/abs/1612.04376))

* Itzykson seminar at IHES,  _Resurgence and Quantization_, April 2017 ([web](https://www.fondation-hadamard.fr/fr/lmh-mathematiques-et-physique/seminaire-itzykson))

### In string theory

* [[Ricardo Schiappa]], _Resurgence Asymptotics in String Theory_, talk at IHES 2017 ([video](https://www.youtube.com/watch?v=8tw4MaBwTEA))

In the [[SYK model]] and [[AdS2-CFT1 duality]]:

* [[Paolo Gregori]], [[Ricardo Schiappa]], *From Minimal Strings towards Jackiw-Teitelboim Gravity: On their Resurgence, Resonance, and Black Holes* ([arXiv:2108.11409](https://arxiv.org/abs/2108.11409))

[[!redirects resurgence]]


retarded product

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## Idea

In [[perturbative quantum field theory]] formalized as [[causal perturbation theory]]/[[perturbative AQFT]] the _retarded products_ ([Glaser-Lehmann-Zimmermann 57](#GlaserLehmannZimmermann57), [Steinmann 71](#Steinmann71)) is a system of [[operator-valued distributions]] which are the [[coefficients]] in the [[formal power series]] expression for the [[quantum observables on interacting fields]].

If the quantum observables are obtained via [[Bogoliubov's formula]] from the [[S-matrix]] induced by [[time-ordered products]], then the retarded products may be expressed in terms of the [[time-ordered products]]. But the retarded products may also be [[axiom|axiomatized]] directly ([Dütsch-Fredenhagen 04](#DuetschFredenhagen04)),
see ([Collini 16, section 2.2](#Collini16)).

## Definition

Let $\Sigma$ be a [[spacetime]] of [[dimension]] $p + 1$ and let $E \overset{fb}{\longrightarrow} \Sigma$ be a [[field bundle]]. Let $\mathbf{L}_{free}\in \Omega^{p+1,0}_\Sigma(E)$ be a [[local Lagrangian density]] for a [[free field theory]] with [[field (physics)|fields]] of type $E$. Let $\mathcal{W}$ be the corresponding [[Wick algebra]] of [[quantum observables]] of the free field, with

$$
  \mathcal{F}_{loc} \overset{:(-):}{\longrightarrow} \mathcal{W}
$$

the corresponding quantization map from [[local observables]] ("[[normal ordering]]").

Let then

$$
  S
  \;\colon\;
  \mathcal{F}_{loc}\langle g,j\rangle
    \longrightarrow
  \mathcal{W}[ [ g/\hbar ] ][ [ j/\hbar ] ]
$$

be a  [[perturbative S-matrix]]. Moreover let

$$
  g_{sw} \mathbf{L}_{int} \in \mathcal{F}_{loc}\langle g\rangle
$$

be an [[adiabatic switching|adiabatically switched]]
[[interaction]] [[Lagrangian density]], so that the full Lagrangian density is

$$
  \mathbf{L} = \mathbf{L}_{free} + g \mathbf{L}_{int}
  \,.
$$

For $A \in \mathcal{F}_{loc}$ a [[local observable]] and $j \in C^\infty_{cp}(\Sigma)$, write

$$
  Z_{\mathbf{L}_{int}}(j A)
  \; \coloneqq \;
  S^{-1}(g_{sw}\mathbf{L}_{int}) S( g_{sw}\mathbf{L}_{int} + j A )
$$

for the [[generating function]] induced by the perturvbative [[S-matrix]].

+-- {: .num_defn #RetardedProductFromPerturbativeSMatrix}
###### Definition
**([[retarded products]] induced from perturbative S-matrix)

It follows from the "perturbation" axiom on the [[S-matrix]] (see there) that there is a system of [[continuous linear functionals]]

$$
  R
    \;\colon\;
  \left(\mathcal{F}_{loc}\langle g\rangle\right)^{\otimes^k}
    \otimes
  (\mathcal{F}_{loc})^{\otimes^l}
    \longrightarrow
  \mathcal{W}[ [ g/\hbar] ] [ [ j/\hbar] ]
$$

for all $k,l \in \mathbb{N}$ such that the [[generating function]] induced by the [[S-matrix]] is expressed as

$$
  Z_{g_{sw} L}(j_{sw} A)
  =
  \underset{k,l \in \mathbb{N}}{\sum} \frac{1}{k! l!}
  R(
    \underset{k \,\text{arguments}}{\underbrace{ g_{sw} L \cdots g_{sw} L } },
    \underset{l \; \text{arguments}}{\underbrace{ j_{sw} A \cdots j_{sw} A }}
  )
  \,.
$$

Similarly there is

$$
  R
    \;\colon\;
  \left(\mathcal{F}_{loc}\langle g \rangle\right)^{\otimes^k}
    \otimes
  \left(\mathcal{F}_{loc}\langle j \rangle\right)
    \longrightarrow
  \mathcal{W}[ [ g ] ] [ [ \hbar ] ]
$$

such that

$$
  \widehat{A}(h)
  =
  \underset{k \in \mathbb{N}}{\sum} \frac{1}{k!}
  r( \underset{k \,\text{arguments}}{\underbrace{g_{sw}L_{int} \cdots g_{sw}L_{int}}}, h A )
  \,.
$$

These are called the (generating) _retarded products_ ([Glaser-Lehmann-Zimmermann 57](#GlaserLehmannZimmermann57), [Epstein-Glaser 73, section 8.1](#EpsteinGlaser73)).

=--

The actual retarded products are, via _[[Bogoliubov's formula]]_, the [[derivatives]] of these generating retarded products with respect to the [[source field]].

## Related concepts

[[!include products in pQFT -- table]]

## References

The concept goes back to

* {#GlaserLehmannZimmermann57} [[Vladimir Glaser]], [[Harry Lehmann]], [[Wolfhart Zimmermann]], _Field operators and retarded functions_, Il Nuovo Cimento 6, 1122-1128 (1957) ([doi:10.1007/bf02747395](http://dx.doi.org/10.1007/bf02747395))

* {#Steinmann71} O. Steinmann, _Perturbation Expansions in Axiomatic Field Theory_, Lecture Notes in Physics, vol. 11, Springer, Berlin and Heidelberg, 1971.

Discussion of retarded products as derived from [[time-ordered products]] in [[causal perturbation theory]] is due to

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], section 8.1 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 ))

Direct axiomatization of retarded products is due to

* {#DuetschFredenhagen04} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Causal perturbation theory in terms of retarded products, and a proof of the Action Ward Identity_, Rev. Math. Phys. 16, 1291 (2004) ([arXiv:hep-th/0403213](https://arxiv.org/abs/hep-th/0403213))

reviewed for instance in

* {#Collini16} [[Giovanni Collini]], section 2.2 of _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

A textbook account is in

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

[[!redirects retarded products]]


retract

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#### Category theory
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#### Idempotents
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#Contents#
* table of contents
{:toc}

## Definition

An [[object]] $A$ in a [[category]] is called a **retract** of an object $B$ if there are [[morphisms]] $i\colon A\to B$ and $r \colon B\to A$ such that $r \circ i = id_A$. In this case $r$ is called a **retraction** of $B$ onto $A$.

$$
  id
    \;\colon\;
  A \underoverset{section}{i}{\to} B \underoverset{retraction}{r}{\to} A
  \,.
$$

Here $i$ may also be called a _[[section]]_ of $r$. (In particular if $r$ is thought of as exhibiting a [[bundle]]; the terminology originates from [[topology]].)

Hence a **retraction** of a [[morphism]] $i  \;\colon\;  A \to B$ is a _left-[[inverse]]_.

In this situation, $r$ is a [[split epimorphism]] and $i$ is a [[split monomorphism]]. The entire situation is said to be a _splitting of  the [[idempotent]]_

$$
  B \stackrel{r}{\longrightarrow} A \stackrel{i}{\longrightarrow} B
  \,.
$$

Accordingly, a **[[split monomorphism]]** is a morphism that *has* a retraction; a **[[split epimorphism]]** is a morphism that *is* a retraction.

## Properties

+-- {: .num_lemma #LeftInverseWithLeftInverseIsLeftInverse}
###### Lemma
**([[left inverse]] with [[left inverse]] is [[inverse]])**

Let $\mathcal{C}$ be a [[category]], and let $f$ and $g$ be [[morphisms]] in $\mathcal{C}$, such that $g$ is a [[left inverse]] to $f$:

$$
  g \circ f = id
  \,.
$$

If $g$ itself has a left inverse $h$

$$
  h \circ g = id
$$

then $h = f$ and $g = f^{-1}$ is an actual (two-sided) [[inverse morphism]] to $f$.

=--

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

Since [[inverse morphisms]] are unique if they exists, it is sufficient to show that

$$
  f \circ g = id
  \,.
$$

Compute as follows:

$$
  \begin{aligned}
    f \circ g
    & =  \underset{ = id}{\underbrace{h \circ g}} \circ f \circ g
    \\
    & =
    h \circ \underset{= id}{\underbrace{g \circ f}} \circ g
    \\
    & = h \circ g
    \\
    & = id
  \end{aligned}
$$

=--

+-- {: .num_remark #RetractsPreservedByFunctor}
###### Remark

Retracts are clearly preserved by any [[functor]].

=--

+-- {: .num_remark #SplitEpisAndMonos}
###### Remark

A [[split epimorphism]] $r; B \to A$ is the strongest of various notions of [[epimorphism]] (e.g., it is a [[regular epimorphism]], in fact an [[abolute limit|absolute]] [[coequalizer]], being the coequalizer of a pair $(e, 1_B)$ where $e = i \circ r: B \to B$ is idempotent). Dually, a [[split monomorphism]] is the strongest of various notions of monomorphism.

=--

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

If an object $B$ has the [[left lifting property]] against a morphism $X \to Y$, then so does every of its retracts $A \to B$:

$$
  \left(
  \array{
     && X
     \\
     & {}^{\mathllap{\exists}}\nearrow& \downarrow
     \\
     A &\to& Y
  }
  \right)
  \;\;\;\;
  :=
  \;\;\;\;
  \left(
  \array{
     && && &&  X
     \\
     &&& {}^{\mathllap{\exists}}\nearrow& && \downarrow
     \\
     A &\to& B &\to& A &\to& Y
  }
  \right)
$$

=--

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

Let $C$ be a [[category]] with [[split idempotent]]s and write $PSh(C) = [C^{op}, Set]$ for its [[presheaf category]]. Then a retract of a [[representable functor]] $F = PSh(C)$ is itself representable.

=--

This appears as ([Borceux, lemma 6.5.6](#Borceux))

## Examples

### To the point

* In a category with [[terminal object]] $*$ every morphism of the form $* \to X$ is a section, and the unique morphism $X \to *$ is the corresponding retraction.

### Of simplices

The inclusion of standard topological [[horn]]s into the topological [[simplex]] $\Lambda^n_k \hookrightarrow \Delta^n$ is a retract in [[Top]].

