Quipper

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### Context
#### Computation
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[[!include constructivism - contents]]
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#### Quantum systems
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[[!include quantum systems -- contents]]
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#### Type theory
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[[!include type theory - contents]]
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#Contents#
* table of contents
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## Idea

_Quipper_ is a [[functional programming language|functional]] [[quantum programming language]], specifically a [[domain specific programming language]] for [[quantum circuits]] which is [[domain specific embedded programming language|embedded]] into [[Haskell]]. As such it is similar to *[[QWIRE]]*.

## Related concepts

* [[QML]]

* [[QPL]]

## References
 {#References}

Quipper landing page:

* [www.mathstat.dal.ca/~selinger/quipper](http://www.mathstat.dal.ca/~selinger/quipper)

### Precursors

Quipper has grown out of developments initiated in

* {#Selinger04} [[Peter Selinger]], _Towards a quantum programming language_, Mathematical Structures in Computer Science **14** 4 (2004) 527–586 [[doi:10.1017/S0960129504004256](https://doi.org/10.1017/S0960129504004256), [pdf](https://www.mathstat.dal.ca/~selinger/papers/qpl.pdf),  [web](https://www.mathstat.dal.ca/~selinger/papers.html#qpl)]

and specifically the [[quantum lambda-calculus]] of:

* [[Benoît Valiron]], *A functional programming language for quantum computation with classical control*, MSc thesis, University of Ottawa (2004) $[$[doi:10.20381/ruor-18372](http://dx.doi.org/10.20381/ruor-18372), [pdf](https://ruor.uottawa.ca/bitstream/10393/26790/1/MR01625.PDF)$]$

* {#SelingerValiron04} [[Peter Selinger]], [[Benoît Valiron]], *A lambda calculus for quantum computation with classical control*, Proc. of TLCA 2005 $[$[arXiv:cs/0404056](https://arxiv.org/abs/cs/0404056), [doi:10.1007/11417170_26](https://doi.org/10.1007/11417170_26)$]$

* {#SelingerValiron09} [[Peter Selinger]], [[Benoît Valiron]], *Quantum Lambda Calculus*, in: *Semantic Techniques in Quantum Computation*, Cambridge University Press (2009) 135-172 $[$[doi:10.1017/CBO9781139193313.005](https://doi.org/10.1017/CBO9781139193313.005), [pdf](https://www.mscs.dal.ca/~selinger/papers/qlambdabook.pdf)$]$

> [[Selinger (2016)](Quipper#Selinger16):] When the [QPL workshop series](https://www.mathstat.dal.ca/~selinger/qpl/) was first founded, it was called "*Quantum Programming Languages*". One year I wasn't participating, and while I wasn't looking they changed the name to "*Quantum Physics and Logic*" --- same acronym!

> Back in those days in the early 21st century we were actually trying to do programming languages for quantum computing $[$[Selinger & Valiron 2004](#SelingerValiron04)$]$, but the sad thing is: In those days nobody really cared. $[...]$

> Now it's 15 years later and several of these parameters have changed: There has been a renewed interest, from government agencies and also from companies who are actually building quantum computers. $[...]$.

> So now people are working on quantum programming languages *again*.

On [[categorical semantics]] on [[von Neumann algebras]]:

* [[Kenta Cho]], *Semantics for a Quantum Programming Language by Operator Algebras*, EPTCS **172** (2014) 165-190 [[arXiv:1412.8545](https://arxiv.org/abs/1412.8545), [doi:10.4204/EPTCS.172.12](https://doi.org/10.4204/EPTCS.172.12)]

### Quipper as such

Exposition of the general idea of [[quantum programming languages]] for [[classically controlled quantum computation]] with an eye towards the `Quipper`-language:

* [[Benoît Valiron]], [[Neil J. Ross]], [[Peter Selinger]], [[D. Scott Alexander]], [[Jonathan M. Smith]], *Programming the quantum future*, Communications of the ACM **58** 8 (2015) 52–61 [[doi:10.1145/2699415](https://doi.org/10.1145/2699415), [pdf](https://www.iro.umontreal.ca/~brassard/cours/6155/Langages%20de%20programmation%20quantique/Programming%20the%20Quantum%20Future.pdf),  [web](https://cacm.acm.org/magazines/2015/8/189851-programming-the-quantum-future/abstract), promo vid:[YT](https://youtu.be/mzce69oFdH0?list=PLn0nrSd4xjjbIHhktZoVlZuj2MbrBBC_f)]

Original articles on Quipper:

* {#GLRSV13} [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: A Scalable Quantum Programming Language_, ACM SIGPLAN Notices **48** 6 (2013) 333-342 [[arXiv:1304.3390](https://arxiv.org/abs/1304.3390), [doi:10.1145/2499370.2462177](https://doi.org/10.1145/2499370.2462177)]

* [[Jonathan Smith]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _Quipper: Concrete Resource Estimation in Quantum Algorithms_, QAPL 2014 ([arXiv:1412.0625](https://arxiv.org/abs/1412.0625))

* [[Neil J. Ross]], Chapters 8-9 in: *Algebraic and Logical Methods in Quantum Computation*, Ph.D. thesis, Dalhousie University (2015) [[arXiv:1510.02198](https://arxiv.org/abs/1510.02198)]

  > (introducing Proto-Quipper)

Introduction and review:

* [[Alexander Green]], [[Peter LeFanu Lumsdaine]], [[Neil Ross]], [[Peter Selinger]], [[Benoît Valiron]], _An Introduction to Quantum Programming in Quipper_, Lecture Notes in Computer Science **7948** Springer (2013) 110-124 [[arXiv:1304.5485](https://arxiv.org/abs/1304.5485), [doi:10.1007/978-3-642-38986-3_10](https://doi.org/10.1007/978-3-642-38986-3_10)]

* {#Selinger16} [[Peter Selinger]], *Introduction to Quipper*, talk at [QPL2016](http://qpl2016.cis.strath.ac.uk) (2016) [[video](https://youtu.be/59frzb__Eqo)]

Example algorithms:

* Safat Siddiqui, Mohammed Jahirul Islam, Omar Shehab, *Five Quantum Algorithms Using Quipper* [[arXiv:1406.4481](https://arxiv.org/abs/1406.4481)]

On quantum [[software verification]] for/with Quipper:

* Linda Anticoli, Carla Piazza, Leonardo Taglialegne, Paolo Zuliani, _Towards Quantum Programs Verification: From Quipper Circuits to QPMC_, In: Devitt S., Lanese I. (eds) Reversible Computation. RC 2016. Lecture Notes in Computer Science, vol 9720. Springer, Cham ([doi:10.1007/978-3-319-40578-0_16](https://doi.org/10.1007/978-3-319-40578-0_16))

### Dependent linear types and categorical semantics
 {#ReferencesDependentLinearTypesAndCategoricalSemantics}

Discussion of some [[dependent linear type theory]] and [[categorical semantics]] for (proto-)[[Quipper]]:

* [[Francisco Rios]], [[Peter Selinger]], *A Categorical Model for a Quantum Circuit Description Language*, EPTCS **266** (2018) 164-178 [[arXiv:1706.02630](https://arxiv.org/abs/1706.02630), [doi:10.4204/EPTCS.266.11](https://doi.org/10.4204/EPTCS.266.11)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil Ross]], [[Peter Selinger]],  _A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper_, in I. Lanese, M. Rawski  (eds.) _Reversible Computation_ RC 2020. Lecture Notes in Computer Science, vol 12227 [[arXiv:2005.08396](https://arxiv.org/abs/2005.08396), [doi:10.1007/978-3-030-52482-1_9](https://doi.org/10.1007/978-3-030-52482-1_9)]

* {#FuKishidaSelinger20} [[Peng Fu]], [[Kohei Kishida]], [[Peter Selinger]], _Linear Dependent Type Theory for Quantum Programming Languages_, LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceJuly 2020 Pages 440–453 ([arXiv:2004.13472](https://arxiv.org/abs/2004.13472), [doi:10.1145/3373718.3394765](https://dl.acm.org/doi/10.1145/3373718.3394765), [pdf](https://depend.cs.uni-saarland.de/lics-icalp/papers/B5.F), [video](https://www.youtube.com/watch?v=GUT8j4V6Nzg))

* [[Francisco Rios]], *On a Categorically Sound Quantum Programming Language for Circuit Description*, Dalhousie University (2021) [[hdl:10222/80771](http://hdl.handle.net/10222/80771)]

* {#Lee22} [[Dongho Lee]], *Formal Methods for Quantum Programming Languages*, Paris Saclay (Dec 2022) [[hal:tel-03895847](https://hal.science/LMF/tel-03895847)]

Semantics on [[quantum CPOs]] (via [[quantum relations]] on [[quantum sets]]) for interpreting Quipper with term recursion:

* [[Andre Kornell]], [[Bert Lindenhovius]], [[Michael Mislove]], *Quantum CPOs*, EPTCS **340** (2021) 174-187 [[arXiv:2109.02196](https://arxiv.org/abs/2109.02196), [doi:10.4204/EPTCS.340.9](https://doi.org/10.4204/EPTCS.340.9)]

### Dynamic lifting
{#ReferencesDynamicLifting}

The issue of "[[dynamic lifting]]" (of "[[bits]]" resulting from [[quantum measurement]] back into the "[[context]]"):

* brief mentioning on [GLRSV13, p. 5](#GLRSV13)

* [[Robert Rand]], *Formally Verified Quantum Programming*, UPenn (2018) [[ediss:3175](https://repository.upenn.edu/edissertations/3175)]

* {#LeeBardinPerrelleValiron2019} [[Dongho Lee]], Sebastien Bardin, [[Valentin Perrelle]], [[Benoît Valiron]], *Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control*, talk at *[Journées Informatique Quantique 2019](https://quantcert.github.io/GT-IQ'19/)* (Nov 2019) [[pdf](https://quantcert.github.io/GT-IQ%2719/slides/Lee.pdf), [[LBPV_QCWithClassicalControl.pdf:file]]]

* [[Dongho Lee]], [[Valentin Perrelle]], [[Benoît Valiron]], Zhaowei Xu, *Concrete Categorical Model of a Quantum Circuit Description Language with Measurement*, Leibniz International Proceedings in Informatics **213** (2021) 51:1-51:20 [[arXiv:2110.02691](https://arxiv.org/abs/2110.02691), [doi:10.4230/LIPIcs.FSTTCS.2021.51](https://doi.org/10.4230/LIPIcs.FSTTCS.2021.51)]

* [[Andrea Colledan]], [[Ugo Dal Lago]], *On Dynamic Lifting and Effect Typing in Circuit Description Languages* [[arXiv:2202.07636](https://arxiv.org/abs/2202.07636)]

* [[Andrea Colledan]], [[Ugo Dal Lago]], *On Dynamic Lifting and Effect Typing in Circuit Description Languages*, talk at *TYPES Workshop*, Nantes (2022)  [[pdf](https://types22.inria.fr/files/2022/06/TYPES_2022_slides_13.pdf), [[ColledanLago-DynamicsLifting.pdf:file]]]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *A biset-enriched categorical model for Proto-Quipper with dynamic lifting*, in proceedings of *[Quantum Physics and Logic 2022](https://www.qplconference.org/)* [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]

* [[Peng Fu]], [[Kohei Kishida]], [[Neil J. Ross]], [[Peter Selinger]], *Proto-Quipper with dynamic lifting*, in proceedings of *[Quantum Physics and Logic 2022](https://www.qplconference.org/)* [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

* [[Dongho Lee]], *Formal Methods for Quantum Programming Languages*, Paris Saclay (Dec 2022) [[hal:tel-03895847](https://hal.science/LMF/tel-03895847)]

* [[Frank (Peng) Fu]], *Proto-Quipper with Dynamic Lifting*, [talk at](CQTS#PengFuOct2023) [[CQTS]] (23 Oct 2023) [video:[YT](https://youtu.be/bBL7rlqbDWM)]

[[!redirects QPL]]

[[!redirects Proto-Quipper]]


Quiv

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

## Idea

__$Quiv$__ or __$DiGraph$__ is the category of [[quivers]] or [[directed graphs]]. It can be viewed as the default categorical model for the concept of a category of graphs.

## Definition

We can define a quiver to be a functor $G\colon X^{op} \to Set$, where $X^{op}$ is the [[category]] with an [[object]] $0$, an object $1$ and two [[morphisms]] $s, t\colon 1 \to 0$, along with [[identity morphisms]].  This lets us define $Quiv$ as the category of [[presheaves]] on $X$, where:

* objects are [[functors]] $G\colon X^{op} \to Set$,
* morphisms are [[natural transformations]] between such functors.

In other words, $Quiv$ is the [[functor category]] from $X^{op}$ to [[Set]].

