six operations

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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# Contents
* table of contents
{: toc}

## Idea

### General

What is known as _(Grothendieck's) six operations_ is a formalization of structure that

* assigns to every [[morphism]] $f$ of suitable [[spaces]] a ([[derived direct image|derived]])[[direct image]]/([[derived inverse image|derived]])[[inverse image]] [[adjunction]] $(f^\ast \dashv f_*)$;

* assigns to every [[separated morphism]] a [[direct image with compact support]]/[[Verdier duality|Verdier dual]] [[adjunction]] $(f_! \dashv f^!)$.

These are _four operations_ and together with

* a [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]] [[tensor product]]/[[internal hom]] [[adjunction]] $((-)\otimes X \dashv [X,-])$

form _six operations_.

(All this is usually interpreted as [[derived functors]]/[[(infinity,1)-functors]] so that for instance in the usual application to [[derived categories]] of [[abelian sheaves]] the last two operations are really [[Tor]] and [[Ext]].)

With a list of compatibility conditions between these (for instance ([Cisinski-Déglise 09, p. x](#CisinskiDeglise09)) this is a structure of _Grothendieck's six operations_.

These consistency conditions include the following:

1. The adjunctions are functorial, hence form [[2-functors]] $f \mapsto f_*$, $f \mapsto f_!$:

1. There is a [[natural transformation]]  $f_! \to f_*$ which is a [[natural equivalence]] when $f$ is a [[proper map]].

1. [[Beck-Chevalley condition]] (base change formulas): given a ([[homotopy pullback|homotopy]]) [[pullback]] [[diagram]]

   $$
     \array{
       && Q_1 \underset{X}{\times} Q_2
       \\
       & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}}
       \\
       Q_1 && && Q_2
       \\
       & {}_{\mathllap{f}}\searrow && \swarrow_{\mathrlap{g}}
       \\
       && X
     }
   $$

   such that $f$ is a [[separated morphism]], then there are [[natural equivalences]]

   $$
     g^\ast \circ f_! \simeq (p_2)_! \circ (p_1)^\ast
   $$

   $$
     f^! \circ g_\ast \simeq (p_1)_\ast\circ (p_2)^!
     \,.
   $$

1. ...

([Cisinski-Déglise 09, p. x](#CisinskiDeglise09))

Morover one imposes a formalization of [[Verdier duality]] with [[dualizing object]]...

([Cisinski-Déglise 09, p. xi](#CisinskiDeglise09))

### Specializations

Often specializations of the general concept play a role:

* [[Wirthmüller context]]: $f^! \simeq f^\ast$ and $f^\ast$ is a strong [[closed monoidal functor]] (hence the [[projection formula]] for $f_!$ holds);

  * [[transfer context]]: in addition the [[projection formula]] for $f_\ast$ holds;

* [[Verdier-Grothendieck context]]: the "projection formula" $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$ holds naturally in $X,Y$:

* [[Grothendieck context]]: $f_! \simeq f_\ast$ and the projection formula holds $Y \otimes f_! X \simeq f_!(f^\ast Y \otimes X)$.

## Mann's approach

Mann ([Mann22](#Mann22)) has developed an abstract framework for a six-functor formalism that combines the approaches of Liu-Zheng and Gaitsgory-Rozenblyum and applies it to [[rigid analytic geometry]], in particular in combination with [[condensed mathematics]] and [[almost mathematics]] to construct an [[$\infty$-category]] of derived solid $\mathcal{O}_{X}^{+}/\pi$ almost modules (see also [Scholze22](#Scholze22)).

\begin{definition}
Let $C$ be an $\infty$-category admitting finite limits and let $E$ be a class of morphisms stable under pullback and composition, and including all isomorphisms. The symmetric monoidal $\infty$-category of correspondences $\mathrm{Corr}(C,E)$ has as its objects the objects in $C$, and has as its morphisms the correspondences in $C$, i.e. for objects $X$ and $Y$ of $C$ a morphism from $X$ to $Y$ is given by an object $W$ of $C$ together with maps

  $$
    \array{
      && W
      \\
      & \swarrow_{f} && \searrow_{g}
      \\
      X &&&& Y
    }
  $$

where $g\in E$. The symmetric monoidal structure comes from the one on $C$.
\end{definition}

\begin{definition}
Let $\mathrm{Cat}_{\infty}$ be the $\infty$-category of $\infty$-categories. A *3-functor formalism* is a lax symmetric monoidal functor
$$\mathcal{D}:\mathrm{Corr}(C,E)\to \mathrm{Cat}_{\infty}$$
\end{definition}