### In arrow categories

Let $\Delta[1] = \{0 \to 1\}$ be the [[interval category]]. For every category $C$ the [[functor category]] $[\Delta[1], C]$ is the [[arrow category]] of $C$.

In the theory of [[weak factorization systems]] and [[model categories]], an important role is played by retracts in $C^{\Delta[1]}$, the [[arrow category]] of $C$.  Explicitly spelled out in terms of the original category $C$, a morphism $f:X\to Y$ is a retract of a morphism $g:Z\to W$ if we have commutative squares

$$
  \array{
     id_X \colon & X & \to & Z & \to & X
     \\
     & f \downarrow  & & g \downarrow & & \downarrow f
     \\
     id_Y \colon & Y & \to & W & \to & Y
  }
$$

such that the top and bottom rows compose to identities.

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

Classes of morphisms in a category $C$ that are given by a left or right [[lifting property]] are preserved under retracts in the [[arrow category]] $[\Delta[1],C]$. In particular the defining classes of a [[model category]] are closed under retracts.

=--

This is fairly immediate, a proof is made explicit [here](ClosurePropertiesOfInjectiveAndProjectiveMorphisms).

This implies:

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

In every category $C$ the class of [[isomorphisms]] is preserved under retracts in the [[arrow category]] $[\Delta[1], C]$

=--

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

This is also checked directly: for

$$
  \array{
    id \colon & a_1 &\to& a_2 &\to& a_1
    \\
    & \downarrow && \downarrow && \downarrow
    \\
    id \colon & b_1 &\to& b_2 &\to& b_1
  }
$$

a retract diagram and $a_2 \to b_2$ an isomorphism, the inverse to $a_1 \to b_1$ is given by the composite

$$
  \array{
    &  & & a_2 &\to& a_1
    \\
    & && \uparrow &&
    \\
    & b_1 &\to& b_2 &&
  }
  \,,
$$

where $b_2 \to a_2$ is the inverse of the middle morphism.

=--

### Retracts of diagrams
 {#RetractsOfDiagrams}

For the following, let $C$ and $J$ be categories and write $J^{\triangleleft}$ for the [[join of quasi-categories|join]] of $J$ with a single [[initial object]], so that [[functor]]s $J^{\triangleleft} \to C$ are precisely [[cone]]s over functors $J \to C$. Write

$$
  i : J \to J^{\triangleleft}
$$

for the canonical inclusion and hence $i^* F$ for the underlying diagram of a cone $F : J^{\triangleleft} \to C$. Finally, write $[J^{\triangleleft}, C]$ for the [[functor category]].

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

If $Id: F_1 \hookrightarrow F_2 \to F_1$ is a [[retract]] in the category $[J^{\triangleleft}, C]$ and $F_2 : J^{\triangleleft} \to C$ is a [[limit]] [[cone]] over the [[diagram]] $i^* F_2 : J \to C$, then also $F_1$ is a limit cone over $i^* F_1$.

=--

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

We give a direct and a more abstract argument.

**Direct argument**. We can directly check the [[universal property]] of the limit: for $G$ any other [[cone]] over $i^* F_1$, the composite $i^* G = i^* F_1 \to i^* F_2$ exhibits $G$ also as a cone over $i^* F_2$. By the pullback property of $F_2$ this extends to a morphism of cones $G \to F_2$. Postcomposition with $F_2 \to F_1$ makes this a morphism of cones $G \to F_1$. By the injectivity of $F_1 \to F_2$ and the universality of $F_2$, any two such cone morphisms are equals.

**More abstract argument**. The limiting cone over a diagram $D : J \to C$ may be regarded as the right [[Kan extension]] $i_* D := Ran_i D$ along $i$

$$
  \array{
    J &\stackrel{D}{\to}& C
    \\
    {}^{\mathllap{i}}\downarrow & \nearrow_{i_* D}
    \\
    J^{\triangleleft}
  }
  \,.
$$

Therefore a cone $F : J^{\triangleleft} \to C$ is limiting precisely if the $(i^* \dashv i_*)$-[[unit of an adjunction|unit]]

$$
  F \stackrel{}{\to} i_* i^* F
$$

is an [[isomorphism]]. Since this unit is a [[natural transformation]] it follows that applied to the retract diagram

$$
  Id : F_1 \hookrightarrow F_2 \to F_1
$$

it yields the retract diagram

$$
  \array{
    Id : & F_1 &\to& F_2 &\to& F_1
    \\
    & \downarrow && \downarrow && \downarrow
    \\
    Id : & i_* i^* F_1 &\to& i_* i^* F_2 &\to& i_* i^* F_1
  }
$$

in $[\Delta[1], [J^{\triangleleft}, C]]$. Here by assumption the middle morphism is an isomorphism. Since isomorphisms are stable under retract, by prop. \ref{RetractOfIso}, also the left and right vertical morphism is an isomorphism, hence also $F_1$ is a limiting cone.

=--

This argument generalizes form limits to [[homotopy limit]]s.

For that, let now $C$ be a [[category with weak equivalences]] and write $Ho(C) : Diagram^{op} \to Cat$ for the corresponding [[derivator]]: $Ho(C)(J) := [J,C](W^J)^{-1}$ is the [[homotopy category]] of $J$-diagrams in $C$, with respect to the degreewise weak equivalences in $C$.

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

Let

$$
  Id : F_1 \to F_2 \to F_1
$$

be a retract in $Ho(C)(J^{\triangleleft})$. If $F_2$ is a [[homotopy limit]] cone over $i^* F_2$, then also $F_1$ is a homotopy limit cone over $i^* F_1$.

=--

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

By the discussion at [[derivator]] we have that

1. $i_* : Ho(C)(J) \to Ho(C)(J^{\triangleleft})$ forms [[homotopy limit]] cones;

1. $F \to i_* i^* F$ is an isomorphism precisely if $F$ is a homotopy limit cone.

With this the claim follows as in prop. \ref{RetractsOfLimits}.

=--

## Related concepts

* [[retractive space]]

* [[deformation retract]], [[neighbourhood retract]]

* [[absolute retract]]

* [[dominant functor]]

## References

* [[Karol Borsuk]], _Theory of retracts_

* [[Sibe Mardešić]], _Absolute Neighborhood  Retracts and Shape  Theory_ ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/mardesic.pdf))

* {#Borceux} [[Francis Borceux]], Def. 1.7.3 and Sec. 6.5 of: _[[Handbook of Categorical Algebra]] I_

See also

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

[[!redirects retracts]]

[[!redirects retraction]]
[[!redirects retractions]]

[[!redirects left inverse]]
[[!redirects left inverses]]
[[!redirects left-inverse]]
[[!redirects left-inverses]]


retract argument

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The "retract argument" is a simple but frequently useful standard lemma in discussion of [[weak factorization systems]]. It asserts that if a [[morphism]] factors as the [[composition]] of two factors such that it is has the left or right [[lifting property]] against its second or first factor, respectively, then it is a [[retract]] (as an [[object]] of the [[arrow category]]) of the respective other factor.

The retract argument is frequently used in the verification of the [[axioms]] of [[model category]] structures.

## Statement

+-- {: .num_lemma #RetractArgument}
###### Lemma
**(retract argument)**

Consider a [[composition|composite]] [[morphism]]

$$
  f \;\colon\; X\stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y
  \,.
$$

Then:

1. If $f$ has the [[left lifting property]] against $p$, then $f$ is a [[retract]] of $i$.

1. If $f$ has the [[right lifting property]] against $i$, then $f$ is a [[retract]] of $p$.

=--

+-- {: .num_remark #RetractsOfMorphisms}
###### Remark

Here by a _retract_ of a [[morphism]] $X \stackrel{f}{\longrightarrow} Y$ in some [[category]] $\mathcal{C}$ is meant a [[retract]] of $f$ as an object in the [[arrow category]] $\mathcal{C}^{\Delta[1]}$, hence a morphism $A \stackrel{g}{\longrightarrow} B$ such that in $\mathcal{C}^{\Delta[1]}$ there is a factorization of the identity on $g$ through $f$

$$
  id_g \;\colon\;
  g \longrightarrow f \longrightarrow g
  \,.
$$

This means equivalently that in $\mathcal{C}$ there is a [[commuting diagram]] of the form

$$
  \array{
    id_A \colon & A &\longrightarrow& X &\longrightarrow& A
    \\
    & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}}
    \\
    id_B \colon & B &\longrightarrow& Y &\longrightarrow& B
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof
(of lemma \ref{RetractArgument})

We discuss the first statement, the second is [[formal duality|formally dual]].

Write the factorization of $f$ as a [[commuting square]] of the form

$$
  \array{
    X &\stackrel{i}{\longrightarrow}& A
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    Y &= & Y
  }
  \,.
$$

By the assumed [[lifting property]] of $f$ against $p$ there exists a diagonal filler $g$ making a [[commuting diagram]] of the form

$$
  \array{
    X &\stackrel{i}{\longrightarrow}& A
    \\
    {}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}}
    \\
    Y &= & Y
  }
  \,.
$$

By rearranging this diagram a little, it is equivalent to

$$
  \array{
    & X &=& X
    \\
    & {}^{\mathllap{f}}\downarrow
     &&
    {}^{\mathllap{i}}\downarrow
    \\
    id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}&
    Y
  }
  \,.
$$

Completing this to the right, this yields a diagram exhibiting the required retract according to remark \ref{RetractsOfMorphisms}:

$$
  \array{
    id_X \colon & X &=& X &=& X
    \\
    & {}^{\mathllap{f}}\downarrow
     &&
    {}^{\mathllap{i}}\downarrow
     &&
     {}^{\mathllap{f}}\downarrow
    \\
    id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}&
    Y
  }
  \,.
$$

=--


retraction > history

> see at _[[retract]]_


retractive space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[parameterized homotopy theory]], by a *retractive space* (older terminology: "ex-space" [[James (1995)](#James95)], cf. footnote 1, [p. 19](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf#page=27) in [May & Sigurdsson (2006)](#MaySigurdsson06)) one means a [[retraction]] in a given [[category]] $\mathcal{S}$ of [[model category|models]] for [[homotopy types]] (usually in [[TopologicalSpaces]] or [[SimplicialSets]]), to be thought of as a *[[bundle]] $p \colon E \to B$ of [[pointed topological spaces]]* over a *base space* $B$, where the [[section]] $i \colon B \to E$ exhibits the [[fiber]]-wise [[base points]].

Just as plain [[pointed topological spaces]] serve as the basis on which to construct [[spectra]], so retractive spaces serve as a basis on which to construct [[parameterized spectra]].