## As a topos

The category $Quiv = Set^{X^{op}}$, being a category of [[presheaves]], is a [[topos]]. The [[representable functors]] $X(-, 0), X(-, 1)$ may be thought of as the "generic figures" (generic vertex, generic edge) that occur in directed graphs:

$$X(-, 0) = \bullet, \qquad X(-, 1) = (x \stackrel{e}{\to} y)$$

From this picture, we can see that:

* $X(-, 0)$ has two subobjects: $\emptyset$ and $\bullet$.
* $X(-, 1)$ has five subobjects: $empty, x, y, (x, y), (x \stackrel{e}{\to} y)$.

The fact that $Quiv$ is a presheaf topos allows us to use [[topos theory]], which give answers to questions like whether finite [[limit|limits]] of directed graphs exist, or how to construct the [[exponential object|exponential]] quiver $Y^X$ of all morphisms $X\to Y$ between two quivers in $Quiv$.

The [[initial object]] $0$ in $Quiv$ is the graph without vertices and edges aka the _empty graph_. The [[terminal object]] is the graph $1$ with one vertex  and one edge. [[point|Points]] of an arbitrary graph $Y$ i.e. maps $1\to Y$ then correpond to _loops_ in $Y$ whence any loopfree graph with a vertex has no points but is not empty ($\neq 0$).

In the following subsections, more of the structure of the topos $Quiv$ will be worked out explicitly.

### The subobject classifier{#Omega_graph}

Knowing the subobjects of the representable functors allows us to calculate the structure of the [[subobject classifier]] $\Omega$, since they correspond to the elements of the value of $\Omega$ at the corresponding objects of $X^{op}$:

* The set of vertices of $\Omega$ is $\Omega(0)=\{\emptyset,\bullet\}$.
* The set of edges of $\Omega$ is $\Omega(1)=\{empty, x, y, (x, y), (x \stackrel{e}{\to} y)\}$.

We can visualize $\Omega$ as the following quiver with two vertices and five edges:

$$\array{
\emptyset & \underoverset{x}{y}{\rightleftarrows} & \bullet \\
\mathllap{empty} \circlearrowleft & & \circlearrowleft \circlearrowleft \mathrlap{x \stackrel{e}{\to} y} \\
 & & \mathllap{(x, y)}
}$$

where the vertex $\emptyset$ has one loop labeled "empty", and the vertex $\bullet$ has two loops with labels $(x, y)$ and ($x \stackrel{e}{\to} y$).

Suppose now that $X\subseteq Y$ is a subgraph and $\chi_X \colon Y\to\Omega$ is its characteristic map. On vertices:

* $\chi_X$ maps vertices of $Y$ not in $X$ to $\emptyset$.
* $\chi_X$ maps vertices in $X$ to $\bullet$.

Thus, a vertex is either contained in a subgraph or not. This is represented in $\Omega$ by two vertices.

For edges, the situation is more complicated since there are five cases, which are represented by five edges in $\Omega$. On edges:

* $\chi_X$ maps edges which have neither source nor target in $X$ to the loop at $\emptyset$.
* $\chi_X$ maps edges with only the source vertex in $X$ to $x$.
* $\chi_X$ maps edges with only the target vertex in $X$ to $y$.
* $\chi_X$ maps edges with both source and target vertices in $X$, but where the edges themselves are not in the subgraph $X$, to $(x,y)$.
* $\chi_X$ maps edges that are in the subgraph $X$ to ($x \stackrel{e}{\to} y$).

### (Double) negation

The [[negation]] $\neg \colon \Omega \to \Omega$ is defined as the characteristic map of $\bot \colon 1 \to\Omega$. It specifies how

$$im(\bot)= \underoverset{{empty}}{\empty}{\circlearrowleft}$$

sits as a subgraph in $\Omega$:

* Since $\bullet$ is not in $im(\bot)$, but $\empty$ is, $\neg$ interchanges the two vertices.
* Thus, all loops at $\bullet$ must go to the loop ${empty}$.
* The edge $empty$ goes to ($x \stackrel{e}{\to} y$), since it is fully contained in $im(\bot)$.
* Now $x$ has only its target vertex in $im(\bot)$, hence it goes to the edge $y$.
* However, $y$ has only its source vertex in $im(\bot)$, so it goes to the edge $x$.

Taking the [[complement]] of a subobject $X\subseteq Y$, i.e. taking the subobject $\neg X$ of $Y$ that is classified by $\neg\circ\chi_X$, then amounts to taking all vertices of $Y$ not in $X$ and all the edges in $Y$ between them.

Hence, the result $\neg\neg X$ of applying $\neg$ twice to $X\subseteq Y$ amounts to adding to $X$ all the edges of $Y$ that have source and target in $X$. This implies that a subgraph $X\subseteq Y$ is [[dense subobject|dense]] for the [[double negation|double negation topology]] given by $\neg\circ\neg \colon\Omega\to\Omega$ precisely when it contains all vertices of $Y$, since taking the complement twice will throw into $\neg\neg X$ all the edges in $Y$ between all the vertices in $X$.

By definition, a quiver $X$ is [[separated object|separated]] for $\neg\neg$ when, for every other quiver $Y$ and dense subobject $i \colon S\hookrightarrow Y$, and any map $f \colon S\to X$, there is at most one $g \colon Y\to X$ such that the following diagram commutes:

$$
\array{
S & & \\
i\downarrow &\searrow &f \\
Y &\underset{g}{\to} & X
}
$$

A separated quiver $X$ is a $\neg\neg$-sheaf when such a unique $g$ always exists.

Suppose that a quiver $X$ has a pair of parallel edges $w,z$. Then the subgraph $i:S\hookrightarrow X$ that is just like $X$ but has $w,z$ omitted is dense in $X$. Let $\tau_{zw}:X\to X$ be the automorphism of $X$ that is just like the identity on $X$ except that it interchanges $w$ and $z$. Then $id_X\circ i=\tau_{zw}\circ i=i$ and one sees that $X$ is not separated.

Conversely, let $X$ be a quiver with at most one edge $x\to y$ between any pair $(x,y)$ of vertices and $f:S\to X$ be a map with $i:S\hookrightarrow Y$ is dense in $Y$. Since $i$ is a bijection on the vertex sets of $S$ and $Y$, if a factorization of $f$ through $g:Y\to X$ and $i$ exists the effect of $g$ on the vertices is uniquely determined by $f$ but since in $X$ there is at most one edge between any pair of vertices the image of any edge $a\to b$ in $Y$ under $g$ is already fixed: it is the unique edge between $g(a)$ and $g(b)$. In particular, one sees that a separated object $X$ is a sheaf precisely when there exists exactly one edge between any pair of vertices since then arbitrary edges in arbitrary factors $Y$ can be mapped to the appropriate edge in $X$. To sum up:

+-- {: .num_prop #negneg_subcats}
###### Proposition
A quiver $X$ is separated for the double negation topology $\neg\neg$ precisely if there exists at most an edge $a\to b$ between any pair $(a,b)$ of vertices. $X$ is a $\neg\neg$-sheaf precisely if there exists a unique edge $a\to b$ between any pair $(a,b)$. $\Box$
=--

The corresponding full subcategories are denoted by $Sep_{\neg\neg}(Quiv)$ and $Sh_{\neg\neg}(Quiv)$ , respectively. By generalities, it follows that $Sep_{\neg\neg}(Quiv)$ is a [[quasitopos]] and $Sh_{\neg\neg}(Quiv)$ is a [[Boolean topos]].

Quivers that have at most one edge between any pair of vertices can be called 'simple' with the caveat that contrary to (the usual concept of) a simple graph they are allowed to have loops. Similarly, $\neg\neg$-sheaves can be called 'complete'.

Since the edges of $\neg\neg$-separated quivers simply encode a binary endorelation on their vertex sets and being a morphism between $\neg\neg$-separated quivers then amounts to preserve that relation one sees that $Sep_{\neg\neg}(Quiv)$ and $Sh_{\neg\neg}(Quiv)$ are equivalent to the categories $Bin$ with objects $(X,\rho)$ where $X$ is a set and $\rho$ a binary endorelation on $X$, and, respectively, the category  $TotalRel$ of sets equipped with the total relation. The latter can be identified with $Set$ since morphisms between sets equipped with the total relation behave just like ordinary functions between sets.

### Some subcategories and adjunctions

The inclusion $Sh_{\neg\neg}(Quiv)\hookrightarrow Sep_{\neg\neg}(Quiv)$ is actually an [[level of a topos|essential localization]], since it corresponds (from the relational perspective) to the [[adjoint triple]] $e\dashv u\dashv t:Set\hookrightarrow Bin$ where $t$ maps a set $X$ to $(X,\tau_X)$ with $\tau_X$ the total relation on $X$, $u$ is the forgetful functor mapping $(X,\rho)$ to $X$ and, $e$ maps a set $X$ to $(X,\empty)$.

Similarly, $Sh_{\neg\neg}(Quiv)\overset{i}{\hookrightarrow} Quiv$ is an essential subtopos: if we identify sheafification $r$ with the functor that maps a quiver to the quiver on the same vertex set with edge set the total relation on the vertex set, then $l\dashv r\dashv i$ where $l$ forgets the edges of a complete quiver.

In particular, it follows then from general properties of the [[double negation|double negation topology]] that $Sh_{\neg\neg}(Quiv)$ is the [[Aufhebung]] of $0\dashv 1$. Whence, there exists indeed a notion of 'codiscreteness' (= an Aufhebung of $0\dashv 1$) for quivers, namely 'completeness', but it does not arise from a right adjoint to the [[section|section functor]] $\Gamma \colon Quiv\to Set$ that maps a quiver to its set of loops. Indeed, the adjoint string $\Pi\dashv\Delta\dashv\Gamma \colon Quiv\to Set$ that comes with the 'discrete' inclusion $\Delta$ that maps a set to the quiver with vertex set $X$ and edge set precisely one loop for every vertex, is not a localisation since $\Pi\dashv\Delta$ is not a [[geometric morphism]] because $\Pi$ fails to preserve products.

Furthermore, since it is a general result for presheaf toposes (cf. [La Palme Reyes et al. 2004](#RRZ04), p.204) that $\Gamma$ has a right adjoint $B$ precisely if a generic figure has a point and in the case of Quiv neither the generic vertex nor the generic edge contains a loop, we see that the functor $\Gamma:Quiv\to Set$ has no right adjoint.

## Related entries

* [[hypergraph]]

* [[graphic category]]

## References

* R. T. Bumby, D. M. Latch, _Categorical Constructions in Graph Theory_ , Internat. J. Math. & Math. Sci. **9** no.1 (1986) pp.1-16. ([web](https://www.hindawi.com/journals/ijmms/1986/791947/))

* {#Law89a} [[F. W. Lawvere]], *Qualitative Distinctions between some Toposes of Generalized Graphs*, Cont. Math. **92** (1989) pp.261-299.

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, *Generic Figures and their Glueings*, Polimetrica Milano 2004. ([pdf](https://www.researchgate.net/profile/Gonzalo-Reyes-8/publication/333579691_Generic_figures_and_their_glueings_A_constructive_approach_to_functor_categories/links/5cf55083a6fdcc847502c695/Generic-figures-and-their-glueings-A-constructive-approach-to-functor-categories.pdf))

* {#Vigna03} S. Vigna, _A Guided Tour in the Topos of Graphs_ , arXiv.0306394 (2003). ([arXiv](https://arxiv.org/abs/math/0306394))

category: category

[[!redirects Quiv]]
[[!redirects DiGraph]]
[[!redirects Digraph]]


quiver

> This entry is about the mathematical notion. For the commutative diagram editor, see *[[quiver (editor)]]*.

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### Context
#### Graph theory
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[[!include graph theory - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

A _quiver_ ([Gabriel 72](#Gabriel72)) is a collection of [[edges]] which may stretch between (ordered) [[pairs]] of "points", called _[[vertices]]_.  Hence a quiver is a kind of [[graph]].

The term "quiver" (German: _Köcher_) --- for what in [[graph theory]] is (and was) known as *[[directed graphs]]* ([[directed graph|directed]] [[pseudograph|pseudo]]-[[multigraphs]], to be precise) --- is due to [Gabriel 1972](#Gabriel72), where it says on the first page:

> For such a 4-tuple $\big[$ $E \underoverset{\underset{t}{\longrightarrow}}{\overset{s}{\longrightarrow}}{} V$ $\big]$ we propose the term _quiver_, and not _graph_, since there are already too many notions attached to the latter word.

{#ConceptWithAnAttitude} Hence the notion "quiver" is a _[[concept with an attitude]]_, indicating that one is interested in certain special constructions with these graphs, distinct from what [[graph theory]] typically cares about: Namely one is interested in their _[[quiver representations]]_. In this way, "quiver" is really a term in [[representation theory]]; see also [Derksen & Weyman 2005, ftn 1](#DerksenWeyman05):

> The underlying motivations of quiver theory are quite different from those in the traditional graph theory. To emphasize this distinction, it is common in our context to use the word "quivers" instead of "graphs".