The above definition formalizes three of the functors in the 6-functor formalism. The derived tensor product comes from the lax symmetric monoidal structure, $f^{*}$ comes from the image of the correspondence $Y\xleftarrow{f}X=X$, and $f_{!}$ comes from the image of the correspondence $X=X\xrightarrow{f}Y$.

\begin{definition}
A *6-functor formalism* is a 3-functor formalism for which $-\otimes A$, $f^{*}$, and $f_{!}$ admit right adjoints.
\end{definition}

## Properties

### Relation to motivic homotopy theory

The [[initial object]] in the [[(infinity,2)-category]] of functors to [[stable (infinity,1)-categories]] which satisfy the _six  operations_ formalism (and a bit more, such that [[motivic homotopy theory|A1-homotopy invariance]]) is _[[stable motivic homotopy theory]]_. See there for more.

## Related concepts

* [[Grothendieck duality]], [[Verdier duality]]

[[!include twisted generalized cohomology in linear homotopy type theory -- table]]

## References

General abstract discussion is in:

* [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf)]

* [[Roy Joshua]], _Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories_ ([[JoshuaDuality.pdf:file]])

* [[Paul Balmer]], [[Ivo Dell'Ambrogio]], [[Beren Sanders]], *Grothendieck-Neeman duality and the Wirthmüller isomorphism*, Compositio Math. **152** (2016) 1740-1776 [[arXiv:1501.01999](http://arxiv.org/abs/1501.01999), [doi:10.1112/S0010437X16007375](https://doi.org/10.1112/S0010437X16007375)]

Lecture notes:

* [[Martin Gallauer]], *An introduction to six-functor formalism*, lecture at *[The Six-Functor Formalism and Motivic Homotopy Theory](https://sites.google.com/view/summer-school-motivic/home)*, Università degli Studi di Milano (Sept. 2021) [[arXiv:2112.10456](https://arxiv.org/abs/2112.10456), [pdf](https://homepages.warwick.ac.uk/staff/Martin.Gallauer/docs/m6ff.pdf)]

* {#Scholze22} [[Peter Scholze]], *Six Functor Formalisms*, [lecture notes ](https://people.mpim-bonn.mpg.de/scholze/Course%20Winter%202223.html) (2022) [[pdf](https://people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf), [[Scholze-SixOperations.pdf:file]]]

An approach to six functor formalism as a bifibered multicategory (multiderivator, when appropriate) over correspondences is in

* Fritz Hörmann, _Fibered derivators, (co)homological descent, and Grothendieck's six functors_ [pdf](http://home.mathematik.uni-freiburg.de/hoermann/overview6fu.pdf);

  _Fibered Multiderivators and (co)homological descent_, [arxiv./1505.00974](http://arxiv.org/abs/1505.00974);

  _Six Functor Formalisms and Fibered Multiderivators_ [arXiv:1603.02146](https://arxiv.org/abs/1603.02146);

  _Derivator Six Functor Formalisms --- Definition and Construction I_ [arXiv:1701.02152](https://arxiv.org/abs/1701.02152)

The traditional applications ([[quasicoherent sheaves of modules]]) are discussed in

* [[Joseph Lipman]], _Notes on derived functors and Grothendieck duality_, in _Foundations of Grothendieck Duality for Diagrams of Schemes_, Lecture Notes in Math. **1960** Springer (2009) 1-259 [[pdf](http://www.math.purdue.edu/~lipman/Duality.pdf)]

* Yves Laszlo, Martin Olsson, _The six operations for sheaves on Artin stacks I: Finite Coefficients_ ([arXiv:math/0512097](http://arxiv.org/abs/math/0512097))

An [[enhanced triangulated category|enhanced]] version of the six operations formalism for [[etale cohomology]] of ([[infinity-stack|higher]]) [[Artin stacks]], using the language of [[stable (infinity,1)-categories]]:

* Yifeng Liu, Weizhe Zheng, _Enhanced six operations and base change theorem for Artin stacks_, [arXiv](http://arxiv.org/abs/1211.5948).