## Definition

A *retractive space* is a [[commuting diagram]] in a [[category]] $\mathcal{S}$ "of spaces", of this form:

\begin{tikzcd}[
    column sep=10pt
  ]
    & E
    \ar[
      d,
      "{ p }"
    ]
    \\
    B
    \ar[r, equals]
    \ar[ur, "i"]
    &
    B
    \mathrlap{\,.}
\end{tikzcd}

Taking the [[category]] $\mathcal{S}_{\mathcal{R}}$ of retractive spaces to have as [[morphisms]] the evident [[commuting diagrams]]

\begin{tikzcd}[
  sep=20pt
]
    B
    \ar[d, "{i}"]
    \ar[rr, "{f}"]
    &&
    B'
    \ar[d, "{ i' }"]
    \\
    E
    \ar[d, "{p}"]
    \ar[rr, "{\widehat f}"]
    &&
    E'
    \ar[d, "{p'}"]
    \\
    B
    \ar[rr, "{f}"]
    &&
    B'
\end{tikzcd}

reflecting [[bundle]]-[[homomorphisms]] respecting the [[base points]], this is [[equivalence of categories|equivalent]] to the [[Grothendieck construction]] on the [[pseudo-functor]]

$$
  \big(\mathcal{S}_{/(-)}\big)^{\ast/}
  \;\colon\;
  \mathcal{S} \longrightarrow Cat
$$

which sends

* spaces $B \in \mathcal{B}$ to the [[pointed category]] $\big(\mathcal{S}_{/B}\big)^{\ast/}$ of [[pointed objects]] in the [[slice category]] of $\mathcal{S}$ over $B$ (i.e. in the category of "[[bundles]]" over the fixed base space $B$),

* morphisms $f \colon B \to B'$ of base spaces to the [[functor]] $f_!$ forming [[pushouts]] of bundles along $f_!$:

\begin{tikzcd}[
    row sep=30pt
  ]
    B
    \ar[d, "{i}"{description}]
    \ar[rr, "{f}"{description}]
    \ar[
      drr,
      phantom,
      "{
        \scalebox{.6}{
          (po)
        }
      }"{pos=.7}
    ]
    &&
    B'
    \ar[d, "{ f_!(i) }"{description}]
    \\
    E
    \ar[d, "{p}"{description}]
    \ar[rr, "{\widehat f}"{description}]
    \ar[
      drr,
      phantom,
      "{
        \scalebox{.6}{
          (po)
        }
      }"{pos=.7}
    ]
    &&
    f_! E
    \ar[d, "{f_!(p)}"{description}]
    \\
    B
    \ar[
      rr,
      "{f}"{description}
    ]
    &&
    B'
\end{tikzcd}

i.e.

$$
  \mathcal{S}_{\mathcal{R}}
  \;\simeq\;
  \int_{B \in \mathcal{S}} \big(\mathcal{S}_{/B}\big)^{\ast/}
  \,.
$$

This follows readily from the definitions, but see also [Braunack-Mayer (2021), Rem. 1.15](#BraunackMayer19); [Hebestreit, Sagave & Schlichtkrull (2020), Lem. 2.14](#HebestreitSagaveSchlichtkrull20).

## Related concepts

* [[retraction]]

* [[parameterized homotopy theory]]

* [[parameterized spectrum]]

* [[model structure for parameterized spectra]]

## References

The terminology "ex-spaces" is due to Ioan James, used for instance in:

* {#James95}  [[Ioan Mackenzie James]], _[[Introduction to fibrewise homotopy theory]]_ in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_ (1995)

Early discussion of their [[model category]] structures includes

* Michele Intermont, [[Mark Johnson]], *Model structures on the category of ex-spaces*, Topology and its Applications **119** 3 (2002) 325-353 [doi:10.1016/S0166-8641(01)00076-1]

Discussion in the context of [[model structures for parameterized spectra]]:

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], Sections 1.3 and 8.5 of: _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006   ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

* {#BraunackMayer21} [[Vincent Braunack-Mayer]], *Combinatorial parametrised spectra*, Algebr. Geom. Topol. **21** (2021) 801-891 [[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), [doi:10.2140/agt.2021.21.801](https://doi.org/10.2140/agt.2021.21.801)]

  > (based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]], 2018)

* {#HebestreitSagaveSchlichtkrull20} [[Fabian Hebestreit]], [[Steffen Sagave]], [[Christian Schlichtkrull]], *Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces*, Forum of Mathematics, Sigma **8** (2020) e16 [[arXiv:1904.01824](https://arxiv.org/abs/1904.01824), [doi:10.1017/fms.2020.11](https://doi.org/10.1017/fms.2020.11)]

* [[Cary Malkiewich]], Section 2.1 in: *Parametrized spectra, a low-tech approach* [[arXiv:1906.04773](https://arxiv.org/abs/1906.04773), user guide: [pdf](https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pdf), [[Malkiewich-ParametrizedSpectraUserGuide.pdf:file]]]

[[!redirects retractive spaces]]

[[!redirects retractive topological space]]
[[!redirects retractive topological spaces]]

[[!redirects ex-space]]
[[!redirects ex-spaces]]

[[!redirects ex-object]]
[[!redirects ex-objects]]


retrocell

## Idea

In a [[double category]] $C$ with [[companions]], a __retrocell__ is …

## Example

In the [[double category]] of [[spans]], a retrocell boils down to a morphism of spans, where the source span is the [[pullback]] of the original source span along the corresponding horizontal arrow.  (Paré, §5.)

## References

* [[Robert Paré]], _Retrocells_, [arXiv:2306.06436](https://arxiv.org/abs/2306.06436).


retrofunctor

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

#Contents#
* table of contents
{:toc}

## Idea ##

A **retrofunctor** (also called a **cofunctor**) is a kind of morphism between [[category|categories]]. In contrast to a [[functor]], the assignment on objects of a retrofunctor goes in the _opposite_ direction to the assignment on morphisms.

Retrofunctors generalise both [[bijective on objects functor|bijective-on-objects functors]] and [[discrete fibration|discrete opfibrations]].

Retrofunctors arise naturally in the study [[internal category in a monoidal category|non-cartesian internal categories]].

In the literature, the terminology "cofunctor" is common. However, terminology is misleading, as a cofunctor is not the [[opposite]] of a functor. We use the terminology "retrofunctor", as this avoids confusion, and aligns with the concept of [[monad retromorphism]] (of which retrofunctors are a motivating example).

## Definition ##

A **retrofunctor** $\varphi : A \nrightarrow B$ from a [[category]] $A$ to a category $B$ consists of a
map sending each [[object]] $a \in A$ to an object $\varphi_{0}a \in B$
and a map sending each pair $(a \in A, u : \varphi_{0}a \to b \in B)$ to a [[morphism]] $\varphi_{1}(a, u) : a \to a'$ in $A$, where $a' = cod(\varphi_{1}(a, u))$, such that

* $\varphi_{1}$ respects codomains: $\varphi_{0}a' = cod(u)$ where $a' = cod(\varphi_{1}(a, u))$,

* $\varphi_{1}$ preserves identity morphisms: $\varphi_{1}(a, 1_{\varphi_{0}a}) = 1_{a}$,

* $\varphi_{1}$ preserves composition: $\varphi_{1}(a, v \circ u) = \varphi_{1}(a', v) \circ \varphi_{1}(a, u)$ where $a' = cod(\varphi_{1}(a, u))$.

\begin{centre}
\begin{tikzcd}
A
\arrow[d, "\varphi"', "/"{anchor=center,sloped}]
& a
\arrow[d, phantom, "\vdots"]
\arrow[r, "{\varphi_{1}(a, u)}"]
& a'
\arrow[d, phantom, "\vdots"]
\\
B
& \varphi_{0} a
\arrow[r, "u"]
& b = \varphi_{0}a'
\end{tikzcd}
\end{centre}

Given a pair of retrofunctors $\varphi : A \nrightarrow B$ and $\gamma : B \nrightarrow C$, their composite retrofunctor $\gamma \circ \varphi \colon A \nrightarrow C$ sends each object $a \in A$ to an object $\gamma_{0}\varphi_{0}a \in C$
and each pair $(a \in A, u : \gamma_{0}\varphi_{0}a \to c \in C)$ to a
morphism $\varphi_{1}(a, \gamma_{1}(\varphi_{0}a, u))$ in $A$.
This defines a category $\mathbf{Cof}$ whose objects are [[small category|small categories]], and whose morphisms are retrofunctors.

## Properties ##

\begin{proposition}
\label{Factorisation}
The category $\mathbf{Cof}$ of small categories and retrofunctors has an [[orthogonal factorization system]] $(Bij^{op}, DOpf)$ which factors each retrofunctor $\varphi : A \nrightarrow B$ into a [[bijective on objects functor]] $A \leftarrow I$ followed by a [[discrete fibration|discrete opfibration]] $I \to B$.
\end{proposition}
\begin{proof}
See ([Garner 2019, Proposition 5](#Garner19))
\end{proof}

\begin{proposition}
Let $\mathbf{Poly}(1, 1)$ be the monoidal category arising from the [[polynomial functor|bicategory of polynomials]] on the singleton set.
Then $\mathbf{Cof}$ is isomorphic to the category of comonoids in $\mathbf{Poly}(1, 1)$.
\end{proposition}
\begin{proof}
Originally proven in ([Ahman-Uustalu 2016](#AhmanUustalu16)). See ([Spivak-Niu 2021, Theorem 6.26](#SpivakNiu21)) for details.
\end{proof}

## Examples ##

* Every [[function]] $A \to B$ yields a retrofunctor $disc(A) \nrightarrow disc(B)$ between [[discrete category|discrete categories]]. This defines a fully faithful functor $\mathbf{Set} \to \mathbf{Cof}$.

* Every [[monoid]] [[homomorphism]] $A \to B$ yields a retrofunctor $B \nrightarrow A$. This defines a fully faithful functor $\mathbf{Mon} \to \mathbf{Cof}^{op}$.

* Every [[bijective on objects functor]] $A \to B$ yields a retrofunctor $B \nrightarrow A$.

* Every [[discrete fibration|discrete opfibration]] $A \to B$ yields a retrofunctor $A \nrightarrow B$.

* Every split [[Grothendieck fibration|Grothendieck opfibration]] has an underlying retrofunctor given by the [[cleavage|splitting]].

* More generally, every [[delta lens]] has an underlying retrofunctor.

* Let $\mathbb{N}$ denote the [[monoid]] of [[natural numbers]] under addition. A retrofunctor $\tau : A \to \mathbb{N}$ is the same as a choice of morphism $\tau(a, 1)$ out of every object in $a \in A$.

* A [[comorphism]] of [[Lie groupoids]] is an [[internal category|internal]] retrofunctor in the category [[Diff]] of [[smooth manifold|smooth manifolds]]; see ([Higgins-Mackenzie 1993, definition 5.12](#HigginsMackenzie93)).