A key result in [[quiver representation]]-theory is _[[Gabriel's theorem]]_, also form [Gabriel 1972](#Gabriel72).

## Definitions

### Slick definition

The **[[walking structure|walking]] quiver**[^1] $X$ is the [[category]] with

* one [[object]] $X_0$, called the object of _vertices_;

* one object $X_1$, called the object of _edges_;

* two [[morphisms]] $s, t\colon X_1 \to X_0$, called
the _source_ and _target_;

* together with [[identity morphisms]].

A **quiver** is a [[functor]] $G\colon X \to$ [[Set]].

More generally, a **quiver [[internalization|in]] a category $C$** is a [[functor]] $G\colon X \to C$.

The category of quivers in $C$, [[Quiv]]$(C)$, is the [[functor category]] $C^{X}$, where:

* objects are functors $G\colon X \to C$,

* morphisms are [[natural transformation|natural transformations]] between such functors.

In the basic case where $C$ is [[Set]], the category Quiv(Set) is equivalent to the category of [[presheaves]] on $X^{op}$.  So: the category of quivers, [[Quiv]], is the category of presheaves on the category $X^{op}$.

[^1]: Also called "the elementary "parallel process" " by [[William Lawvere|Lawvere]] in [p. 272](#GeneralizedGraphs), "a category that has no compositions in it" by [[William Lawvere|Lawvere]], in his lecture at [[Como]].

### Nuts-and-bolts definitions

A __quiver__ $G$ consists of two sets $E$ (the set of _edges_ of $G$), $V$ (the set of _vertices_ of $G$) and two functions

$$s, t\colon E \rightrightarrows V$$

(the source and target functions). More generally, a __quiver internal to a category__ (more simply, *in* a category) $C$ consists of two objects $E$, $V$ and two morphisms $s, t\colon E \rightrightarrows V$.

If $G = (E, V, s, t)$ and $G' = (E', V', s', t')$ are two quivers in a category $C$, a __morphism__ $g\colon G \to G'$ is a pair of morphisms $g_0\colon V \to V'$, $g_1\colon E \to E'$ such that $s' \circ g_1 = g_0 \circ s$ and $t' \circ g_1 = g_0 \circ t$.

In [[graph theory]], a quiver is often (cf. [p. 4](#DG2nd) or [Figure 2-2](#White84)) called a __directed pseudograph__ (or some variation on that theme), but category theorists often just call them __directed graphs__.

## Remarks

Let $G_0 = G(X_0)$ and $G_1 = G(X_1)$.

* A quiver in $C$ is a [[presheaf]] on $X^{op}$ with values in $C$.

* A quiver is a [[globular set]] which is concentrated in the first two degrees.

* A quiver can have distinct edges $e,e'\in G_1$ such that $s(e) = s(e')$ and $t(e) = t(e')$. A quiver can also have loops, namely, edges with $s(e) = t(e)$.

* A quiver is **[[complete graph|complete]]** if for any pair of vertices $v,v'\in G_0$, there exists a unique directed edge $e\in G_1$ with $s(e) = v, t(e) = v'$.

## Terminology

Saying _quiver_ instead of _directed (multi)graph_ indicates focus on a certain set of operation intended on that graph. Notably there is the notion of a [[quiver representation]].

Now, one sees that a _representation_ of a graph $G$ in the sense of quiver representation is nothing but a [[functor]] $\rho\colon Q := F(G) \to Vect$ from the _free category_ $F(G)$ on the quiver $G$:

Given a graph $G$ with collection of vertices $G_0$ and collection of edges $G_1$, there is the free category $F(G)$ on the graph whose collection of objects coincides with the collection of vertices, and whose collection of morphisms consists of finite [[sequence]]s of edges in $G$ that fit together head-to-tail (also known as _[[path]]s_). The composition operation in this free category is the concatenation of sequences of edges.

Here we are taking advantage of the [[adjoint functor|adjunction]] between [[Cat]] (the category of small categories) and [[Quiv]] (the category of directed graphs).  Namely, any category has an underlying directed graph:

$$U\colon Cat \to Quiv$$

and the left adjoint of this functor gives the free category on a directed graph:

$$F\colon Quiv\to Cat $$

Since this is the central operation on quivers that justifies their distinction from the plain concept of directed graph, we may adopt here the point of view that _quiver_ is synonymous with _free category_.

So a [[representation]] of a quiver $Q = F(G)$ is a functor

$$R\colon Q \to Vect $$

Concretely, such a thing assigns a vector space to each vertex of the graph $G$, and a linear operator to each edge.  Representations of quivers are fascinating things, with connections to ADE theory, quantum groups, string theory, and more.

## Properties

### Relation to free categories

It may be handy to *identify* a quiver with its free category.  This can be justified in the sense that the functor $F\colon Quiv \to Cat$ is an embedding ($k$-[[k-surjectivity|surjective]] for all $k \gt 0$) on the [[cores]].  In other words, [[isomorphisms]] between quivers may be identified with [[equivalence of categories|equivalences]] between free categories with no ambiguity.

However, at the level of noninvertible morphisms, this doesn\'t work; while $U$ is [[faithful functor|faithful]], it is *not* [[full functor|full]].  In other words, there are many [[functors]] between free categories that are not morphisms of quivers.

Nevertheless, if we fix a quiver $G$ and a category $D$, then a [[quiver representation|representation]] of $G$ in $D$ is precisely a functor from $F(G)$ to $D$ (or equivalently a quiver morphism from $G$ to $U(D)$), and we may well want to think of this as being a morphism (a [[heteromorphism]]) from $G$ to $D$.  As long as $D$ is not itself a free category, this is unlikely to cause confusion.

### Relation to representation theory of algebras

For $Q$ a quiver, write $k Q$ for the _path algebra_ of $Q$
over a ground field $k$.  That is, $k Q$ is an algebra with $k$-basis given by finite composable sequences of arrows in $Q$, including a "lazy path" of
length zero at each vertex. The product of two paths composable paths is their composite, and the product of non-composable paths is zero.

A module over $k Q$ is the same thing as a representation of $Q$, so the theory of representations of quivers can be viewed within the broader context of representation theory of (associative) algebras.

If $Q$ is acyclic, then $k Q$ is finite-dimensional as a vector space, so in studying representations of $Q$, we are really studying representations of a finite dimensional algebra, for which  many interesting tools exist
(Auslander-Reiten theory, tilting, etc.).

### Classification
 {#Classification}

[[Gabriel's theorem]] ([Gabriel 72](#Gabriel72)) says that connected quivers with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]]are precisely the _[[Dynkin quivers]]_: those whose underlying [[undirected graph]] is a [[Dynkin diagram]] in the [[ADE classification|ADE series]], and that the [[indecomposable object|indecomposable]] [[quiver representations]] are in [[bijection]] with the positive [[root (in representation theory)|roots]] in the [[root system]] of the Dynkin diagram. ([Gabriel 72](#Gabriel72)).

## Generalizations

### Enriched quivers

Let $V$ be a [[category]] (or a [[(infinity,1)-category]]). A quiver $Q$ is a __$V$-enriched quiver__ if it has a collection of objects $Ob(Q)$ and a $V$-valued functor $Mor: Ob(Q) \times Ob(Q) \to V$ for all objects $a, b \in Ob(Q)$. Quivers in the usual sense are enriched in [[Set]], while [[graph|loop directed graphs]]/[[binary relation|binary]] [[relation|endorelations]] are quivers enriched in the category of [[truth values]] $\Omega$. [[n-quiver|$n$-quivers]] are quivers enriched in the category of $(n-1)$-quivers.

In [[dependent type theory]] with [[type of types|universes]], a [[type]] equipped with an [[identity type]] is a quiver type enriched in a universe [[Type]].

## Related concepts

* [[quiver representation]]

* [[McKay quiver]], [[McKay correspondence]]

* [[quiver gauge theory]]

* [[knots-quivers correspondence]]

* [[zigzag persistence]]

[[!include oidification - table]]

## References

The concept of quiver in the context of [[quiver representation]] (and their classification in [[Gabriel's theorem]]) originates with

* {#Gabriel72} [[Peter Gabriel]], _Unzerlegbare Darstellungen. I_, Manuscripta Mathematica 6: 71–103, (1972) ([doi:10.1007/BF01298413](https://link.springer.com/article/10.1007/BF01298413), [MR332887](http://www.ams.org/mathscinet-getitem?mr=332887) [doi](http://dx.doi.org/10.1007/BF01298413))

Some general-purpose references include

* {#DerksenWeyman05} Harm Derksen, Jerzy Weyman, *Quiver representations*, Notices of the AMS  (Feb 2005) [[pdf](http://www.ams.org/notices/200502/fea-weyman.pdf), full:[pdf](http://www.ams.org/notices/200502/200502FullIssue.pdf)]

* William Crawley-Boevey, _Lectures on quiver representations_ ([pdf](https://www.math.uni-bielefeld.de/~wcrawley/quivlecs.pdf)).

* Alistair Savage, _Finite-dimensional algebras and quivers_ ([arXiv:math/0505082](http://www.arxiv.org/abs/math/0505082)), _Encyclopedia of Mathematical Physics_, eds. J.-P. Françoise, G.L. Naber and Tsou S.T., Oxford, Elsevier, 2006, volume 2, pp. 313-320.

Quivers (referred to as _directed pseudographs_) were a tool in parts of the work of Ringel and Youngs in the second half the twentieth century to prove Heawood's formula for every finite genus, cf. e.g. Fig. 2.3 the monograph

* {#Ringel1974} Gerhard Ringel: _Map Color Theorem_. Springer. Grundlehren Band 209. 1974

Beware that, strictly speaking, for Ringel, "quiver" means "embedded quiver" (into a given surface);  in particular the author distinguishes between the two possible  orientations of an embedded loop.

Quivers embedded in surfaces are studied in:

* {#White84} Arthur T. White: _Graphs, Groups and Surfaces_. North Holland. Completely revised and enlarged edition (1985)

A special kind of quiver (finite, no loops, no parallel arcs) is treated in

* {#DG2nd} Gregory Gutin, Jørgen Bang-Jensen: _Digraphs: Theory, Algorithms and Applications_. Springer Monographs in Mathematics. Second Edition (2009)

* {#GeneralizedGraphs} [[William Lawvere]]: _Qualitative Distinctions Between Some Toposes of Generalized Graphs_, Contemporary Mathematics 92 (1989)

Some introductory material on the relation between quivers (there called _multigraphs_) and categories can be found in

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_. ([arXiv](https://arxiv.org/abs/1912.10642))

[[!redirects quiver]]
[[!redirects quivers]]

[[!redirects quiver theory]]
[[!redirects quiver theories]]

[[!redirects directed pseudograph]]
[[!redirects directed pseudographs]]

[[!redirects weak quiver]]
[[!redirects weak quivers]]
[[!redirects strict quiver]]
[[!redirects strict quivers]]


quiver (editor)

> This entry is about the commutative diagram editor. For the mathematical notion, see *[[quiver]]*.

\begin{imagefromfile}
  "file_name": "quiver.png",
  "width": 640
\end{imagefromfile}

**quiver** ([q.uiver.app](https://q.uiver.app/)) is an open source web-based editor, created by [[varkor]], for [[commutative diagrams]] and [[pasting diagrams]]. It exports to LaTeX via [tikz-cd](https://github.com/astoff/tikz-cd).

The name "quiver" is based on the term *[[quiver]]* used for [[directed graphs]] in [[representation theory]].

For the purpose of editing diagrams here on the nLab, see also at *[[HowTo]]* the section on *[TikZ](HowTo#TikZ)*.

## Links

* [**quiver**](https://q.uiver.app/)

* [quiver GitHub repository](https://github.com/varkor/quiver)

category: software

[[!redirects quiver editor]]


quiver algebra

The **quiver algebra** or **path algebra** of a [[quiver]] $G$ is the [[category algebra]] of the [[free category]] over $G$.

[[!redirects quiver algebra]]
[[!redirects quiver algebras]]

[[!redirects path algebra]]
[[!redirects path algebras]]


quiver gauge theory

+-- {: .rightHandSide}
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

A _quiver gauge theory_ is a [[gauge theory]] -- usually a [[super Yang-Mills theory]] -- whose [[field (physics)|field]] content is determined by a [[quiver]] in that to each node $x_i$ of the quiver is assigned a [[gauge group]] $U(n_i)$ and to each edge a [[fermion]] field species which is charged as the [[fundamental representation]] under the gauge group of the target and the anti-fundamental representation of the domain of the quiver edge.