The six functor formalism for [[motivic homotopy theory]]:

* [[Joseph Ayoub]], _Les six opérations de Grothendieck et le formalisme des cycles évanescants dans le monde motivique_ PhD thesis, Paris [[pdf](http://www.math.uiuc.edu/K-theory/0761/THESE.pdf), [K:0761](https://faculty.math.illinois.edu/K-theory/0761)]

* [[Joseph Ayoub]], *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I)*, Astérisque **314** (2007) [[numdam:AST_2007__314__R1_0](http://www.numdam.org/issues/AST_2007__314__R1_0)]

* [[Joseph Ayoub]], *Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II)*, Astérisque **315** (2007) [[numdam:AST_2007__315__1_0](http://www.numdam.org/issues/AST_2007__315__1_0)]

* {#CisinskiDeglise09} [[Denis-Charles Cisinski]], [[Frédéric Déglise]], section A.5 of: *Triangulated categories of mixed motives*, Springer Monographs in Mathematics, Springer (2019) [[arXiv:0912.2110](http://arxiv.org/abs/0912.2110), [doi:10.1007/978-3-030-33242-6](https://doi.org/10.1007/978-3-030-33242-6)]

and in the context of [[rigid analytic geometry]]:

* [[Joseph Ayoub]], [[Martin Gallauer]], [[Alberto Vezzani]], *The six-functor formalism for rigid analytic motives*, Forum of Mathematics, Sigma **10** (2022) E61 [[doi:10.1017/fms.2022.55](https://doi.org/10.1017/fms.2022.55), [arXiv:2010.15004](https://arxiv.org/abs/2010.15004)]

Enhancement to [[equivariant motivic homotopy theory]]:

* [[Marc Hoyois]], *The six operations in equivariant motivic homotopy theory*, Adv. Math. **305** (2017) 197-279 [[arXiv:1509.02145](https://arxiv.org/abs/1509.02145), [doi:10.1016/j.aim.2016.09.031](https://doi.org/10.1016/j.aim.2016.09.031)]

Discussion of six-functor formalism for [[abelian sheaves]] on [[manifolds]]/[[topological spaces]]:

* [[Masaki Kashiwara]], [[Pierre Schapira]], Chapters II and III of: _Sheaves on manifolds_, Grundlehren **292**, Springer (1990) [[doi:10.1007/978-3-662-02661-8](https://doi.org/10.1007/978-3-662-02661-8)]

* [[Marco Volpe]], *The six operations in topology* [[arXiv:2110.10212](https://arxiv.org/abs/2110.10212)]

See also:

* Brad Drew, Martin Gallauer, *The universal six-functor formalism* [[arXiv:2009.13610](https://arxiv.org/abs/2009.13610)]

Discussion for pull-push of ([[holonomic D-module|holonomic]]) [[D-modules]] is in

* [[Joseph Bernstein]], around p. 18 of _Algebraic theory of D-modules_ ([[BernsteinDModule.pdf:file]], [ps](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps), [dvi](http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.dvi))

reviewed for instance in

* [[Pavel Etingof]], _Formalism of six functors on all (coherent) D-modules_ ([pdf](http://www-math.mit.edu/~etingof/dmodfactsheet.pdf))

* [[David Ben-Zvi]], [[David Nadler]], section 3 of _The Character Theory of a Complex Group_ ([arXiv:0904.1247](http://arxiv.org/abs/0904.1247))

The six operations for [[derived categories]] of [[quasi-coherent sheaves]], [[ind-coherent sheaves]], and [[D-modules]] on [[derived stacks]] are developed in

* [[Dennis Gaitsgory]], [[Notes on geometric Langlands]], [web](http://www.math.harvard.edu/~gaitsgde/GL/).