## References ##

The notion of retrofunctor first appeared under the name _comorphism_ in the paper:

* {#HigginsMackenzie93} [[Philip J. Higgins]], [[Kirill C. H. Mackenzie]], _Duality for base-changing morphisms of vector bundles, modules, Lie algebroids and Poisson structures_, Mathematical Proceedings of the Cambridge Philosophical Society, 114, 1993 ([doi:10.1017/S0305004100071760](http://dx.doi.org/10.1017/S0305004100071760))

The definition of [[internal category|internal]] retrofunctor between [[internal category in a monoidal category|non-cartesian internal categories]] was introduced in Section 4.4 of the thesis:

* {#Aguiar97} [[Marcelo Aguiar]], _Internal categories and quantum groups_, PhD thesis, Cornell University, 1997 ([pdf](http://pi.math.cornell.edu/~maguiar/thesis2.pdf))

The characterization of retrofunctors as morphisms between directed [[polynomial functor|containers]] is developed in the papers:

* {#AhmanUustalu16} [[Danel Ahman]], [[Tarmo Uustalu]], _Directed Containers as Categories_, EPTCS, 207, 2016 ([doi:10.4204/EPTCS.207.5](http://dx.doi.org/10.4204/EPTCS.207.5))

* {#AhmanUustalu17} [[Danel Ahman]], [[Tarmo Uustalu]], _Taking updates seriously_, CEUR Workshop Proceedings, 1827, 2017 ([pdf](http://ceur-ws.org/Vol-1827/paper11.pdf))

The link between retrofunctors, [[delta lenses]], and split [[Grothendieck fibration|Grothendieck opfibrations]] is developed in the papers:

* {#Clarke20a} [[Bryce Clarke]], _Internal lenses as functors and cofunctors_, EPTCS, 323, 2020 ([doi:10.4204/EPTCS.323.13](http://dx.doi.org/10.4204/EPTCS.323.13))

* {#Clarke20b} [[Bryce Clarke]], _Internal split opfibrations and cofunctors_, Theory and Applications of Categories, 35, 2020 ([link](http://www.tac.mta.ca/tac/volumes/35/44/35-44abs.html))

Retrofunctors between [[groupoid|groupoids]] and the link with [[inner automorphism|inner automorphisms]] of groupoids is explored in the paper:

* {#Garner19} [[Richard Garner]], _Inner automorphisms of groupoids_, preprint, 2019 ([arXiv:1907.10378](https://arxiv.org/abs/1907.10378))

The notion of retrofunctor between partite [[internal category|internal categories]] is introduced in Definition 5.5 of the paper:

* {#CockettGarner20} [[Robin Cockett]], [[Richard Garner]], _Generalising the étale groupoid--complete pseudogroup correspondence_, preprint, 2020 ([arXiv:2004.09699](https://arxiv.org/abs/2004.09699))

A detailed account of the relationship between retrofunctors and polynomials appears in Chapter 6 of the draft textbook:

* {#SpivakNiu21} [[David Spivak]], Nelson Niu, _Polynomial Functors: A General Theory of Interaction_, 2021 ([pdf](https://topos.site/poly-book.pdf))

Enrichment in the category of $\mathbf{Cof}$ of categories and retrofunctors is considered in the paper:

* [[Kristopher Brown]], [[David I. Spivak]], _Dynamic Tracing: a graphical language for rewriting protocols_, 2023. ([arXiv:2304.14950](https://arxiv.org/abs/2304.14950))

The terminology _retrofunctor_ was introduced in:

* [[Matthew Di Meglio]], _The category of asymmetric lenses and its proxy pullbacks_. Master’s thesis. Macquarie University, 2021. doi: 10.25949/20236449, [pdf](https://mdimeglio.github.io/papers/The_category_of_asymmetric_lenses_and_its_proxy_pullbacks.pdf)

The terminology is justified by the fact that retrofunctors are [[monad retromorphisms]] in the double category $Span(Set)$.

[[!redirects cofunctor]]
[[!redirects cofunctors]]
[[!redirects retrofunctor]]
[[!redirects retrofunctors]]


Revaz Gamkrelidze

* [Wikipedia entry](https://en.wikipedia.org/wiki/Revaz_Gamkrelidze)

* _Revaz Gamkrelidze – 90_,  Mem. Differential Equations Math. Phys. 70 (2017), 1–6 ([pdf](https://www.emis.de/journals/MDEMP/vol70/vol70-1.pdf))

## Selected writings

On [[Lev Pontryagin]]'s work:

* {#Gamkrelidze86} [[Revaz Gamkrelidze]] (ed.), _L. S. Pontryagin -- Selected Works_, Volume 1: _Selected Research Papers_,  [Classics of Soviet Mathematics](https://www.routledge.com/Classics-of-Soviet-Mathematics/book-series/CRCCLASOVMTH), Gordon and Breach 1986, CRC Press 2019 ([ISBN:9782881241055](https://www.routledge.com/Selected-Research-Papers/Pontryagn/p/book/9782881241055))

category: people


reverse lattice of filters

It is convenient for the [[poset]] $\mathfrak{F}(A)$ of [[filters]] on a given [[set]] $A$ to have the reverse order $\sqsubseteq$ to set-theoretic [[subset|inclusion]]:
$$ a \sqsubseteq b \Leftrightarrow a \supseteq b .$$
We will call it the __reverse lattice of filters__ (since it is a [[lattice]]); compare [[reverse poset]].

We can denote the lattice operations on $\mathfrak{F}(A)$ as $\sqsubseteq$, $\sqcap$, $\sqcup$, $⨅$, $\bigsqcup$. By $0^{\mathfrak{F}(A)}$ and $1^{\mathfrak{F}(A)}$ we denote the [[bottom|minimal]] and [[top|maximal]] elements of this poset. So set-theoretically, $\mathcal{X} \sqcap \mathcal{Y}$ is the filter generated by the [[union]] $X \cup Y$, $\mathcal{X} \sqcup \mathcal{Y}$ is the [[intersection]] $X \cap Y$, and similarly for $⨅_i \mathcal{X}_i$ and $\bigsqcup_i \mathcal{X}_i$, while $0^{\mathfrak{F}(A)}$ is the [[improper filter]] (the [[power set]]) $\mathcal{P}(A)$ and $1^{\mathfrak{F}(A)}$ is the filter generated by the [[empty set]], which is $\{A\}$.

[[!redirects reverse poset of filters]]
[[!redirects reverse posets of filters]]
[[!redirects reverse lattice of filters]]
[[!redirects reverse lattices of filters]]


reverse mathematics

# Reverse mathematics
* table of contents
{: toc}

## Idea

_Reverse mathematics_ refers mainly to a program introduced by Harvey Friedman and Stephen Simpson, which aims to establish for many theorems of classical analysis exactly which set existence principles they rely on. They consider the formalization of analysis based on Polish spaces in the language of second order arithmetic (hence the need to focus on separable spaces). One then tries to show that various theorems are equivalent to a set existence axiom relative to a weak base theory, usually $\mathrm{RCA}_0$ (a finitistically reducible system axiomatizing recursive set comprehension only). The implication from a set existence axiom to a theorem is the usual (forward) direction of mathematics, while the implication from a theorem back to the axiom is _reverse_ direction of reverse mathematics.

There are other uses of the term reverse mathematics, notably _constructive reverse mathematics_ (see below).

## The "Big Five" subsystems

Work on reverses mathematics has isolated five subsystems of second order arithmetic that correspond to many classical theorems. For more information see [[ordinal analysis]].

These are in increasing order: $\mathrm{RCA}_0$, $\mathrm{WKL}_0$, $\mathrm{ACA}_0$, $\mathrm{ATR}_0$ and $\Pi^1_1{-}\mathrm{CA}_0$. The first two are finitistically reducible, but $\mathrm{WKL}_0$ introduces non-recursive sets. The system $\mathrm{ACA}_0$ has the same strength as first-order Peano arithmetic, $\mathrm{PA}$. The system $\mathrm{ATR}_0$ is predicatively reducible, while the system $\Pi^1_1{-}\mathrm{CA}_0$ corresponds to the first-order system $\mathrm{ID}_{\lt\omega}$ of finitely many generalized inductive definitions.

## Constructive reverse mathematics

_Constructive reverse mathematics_ refers to a similar program initiated by Ishihara which aims to calibrate non-constructive theorems with respect to non-constructive axioms over a constructive base theory. The non-constructive axioms he considers are

* Omniscience principles, in decreasing strength: LPO (limited principle of omniscience), WLPO (weak LPO), LLPO (lesser LPO).

* Variations of Markov's Principles (MP): WMP (Weak MP), MP$^{\vee}$ (disjunctive MP).

* Principles from Brouwer's intuitionism: FAN$_\Delta$ (detachable fan principle), BD-N (boundedness principle).

## References

* Stephen G. Simpson, _Subsystems of Second Order Arithmetic_, 2nd edition, Cambridge University Press, 2010.

* Wikipedia, _[Reverse mathematics](http://en.wikipedia.org/wiki/Reverse_mathematics)_

* _Natural examples of Reverse Mathematics outside classical analysis?_, [MO:q/185941](http://mathoverflow.net/q/185941/381)

In the context of [[constructive mathematics]]:

* [[Hajime Ishihara]], _Reverse Mathematics in Bishop's Constructive Mathematics_, Philosophia Scientiæ, CS 6 (2006) ([doi:10.4000/philosophiascientiae.406](https://doi.org/10.4000/philosophiascientiae.406),  [pdf](https://philosophiascientiae.revues.org/pdf/406))

* [[Hannes Diener]], *Constructive Reverse Mathematics*, 2018 ([arXiv:1804.05495](https://arxiv.org/abs/1804.05495), [dspace:ubsi/1306](https://dspace.ub.uni-siegen.de/handle/ubsi/1306))

* Takayuki Kihara, *Degrees of incomputability, realizability and constructive reverse mathematics* [[arXiv:2002.10712](https://arxiv.org/abs/2002.10712)]

[[!redirects constructive reverse mathematics]]


reverse monoidal category

[[!redirects reverse category]]
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###Context###

#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
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#Contents#
* table of contents
{:toc}

## Definition
The **reverse monoidal category** of a [[monoidal category]], $\mathcal{C}$, has the same underlying category and unit as $\mathcal{C}$ but reversed monoidal product, $X \otimes^{rev} Y = Y \otimes X$, and similarly for *tensors of* morphisms. The associator in the reverse category is $\alpha^{rev}_{X,Y,Z} = \alpha^{-1}_{Z,Y,X}$.

$V^{rev}$ is equivalent to $(\Sigma V)^{op}$ where $\Sigma V$ is the [[delooping]] of $V$, i.e. $V$ viewed as a one-object [[bicategory]] where ${op}$ is the [[opposite 2-category|opposite on 1-cells]].

Note that this notion is different from that of the [[opposite category]], which involves reversing the order of the arrows.

## Related pages
- [[opposite category]]
- [[opposite 2-category]]

[[!redirects reverse category]]
[[!redirects rev]]


reversible computation

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### Context
#### Computing
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In as far as a [[computation]] is a [[function]] from input data to output result, a *reversible computation* is an [[invertible]] function, where the input may be recovered from knowledge of the output.
More generally, as far as a [[computation]] is any [[morphism]] (under the [[computational trilogy]]), a reversible computation is an *[[isomorphism]]*.