Such data naturally arises in [[geometric engineering of quantum field theory]] from [[D-brane]] models in [[string theory]] with D-branes located at certain [[singularities]] of a [[Calabi-Yau variety]]: the nodes $x_i$ of the quiver for a $(d+1)$-dimensional quiver gauge theory correspond to $n_i$ coincident $d$-[[D-branes]] and the edges of the quiver to [[open string]] states stretching between these branes. Under this interpretation, a [[representation]] of the quiver corresponds to giving all these fields a [[vacuum expectation value]].

### As fixed point locus

Consider a $U(N)$ [[Chan-Paton gauge field]] theory on [[D-branes]] at a $G$-[[orbifold]] [[singularity]] $\mathbb{C}^n\sslash G$ for $\Gamma$ a [[finite group]].

Then the [[fermions]] transform in $\mathbf{c} \otimes (N \mathbf[r]) \otimes (N \mathbf{r}^\ast)$, where $\mathbf{r}$ denotes the [[regular representation]] of $\Gamma$ and $\mathbf{c}$ denotes the representation which defines the [[orbifold]] [[action]].

Then the $\Gamma$-[[fixed point space]] inside this representation is

$$
  \big(
    \mathbf{c} \otimes (N \mathbf{r}) \otimes (N \mathbf{r}^\ast)
  \big)^\Gamma
$$

is the corresponding weighted [[McKay quiver]] matrix.

This observation is due to [Lawrence-Nekrasov-Vafa 98, Section 2.1](#LawrenceNekrasovVafa98)

## Properties

### Relation between coherent sheaves and quiver representations

The [[representation theory|representation theoretic]] aspects of the [[gauge theory]] thus obtained depend only on aspects which are already seen by the [[B-model]] [[topological string]], which in this context plays the role of a simplified version of the physical [[superstring]] itself. By the discussion at _[[TCFT]]_ this [[topological field theory]] is essentially determined by its [[derived category]] of [[B-branes]], which is a [[derived category]] of [[coherent sheaves]] on the given [[target space|target]] [[spacetime]]. Localized to suitable [[singularities]] the [[B-branes]] decompose into [[fractional branes]] which is mathematically reflected by the existence of [[exceptional collections]] of objects in the derived category. The [[endomorphism algebra]] of the [[direct sum]] of the objects in the exceptional collection turns out to be the [[path algebra]] of the corresponding [[quiver]] and under this identification the [[derived categories]] of [[coherent sheaves]] and of [[quiver representations]] are [[equivalence of categories|equivalent]] ([Bondal 90](#Bondal90)). This equivalence to quiver representations gives one of the ways of getting a concrete combinatorial handle on derived categories of [[B-branes]] and hence on properties of quiver gauge theory.

### Relation to Seiberg duality

Under [[geometric engineering of quantum field theory]] via [[D-branes]] situated at [[ADE-singularities]] in non-compact [[Calabi-Yau varieties]] as above, for instance [[Seiberg duality]] of the corresponding quiver gauge field theories may be understood in terms of [[equivalences of categories]] of derived quiver representations corresponding to [[mutations of exceptional collections]] etc. ([Robles-Llana & Rocek 04](#RoblesLlanaRocek04)).

## Related concepts

* [[McKay correspondence]]

* [[knots-quivers correspondence]]

* [[AGT correspondence]]

* [[fractional D-brane]]

## References

### General

An original article is

* {#DouglasMoore96} [[Michael Douglas]], [[Gregory Moore]], _D-branes, Quivers, and ALE Instantons_ ([arXiv:hep-th/9603167](http://arxiv.org/abs/hep-th/9603167))

The understanding of the [[McKay quiver]] as the [[fixed point space]] in the [[fermion]]-representation under [[R-symmetry]] is due to

* {#LawrenceNekrasovVafa98} A. Lawrence, [[Nikita Nekrasov]], [[Cumrun Vafa]], _On Conformal Theories in Four Dimensions_, Nucl.Phys. B533 (1998) 199-209 ([arXiv:hep-th/9803015](https://arxiv.org/abs/hep-th/9803015))

In

* {#Bondal90} [[Alexei Bondal]], _Helices, representations of quivers and Koszul algebras_, in Helices and vector bundles, vol. 148 of London Math. Soc. Lecture Note Ser., pp. 75–95. Cambridge Univ. Press, Cambridge, 1990.

an [[exceptional collection]] in the [[derived category]] of [[coherent sheaves]] on a suitable [[Calabi-Yau variety]] [[cone]] is used to induce a [[quiver]] and make the derived category be equivalent to that of its [[quiver representations]].

Review and discussion of further details:

* {#He04} [[Yang-Hui He]], _Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities_ ([arXiv:hep-th/0408142](https://arxiv.org/abs/hep-th/0408142))

* {#He18} [[Yang-Hui He]], _Quiver Gauge Theories: Finitude and Trichotomoty_, Mathematics 2018, 6(12), 291 ([doi:10.3390/math6120291](https://doi.org/10.3390/math6120291))

* {#BergmnProudfoot04} [[Aaron Bergman]], [[Nicholas Proudfoot]], _Moduli Spaces for D-branes at the Tip of a Cone_, JHEP0603:073, 2006 ([arXiv:hep-th/0510158](http://arxiv.org/abs/hep-th/0510158))

* {#Katz04} [[Sheldon Katz]], _ADE Quiver representations and branes_ ([pdf](http://math.tecnico.ulisboa.pt/galg/WAGP04/Katz-lecture2.pdf)), lecture 2 of _ADE Geometry and dualities_  ([pdf](http://math.tecnico.ulisboa.pt/galg/WAGP04/Katz-lecture1.pdf))

* [[Taro Kimura]], *Quiver Gauge Theory*, Chapter 2 in: *Instanton Counting, Quantum Geometry and Algebra*, Mathematical Physics Studies, Springer (2021) [[doi:10.1007/978-3-030-76190-5_4](https://doi.org/10.1007/978-3-030-76190-5_4)]

In [[heterotic string theory]] in relation to [[Donaldson-Thomas theory]]:

* {#HeLee12} [[Yang-Hui He]], Seung-Joo Lee, _Quiver Structure of Heterotic Moduli_, J. High Energ. Phys. (2012) 2012: 119 ([arXiv:1208.3004](https://arxiv.org/abs/1208.3004))

With emphasis on [[string phenomenology]]:

* [[Angel Uranga]], _From quiver diagrams to particle physics_ ([arXiv:hep-th/0007173](https://arxiv.org/abs/hep-th/0007173))

Discussion from the point of view of [[worldsheet]] [[2d CFT]] is in

* [[Wolfgang Lerche]], A. Lutken, [[Christoph Schweigert]], _D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories_, Nucl.Phys. B622 (2002) 269-278 ([arXiv:hep-th/0006247](http://arxiv.org/abs/hep-th/0006247))

Discussion for [[Sasakian manifolds]] includes

* [[Olaf Lechtenfeld]], Alexander Popov, [[Richard Szabo]], _Sasakian quiver gauge theories and instantons on Calabi-Yau cones_ ([arXiv:1412.4409](http://arxiv.org/abs/1412.4409))

### Seiberg duality

* David Berenstein, [[Michael Douglas]], _Seiberg Duality for Quiver Gauge Theories_ ([arXiv:hep-th/0207027](http://arxiv.org/abs/hep-th/0207027))

* Subir Mukhopadhyay, Koushik Ray, _Seiberg duality as derived equivalence for some quiver gauge theories_ ([arXiv:hep-th/0309191](http://arxiv.org/abs/hep-th/0309191))

* {#RoblesLlanaRocek04} Daniel Robles-Llana, Martin Rocek, _Quivers, Quotients, and Duality_ ([arXiv:hep-th/0405230](http://arxiv.org/abs/hep-th/0405230))

On an [[algorithm]] for constructing [[S-duality|S-dual]] [[quiver gauge theories]]:

* Chiung Hwang, [[Sara Pasquetti]], Matteo Sacchi, *Rethinking mirror symmetry as a local duality on fields*, Phys. Rev. D **106** (2022) 105014 [[arXiv:2110.11362](https://arxiv.org/abs/2110.11362), [doi:10.1103/PhysRevD.106.105014](https://doi.org/10.1103/PhysRevD.106.105014)]

* Riccardo Comi, Chiung Hwang, Fabio Marino, [[Sara Pasquetti]], Matteo Sacchi, *The $SL(2, \mathbb{Z})$ dualization algorithm at work* [[arXiv:2212.10571](https://arxiv.org/abs/2212.10571)]

[[!redirects quiver gauge theories]]


quiver representation

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[quiver]] $Q$, a *[[linear representation]]* of $Q$ (over some [[ground field]] $\mathbb{K}$) is:

1. a $\mathbb{K}$-[[vector space]] $V_v$ for each [[vertex]] $v \in Q_0$,

1. a [[linear map]] $V_v \xrightarrow{\rho(e)} V_{v'}$ for each [[edge]] $e \in Q_1$.

Notice that there is *no* further compatibility condition, in particular if there are edges forming a [[triangle]], then the associated [[linear maps]] are *not required* to be related under [[composition]].

A [[homomorphism]] between two quiver representations is a [[linear map]] $V_v \xrightarrow{\phi_v} V'_v$ for each [[vertex]], such that for each [[edge]] $v \xrightarrow{e} v'$ the following [[commuting diagram|diagram commutes]] in [[Vect]]${}_{\mathbb{K}}$:

\begin{tikzcd}
  V_v
  \ar[r, "{ \rho(e)}"]
  \ar[d, "{\phi_v}"]
  &
  V_{v'}
  \ar[d, "{\phi_{v'}}"]
  \\
  V'_v
  \ar[r, "{\rho'(e)}"]
  &
  V'_{v'}
  \mathrlap{\,.}
\end{tikzcd}

This makes a [[category]] of quiver representations of $Q$ over $\mathbb{K}$, typically denoted $Rep_{\mathbb{K}}(Q)$, or similar.

In other words, if one regards $Q$ as the [[directed graph]] that it is, and considers its [[free category]] $FrCat(Q)$, then

1. a quiver representation is a [[functor]]

   $$
     FrCat(Q) \xrightarrow{\rho} Vect_{\mathbb{K}}
     \,,
   $$

1. a [[morphism]] of quiver representations is a [[natural transformation]]

   $$
     \phi \;\colon\; \rho \Rightarrow \rho'
     \,,
   $$

1. and the [[category of quiver representations]] is [[equivalence of categories|equivalently]] the [[functor category]] from the [[free category]] of the quiver to the [[category of vector spaces]]:

   $$
     Rep_{\mathbb{K}}(Q)
     \;\simeq\;
     Func
     \big(
       FrCat(Q)
       ,\,
       Vect_{\mathbb{K}}
     \big)
     \,.
   $$

## Related concepts

* [[Gabriel's theorem]]

## References

Review:

* Harm Derksen, Jerzy Weyman, *Quiver Representations*, Notices of the AMS **52** 2 (2005) $[$[pdf](https://www.ams.org/notices/200502/fea-weyman.pdf)$]$

Textbook accounts:

* [[Ralf Schiffler]], *Quiver Representations*, CMS Books in Mathematics, Springer  (2014) $[$[doi:10.1007/978-3-319-09204-1](https://doi.org/10.1007/978-3-319-09204-1)$]$

Discussion for A-type quivers in the context of ([[zigzag persistence|zigzag]]) [[persistent homology]] and [[topological data analysis]]:

* {#CarlssonDeSilva10} [[Gunnar Carlsson]], [[Vin de Silva]], *Zigzag Persistence*, Found Comput Math **10** (2010) 367–405 $[$[arXiv:0812.0197](https://arxiv.org/abs/0812.0197), [doi:10.1007/s10208-010-9066-0](https://doi.org/10.1007/s10208-010-9066-0)$]$

* {#Oudot15} [[Steve Y. Oudot]], *Persistence Theory: From Quiver Representations to Data Analysis*, Mathematical Surveys and Monographs **209** AMS (2015) $[$[ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/), [pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf)$]$

[[!redirects quiver representations]]

[[!redirects quiver representation theory]]
[[!redirects quiver representation theories]]

[[!redirects representation of a quiver]]
[[!redirects representations of a quiver]]
[[!redirects representations of quivers]]

[[!redirects category of quiver representations]]
[[!redirects categories of quiver representations]]


quiver variety

#Contents#
* table of contents
{:toc}

## Idea

The term **quiver variety** refers to a number of [[algebraic varieties]] constructed as [[moduli spaces]] of [[quiver representations]].

The most important are the _Nakajima quiver varieties_ ([Nakajima 98](#Nakajima98)), which give the moduli of stable representations of [[preprojective algebras]].
These are notable for allowing a geometrical construction of the [[universal enveloping algebra]] of [[Kac-Moody algebras]] [[action|acting]] on the cohomology of quiver varieties and related spaces.