Six operations in the setup of [[o-minimal structure]]s is discussed in

* Mario J. Edmundo, Luca Prelli, _The six Grothendieck operations on o-minimal sheaves_, Comptes Rendus Mathematique 352:6 (2014) 455-458 [arxiv/1401.0846](http://arxiv.org/abs/1401.0846) [doi](https://doi.org/10.1016/j.crma.2014.03.021)

Grothendieck six operations formalism using diagrams of topoi

* Alexey Kalugin, _On categorical approach to Verdier duality_, [arxiv/1505.06922](http://arxiv.org/abs/1505.06922)

Six operations formalism for [[rigid analytic geometry]] is in

* {#Mann22} Lucas Mann, _A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry_ ([arXiv:2206.02022](https://arxiv.org/abs/2206.02022))

The abstract approach behind Mann's work is the subject of [Scholze (2022)](#Scholze22).

[[!redirects six operations]]

[[!redirects Grothendieck's six operations]]
[[!redirects Grothendieck's yoga of six operations]]
[[!redirects Grothendieck yoga of six operations]]

[[!redirects Grothendieck\'s six operations]]
[[!redirects Grothendieck's six operations]]
[[!redirects Grothendieck six operations]]
[[!redirects yoga of six functors]]
[[!redirects yoga of six operations]]

[[!redirects six operation yoga]]
[[!redirects six operations yoga]]

[[!redirects motivic yoga]]


Siye Wu

* [webpage](http://www.colorado.edu/math/people/professors/wu.html)

## Selected writings

On [[analytic torsion]] and [[twisted de Rham cohomology]]:

* [[Varghese Mathai]], [[Siye Wu]], _Analytic torsion for twisted de Rham complexes_, J. Diff. Geom. 88:297-332, 2011 ([arXiv:0810.4204](https://arxiv.org/abs/0810.4204))

category: people


size issues

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#### Foundations
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\tableofcontents

## Overview

There are various ways to deal with size issues in the [[foundations]] of [[mathematics]]. All of them involve the notion of [[universe]] in one way or another.

* In [[set theory]], one usually either uses [[Grothendieck universes]], or some internal model of [[set theory]] or [[class theory]], whether [[well-pointed category|well-pointed]] [[Heyting categories]], [[well-pointed category|well-pointed]] [[division allegories]], [[sets]] with [[extensional relation|extensional]] [[well-founded relation|well-founded]] [[transitive relation|transitive]] relations, [[categories with class structure]], et cetera.

* In [[class theory]], the notion of universe is already in the theory via the [[universal class]], which is by definition a [[class]] [[universe]].

* In [[dependent type theory]], one uses [[univalent universe|univalent]] [[Russell universes]] or [[Tarski universes]], which are basically internal models of [[dependent type theory]].

In any case, given a universe $U$, we say that a [[collection]] is **$U$-small** if it is in $U$, a [[collection]] is **$U$-large** if it is not in $U$, any subcollection of $U$ is a **class**, and a subcollection of $U$ is a **proper class** if it is not a [[singleton]] subcollection of $U$.

## See also

*  [[set]], [[proper class]], [[Grothendieck universe]]
*  [[small category]], [[essentially small category]], [[large category]]
*  [[Set]], [[Cat]], [[michaelshulman:category of all sets]]
*  [[small presheaf]], [[accessible category]], [[presentable category]]

[[!redirects size issue]]
[[!redirects size issues]]


Sjoerd Crans

Sjoerd Crans has mainly worked on aspects of the [[algebraic definition of higher category|algebraic definition of higher categories]], notably on clarifying the right [[monoidal category|monoidal structure]] on [[geometric shapes for higher structures|globular]] [[omega-category|∞-categories]] that achieves the analogous effect that the simple [[cartesian monoidal category|cartesian]] product does on [[simplicial set]]s. This is now known as the [[Crans-Gray tensor product]].

Crans also made an influential contribution to the theory of [[models for ∞-stack (∞,1)-toposes]] in his study of the [[model structure on simplicial sheaves]].