Typical operations of a [[classical computer]] are not reversible: For example standard logical gates such as [[AND]], [[XOR]] etc. are [[functions]] of the form $\{0,1\}^2 \longrightarrow \{0,1\}$ and as such not [[invertible]]. Standard classical gates which are invertible include the [[NOT]]-gate $\{0,1\} \xrightarrow{\sim} \{0,1\}$ (which switches its single input bit) and the [[CNOT]]-gate $\{0,1\}^2 \xrightarrow{\sim} \{0,1\}^2$ (which keeps the first "control" bit and switches the second bit if the first equals 1).

{#ExampleOfIncrementalAddition} Using just these reversible NOT- and CNOT-gates one may for instance implement the operation of *incremental [[addition]]* $(-)+n \;\colon\; \mathbb{Z} \xrightarrow{\sim} \mathbb{Z}$ by a fixed [[natural number]] $n$ (e.g. [Kaye 2004](#Kaye04)). As opposed to binary addition $(-) + (-) \,\colon\, \mathbb{Z}^2 \xrightarrow{\;} \mathbb{Z}$, this operation is invertible, the [[inverse]] being the corresponding incremental [[subtraction]] $(-)-n \;\colon\; \mathbb{Z} \xrightarrow{\sim} \mathbb{Z}$.

This state of affairs was reflected, for instance, in [[Gottfried Leibniz]]'s *[stepped reckoner](https://en.wikipedia.org/wiki/Stepped_reckoner)* machine, which performs incremental addition when its crank is turned in one direction, and the inverse incremental subtraction when the crank is turned back in the opposite direction.

\linebreak

But the NOT- and CNOT-gates by themselves are not universal.
On the other hand, the [[CCNOT]] gate $\{0,1\}^3 \xrightarrow{\sim} \{0,1\}^3$ (which regards the first two inputs as control bits and switches the third input if these two are both 1) is both invertible as well as universal (e.g. [Nielsen & Chuang 2000, §1.4.1](#NielsenChuang00)), in that any function of the form $\{0,1\}^{n_1} \xrightarrow{\;} \{0,1\}^{n_2}$ is equal to a circuit of CCNOT gates pre-composed with some cartesian power of $\{0\} \xhookrightarrow{\;} \{0,1\}$ (introducing auxiliary "bath" or "ancilla" bits) and post-composed with some cartesian power of $\{0,1\} \to \ast$ (deleting auxiliary bits). In other words: Every irreversible classical computation on some computing system may be understood as a reversible computation on the system coupled to an ambient "bath" together with a loss of information of computation output into this bath.

Indeed, in terms of [[physics|physical]] implementation of computation, the distinction between reversible and irreversible computation is essentially that of reversible or irreversible dynamical processes as known from [[thermodynamics]]: Fundamentally and under idealized conditions, physical processes are invertible, but "at finite temperature" there is [[entropy]]-increase associated with most processes, making them irreversible. This relation between (non-)reversible computation and [[thermodynamics]] is known as *Landauer's principle* (due to [Landauer 1961](#Landauer61), review e.g. in [Nielsen & Chuang 2000, §3.2.5](#NielsenChuang00)).

While the notion of reversible computation was originally conceived in the context of classical computation, more recently its key example is understood to be *[[quantum computation]]*: Here the logical gates (*ex*cluding the final [[quantum measurement]] which [[wave function collapse|collapses]] the [[quantum state]]) are [[linear functions]] which are [[unitary operator]] and hence necessarily [[invertible]] (see [Frank 2003](#Frank03) for emphasis).

Therefore, the practical problem of implementing quantum computation faces analogous [[thermodynamics|thermodynamical]] issues as embodied by Landauer's principle, which are the reason for the need of [[quantum error correction]] and/or [[topological quantum computation|topological quantum stabilization]].

## Related concepts

* [[quantum computation]], [[quantum circuit]]

* [[nondeterministic computation]]

* [[parallel computing]]

## References

Original articles:

* {#Landauer61} [[Rolf Landauer]], *Irreversibility and Heat Generation in the Computing Process*,  IBM Journal of Research and Development **5** 3 (1961) 183–191 [[doi:10.1147%2Frd.53.0183](https://doi.org/10.1147%2Frd.53.0183)]

* [[Charles H. Bennett]], *Logical Reversibility of Computation*, IBM Journal of Research and Development  **17** 6  (1973) [[doi:10.1147/rd.176.0525](https://doi.org/10.1147/rd.176.0525)]

Review:

* [[Charles H. Bennett]], *The thermodynamics of computation -- a review*, International Journal of Theoretical Physics **21** (1982) 905–940 [[doi:10.1007/BF02084158](https://doi.org/10.1007/BF02084158)]

* [[Rolf Landauer]], [[Charles H. Bennett]]: *[The Fundamental Physical Limits of Computation](https://www.scientificamerican.com/article/the-fundamental-physical-limits-of-computation/)*, Scientific American **253** 1 (1985)

* [[Rolf Landauer]]: *Computation: A Fundamental Physical View*, Phys. Scr. **35** (1987) 88 [[doi:10.1088/0031-8949/35/1/021](https://iopscience.iop.org/article/10.1088/0031-8949/35/1/021)]

* [[Charles H. Bennett]], *Notes on the history of reversible computation*,  IBM Journal of Research and Development *32* 1 (1988) [[doi:10.1147/rd.321.0016](https://doi.org/10.1147/rd.321.0016)]

* [[Charles H. Bennett]], *Notes on Landauer's principle, reversible computation, and Maxwell's Demon*, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, **34** 3 (2003) 501-510 (doi:10.1016/S1355-2198(03)00039-X)

Review of current developments (as of 2020):

* B. Aman et al., *Foundations of Reversible Computation*, in: *Reversible Computation: Extending Horizons of Computing. RC 2020* Lecture Notes in Computer Science **12070**, Springer (2020) [[doi:10.1007/978-3-030-47361-7_1](https://doi.org/10.1007/978-3-030-47361-7_1)]

See also:

* Wikipedia, *[Reversible computing](https://en.wikipedia.org/wiki/Reversible_computing)*

* Wikipedia, *[Landauer's principle](https://en.wikipedia.org/wiki/Landauer%27s_principle)*

Discussion of reversible computation in view of [[quantum computation]]:

* {#Kitaev97} [[Alexei Kitaev]], §2.2 of: *Quantum computations: algorithms and error correction*, Russian Mathematical Surveys, **52** 6 (1997) [[doi:10.1070/RM1997v052n06ABEH002155](https://iopscience.iop.org/article/10.1070/RM1997v052n06ABEH002155), pdf]

* {#NielsenChuang00} [[Michael A. Nielsen]], [[Isaac L. Chuang]], §1.4.1 and §3.2.5 in: *Quantum computation and quantum information*, Cambridge University Press (2000) $[$[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]$]$

* {#Frank03} [[Michael P. Frank]], *Reversible Computing -- Quantum Computing’s Practical Cousin*, Simons Conference Lecture, Stony Brook (2003) [[pdf](http://insti.physics.sunysb.edu/conf/simons-qcomputation/talks/frank.pdf), [[Frank-ReversibleComputing.pdf:file]]]

* {#Kaye04} [[Phillip Kaye]], *Reversible addition circuit using one ancillary bit with application to quantum computing* [[quant-ph/0408173](https://arxiv.org/abs/quant-ph/0408173)]

* [[Phillip Kaye]], [[Raymond Laflamme]], [[Michele Mosca]], §1.5 in: *An Introduction to Quantum Computing*, Oxford University Press (2007) [[pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149125645.pdf), [ISBN:9780198570493](https://global.oup.com/academic/product/an-introduction-to-quantum-computing-9780198570493?cc=at&lang=en&)]

* [[Michael P. Frank]], Karpur Shukla, *Quantum Foundations of Classical Reversible Computing*, Entropy **23** 6 (2021) 701  [[arXiv:2105.00065](https://arxiv.org/abs/2105.00065), [doi:10.3390/e23060701](https://doi.org/10.3390/e23060701)]

See also:

* S.Saniya Samrin, Rachamma Patil, Sumangala Itagi, Smita C. Chetti, Afiya Tasneem, *Design of logic gates using reversible gates with reduced quantum cost*, Global Transitions Proceedings **3** 1 (2022) 136-141 [[doi:10.1016/j.gltp.2022.04.011](https://doi.org/10.1016/j.gltp.2022.04.011)]

[[!redirects reversible computations]]

[[!redirects reversible computing]]

[[!redirects Landauer's principle]]
[[!redirects Landauer principle]]


rewriting

# Rewriting
* table of contents
{: toc}

## Idea

Given an alphabet of letters and symbols (perhaps '[[type]]d', so that certain symbols are declared to be 'symbols for functions', etc., e.g. '$+$' is a symbol for a binary operation), then one can form [[list|strings]] of letters to give words or more generally (well formed) formulae involving symbols of all types.

In a _rewriting system_, one specifies a set of rules that describe valid replacements of subformulae by other ones. On some formulae of a rewriting system, the rewriting rules may produce conflicts, when two or more rules can be applied.

+-- {: .un_example}
###### Example type
One type of example of a rewriting system is given by a [[group presentation]], $(X,R)$, where $X$ is a set of generators for a group $G$ and $R$ is a subset of pairs of elements in the free group on $X$.  As a definite example, take

* $X = \{a,b\}$

* $R = \{(a a a,1),(b b,1),(a b a b,1)\}$

(and the group being presented is the symmetric group, $S_3$ on three symbols).
We think of $(a a a,1)$ as a 'rewrite rule': ''replace $a a a$ by 1'', often written $a a a \rightsquigarrow 1$ (or just $a a a \to 1$).

Working with the presentation, we can then  start generating words in the generators, for instance $a a a b a b$ and then see how that word can be 'rewritten' using the rewrite rules to $b a b$ using the first rule and to $a a$ using the last. There is thus a potential conflict between these rewritten forms of the word.
=--

It is usual to choose a '[[normal form]]' for each word.  In the example, in the symmetric group and the above presentation the elements of the group are often listed as $1, a, a^2, b, a b, a^2 b$.  These might be our choice of normal forms for the elements. In other words any element has a unique representative in the form $a^i b^j$, and we _choose_ to label them that way. With the two uses of the rules applied to $a a a b a b$, the first did not gave a normal form, the second did.

In order to transform a rewriting system into a computation [[algorithm]], one needs to apply the rules in a deterministic way, using a reduction strategy. We also need to know that there is a unique normal form that can be found for each word and that the 'algorithm' will _terminate_, that is it really _is_ an algorithm!.

We therefore will need to discuss [[confluence]], [[normal form]], [[termination]] and [[reduction strategy|reduction strategies]].

## Related entries

* [[Globular]]

* [[abstract rewriting system]]

## References

### General

* Axel Thue,  Probleme über Veränderungen von Zeichenreihen nach gegebenen Regeln., Kristiania Vidensk. Selsk, Skr. (1914), no. 10, 493–524.