## Related concepts

[[!include moduli spaces -- contents]]

## References

* {#Nakajima94} [[Hiraku Nakajima]], _Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras._ Duke Math. J. 76 (1994), no. 2, 365--416  [MR1302318](https://mathscinet.ams.org/mathscinet/article?mr=MR1302318)

*  {#Nakajima98} [[Hiraku Nakajima]], _Quiver varieties and Kac-Moody algebras._ Duke Math. J. 91 (1998), no. 3, 515--560 [MR1604167](https://mathscinet.ams.org/mathscinet/article?mr=MR1604167)

* [[Bernard Leclerc]], _Quantum loop algebras, quiver varieties, and cluster algebraṣ_, 117--152 in in A. Skowronski, K. Yamagata K. (eds.), Representations of algebras and related topics, Eur. Math. Soc. 2011

* H. Nakajima, _Quiver varieties and finite-dimensional representations of quantum affine algebras_, J. Amer. Math. Soc. 14 (2001), no. 1, 145--238 [MR1808477](https://mathscinet.ams.org/mathscinet/article?mr=MR1808477)
* H. Nakajima, _Quiver varieties and $t$-analogs of $q$-characters of quantum affine algebras_, Ann. of Math. (2) 160 (2004), no. 3, 1057--1097 [MR2144973](https://mathscinet.ams.org/mathscinet/article?mr=MR2144973)

On [[equivariant elliptic cohomology]] of [[quiver varieties]] in relation to the [[AGT correspondence]]:

* [[Mina Aganagic]], [[Andrei Okounkov]], _Elliptic stable envelopes_ ([arXiv:1604.00423](https://arxiv.org/abs/1604.00423))

* [[Andrei Okounkov]], _Inductive construction of stable envelopes and applications, I. Actions of tori. Elliptic cohomology and K-theory_ ([arXiv:2007.09094](https://arxiv.org/abs/2007.09094))

following the analogous non-elliptic discussion in:

* [[Davesh Maulik]], [[Andrei Okounkov]], _Quantum groups and quantum cohomology_, Asterisque 408(2019) ([arXiv:1211.1287](https://arxiv.org/abs/1211.1287), [ISBN: 978-2-85629-900-5](https://bookstore.ams.org/ast-408))

Review in:

* Andrey Smirnov, _Stable envelopes for $A_n$, $\widehat A_n$-quiver varieties_, 2019 ([pdf](https://math.mit.edu/events/quantum/a_smirnov.pdf))

[[!redirects quiver varieties]]


quiver with potential

Given a [[quiver]] $Q$, a potential for $Q$ is a cyclically invariant element in its path algebra or its completion. Cyclic invariance means that it is a linear combination of cyclic paths; cyclic invariance can be in a ${\mathbf Z}_2$-graded sense in which case it is called a superpotential. This is in agreement with another notion of a superpotential in [[Donaldson-Thomas theory]]. The space of superpotentials has a [[necklace Lie algebra]] structure.

It is introduced in

* H. Derksen, J. Weyman, [[A. Zelevinsky]], _Quivers with potentials and their representations I: Mutations_, Sel. math., New ser. __14__, 59–119 (2008) [doi](https://doi.org/10.1007/s00029-008-0057-9) _Quivers with potentials and their representations II: Applications to cluster algebras_, J. Amer. Math. Soc. __23__ (2010) 749-790 [doi](https://doi.org/10.1090/S0894-0347-10-00662-4)

(What is a relation to Paul S. Aspinwall, Lukasz M. Fidkowski, _Superpotentials for quiver gauge theories_, [hep-th/0506041](https://arxiv.org/abs/hep-th/0506041) ?)

* Raf Bocklandt, [[Travis Schedler]], Michael Wemyss, _Superpotentials and higher order derivations_, J. Pure and Appl. Alg. __214__:9 (2010) 1501-1522 [doi](https://doi.org/10.1016/j.jpaa.2009.07.013)

Relation to generalized cluster categories is in

* Claire Amiot, _Cluster categories for algebras of global dimension 2 and quivers with potential_, Ann. Inst. Fourier __59__:6 (2009) 2525-2590 [link](http://aif.cedram.org/item?id=AIF_2009__59_6_2525_0)

An analytic version of the theory is in

* Zheng Hua, [[Bernhard Keller]], _Quivers with analytic potentials_, [arxiv:1909.13517](https://arxiv.org/abs/1909.13517)
* Okke van Garderen, _Cyclic A-infinity algebras and Calabi-Yau structures in the analytic setting_, [arxiv:2306.00771](https://arxiv.org/abs/2306.00771)

The proof that an important class of "exact" $d$-[[Calabi-Yau algebra]]s are derived from superpotentials is in

* [[Michel Van den Bergh]], _Calabi-Yau algebras and superpotentials_, Sel. Math. New Ser. __21__, 555–603 (2015). [doi](https://doi.org/10.1007/s00029-014-0166-6)

An important counterexample of a Calabi-Yau algebras which are not of this form (but are related to topology) is exhibited in

* B. Davison, _Superpotential algebras and manifolds_, Adv. Math. 231(2), 879–912 (2012)

* B. Davison, _The critical CoHA of a quiver with potential_, The Quarterly Journal of Mathematics 68(2):635–703, 2017

* Ben Davison, Sven Meinhardt, _Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras_, [arXiv:1601.02479](https://arxiv.org/abs/1601.02479)


Quoc Ho

* [Webpage](https://quoc-ho.github.io)

## Selected writings

On [[factorization homology]]:

* [[Quoc Ho]], _Densities and stability via factorization homology_, [arXiv:1802.07948](https://arxiv.org/abs/1802.07948)

Expressing the [[rational cohomology]] of [[ordered configuration spaces of points]] via [[factorization homology]] and [[Ran spaces]], and relation to [[representation stability]]:

* [[Quoc P. Ho]], _Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras_ [[arXiv:2004.00252](https://arxiv.org/abs/2004.00252)]

## Related entries

* [[factorization homology]]

category: people

[[!redirects Quoc P. Ho]]


Quot scheme

__Quot scheme__ is certain moduli space of [[coherent sheaf|coherent sheaves]], introduced by [[Grothendieck]] in [[FGA]]. A modern exposition is in [[FGA explained]]:

* Nitin Nitsure, _Construction of Hilbert and Quot schemes_, [[FGA explained]], 105--137, MR2223407; [math/0504590](https://arxiv.org/abs/math/0504590)

[[!redirects Quot schemes]]


quotient bialgebra

# Quotient bialgebras
* table of contents
{: toc}

## Geometric motivation

Given a [[field]] $k$, the $k$-valued [[functions]] on a [[finite group]] form a [[Hopf algebra]]. Given a [[subgroup]] $B\subset G$, there is an induced map of Hopf algebras $k[G]\to k[B]$, which is a [[surjective]] [[homomorphism]] of [[commutative Hopf algebra]]s. Similarly, for Hopf algebras of [[regular function]]s on an [[algebraic group]] over a field.

The generalization to noncommutative Hopf algebras hence may be viewed as describing the notion of a [[quantum subgroup]], or in the [[bialgebra]] version of a quantum sub[[semigroup]].

However there is also a weaker notion of a quantum subgroup, and also a dual notion (e.g. via [[coideal]] subalgebras).

## Definition

Given a $k$-[[bialgebra]] $H$, a __quotient bialgebra__ is a bialgebra $Q$ equipped with an [[epimorphism]] of bialgebras $\pi: H\to Q$.

If both bialgebras are [[Hopf algebras]] then the epimorphism will automatically preserve the [[antipode]].

## Quotient bialgebras from bialgebra ideals

A __[[bialgebra ideal]]__ is an [[ideal]] in the sense of [[associative unital algebras]] which is also a [[coideal]] of [[coassociative coalgebra]]s.

A __[[Hopf ideal]]__ is a bialgebra ideal which is invariant under the [[antipode]] map.

If $H$ is a bialgebra and $I\subset H$ a bialgebra ideal then the quotient associative algebra $H/I$ has a natural structure of a bialgebra. Moreover, if $H$ is a Hopf algebra and $I\subset H$ is a Hopf ideal then the projection $H\to H/I$ will be an epimorphism of Hopf algebras.

[[!redirects quotient bialgebra]]
[[!redirects quotient bialgebras]]
[[!redirects quotient Hopf algebra]]
[[!redirects quotient Hopf algebras]]


quotient bimodule

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## Idea

The [[quotient]] of a [[bimodule]] by a [[subbimodule]].

## Definition

Given [[rings]] $R$ and $S$, a $R$-$S$-bimodule $B$, and a sub-$R$-$S$-bimodule $I$ with a $R$-$S$-[[bimodule monomorphism]] $i:I \hookrightarrow B$, the [[quotient]] of $B$ by $I$ is the [[initial object|initial]] $R$-$S$-bimodule $B/I$ with a $R$-$S$-[[bimodule homomorphism]] $h:B \to B/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-$S$-[[bimodule]] $A$ with a $R$-$S$-bimodule homomorphism $k:B \to A$ such that for every element $a \in I$, $k(i(a)) = 0_A$, there is a unique $R$-$S$-bimodule homomorphism $l:B/I \to A$ such that $l \circ h = k$.

## Related concepts

* [[quotient group]]

* [[quotient module]]

* [[quotient ring]]

[[!redirects quotient bimodules]]


quotient category

#Contents#
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## Idea

The term **quotient category** has more than one meaning.

1. Sometimes it refers to the [[Serre quotient category]] by a [[thick subcategory]].  In this case it is not the category that *is* a quotient, but its *objects* are quotients.  This is an instance of a [[localization functor]], often a [[strict localization functor]].

1. Other times, it refers to an actual quotient of some kind in [[Cat]].  There are different possible meanings of this, but one of the more useful is the quotient of the homsets of a category by equivalence relations compatible with composition.

## References

* [[Saunders MacLane]], §II.8 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects quotient functor]]
[[!redirects quotient functors]]
[[!redirects quotient categories]]


quotient chain complex

If $i : X^\bullet \hookrightarrow Y^\bullet$ is an inclusion of [[chain complexes]] in an [[additive category]] $A$, the **quotient chain complex** $Y^\bullet/X^\bullet$ is a [[cofibre]] of $i$ in the (discrete) category $Comp(A)$.
Hence
  $$ Y^\bullet / X^\bullet \simeq Y^\bullet \oplus_{X^\bullet} 0. $$
(Recall that $Comp(A)$ admits [[pushouts]].)


quotient equivalence relation

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#### Relations
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# Contents
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## Idea

Similar to the construction of a [[quotient set]] from a [[set]] (i.e. [[h-set]]) with an [[equivalence relation]] as a [[higher inductive type]] in [[homotopy type theory]], there is a construction of a "quotient equivalence relation" from a type with two equivalence relations, as an [[higher inductive type]] family.

This construction is reminiscent of the quotient set construction in set theory, because equality in set theory is defined in the foundations as a equivalence relation. The difference is that here we treat both equivalence relations on the same level, while in set theory one of the equivalence relations is given primacy as the default equality of the foundations.

## Definition

Recall that an [[equivalence relation]] on a type $A$ is a type family $\sim$ consisting of dependent types $a \sim b$ indexed by objects $a:A$ and $b:A$, with witnesses of reflexivity, symmetry, and transitivity, and (-1)-truncatedness
$$a:A \vdash \mathrm{refl}(a):(a \sim a)$$
$$a:A, b:A \vdash \mathrm{sym}(a, b):(a \sim b) \to (b \sim a)$$
$$a:A, b:A, c:A \vdash \mathrm{trans}(a, b, c):(a \sim b) \times (b \sim c) \to (a \sim c)$$
$$a:A, b:A \vdash \mathrm{proptrunc}(a, b):\prod_{p:a \sim b} \prod_{q: a \sim b} p =_{a \sim b} q$$

Given a type $A$ with two equivalence relations $\equiv_A$ and $\sim_{A}$, the quotient equivalence relation on $(A, \equiv_{A}, \sim_{A})$ is the higher inductive type family $\equiv^{'}_{A}$ inductively generated by the following constructors:

* For each element $a:A$ and $b:A$, a function
$$f_{\equiv_A}(a, b):(a \equiv_{A} b) \to (a \equiv^{'}_{A} b)$$

* For each element $a:A$ and $b:A$, a function
$$f_{\sim_A}(a, b):(a \sim_{A} b) \to (a \equiv^{'}_{A} b)$$

* For each element $a:A$, a witness
$$\mathrm{refl}(a):(a \equiv^{'}_{A} a)$$

* For each element $a:A$ and $b:A$, a witness
$$\mathrm{sym}(a, b, c):(a \equiv^{'}_{A} b) \to (b \equiv^{'}_{A} a)$$

* For each element $a:A$, $b:A$, and $c:A$, a witness
$$\mathrm{trans}(a, b, c):(a \equiv^{'}_{A} b) \times (b \equiv^{'}_{A} c) \to (a \equiv^{'}_{A} c)$$

* For each element $a:A$ and $b:A$, a witness
$$\mathrm{proptrunc}(a, b):\prod_{p:a \equiv^{'}_{A} b} \prod_{q: a \equiv^{'}_{A} b} p =_{a \equiv^{'}_{A} b} q$$

## See also

* [[higher inductive type]]
* [[inductive-inductive type]]
* [[quotient set]]
* [[equivalence relation]]

[[!redirects quotient equivalence relations]]


quotient group

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## Idea

A _quotient group_ is a [[quotient object]] in the [[category]] [[Grp]] of [[groups]].