## Selected writings

On [[braided monoidal 2-categories]] and [[sylleptic monoidal 2-categories]]:

* [[Sjoerd E. Crans]], *Generalized Centers of Braided and Sylleptic Monoidal 2-Categories*, Advances in Mathematics, Volume 136, Issue 2, 25 June 1998, Pages 183-223 ([doi:10.1006/aima.1998.1720](https://doi.org/10.1006/aima.1998.1720))

A collection of articles by Sjoerd Crans is here:

* [Articles by Sjoerd Crans](http://home.tiscali.nl/secrans/papers/) (the site seems to be down, see the [cached version of the site](https://web.archive.org/web/20130514042310/http://home.tiscali.nl/secrans/papers/))

# Related $n$Lab entries #

* [[strict ∞-category]]

  * [[globe category]]

  * [[cube category]]

* [[Crans-Gray tensor product]]

* [[model structure on simplicial sheaves]]

* [[model structure on presheaves of simplicial groupoids]]

category: people

[[!redirects Sjoerd E. Crans]]


skeletal > history

< [[skeletal]]
[[!redirects skeletal -- history]]


skeletal geometric morphism

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

## Idea

**Skeletal geometric morphisms** are those [[geometric morphisms]] that preserve [[double negation|double negation sheaves]] and therefore play a role in the descriptions of classes of toposes like e.g. [[Boolean topos|Boolean]] or [[De Morgan toposes]] in whose definitions the negation participates.

The notion of a skeletal geometric morphism can be viewed as a weakening of the notion of [[open geometric morphism]].

## Definition

A [[geometric morphism]] $f :  \mathcal{F} \to \mathcal{E}$ is called _skeletal_ if the following equivalent conditions hold:

* $f$ restricts to a geometric morphism $Sh_{\neg\neg}(\mathcal{F}) \to Sh_{\neg\neg}(\mathcal{E})$ .

* The [[inverse image]] $f^\ast$ preserves $\neg\neg$-dense monomorphisms.

* For any subobject $A'\rightarrowtail A$ in $\mathcal{E}$: $\neg\neg f^\ast(A')=\neg f^\ast(\neg A')$ in $Sub_\mathcal{F}(f^\ast(A))$.

## Example

* Inverse images of [[open geometric morphism|open geometric morphisms]] are [[Heyting functors]], hence commute with $\neg$ and, therefore, _open geometric morphisms are skeletal_. In particular, geometric morphisms with [[Boolean topos|Boolean]] codomain are open ([Johnstone 2002, p.612](#J02)), hence skeletal.

* [[dense subtopos|Dense subtoposes]] $i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E}$ are precisely those subtoposes with $Sh_{\neg\neg}(Sh_j(\mathcal{E}))=Sh_{\neg\neg}(\mathcal{E})$ (cf. [this proposition](dense+subtopos#negdense)) and, therefore, are skeletal.

## Properties

The following two propositions concern skeletal [[geometric embedding|inclusions]] (cf. Johnstone ([2002, p.1007](#J02))):

+-- {: .num_prop}
###### Proposition
An inclusion $i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E}$ is skeletal iff $ext(j)$ , the $j$-closure of $0\rightarrowtail 1$ , is a $\neg\neg$-closed [[subterminal object]] of $\mathcal{E}$.
=--

**Proof**: First, notice that in general for a topology $j$ a subterminal $U$ is $j$-closed iff it is a $j$-sheaf since $1$ is always a $j$-sheaf. Hence $0$ is $\neg\neg$-closed precisely when it is a $\neg\neg$-sheaf.

Now assume $i$ skeletal. Since $ext(j)$ is a $\neg\neg$-sheaf in $Sh_j(\mathcal{E})$ and $i$ preserves them, it is also a $\neg\neg$-sheaf in $\mathcal{E}$.