* [[Max H. A. Newman]], *On theories with a combinatorial definition of "equivalence"*, Annals of Mathematics **43** 2 (1942) 223-243 [[doi:10.2307/1968867](https://doi.org/10.2307/1968867)]

  > (proves [[Newman's lemma]])

*  Ronald V. Book, Friedrich Otto, _String-rewriting systems_, Texts and Monographs in Computer Science,
Springer-Verlag, 1993.

* [[Fabio Gadducci]], _On the algebraic approach to concurrent term rewriting_, PhD thesis (1996) [[pdf](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1277313dd2c4fce640780b7472221156579e392f), [[Gadducci-AlgebraicRewriting.pdf:file]]]

* Franz Baader, Tobias Nipkow, _Term rewriting and all that_, Cambridge University Press (1998) [[webpage](https://www21.in.tum.de/~nipkow/TRaAT/)]

* [[Andrea Corradini]], [[Fabio Gadducci]], _An algebraic presentation of term graphs, via gs-monoidal categories_, Applied Categorical Structures **7** (1999) 299-331 [[doi:10.1023/A:1008647417502](https://doi.org/10.1023/A:1008647417502)]

  > (introducing [[gs-monoidal categories]] for term rewriting)

*   _Terese_, Term rewriting systems, Cambridge Tracts in Theoretical Computer Science, vol. 55, Cambridge University Press, 2003.

See also:

* Wikipedia, *[Rewriting](http://en.wikipedia.org/wiki/Rewriting)*

On [[rewriting]] in [[group presentations]]:

* [[Nick Gilbert]], Emma Antonia Mcdougall, _Groupoids and the algebra of rewriting in group presentations_, Beiträge zur Algebra und Geometrie **62** 3 (2021) 1-17 [[doi:10.1007/s13366-020-00531-6](https://doi.org/10.1007/s13366-020-00531-6), [arXiv:1901.04348](https://arxiv.org/abs/1901.04348)]

### Higher dimensional rewriting

For the higher dimensional forms of rewriting, one source is in the work of Guiraud and Malbos:

* [[Yves Guiraud]], [[Philippe Malbos]], _Identities among relations for higher-dimensional rewriting systems_, Séminaires et Congrès, SMF, (to appear), [arXiv:0910.4538](http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.4538v2.pdf); _Higher-dimensional categories with finite derivation type_, Theory Appl. Categ. __22__
(2009), No. 18, 420–478, [tac](http://www.tac.mta.ca/tac/volumes/22/18/22-18abs.html); pdf slides [Homotopical methods in polygraphic rewriting](http://ljk.imag.fr/membres/Dominique.Duval/CCS09/CCS09-Malbos.pdf), [n-Catégories de type de dérivation fini](http://www.loria.fr/~guiraudy/exposes/ntdf-luminy.pdf).

This also includes emerging _homotopical theory of computation_ based on [[polygraph]]s (introduced by [[Albert Burroni]]) and polygraphic resolutions (introduced by [[François Métayer]]):

* Yves Lafont, _Algebra and geometry of rewriting_, Applied Categorical Structures
August __15__:4, 2007, pp 415-437  [doi](https://doi.org/10.1007/s10485-007-9083-6)

Again within the context of higher dimensional forms of rewriting, [[Tibor Beke]] has given a [[categorification]] of certain rewriting procedures of Knuth. This is of relevance here as it contains a strong result on coherence theory:

* [[Tibor Beke]], _Categorification, term rewriting and the Knuth-Bendix procedure_, J. Pure and Applied Algebra 215:5 (2011) 728-740 ([pdf](http://faculty.uml.edu/tbeke/knuth.pdf) at author's page) [doi:10.1016/j.jpaa.2010.06.019](https://doi.org/10.1016/j.jpaa.2010.06.019)

An approach via diagrammatic sets is in

* [[Amar Hadzihasanovic]], _Diagrammatic sets and rewriting in weak higher categories_, ([arXiv:2007.14505](https://arxiv.org/abs/2007.14505))

with slides (for GeoCat 2020), [here](https://www.irif.fr/~ahadziha/files/talk-geocat-rewriting.pdf).

* [[Samuel Mimram]], _Towards 3-dimensional rewriting theory_, ([arXiv:1403.4094](https://arxiv.org/abs/1403.4094))

See also:

* [[Neil Ghani]], C. Lüth, _Rewriting via coinserters_,  Nordic Journal of Computing 10 (2003) 290–312 [pdf](http://www.informatik.uni-bremen.de/~clueth/awe/papers/njc04b.pdf)

On the online proof assistant *[[Globular]]* for higher dimensional [[rewriting]] via [[semistrict]] [[globular set|globular]] [[higher categories]]:

* [[Krzysztof Bar]], [[Aleks Kissinger]], [[Jamie Vicary]], *Globular: an online proof assistant for higher-dimensional rewriting*,  Logical Methods in Computer Science **14** 1 (2018) [[doi:10.23638/LMCS-14(1:8)2018](https://doi.org/10.23638/LMCS-14(1:8)2018), [arXiv:1612.01093](https://arxiv.org/abs/1612.01093)]

See also:

* Nicolas Behr, [[Joachim Kock]], _Tracelet Hopf algebras and [[decomposition spaces]]_, [arXiv:2105.06186](https://arxiv.org/abs/2105.06186)

> Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.

The following [[machine learning]]-inspired rewriting systems for meta/hyper-graphs claim to be related to [[homotopy type theory]]

* [[Ben Goertzel]], _Reflective metagraph rewriting as a foundation for an AGI "Language of Thought"_, [arXiv:2112.08272](https://arxiv.org/abs/2112.08272)
* Xerxes D Arsiwalla, Jonathan Gorard, _Pregeometric spaces from Wolfram model rewriting systems as homotopy types_ [arXiv:2111.03460](https://arxiv.org/abs/2111.03460)

category: combinatorics, algebra, computer science, logic

[[!redirects rewriting]]
[[!redirects rewriting system]]
[[!redirects rewriting systems]]
[[!redirects rewriting procedure]]
[[!redirects rewriting procedures]]

[[!redirects rewrite rule]]
[[!redirects rewrite rules]]


Rezk completion

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea ##

There is a canonical way to turn any [[category]] in [[homotopy type theory]] into a [[weak equivalence|weakly equivalent]] [[univalent category]]. This can be thought of as an analogue of [[univalence]] but for [[isomorphisms]] instead of [[equivalence]].

Intuitively, the Rezk completion is a "strictification via Yoneda" type result, in the style of [[Yoneda lemma for bicategories|strictification for bicategories via Yoneda]]. One starts with a category that may not have nice strictness properties, embeds it into the category of presheaves, which does have the nice strictness properties, and then restricts to representables, which gives something equivalent to the original category, but retains the nice strictness properties.

Another intuition is that the Rezk completion is a [[vertical
categorification]] of the construction of a [[quotient]] of a type by
an [[equivalence relation]] by taking [[equivalence classes]]. That
is, we can think of a type $X$ equipped with an equivalence relation
as a [[boolean]]-enriched groupoid, and a boolean-valued presheaf is
equivalently a predicate on $X$ that respects the equivalence
relation. Then the representable presheaves are those predicates that
are furthermore inhabited, i.e., precisely equivalence classes.

## Construction ##

We work in a [[dependent type theory]] where [[UIP]] or [[axiom K]] cannot be proven. These results are from [UFP13](#UFP13). Note: [UFP13](#UFP13)  calls a [[category]] a "[[precategory]]" and a [[univalent category]] a "category", but here we shall refer to the standard terminology of "category" and "univalent category" respectively.

\begin{theorem}\label{Theorem995}
**(Theorem 9.9.5 in [UFP13](#UFP13))**
For any [[category]] $A$, there is a [[univalent category]] $\hat{A}$ and a [[weak equivalence]] $A\to \hat{A}$.
\end{theorem}
\begin{proof}
Let $\hat{A}_0$ be the type of [[representable]] objects of $\mathit{Set}^{A^{\mathrm{op}}}$, with hom-sets inherited from $\mathit{Set}^{A^{\mathrm{op}}}$. Then the inclusion $\hat{A} \to \mathit{Set}^{A^{\mathrm{op}}}$ is [[fully faithful]] and an [[embedding]] on [[objects]]. Since $\mathit{Set}^{A^{\mathrm{op}}}$ is a category by Theorem 9.2.5 (see [[functor category]]), $\hat{A}$ is also a category. Let $A \to \hat{A}$ be the [[Yoneda embedding]]. This is [[fully faithful]] by corollary 9.5.6 (See [[Yoneda embedding]]), and [[essentially surjective]] by definition of $\hat{A}_0$. Thus it is a [[weak equivalence]].
\end{proof}

* $\hat{A}_0\equiv \sum_{(F : \mathit{Set}^{A^{\mathrm{op}}})} \sum_{(a:A)} \mathbf{y} a \cong F$

This has the unfortunate side affect of raising the [[universe]] level. If $A$ is a [[univalent category]] in a universe $\mathcal{U}$, then in this proof $\mathit{Set}$ must at least be as large as $\mathit{Set}_{\mathcal{U}}$. Hence the univalent category $\mathit{Set}_{\mathcal{U}}$ is in a higher universe than $\mathcal{U}$ hence $\mathit{Set}^{A^{\mathrm{op}}}$ must also be in a higher universe and finally $\hat{A}$ is also in a higher universe than $\mathcal{U}$.

Now this can all be avoided by constructing a [[higher inductive type]] $\hat{A}$ with constructors:

* A function $i : A_0 \to \hat{A}_0$.
* For each $a,b:A$ and $e: a \cong b$, an equality $j e : i a = i b$
* For each $a : A$, an equality $j(1_a)=refl_{i a}$.
* For each $a,b,c:A$, $f : a \cong b$, and $g : b \cong c$, an equality $j(g \circ f)=j(f) \cdot j(g).$
* 1-truncation: for all $x,y:\hat{A}_0$ and $p,q: x = y$ and $r,s : p = q$, an equality $r=s$.

If we ignore the last constructor we could also write the above as $\| \hat{A}_0 \|_1$.

We now go on to build a univalent category $\hat{A}$ with a [[weak equivalence]] $A \to \hat{A}$ by taking the type of objects as $\hat{A}_0$ and defining hom-sets by double [[induction]]. The advantage of this approach is that it preserves [[universe]] levels, there are a lot of things to check but it is an easy proof. The kind of proof that is well suited to a proof assistant. For the complete proof see Theorem 9.9.5 of the [[HoTT book]].