## Definition

For $G$ a [[group]] and $H \hookrightarrow G$ a [[normal subgroup]], the [[quotient group]] $G/H$ is the set of [[cosets]], equipped with a group structure induced from $G$.

## Examples

* For $A \hookrightarrow B$ a morphism between [[abelian groups]] the quotient $B/A$ is equivalently the [[cokernel]] of the inclusion.

* The quotient groups of any group by itself is the [[trivial group]]: $G/G = 1$.

## Related concepts

* [[quotient module]]

* [[quotient bimodule]]

* [[quotient ring]]

[[!redirects quotient group]]
[[!redirects quotient groups]]


quotient inductive type

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## Idea

Quotient inductive types are [[higher inductive types]] with a [[0-truncation]] constructor.

## Examples

All quotient inductive types described below are given together with some pseudo-[[Coq]] code, which would implement that QIT if Coq supported QITs natively.

### Integers

A definition of the set of [[integers]] as a quotient inductive type.

    Inductive int : Type :=
    | zero : int
    | succ : int -> int
    | pred : int -> int
    | sec : forall (x : int) pred succ x == x
    | ret : forall (x : int) succ pred x == x
    | contr1 : forall (x y : int) (p q : x == y), p == q.

Another definition of the integers as a quotient inductive type:

    Inductive int : Type :=
    | zero : int
    | succ : int -> int
    | neg : int -> int
    | axiom1 : neg zero  == zero
    | axiom2 : forall (x : int) succ neg succ x == neg x
    | contr1 : forall (x y : int) (p q : x == y), p == q.

### Dyadic rational numbers

A minimal inductive definition of the [[dyadic rational numbers]] as a quotient inductive type:

    Inductive dyadic : Type :=
    | zero : dyadic
    | succ : dyadic -> dyadic
    | pred : dyadic -> dyadic
    | act : nat -> dyadic -> dyadic
    | sec : forall (x : int) pred succ x == x
    | ret : forall (x : int) succ pred x == x
    | initfunc : forall (x : dyadic) act 0 x == succ x
    | funcsqrt : forall (n : nat) forall (x : dyadic) act (succ n) (act (succ n) x) == act n x
    | contr1 : forall (x y : dyadic) (p q : x == y), p == q.

A definition of the dyadic rationals as a [[symmetric midpoint algebra]]

    Inductive dyadic : Type :=
    | zero : dyadic
    | succ : dyadic -> dyadic
    | neg : dyadic -> dyadic
    | mid : dyadic -> dyadic -> dyadic
    | inv : forall (x : dyadic) neg neg x == x
    | idem : forall (x : dyaduc) mid x x == x
    | comm : forall (x y : dyadic) mid x y == mid y x
    | medial : forall (w x y z : dyadic) mid (mid w x) (mid y z) = mid (mid w y) (mid x z)
    | neutr : forall (x : dyadic) mid (neg x) x == zero
    | distneg : forall (x y : dyadic) mid (neg x) (neg y) == neg mid x y
    | distsucc : forall forall (x y : dyadic) mid (succ x) (succ y) == succ mid x y
    | twicesucc : forall forall (x y : dyadic) mid (succ succ x) y == succ mid x y
    | contr1 : forall (x y : dyadic) (p q : x == y), p == q.

### Classically extended natural numbers

The [[set]] of [[extended natural numbers]] as it behaves in [[classical mathematics]] as a quotient inductive type

    Inductive extnat : Type :=
    | zero : extnat
    | succ : extnat -> extnat
    | inf : extnat
    | fixpoint : succ inf == inf
    | contr1 : forall (x y : int) (p q : x == y), p == q.

### Polynomial rings over integers

A definition of the [[polynomial ring]] over the integers with a set of indeterminants $A$.

    Inductive intpoly (A: Type)  :=
    | indet : A -> (intpoly A)
    | zero : intpoly A
    | one : intpoly A
    | neg : (intpoly A) -> (intpoly A)
    | add : (intpoly A) -> (intpoly A) -> (intpoly A)
    | mult : (intpoly A) -> (intpoly A) -> (intpoly A)
    | alunital : forall (x : intpoly A) add zero x == x
    | arunital : forall (x : intpoly A) add x zero == x
    | aassoc : forall (x y z : intpoly A) add x (add y z) == add (add x y) z
    | acomm : forall (x y : intpoly A) add x y = add y x
    | alinv : forall (x : intpoly A) add (neg x) x == zero
    | arinv : forall (x : intpoly A) add x (neg x) == zero
    | ldist : forall (x y z : intpoly A) mult x (add y z) == add (mult x y) (mult x z)
    | rdist : forall (x y z : intpoly A) mult (add x y) z == add (mult x z) (mult y z)
    | mlunital : forall (x : intpoly A) mult one x == x
    | mrunital : forall (x : intpoly A) mult x one == x
    | massoc : forall (x y z : intpoly A) mult x (mult y z) == mult (mult x y) z
    | mcomm : forall (x y : intpoly A) mult x y = mult y x
    | contr1 : forall (x y : intpoly A) (p q : x == y), p == q.

### Quotients of sets

The [[quotient]] of an [[hProp]]-valued or (-1)-truncated [[equivalence relation]], yielding an [[hSet]] or [[0-truncated]] type:

    Inductive quotient (A : Type) (R : A -> A -> hProp) : Type :=
    | proj : A -> quotient A R
    | relate : forall (x y : A), R x y -> proj x == proj y
    | contr1 : forall (x y : quotient A R) (p q : x == y), p == q.

### Sierpinski space

A definition of [[Sierpinski space]]:

    Inductive sierpinski : Type :=
    | top : sierpinski
    | meet : sierpinski -> sierpinski -> sierpinski
    | meet-left-unital : forall (a : sierpinski) meet top a = a
    | meet-right-unital : forall (a : sierpinski) meet a top = a
    | meet-associative : forall (a b c : sierpinski) meet meet a b c = meet a meet b c
    | meet-commutative : forall (a b : sierpinski) meet a b = meet b a
    | meet-idempotent : forall (a : sierpinski) meet a a = a
    | bottom : sierpinski
    | join : sierpinski -> sierpinski -> sierpinski
    | join-left-unital : forall (a : sierpinski) join bottom a = a
    | join-right-unital : forall (a : sierpinski) join a bottom = a
    | join-associative : forall (a b c : sierpinski) join join a b c = join a join b c
    | join-commutative : forall (a b : sierpinski) join a b = join b a
    | join-idempotent : forall (a : sierpinski) join a a = a
    | meet-join-absorption : forall (a b : sierpinski) meet a join a b = a
    | join-meet-absorption : forall (a b : sierpinski) join a meet a b = a
    | sequence-join : (nat -> sierpinski) -> sierpinski
    | sequence-bounded-above : forall (n : nat) (s : nat -> sierpinski) meet s n sequence-join s = s n
    | sequence-least-upper-bound : forall (a : sierpinski) (s : nat -> sierpinski) (forall (n : nat) meet s n a = s n) -> (meet sequence-join s x = sequence-join s)
    | element-sequence-meet : (sierpinski) -> (nat -> sierpinski) -> (nat -> sierpinski)
    | p : forall (a : sierpinski) (s : nat -> sierpinski) element-sequence-meet a s n = meet a s n
    | distributive : forall (a : sierpinski) (s : nat -> sierpinski) meet a sequence-join s = sequence-join element-sequence-meet a s
    | contr1 : forall (x y : int) (p q : x == y), p == q.

### Cumulative hierarchy

According to [[Homotopy Type Theory -- Univalent Foundations of Mathematics]] the [[cumulative hierarchy]] of material sets could be constructed from a type universe as a quotient inductive type.

### Cauchy complete real numbers

According to [[Homotopy Type Theory -- Univalent Foundations of Mathematics]] the [[Cauchy complete]] [[real numbers]] could be constructed from a type universe as a quotient [[inductive-inductive type]].

### Permutable trees

Altenkirch et al. gives the following definition of a permutable tree:

    Inductive tree (A : Type) :=
    | leaf : tree A
    | node : (A -> tree A) -> tree A
    | mix : (forall (e : A -> A) isequiv(e)) -> (forall (e : A -> A) forall (f: A -> tree A) node f = node(f \circ e)
    | contr1 : forall (x y : tree A) (p q : x == y), p == q.

### Free posets

A definition of a [[free object|free]] [[poset]] as a quotient inductive-inductive type:

    Inductive poset (A: Type) :=
    | inj: A -> poset A
    | antisym: (forall x y : poset A) (x \leq y) -> (y \leq x) -> (x == y)
    | setcontr: forall (x y: poset A) (p q : x == y), p == q.

    Inductive \leq {poset A}: poset A -> poset A -> Type :=
    | refl: (forall x: poset A) x \leq x
    | trans: (forall x y z: poset A) (x \leq y) -> (y \leq z) -> (x \leq z)
    | pordercontr: forall (x y : poset A) (p q : x \leq y), p == q.

### Free pointed omega-complete partial orders
 {#FreePointedOmegaCompletePartialOrders}

It is possible to define the [[free object|free]] [[pointed object|pointed]] [[countable ordinal|omega]]-complete [[partial order]] on a 0-truncated type $A$ as a quotient inductive type. They are usually abbreviated as free pointed $\omega$-cpos (see [Altenkirch, Danielsson, and Kraus 2017,  §3.2](#ADK2017))

    Inductive omegacpo (A: Type) :=
    | inj: A -> omegacpo A
    | bottom: omegacpo A
    | denumjoin: (exists a: nat -> omegacpo A) (forall n: nat) a n \leq a succ n
    | antisym: (forall x y : omegacpo A) (x \leq y) -> (y \leq x) -> (x == y)
    | setcontr: forall (x y: omegacpo A) (p q : x == y), p == q.

    Inductive \leq {omegacpo A}: omegacpo A -> omegacpo A -> Type :=
    | refl: (forall x: omegacpo A) x \leq x
    | trans: (forall x y z: omegacpo A) (x \leq y) -> (y \leq z) -> (x \leq z)
    | init: (forall x : omegacpo A) bottom \leq x
    | upperbound: (forall a: nat -> omegacpo A) ((forall n: nat) a n) \leq denumjoin a p
    | leastupper: (forall x : omegacpo A) ((forall a: nat -> omegacpo A) ((forall n: nat) a n) \leq x) -> (forall a: nat -> omegacpo A) denum a p \leq x)
    | pordercontr: forall (x y : poset A) (p q : x \leq y), p == q.

Examples include the [[Sierpinski space]] $1_\bot$.

### Internal type theory

From Altenkirch and Kaposi 2017

## Related concepts

* [[higher inductive type]]

## References

* [[Univalent Foundations Project]], section 11 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_ (2013)

* {#ADK2017} [[Thorsten Altenkirch]], [[Nils Anders Danielsson]], [[Nicolai Kraus]], *Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type*, FoSSaCS 2017, Lecture Notes in Computer Science **10203** Springer (2017) [[arXiv:1610.09254](https://arxiv.org/abs/1610.09254)]

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Fredrik Nordvall Forsberg]], *Quotient inductive-inductive types*, [arXiv](https://arxiv.org/abs/1612.02346), 2016

* [[Thorsten Altenkirch]], [[Ambrus Kaposi]], *Type Theory in Type Theory using Quotient Inductive Types*, POPL17, 2017 [PDF](http://www.cs.nott.ac.uk/~psztxa/publ/tt-in-tt.pdf)

* [[Thorsten Altenkirch]], [[Luis Scoccola]], *The Integers as a Higher Inductive Type*, [arXiv](https://arxiv.org/abs/2007.00167)

[[!redirects quotient inductive types]]


quotient module

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## Idea

For $N$ a [[module]] (over some [[ring]] $R$) and $S \hookrightarrow N$ a submodule, then the corresponding _quotient module_ $N/S$ is the module where all elements in $N$ that differ by an element in $S$ are identified.