Conversely, assume $ext(j)$ is a $\neg\neg$-sheaf in $\mathcal{E}$. Since it is also a $j$-sheaf, it is contained in $Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E})$ but this coincides with $Sh_{\neg\neg}(Sh_j(\mathcal{E}))$ because as a subtopos of the Boolean $Sh_{\neg\neg}(\mathcal{E})$ the intersection $Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E})$ is Boolean and $Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E})$, since it contains $ext(j)$, is dense in $Sh_j(\mathcal{E})$ and there can be only one such dense Boolean subtopos. $\qed$

+-- {: .num_prop}
###### Proposition
The class $\Sigma$ of skeletal [[geometric embedding|inclusions]] is the smallest class $\Gamma$ of geometric morphisms such that:

* $\Gamma$ contains [[open subtopos|open inclusions]] and,

* $\Gamma$ is closed under precomposition with [[dense subtopos|dense inclusions]]: from $g$ dense, $f\in\Gamma$ and $f,g$ composable, follows $fg\in\Gamma$. $\qed$
=--

The following exhibits the link between skeletal morphisms and Booleanness:

+-- {: .num_prop #Boolean_skeletal}
###### Proposition
A topos $\mathcal{E}$ is Boolean iff all geometric morphisms $\mathcal{F}\to\mathcal{E}$ to $\mathcal{E}$ are skeletal.
=--

**Proof**:
When $\mathcal{E}$ is Boolean it coincides with $Sh_{\neg\neg}(\mathcal{E})$ hence $\neg\neg$-sheaves of $\mathcal{F}$ trivially have to land there.

Conversely, assume all $\mathcal{F}\to\mathcal{E}$ are skeletal. By [[Barr's theorem]], $\mathcal{E}$ receives a surjective $f:\mathcal{B}\to\mathcal{E}$ from a Boolean topos. $f$ being skeletal and surjective implies that $im(f)=\mathcal{E}$ is Boolean. $\qed$

A pullback characterisation of [[open geometric morphisms]] from Johnstone ([2006, cor. 4.9](#J06)):

+-- {: .num_prop}
###### Proposition
A [[geometric morphism]] $f:\mathcal{F}\to\mathcal{E}$ is open iff its pullback along any [[bounded geometric morphism]] with codomain $\mathcal{E}$ is skeletal. $\qed$
=--

## Remark

Skeletal morphisms between [[frame|frames]] are studied in [Banaschewski-Pultr (1994](#Banaschewski_Pultr94),[1996](#Banaschewski_Pultr96)), called _weakly open_ there.

The equivalent concept for topological spaces appears in [Mioduszewski-Rudolf (1969)](#Mioduszewski_Rudolf69).

It is possible to define an analogous concept of _m-skeletal geometric morphism_ using the [[De Morganization|De Morgan topology]] on a topos $\mathcal{E}$ instead of $\neg\neg$.

## Related entries

* [[open geometric morphism]]

* [[dominant geometric morphism]]

* [[proper geometric morphism]]

* [[double negation]]

* [[De Morganization]]

* [[De Morgan topos]]

## References

* {#Banaschewski_Pultr94} [[B. Banaschewski]], A. Pultr, *Variants of openness*, Appl. Cat. Struc. **2** (1994) 1-21 [[doi:10.1007/BF00873038](https://doi.org/10.1007/BF00873038)]

* {#Banaschewski_Pultr96} [[B. Banaschewski]], A. Pultr, *Booleanization*, Cah. Top. Géom. Diff. Cat. **XXXVII** **1** (1996)  41-60 [[numdam:CTGDC_1996__37_1_41_0](http://numdam.mathdoc.fr/numdam-bin/fitem?id=CTGDC_1996__37_1_41_0)]

* [[Peter Johnstone]], _Factorization theorems for geometric morphisms II_ , pp.216-233 in LNM **915** Springer Heidelberg 1982.

* {#J02}[[Peter Johnstone]], _[[Sketches of an Elephant]] vol.II_ , Oxford UP 2002. (section D4.6, pp.1006-1010)

* {#J06}[[Peter Johnstone]], _Complemented sublocales and open maps_ , Annals of Pure and Applied Logic **137** (2006) pp.240–255.

* {#Johnstone13}[[Peter Johnstone]], _The Gleason Cover of a Realizability Topos_ , TAC **28** no.32 (2013) pp.1139-1152. ([abstract](http://www.tac.mta.ca/tac/volumes/28/32/28-32abs.html))

* {#Mioduszewski_Rudolf69}J. Mioduszewski, L. Rudolf, _H-closed and extremally disconnected Hausdorff spaces_ , Dissertationes Math. **66** 1969. ([toc](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-99c47193-1b38-4485-9b3f-62fc71f8f83b))

[[!redirects skeletal geometric morphisms]]
[[!redirects skeletal morphism]]


skeletal groupoid

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

## Idea

A *skeletal groupoid* is a (usually: [[small category|small]]) [[strict category]] (i.e. equipped with a definite [[set]] of [[objects]]) which is both a [[groupoid]] and [[skeletal category]].