## Related concepts

* [[Segal completion]]

* [[category theory]]

* [[Yoneda lemma]]

* [[weak equivalence]]

* [[higher inductive type]]

## References ##

The relation between [[complete Segal space|Segal completeness]] (now often "Rezk completion") for [[internal categories in HoTT]] and the [[univalence axiom]] had been pointed out in:

* [[Urs Schreiber]], _[Segal completenes and Univalence](https://golem.ph.utexas.edu/category/2012/05/segalcompleteness_and_univalen.html)_ (2012)

This was developed in

* {#AhrensKapulkinShulman13} [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], _Univalent categories and the Rezk completion_, Mathematical Structures in Computer Science **25** 5 (*From type theory and homotopy theory to Univalent Foundations of Mathematics*),  (2015) 1010-1039   $[$[arXiv:1303.0584](https://arxiv.org/abs/1303.0584), [doi:10.1007/978-3-319-21284-5_14](https://doi.org/10.1007/978-3-319-21284-5_14)$]$

See also:

* {#UFP13} [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

category: category theory


Rezk type

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## Definition

In [[simplicial type theory]], a [[Segal type]] $A$ is a **Rezk type** if for all elements $x:A$ and $y:A$ there is an equivalence between the [[identity type]] $x =_A y$ and the [[type of isomorphisms in a Segal type]] $x \cong_A y$:

$$x:A, y:A \vdash \mathrm{RezkComplete}(x, y):\mathrm{isEquiv}(\mathrm{idToIso}(x, y))$$

where

$$
  \mathrm{idToIso}(x, y)
  \;\colon\;
  (x =_A y)
  \;
  \to
  \;
  (x \cong_A y)
$$

$$
  \mathrm{idToIso}(x, y)
  \big(
    \mathrm{refl}_A(x)
  \big)
    \;\coloneqq\;
  \mathrm{id}_A(x)
$$

## Related concepts

* [[Segal type]]
* [[isomorphism in a Segal type]]
* [[discrete Segal type]]
* [[univalent category]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$


rheonomy modality

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## Idea

In [[higher supergeometry]] the [[bosonic modality]] $\stackrel{\rightsquigarrow}{(-)}$ which sends [[supermanifolds]] to their underlying ordinary [[boson|bosonic]] [[smooth manifolds]] has a further [[right adjoint]] $\Rh$, see at _[[super smooth infinity-groupoid]]_.

This means that if $\mathbf{Fields}$ is a [[moduli stack]] of [[field (physics)|fields]], for instance for [[supergravity]], then $Rh(\mathbf{Fields})$ is such that for $\hat X$ any [[supermanifold]] with underlying manifold $X \to \hat X$, then maps

$$
  \hat X \longrightarrow Rh(\mathbf{Fields})
$$

are equivalently maps

$$
  X \longrightarrow \mathbf{Fields}
$$

hence are fields configurations on the underlying ordinary manifold $X$.

In the [[supergeometry]] formulation of [[supergravity]] this is what goes into the [rheonomy superspace constraint](D%27Auria-Fre+formulation+of+supergravity#Rheonomy) which demands that on-shell super-field configurations $\hat X \to \mathbf{Fields}$ have to be uniquely determined by their restriction along $X \to \hat X$. Therefore the space $Rh(\mathbf{Fields})$ contains the _rheonomic field configurations_ among all the field configurations modulated by $\mathbf{Fields}$.

## See also

[[!include cohesion - table]]

[[!redirects rheonomic modality]]
[[!redirects rheonomic modalities]]


RHIC

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{:toc}

## Idea

An [[experiment]] in [[particle physics]].

## Related entries

* [[LHC]], [[LIGO]]

* [[Super-Kamiokande]]

* [[EDGES]]

## References

### General

* [RHIC website](https://www.bnl.gov/rhic/)

See also

* Wikipedia, _[Relativistic Heavy Ion Collider](https://en.wikipedia.org/wiki/Relativistic_Heavy_Ion_Collider)_

### Quark-gluon plasma

Realization of the [[quark-gluon plasma]] at the [[RHIC]] [[experiment]] has tentatively been claimed in

* John Adams, et al. _Experimental and theoretical challenges in the search for the quark–gluon plasma: The STAR Collaboration's critical assessment of the evidence from RHIC collisions_. Nuclear Physics A 757.1-2 (2005): 102-183.

* K. Adcox, et al., _Formation of dense partonic matter in relativistic nucleus–nucleus collisions at RHIC: experimental evaluation by the PHENIX collaboration_ Nuclear Physics A 757.1-2 (2005): 184-283.

Exposition and review:

* Maria Elena Tejeda-Yeomans, _Heavy-ion physics: freedom to do hot, dense, exciting QCD_ ([arXiv:2004.13812](https://arxiv.org/abs/2004.13812))

[[!redirects Relativistic Heavy Ion Collide]]


rho meson

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## Idea

In [[nuclear physics]], specifically in the [[chiral perturbation theory]] of [[quantum chromodynamics]], the _rho-meson_ is the [[isospin]]-triplet [[vector meson]] [[field (physics)|field]] in the first-[[generation of fermions]], i.e. a [[bound state]] of an [[up quark]] and a [[down quark]] (a [[light meson]]), the [[chiral partner]] of the [[a1-meson]];

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

\linebreak

[[!include flavors of fundamental particles -- table]]

## Properties

### Relation to Skyrmions

The [[skyrmion]]-[[model (in theoretical physics)|model]] (see there) realizes [[baryons]] as [[solitons]]/[[instantons]] in a cloud of [[pions]]. If one adds to this also the ρ one gets also good Skyrmion models of light [[atomic nuclei]] ([Naya=Sutcliffe 18](#NayaSutcliffe18))

## Related concepts

* [[B-meson]]

* [[pion]], [[omega-meson]]

## References

### General

See also

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

* Paul Singer, _Radiative  $\rho$-Meson Decay_, Phys. Rev. 130, 2441 (1963) ([doi:10.1103/PhysRev.130.2441](https://doi.org/10.1103/PhysRev.130.2441))

Via [[holographic light front QCD]]:

* Satvir Kaur, Chandan Mondal, Harleen Dahiya, _Light-front holographic $\rho$-meson distributions in the momentum space_ ([arXiv:2009.04288](https://arxiv.org/abs/2009.04288))

[[!include Skyrme hadrodynamics with vector mesons -- references ]]

[[!redirects rho mesons]]

[[!redirects rho-meson]]
[[!redirects rho-mesons]]

[[!redirects ρ meson]]
[[!redirects ρ mesons]]

[[!redirects ρ-meson]]
[[!redirects ρ-mesons]]

[[!redirects A1-meson]]
[[!redirects A1-mesons]]


ribbon category

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## Idea

A **ribbon category** [[Reshetikhin & Turaev (1990)](#ReshetikhinTuraev90)] (also called a **tortile category** [[Joyal & Street (1993)](#JoyalStreet93)] or **balanced rigid braided tensor category**) is a [[monoidal category]] $(\mathcal{C}, \otimes, \mathbb{1}, \alpha, l, r)$ equipped with [[braiding]] $\beta=\{\beta_{X,Y}\}$, twist $\theta=\{\theta_X\}$ and duality $(\vee, b, d)$ that satisfy some compatibility conditions.

## Definition

A [[braided monoidal category]] is a [[monoidal category]] $\mathcal{C}$ equipped with a [[braiding]] $\beta$, which is a [[natural isomorphism]] $\beta_{X,Y}\colon X \otimes Y \to Y \otimes X$ obeying the [[hexagon identities]].

A braided monoidal category is *[[rigid monoidal category|rigid]]* if, for every [[object]] $X$, there exist objects $X^{\vee}$ and ${^{\vee}}X$ (called its [[dualisable object|right and left dual]]) and associated morphisms
$$b_X:\mathbb{1}\to X\otimes X^{\vee}, d_X: X^{\vee}\otimes X\to \mathbb{1}$$
$$b_X:\mathbb{1}\to {^{\vee}}X\otimes X, d_X: X\otimes {^{\vee}}X\to \mathbb{1}$$
obeying the zig-zag identities.

A **twist** on rigid braided monoidal category is a [[natural isomorphism]] from the [[identity functor]] to itself, with components $\theta_X \colon X \to X$ for which
$$\theta_{X\otimes Y}=\beta_{Y,X}\beta_{X,Y}\theta_{X}\otimes \theta_{Y},$$
$$\theta_{\mathbb{1}}=\mathrm{id},$$
$$\theta_{X^{\vee}}=\theta_{X}^{\vee}.$$

A ribbon category is a rigid braided monoidal category equipped with such a twist.

## Related concepts

* [[modular tensor category]]

## Reference

* {#ReshetikhinTuraev90} [[Nicolai Reshetikhin]], [[Vladimir Turaev]], *Ribbon graphs and their invariants derived from quantum groups*, Commun. Math. Phys. **127** 1 (1990) [[doi:10.1007/BF02096491](https://doi.org/10.1007/BF02096491)]

* {#JoyalStreet93} [[André Joyal]], [[Ross Street]], *Braided tensor categories*, Advances in Mathematics **102** (1993) 20–78 [[doi:10.1006/aima.1993.1055](https://doi.org/10.1006/aima.1993.1055)]

* [[Mei Chee Shum]], *Tortile tensor categories*, Journal of Pure and Applied Algebra **93** 1 (1994) 57-110 [10.1016/0022-4049(92)00039-T]

[[!redirects ribbon category]]
[[!redirects ribbon categories]]
[[!redirects tortile category]]
[[!redirects tortile categories]]
[[!redirects tortile monoidal category]]
[[!redirects tortile monoidal categories]]


ribbon graph

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## Idea

A _ribbon graph_ (also called _fat graph_ ) is a (typically finite and connected) [[graph]] equipped with a cyclic ordering on the half edges incident to each vertex.

To each ribbon graph, one can associate an [[orientation|oriented]] [[surface]] with [[boundary]] by replacing edges by thin oriented rectangles (ribbons) and vertices by disks, and pasting rectangles to disks according to the chosen cyclic orders at the vertices. (For some illustrations of this, see [Mulase-Penkava](#MulPen).)

## Definition

The following definition of an ordinary [[graph]] (allowing loops and multiple edges) is not the standard one used for ordinary graphs, but is easily seen to be equivalent to it, while being the right starting point for describing ribbon graphs.

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

A **[[graph]]** is a quadruple $(V, H, s, i)$, where

* $V$ is a [[set]], called the **set of vertices**;

* $H$ is a [[set]], called the **set of half-edges**

* $s : H \to V$ is a [[function]], thought of as sending each half-edge to the vertex that it is _incident_ on;

* $i : H \to H$ is a [[involution|involutive]] [[function]] without fixed points, thought of as sending each half-edge to its other half.

The set of [[cycle]]s of $i$ is the set $E$ of full edges.

A [[homomorphism]] of graphs is a pair of [[function]]s $V_1 \to V_2$ and $H_1 \to H_2$ commuting with the $i$- and $s$-maps.

=--

An edge is a *loop* if its constituent half-edges are incident on the same vertex.

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

A **fat graph** or **ribbon graph** is a graph $(V,H,i,s)$ equipped with a [[bijection]] $\sigma : H \to H$ whose [[cycle]]s correspond to the sets $s^{-1}(v)$ of half-edges incident on vertices $v \in V$.