If the ring $R$ is a [[field]] then $R$-modules are called _[[vector spaces]]_ and quotient modules are called _quotient vector spaces_.

## Definition

Thoughout let $R$ be some [[ring]]. Write $R$[[Mod]] for the [[category]] of [[module]]s over $R$. Write $U:R Mod \to$ [[Set]] for the [[forgetful functor]] that sends a module to its underlying [[set]].

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

For $i : K \hookrightarrow N$ a [[submodule]], the **quotient module** $\frac{N}{K}$ is the [[quotient group]] of the underlying groups, equipped with the $R$-[[action]] induced by that on $N$.

=--

## Properties

### Equivalent characterizations

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

The quotient module is equivalently the [[cokernel]] of the inclusion in $R$[[Mod]]

$$
  \frac{N}{K} \simeq coker(i)
  \,.
$$

=--

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

The quotient module is equivalently the [[quotient object]] of the [[congruence]] $N \oplus K \to N \oplus N$ given by projection on the first factor and by addition in $N$.

=--

## Related concepts

* [[quotient group]]

* [[quotient bimodule]]

* [[quotient ring]]

* [[quotient object]]

[[!redirects  quotient modules]]

[[!redirects quotient vector space]]
[[!redirects quotient vector spaces]]


quotient norm

## Idea

In the context of [[Banach spaces]]: induced [[norm]] on [[quotient]] of [[Banach spaces]]

## References

* [[Vladimir Berkovich]], section 2.1 of _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

[[!redirects quotient norms]]


quotient object

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## Definition

The **quotient object** $Q$ of a [[congruence]] (an [[internalization|internal]] [[equivalence relation]]) $E$ on an [[object]] $X$ in a [[category]] $C$ is the [[coequalizer]] $Q$ of the induced pair of morphisms $E \,\rightrightarrows\, X$.

If $E$ is additionally the [[kernel pair]] of the map $X \to Q$, then $Q$ is called an **effective quotient** -- and $E$ is called an **effective** congruence, with the map $X \to Q$ being an [[effective epimorphism]] (terminology as for _[[effective group action]]_).

Sometimes the term is used more loosely to mean an arbitrary [[coequalizer]].  It may also refer to a co-[[subobject]] of $X$ (that is, a subobject of $X$ in the [[opposite category]] $C^\op$), without reference to any congruence on $X$.  Note that in a [[regular category]], any [[regular epimorphism]] (i.e. a "regular quotient" in the co-subobject sense) is in fact the quotient (= coequalizer) of its [[kernel pair]] (actually, we can prove this under weaker hypotheses; see below).

## Galois connection between quotients and relations

As we have said, there are various notions of quotient object. Let us consider the most general one, so that $Quot(X)$ of an object $X$ denotes the poset of co-subobjects of $X$, in other words the [[poset|posetal]] [[reflection]] of the [[preorder]] of epis $X \to Q$ which is a [[full subcategory]] of the [[co-slice category]] $X \downarrow \mathbf{C}$. A *regular quotient* then refers to a regular epi $X \to Q$.

On the other hand, let $Rel(X)$ be the poset of [[relations]] on $X$, i.e., the
poset of subobjects of $X \times X$, or in other words the posetal reflection of the preorder of monos $i = \langle e_1, e_2 \rangle: E \rightarrowtail X \times X$ which is a full subcategory of the [[slice category]] $\mathbf{C} \downarrow X \times X$.

Between $Quot(X)$ and $Rel(X)$ there is a relation $\perp$ where $q \perp \langle e_1, e_2 \rangle$ means exactly $q \circ e_1 = q \circ e_2$. If the coequalizer $coeq(e_1, e_2)$ of the parallel pair $e_1, e_2: E \rightrightarrows X$ exists, then by definition we have $coeq(e_1, e_2) \leq q$ iff $q e_1 = q e_2$. On the other hand, if the [[kernel pair]] $\ker(q)$ of $q$ exists, then by definition we have $q e_1 = q e_2$ iff $\langle e_1, e_2 \rangle \leq \ker(q)$.

This indicates in a category which admits coequalizers and kernel pairs, we have

+-- {: .num_prop}
###### Proposition
$coeq: Rel(X) \to Quot(X)$ is [[left adjoint]] to $\ker: Quot(X) \to Rel(X)$.
=--

Or, in other words, that $\ker$ and $coeq$ set up a [[Galois connection]] between $Quot(X)^{op}$ and $Rel(X)$.

Restricting consideration to kernel pairs only of epis, or coequalizers only of jointly monic pairs, is no real restriction in the presence of epi-mono factorizations:

+-- {: .num_lemma #image}
###### Lemma
In a category where every morphism $f: A \to B$ has an [[image|epi-mono factorization]] $f = i \circ q$, we have $\ker(f) = \ker(q)$. Similarly, for a pair $f, g: X \rightrightarrows Y$, we have $coeq(f, g) = coeq(e_1, e_2)$ where $\langle f, g \rangle: X \to Y \times Y$ factors as an epi $p: X \to E$ followed by a mono $\langle e_1, e_2 \rangle: E \to Y \times Y$.
=--

+-- {: .proof}
###### Proof
We prove just the first statement; the second is proven similarly. It suffices to observe that the same class of jointly monic pairs $(e_1, e_2)$ are coequalized by $f$ as by $q$; the kernel pair is by definition the maximum of this class. If $q e_1 = q e_2$, then by applying $i$ to both sides we deduce $f e_1 = f e_2$. If $f e_1 = f e_2$, i.e., if $i q e_1 = i q e_2$, then $q e_1 = q e_2$ by monicity of $i$.
=--

+-- {: .num_prop #galois}
###### Proposition
Suppose $\mathbf{C}$ is a category with coequalizers and kernel pairs and where every morphism has an epi-mono factorization. Then every regular epi $q$ is the coequalizer of its kernel pair: $q = coeq \circ \ker(q)$. And every kernel pair is the kernel pair of its coequalizer: $i = \ker \circ coeq(i)$.
=--

+-- {: .proof}
###### Proof
We just prove the first statement; the second is proved similarly. We have of course a [[counit]] $coeq \circ \ker(q) \leq q$. On the other hand, if $q = coeq(f, g)$ (where we may assume $\langle f, g \rangle$ is monic by the lemma), then we have a unit $\langle f, g \rangle \leq \ker \circ coeq(f, g) = \ker(q)$; applying $coeq$ to each side, we have $q \leq coeq \circ \ker(q)$, as desired.
=--

## In toposes
 {#InToposes}

Constructing quotient objects in an [[elementary topos]] $\mathbf{E}$, starting from one or another standard definition (e.g., [[finitely complete category]] with [[power objects]]) that doesn't already mention colimits, is not trivial.

The standard approach seen in textbooks (see for example _[[Sheaves in Geometry and Logic]]_), apparently first introduced by C.J. Mikkelsen but first published by [Paré](#RP), is to prove that the contravariant [[power object]] functor $P \colon \mathbf{E}^{op} \to \mathbf{E}$ is [[monadic functor|monadic]]. It follows that $P$ [[reflected limit|reflects]] [[finite limits]] in $\mathbf{E}^{op}$ from limits in $\mathbf{E}$, but finite limits in $\mathbf{E}^{op}$ are of course finite colimits in $\mathbf{E}$.

This elegant approach does involve a fair amount of categorical machinery (a [[monadicity theorem]], [[Beck-Chevalley conditions]], and consideration of up to six applications of the power object functor), making it a challenge to get across intuitively in say an undergraduate course.

Other approaches that are closer to naive or common sense set-theoretic reasoning are possible. In the case of quotient objects, to form the coequalizer of a parallel pair $f, g: X \rightrightarrows Y$, we outline a possible path to take (see also at _[quotient type -- from univalence](quotient+type#FromUnivalence)_):

* Construct enough of the [[internal logic]] to make available logical operators $\wedge, \Rightarrow, \forall$.

* Construct an internal [[intersection]] operator $\bigcap: P P X \to P X$ via the formula $\bigcap \Phi = \{x: X\; |\; \forall_{S: P X} \Phi \ni S \Rightarrow S \ni x\}$.

* Construct the [[image]] of a map $f: X \to Y$ by taking the internal intersection of all [[subobjects]] of $Y$ through which $f$ factors.

* To construct the [[coequalizer]] of $f, g: X \rightrightarrows Y$:
  * Take the image of $\langle f, g \rangle: X \to Y \times Y$ to get a relation $\langle p_1, p_2 \rangle: R \hookrightarrow Y \times Y$. According to Lemma \ref{image}, a coequalizer of $(p_1, p_2)$ is a coequalizer of $(f, g)$.
  * Then take the equivalence relation $\langle e_1, e_2 \rangle: E \hookrightarrow Y \times Y$ generated by $R$, by taking the intersection of all equivalence relations containing $R$; a coequalizer of $(e_1, e_2)$ is a coequalizer of $(p_1, p_2)$ (akin to the fact that any coequalizer must be the coequalizer of its kernel pair, as in Proposition \ref{galois}).
  * Take the image factorization of the map $\chi_E: Y \to P Y$ that classifies the relation $E \hookrightarrow Y \times Y$. That is, factor $\chi_E$ as an epi $q: Y \to Q$ followed by a mono $Q \to P Y$. Then $q$ is the desired coequalizer of $(e_1, e_2)$.

Notice that what these steps collectively do is form the object $Q$ of equivalence classes of the equivalence relation generated by the relation $f(x) \sim g(x)$, which is exactly what we would do in ordinary set theory. Full details will appear elsewhere.

## In higher category theory

These notions have generalizations when $C$ is an [[(∞,1)-category]]:

* an equivalence relation is then a [[groupoid object in an (∞,1)-category]]

* it has an "effective quotient" if it is [[delooping|deloopable]].

For instance an [[action groupoid]] is a quotient of a group action in 2-category theory.

## Examples

* [[quotient set]]

* [[quotient group]]

* [[quotient module]], [[quotient bimodule]]

* [[quotient ring]]

([[quotient norm]])

## Related concepts

* [[fraction]]

* [[subquotient]]

* [[quotient space]], [[geometric invariant theory]]

* [[quotient stack]]

* In [[type theory]]/[[homotopy type theory]] the analogous concept is that of [[quotient types]].

## References

* {#Pare74} [[Robert Paré]], _Colimits in Topoi_, Bull. AMS 80 (1974) pp.556-561. ([pdf](http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13497-X/S0002-9904-1974-13497-X.pdf))
 {#RP}

[[!redirects quotient object]]
[[!redirects quotient objects]]
[[!redirects quotient]]
[[!redirects quotients]]

[[!redirects regular quotient]]
[[!redirects regular quotients]]

[[!redirects effective quotient]]
[[!redirects effective quotients]]

[[!redirects quotient map]]
[[!redirects quotient maps]]
[[!redirects quotient morphism]]
[[!redirects quotient morphisms]]


quotient object coclassifier

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

# Contents
* table of contents
{:toc}

## Idea

The notion of a *quotient object coclassifier* in a [[finitely cocomplete category]] is [[abstract duality|dual]] to that of a *[[subobject classifier]]* in a [[finitely complete category]].

## Definition

In a category $C$ with [[finite colimits]], a quotient object coclassifier is an object $Q$ with an [[epimorphism]] $\epsilon:Q \to \mathbb{0}$ into the [[initial object]] $\mathbb{0}$, such that for every [[epimorphism]] $f:X \to U$ there is a unique morphism $\digamma_U:Q \to X$ such that there is a [[pushout]] [[diagram]] of the form

$$
  \array{
    Q
      &\overset{\epsilon}{\longrightarrow}&
    \mathbb{0}
    \\
    \big\downarrow {}^{\mathrlap{\digamma_U}}
    &&
    \big\downarrow {}^{\mathrlap{\exists !}}
    \\
    X &\underset{f}{\longrightarrow}& U
  }
  \,.
$$

## See also

* [[subobject classifier]]

* [[finitely cocomplete category]]

* [[cotopos]]


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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #QuotientProjectionsOutOfCompactHausdorffSpacesAreClosedPreciselyIfTheCodomainIsHausdorff}
###### Proposition
**(quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff)**

Let

$$
  \pi \;\colon\; (X, \tau_X) \longrightarrow (Y, \tau_Y)
$$

be a [[continuous function]] between [[topological spaces]] such that

1. $(X, \tau)$ is a [[compact Hausdorff topological space]];

1. $\pi$ is a [[surjection]] and $\tau_Y$ is the corresponding [[quotient topological space|quotient topology]].

Then the following are equivalent

1. $(Y, \tau_Y)$ is itself a [[Hausdorff topological space]];

1. $\pi$ is a [[closed map]].