=--

This means that on all sets $s^{-1}(v)$ of jointly incident half-edges there is induced a cyclic ordering.  When drawing ribbon graphs on paper this cyclic ordering is identified with the ordering induced from the standard orientation of the plane.

Note that this definition (which appears in [Igusa 02](#Igusa02)) is essentially equivalent to that of an oriented [[combinatorial map]], except that the usual definition of combinatorial map does not make the set $V$ and the function $s : H \to V$ explicit, instead identifying vertices with the cycles of $\sigma$.  Under this style of definition, the extra condition (often assumed) that the underlying graph be connected corresponds to the condition that the group generated by the permutations $i$ and $\sigma$ acts [[transitive action|transitively]] on the set of half-edges $H$.

### Boundary of a fat graph

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

Write

$$
  \partial : FatGraph \to Graph
$$

for the [[functor]] -- called the **boundary graph functor** -- that sends a fat graph $\Gamma = (V, H, s, i, \sigma)$ to the graph

$$
  \partial \Gamma = \left(
     \coprod_{v \in V} \coprod_{h \in s^{-1}(v)} \{v_h\},\,
     \coprod_{h \in H} \{h_0, h_1\},\,
     s_{\partial_\Gamma},\,
     i_{\partial_\Gamma}
  \right)
$$

with $s_{\partial \Gamma}$ given by

$$
  s_{\partial \Gamma}(h_0) = v_h
$$

$$
  s_{\partial \Gamma}(h_1) = \sigma(v_h)
$$

and $i_{\partial \Gamma}$ given by

$$
  i_{\partial \Gamma}(h_0) = i(h)_1
$$

$$
  i_{\partial \Gamma}(h_1) = i(h)_0
  \,.
$$

=--

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

The set of [[boundary]] components of $\Sigma_\Gamma$ is naturally identified with the [[cycle]]s of $\sigma \circ i : H \to H$, or equivalently with the cycles of $i \circ \sigma$.

=--

Write $S : FatGraph \to Graph$ for the evident [[forgetful functor]] sending fat graphs to their underlying graph.

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

There is a [[natural transformation]]

$$
  c : \partial \to S
$$

from the boundary graph functor to the forgetful functor -- called the **collapse map**, whose components morphism on a fat graph $\Gamma$ sends on vertices

$$
  v_h \mapsto s(h)
$$

and on half-edges

$$
  h_{0,1} \mapsto h
  \,.
$$

=--

### Further constructions

Given any edge $e$ in a graph $\Gamma$,
we can collapse it to a new graph $\Gamma/e$ by merging the two vertices
containing the edge and removing both half edges.

A forest $F$ (disjoint union of trees) spans a graph $\Gamma$ if it contains all of its vertices. One can collapse $\Gamma$ to $\Gamma/F$ by collapsing each tree to a separate vertex. Then the edges in $\Gamma/F$ are the edges of $\Gamma$ which do not lie in any tree in $F$; the vertices of $\Gamma/F$ correspond to the sets of leaves of the component trees of $F$.

A morphism of graphs $\Gamma_0\to\Gamma_1$ is defined as an isomorphism $\Gamma_0/F\to\Gamma_1$ where $F$ is a spanning forest of $\Gamma_0$. Morphisms of such graphs are both epi and mono.

For  a ribbon graph, the set of leaves of any tree inherits a cyclic order. Thus $\Gamma/F$ has a canonical structure a ribbon graph. This makes the following definition possible: A morphism of ribbon graphs is the morphism of underlying graphs which respects the ordering on the vertices; i.e.  $\Gamma_0/F\to\Gamma_1$ is an isomorphism of ribbon graphs. Every endomorphism of a ribbon graph is an automorphism and for any two ribbon graphs $\Gamma$ and $\Gamma'$, the set $Aut(\Gamma)$ acts freely on $Hom(\Gamma,\Gamma')$ on the right and $Aut(\Gamma')$ acts freely on $Hom(\Gamma,\Gamma')$ from the left.

Let $\mathcal{Fat}$ denote the category of ribbon graphs and ribbon automorphisms. Strebel proved that the geometric realization $|\mathcal{Fat}|$ of the category of ribbon graphs decomposes into a direct sum of classifying spaces of all [[mapping class group]]s $M^s_g$

$$
|\mathcal{Fat}| = \coprod_{g,s} BM^s_g
$$

$M^s_g$ is a mapping class group of a surface of genus $g$ with $s$ punctures. Alternatively, one can instead of the geometric realization take a moduli space of ribbon graphs with metric. A **metric** on a ribbon graph is a positive real valued function on the set of edges.

## Properties

For the following we represent a fat graph by

* a set $V$ of vertices;

* a set $H$ of half-edges;

* a source map $s : H \to V$;

* an involution $i :  H \to H$ which sends each half-edge to its partner half-edge;

* an permutation $\sigma : H \to H$ which gives the cyclic order on the edges.

### Geometric realization

For $\Gamma$ a fat graph, write $\Sigma_\Gamma$ for the [[surface]] that it defines.

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

The [[genus]] of $\Sigma_\Gamma$ is

$$
  g = \frac{1}{2}
   \left(
     2 - \vert V\vert + \vert E\vert - \vert boundary components\vert
   \right)
  \,.
$$

=--

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

For every surface $S$ with boundary, there is a fat graph $\Gamma$ and a [[homeomorphism]]

$$
  S \stackrel{\simeq}{\to}  \Sigma_\Gamma
  \,.
$$

=--

### Moduli space of surfaces / classifying spaces of mapping class groups

Write $FatGraph^c_3$ for the [[full subcategory]] of fat graphs on the connected fat graphs for which every vertex has valence at least 3. Write $\vert FatGraph^c_3 \vert \in $ [[Top]] for the [[geometric realization]] of this [[category]]. For $\Sigma$ a [[surface]], write $\Gamma_Sigma$ for its [[mapping class group]] and $B \Gamma_\Sigma$ for the corresponding [[classifying space]].

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

There is a [[weak homotopy equivalence]]

$$
  \vert FatGraph^c_3 \vert \simeq
  \coprod_{[\Sigma]} B \Gamma_\Sigma
  \,,
$$

where on the right thr coproduct ranges over isomorphism classes of orientable, closed 2-dimensional manifolds with $n \geq 1$ marked points, except those for which $(g,n) = (0,1)$ or  $(g, n) = (0,2)$.

=--

In various guises this theorem has been proven by [Costello](#Costello), [Kontsevich92](#Kontsevich92), [Igusa02](#Igusa02), [Penner](#Penner), [Strebel](#Strebel).

The restriction to valence $\geq 3$ can be dropped:

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

There is a [[weak homotopy equivalence]]

$$
  \vert FatGraph^c \vert \simeq
  B U(1) \coprod B U(1) \coprod_{[\Sigma]} B \Gamma_\Sigma
  \,,
$$

where again on the right the [[coproduct]] ranges over [[isomorphism class]]es of orientable, closed 2-dimensional manifolds with $n \geq 1$ marked points, except those for which $(g,n) = (0,1)$ or  $(g, n) = (0,2)$. The two extra copies of $B U(1)$ corespond to these two exceptional cases.

=--

This has been shown in ([Godin](#Godin)). One of the $B U(1)$-summands is also produced in ([Igusa02](#Igusa02)). A detailed complete proof appears as ([Kupers, theorem 3.59](#Kupers)).

## Related concepts

* [[stringy weight system]]

* [[graph]]

* [[directed graph]]

  * [[directed n-graph]]

* **ribbon graph**

* [[topological map]]

* [[combinatorial map]]

* [[ribbon category]]

## References

Survey in the context of [[string topology]]:

* [[Sander Kupers]], Chapter 3 of: _String topology operations_ MSc thesis (2011) ([dspace:1874/206062](https://dspace.library.uu.nl/handle/1874/206062))

Survey in the context of [[weight systems]] on [[chord diagrams]]/[[Jacobi diagrams]]:

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 7 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

Original articles:

* {#Kontsevich92}[[Maxim Kontsevich]], _Intersection theory on the moduli space of curves and the matrix Airy function_ , Commun.
Math. Phys. (1992), no. 147, 1-23.

* {#Igusa02}[[Kiyoshi Igusa]], _Higher Franz Reidemeister torsion_ , IP Studies in Advanced Mathematics, American Mathematical Society, 2002.

* {#Igusa04}[[Kiyoshi Igusa]], _Graph cohomology and Kontsevich cycles_, Topology __43__ (2004), n. 6, p. 1469-1510, [MR2005d:57028](http://www.ams.org/mathscinet-getitem?mr=2005d:57028), [doi](https://doi.org/10.1016/j.top.2004.03.004)

* {#Strebel} K. Strebel, _Quadratic Differentials_, Springer, Berlin, 1984, [MR86a:30072](https://www.ams.org/mathscinet-getitem?mr=86a:30072)

* {#Penner}R. C. Penner, _The decorated Teichmüller space of punctured surfaces_, Commun. Math. Phys. __113__ (2) (1987) 299--339. [MR89h:32044](https://www.ams.org/mathscinet-getitem?mr=89h:32044)

* John Harer, _The cohomology of the moduli space of curves_, Lec. Notes in Math. __1337__, 138--221 (1988)

* {#MulPen} M. Mulase, M. Penkava, _Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over_ $\overline{\mathbb{Q}}$. Mikio Sato: a great Japanese mathematician of the twentieth century.  Asian J. Math. __2__ (1998),  no. 4, 875--919. ([math-ph/9811024](https://arxiv.org/abs/math-ph/9811024))

* [[Maxim Kontsevich]], _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121, [pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf)

* [[Ib Madsen]], [[Michael Weiss]], _The stable moduli space of Riemann surfaces: [[Mumford's conjecture]]_,  Ann. of Math. (2) __165__ (2007), no. 3, 843--941, [MR2009b:14051](http://www.ams.org/mathscinet-getitem?mr=2009b:14051), [doi](http://dx.doi.org/10.4007/annals.2007.165.843)

* J. Conant, [[Karen Vogtmann]], _On a theorem of Kontsevich_, [math.QA/0208169](https://arxiv.org/abs/math/0208169)

* Gabriele Mondello, _Riemann surfaces, ribbon graphs and combinatorial classes_, [pdf](http://www1.mat.uniroma1.it/people/mondello/ricerca/papers/ober-definitive.pdf) [arxiv:0705.1792](https://arxiv.org/abs/0705.1792)

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

* [[Kevin Costello]]. _A dual version of the ribbon graph decomposition of moduli space_, Geometry & Topology 11 (2007) 1637--1652 [doi](https://doi.org/10.2140/gt.2007.11.1637)

Further discussion

* MathOverflow: [why does ribbon graph cohomology compute cohomology of mapping class group](http://mathoverflow.net/questions/51978/why-does-ribbon-graph-cohomology-compute-cohomology-of-mcg)

[[!redirects ribbon graphs]]

[[!redirects fat graph]]
[[!redirects fat graphs]]

[[!redirects fatgraph]]
[[!redirects fatgraphs]]