=--

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

The implicaton $\left( (Y, \tau_Y)\, \text{Hausdorff} \right) \Rightarrow \left( \pi \, \text{closed}  \right)$
follows since [[maps from compact spaces to Hausdorff spaces are closed and proper|maps from compact spaces to Hausdorff spaces are closed]]. We need to show the converse.

Hence assume that $\pi$ is a closed map. We need to show that for every pair of
distinct points $y_1 \neq y_2 \in Y$ there exist [[open neighbourhoods]] $U_{y_1}, U_{y_2} \in \tau_Y$
which are disjoint, $U_{y_1} \cap U_{y_2} = \emptyset$.

First notice that the [[singleton]] subsets $\{x\}, \{y\} \in Y$ are closed. This is because they are images
of singleton subsets in $X$, by surjectivity of $f$, and because singletons in a Hausdorff space are closed, and because images under $f$ of
closed subsets are closed, by the assumption that $f$ is a closed map.

It follows that the [[pre-images]]

$$
  C_1 \coloneqq \pi^{-1}(\{y_1\})
  \phantom{AA}
  C_2 \coloneqq \pi^{-1}(\{y_2\})
  \,.
$$

are [[closed subsets]] of $X$.

Now since [[compact Hausdorff spaces are normal]] it follows that we may find disjoint open subset $U_1, U_2 \in \tau_X$
such that

$$
  C_1 \subset U_1 \phantom{AAA} C_2 \subset U_2
  \,.
$$

Moreover, by [this lemma](saturated+subset#SaturatedOpenNeighbourhoodsOfSaturatedClosedSubsets) we may find these $U_i$ such that they
are both [[saturated subsets]]. Therefore finally [this lemma](saturated+subset#DetectViaSaturatedSubsetsContinuousQuotientMap)
says that the images $\pi(U_i)$ are open in $(Y,\tau_Y)$. These are now clearly disjoint open neighbourhoods of $y_1$ and $y_2$.

=--

## Applications

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

Consider the function

$$
  \array{
     [0,2\pi]/\sim &\longrightarrow& S^1 \subset \mathbb{R}^2
     \\
     t &\mapsto& (cos(t), sin(t))
  }
$$

* from the [[quotient topological space]] of the [[closed interval]]
  by the [[equivalence relation]] which identifies the two endpoints

  $$
    (x \sim y) \;\Leftrightarrow\; \left( \left( x = y\right) \,\text{or}\, \left( \left( x \in \{0,2\pi\}\right)  \,\text{and}\, \left( y\in \{0, 2\pi\} \right)   \right) \right)
  $$

* to the unit [[circle]] $S^1 = S_0(1) \subset \mathbb{R}^2$
regarded as a [[topological subspace]] of the 2-dimensional [[Euclidean space]]  equipped with its
[[metric topology]] .

This is clearly a [[continuous function]] and a [[bijection]] on the underlying sets.
Moreover, since [[continuous images of compact spaces are compact]]
and since the closed interval $[0,1]$ is compact we also obtain another proof that
the [[circle]] is compact.

Hence since [[continuous bijections from compact spaces to Hausdorff spaces are homeomorphisms]] the above map is in fact a [[homeomorphism]]

$$
  [0,2\pi]/\sim \;\simeq\; S^1
  \,.
$$

=--


quotient ring

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

# Contents
* table of contents
{: toc}

## Idea

The [[quotient]] of a [[ring]] by an [[ideal]].

## Definition

### Quotient by two-sided ideals

Given a [[ring]] $R$ and a [[two-sided ideal]] $I$ with canonical $R$-$R$-[[bimodule monomorphism]] $i:I \hookrightarrow R$, the quotient of $R$ by $I$ is the [[initial object|initial]] two-sided $R$-[[associative unital algebra|algebra]] $R/I$ with canonical [[ring homomorphism]] $h:R \to R/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-algebra $S$ with canonical ring homomorphism $k:R \to S$ such that for every element $a \in I$, $k(i(a)) = 0_S$, there is a unique ring homomorphism $l:R/I \to S$ such that $l \circ h = k$.

### Quotient by left ideals

Given a [[ring]] $R$ and a [[left ideal]] $I$ with canonical left $R$-[[module]] [[monomorphism]] $i:I \hookrightarrow R$, the quotient of $R$ by $I$ is the [[initial object|initial]] left $R$-[[associative unital algebra|algebra]] $R/I$ with canonical [[ring homomorphism]] $h:R \to R/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-algebra $S$ with canonical ring homomorphism $k:R \to S$ such that for every element $a \in I$, $k(i(a)) = 0_S$, there is a unique ring homomorphism $l:R/I \to S$ such that $l \circ h = k$.

### Quotient by right ideals

Given a [[ring]] $R$ and a [[right ideal]] $I$ with canonical right $R$-[[module]] [[monomorphism]] $i:I \hookrightarrow R$, the quotient of $R$ by $I$ is the [[initial object|initial]] right $R$-[[associative unital algebra|algebra]] $R/I$ with canonical [[ring homomorphism]] $h:R \to R/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-algebra $S$ with canonical ring homomorphism $k:R \to S$ such that for every element $a \in I$, $k(i(a)) = 0_S$, there is a unique ring homomorphism $l:R/I \to S$ such that $l \circ h = k$.

## Related concepts

* [[associative unital algebra]]

* [[quotient group]]

* [[quotient module]]

* [[quotient bimodule]]

* [[localization of a ring]]

* [[completion of a ring]]

* [[Koszul complex]]

## References

See also

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

[[!redirects quotient rings]]

[[!redirects quotient algebra]]
[[!redirects quotient algebras]]


quotient set

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

# Quotient sets
* table of contents
{: toc}

## Definitions

### In set theory

Given a [[set]] $S$ and an [[equivalence relation]] $\equiv$ on $S$, the __quotient set__ of $S$ by $\equiv$ is the set $S/{\equiv}$ whose elements are the elements of $S$ but where two elements are now considered [[equality|equal]] if in $S$ they were merely equivalent.

If changing the definition of equality like this is not allowed in the given [[foundations]] of mathematics, then one can still define $S/{\equiv}$ as a [[subset]] of the [[power set]] of $S$.

* In [[material set theory]] the definition is as follows:
\[\label{setdef}
A \in S/{\equiv} \;\Leftrightarrow\; \exists (x: S),\; A = \{y: S \;\mid\; x \equiv y\}\]
* In [[structural set theory]] the definition is slightly different, because sets are not elements of sets and one cannot in general compare sets for strict equality.

If there are also no (extensional) power sets in the foundations of mathematics, then there are two alternatives:

* One could use [[setoids]] (in which case, perhaps one would do better to change your terminology as described there).
* One could just include the existence of [[quotient objects]] in [[Set]] as an axiom (the __axiom of quotient sets__.

The latter is the case if one assumes that $\Set$ is a [[pretopos]] or a [[Grothendieck topos]] as given by [[Giraud's axioms]]). There are actually two possible axiom of quotient sets, depending on what definition of relation one uses.

* The weaker version of the axiom of quotient sets is the one which states that the [[coherent category]] $Set$ is a [[pretopos]]: that every [[internal equivalence relation]] on a set $A$, defined as a subset $\equiv$ of the Cartesian product $A \times A$ satisfying reflexivity, transitivity, and symmetry, has a quotient set $A/\equiv$.

* The stronger version of the axiom of quotient sets states that every [[proposition]] $x \equiv y$ in the [[context]] of the [[variables]] $x \in A$ and $y \in A$ which satisfies reflexivity, transitivity, and symmetry has a quotient set $A/\equiv$. In some presentations of [[set theory]], this is an [[axiom schema]] of quotient sets.

In any case, the element of $S/{\equiv}$ that comes from the element $x$ of $S$ may be denoted $[x]_{\equiv}$, or simply $[x]$ if ${\equiv}$ is understood, or simply $x$ if there will be no confusion as to which set it is an element of.  This $[x]$ is called the __[[equivalence class]]__ of $x$ with respect to $\equiv$; the term 'class' here is an old word for 'set' (in the sense of 'subset') and refers to the definition (eq:setdef) above, where $[x]$ is literally the set $A$.

#### As initial objects in a category

Let $S$ be a [[set]] and let $\equiv$ be an [[equivalence relation]] on $S$. Let us define a *quotient set algebra of $S$ and $\equiv$* to be a set $A$ with a function $\iota S \to A$ such that for all $a \in S$ and $b \in S$, $(a \equiv b)$ implies $(\iota(a) = \iota(b))$.

A *quotient set algebra homomorphism of $S$ and $\equiv$* is a function $f: A \to B$ between two quotient set algebras $A$ and $B$ such that for all $a \in S$, $f(\iota_A(a)) = \iota_B(a)$.

The *category of quotient set algebras of $S$ and $\equiv$* is the category $QSA(S, \equiv)$ whose objects $Ob(QSA(S, \equiv))$ are quotient set algebras of $S$ and $\equiv$ and whose morphisms $Mor(A, B)$ for $A \in Ob(QSA(S, \equiv))$ and $B \in Ob(QSA(S, \equiv))$ are quotient set algebra homomorphisms of $S$ and $\equiv$. The **quotient set** of $S$ and $\equiv$, denoted $S/\equiv$, is defined as the initial object in the category of quotient set algebras of $S$ and $\equiv$.

### In dependent type theory

Similar to set theory, there are multiple ways of defining quotient sets in [[dependent type theory]]. If the type theory has a [[univalent]] [[type of all propositions]] $(\mathrm{Prop}, \mathrm{El})$ and thus [[power sets]] $\mathcal{P}T \coloneqq T \to \mathrm{Prop}$ of a type $T$, then given a type $T$ with an equivalence relation $(-)\equiv(-):\mathcal{P}(T \times T)$, one could construct the quotient set $T/\equiv$ as a [[subtype]] of the [[power set]], as follows: we first define the [[predicate]] on the powerset $\mathcal{P}T$:

$$P:\mathcal{P}T \vdash \mathrm{isEquivalenceClass}(P) \coloneqq \left[\sum_{x:T} \prod_{y:T} P(x) =_\mathrm{Prop} (x \equiv y) \right]$$

since by [[univalence]] there is an equivalence between the types $P(x) =_\mathrm{Prop} (x \equiv y)$ and $\mathrm{El}(P(x)) \simeq \mathrm{El}(x \equiv y)$, which for propositions is the same as $\mathrm{El}(P(x)) \iff \mathrm{El}(x \equiv y)$. Then we define the subtype

$$T/\equiv \coloneqq \sum_{P:\mathcal{P}T} isEquivalenceClass(P)$$

Alternatively, if the [[dependent type theory]] does not have a type of all propositions, then the quotient set $T/\equiv$ can still nevertheless be constructed. The analogue of the axiom of quotient sets in dependent type theory is the rules for the quotient set [[higher inductive type]]. In particular, the quotient set $T/\equiv$ is a [[higher inductive type]] inductively generated by the following:

* a function $\iota: T \to T/\equiv$

* a family of dependent terms

  $$
    a \colon T
    ,\;
    b \colon T
    \;\;\; \vdash \;\;\;
    eq_{T/\equiv}(a, b)
    \,\colon\,
    (a \equiv b)
    \;\to\;
    (\iota(a) =_{T/\equiv} \iota(b))
  $$

* a family of dependent terms
$$a:T/\equiv, b:T/\equiv \vdash \tau(a, b): isProp(a =_{T/\equiv} b)$$

In general, if one doesn't have impredicative quotient sets or quotient sets as a higher inductive type, than one could only construct quotient sets of equivalence relations with a choice of unique representatives, which for equivalence relation $R(x, y)$ indexed by $x:A$ and $y:A$ consists of a type family $C(x)$ indexed by $x:A$, and an element of type

$$\prod_{x:A} \exists!y:A.C(x) \times R(x, y)$$

Then the type $\sum_{x:A} C(x)$ is the quotient set of $A$ with respect to $R$.

Equivalently, one could use functions instead of type families, and say that a choice of unique representatives is a type $C$ with a function $f:C \to A$ and an element of type

$$\prod_{x:A} \exists!y:A.\left(\sum_{z:C} f(z) =_A x\right) \times R(x, y)$$

Then
$$C \simeq \sum_{x:A} \sum_{z:C} f(z) =_A x$$
is the quotient set of $A$ with respect to $R$.

## Generalisations

Quotient sets in [[Set]] generalise to [[quotient object]]s in other categories. In particular, an [[exact category]] is a [[regular category]] in which every [[congruence]] on every object has an effective quotient object.

## See also

* [[equivalence relation]]

* [[setoid]]

* [[quotient type]]

* [[higher inductive type]]

## Rereferences

For quotient sets in [[dependent type theory]], see:

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

See the references at *[[quotient type]]*.

category: foundational axiom

[[!redirects quotient set]]
[[!redirects quotient sets]]

[[!redirects axiom of quotient sets]]
[[!redirects axiom of quotients]]