In other words, this means that a skeletal groupoid is a ([[strict groupoid|strict]]) [[groupoid]] for which, equivalently:

* for every [[morphism]] $\phi$ its [[source]] [[equality|equals]] its [[target]]: $s(\phi) = t(\phi)$,

* all morphisms are [[automorphisms]] of some object,

* all [[connected components]] are [[delooping groupoids]] $\mathbf{B} G$, namely of the [[automorphism group]] $G$ on the unique object in that connected components,

and so skeletal groupoids $\mathcal{X}$ are exactly (namely: up to [[isomorphism]]) the [[disjoint unions]] of [[delooping groupoids]]:

$$
  \mathcal{X} \,\in\, Grpd_{strct}
  \;\;\;\;\;\;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;\;\;\;\;\;
  \mathcal{X} \;\text{skeletal}
    \;\;\;\;\;\;\;\;
    \Leftrightarrow
    \;\;\;\;\;\;\;\;\;
  \mathcal{X}
  \;\;\underset{\color{red}iso}{\simeq}\;\;
  \underset{ x \in Obj(\mathcal{X}) }{\coprod}
  \mathbf{B} Aut_{\mathcal{X}}(x)
  \,.
$$

As discussed at *[[skeletal category]]* (and in much detail [here](Introduction+to+Topology+--+2#EveryGroupoidIsEquivalentToDisjointUnionOfGroupDeloopings) at *[[Introduction to Topology -- 2]]*), if the [[axiom of choice]] holds in the underlying category of [[Sets]] then every groupoid is [[equivalence of categories|equivalent as a category]] --- hence [[homotopy equivalence|homotopy equivalent]] as a [[homotopy 1-type]] --- to a skeletal groupoid, and to an essentially unique one, up to [[isomorphism]] of strict groupoids:

$$
  \mathcal{X} \,\in\, Grpd_{strct}
  \;\;\;\;\;\;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;\;\;\;\;\;
  \mathcal{X}
  \;\;\;\;
    \underset{\color{red}equiv}{\simeq}
  \;\;\;\;
  \underset{ x \in \pi_0(\mathcal{X}) }{\coprod}
  \mathbf{B} \pi_1(\mathcal{X}, x)
  \,.
$$

As indicated on the right, this skeletalization of $\mathcal{X}$ extracts its [[homotopy groups]]: The set of objects of the skeleton is the set $\pi_0(\mathcal{X})$ of [[connected components]], and the [[automorphism group]] at a given object is the [[fundamental group]] $\pi_1(\mathcal{X},x)$ at that basepoint.

## Properties

### Relation to groups

The [[1-category]] of skeletal groupoids is (see [there](free+coproduct+completion#SkeletalGroupoids)) [[equivalence of categories|equivalently]] the [[free coproduct completion]] of the category [[Grp]] of [[groups]].

## Related entries

* [[skeleton]]

* [[delooping groupoid]]

## References

See also the general references at *[[skeletal category]]*, such as

* {#MacLane71} [[Saunders MacLane]], p. 91 of: _[[Categories for the Working mMathematician]]_, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

Discussion of [[colimits]] over certain [[diagrams]] of the shape of skeletal groupoids and regarded as generalized [[coproducts]] (*[[quasi-coproducts]]*):

* [[Hongde Hu]], [[Walter Tholen]], *Quasi-coproducts and accessible categories with wide pullbacks*, Appl Categor Struct **4** (1996) 387–402 [[doi:10.1007/BF00122686](https://doi.org/10.1007/BF00122686)]

Characterization of skeletal groupoids as the "trivial objects" with respect to a [[pretorsion theory]] on [[Cat]]:

* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]

[[!redirects skeletal groupoids]]


skeleton

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

> This entry is about the notion of _skeleton_ in [[category theory]]. For the notion of (co)skeletal [[simplicial sets]] see at _[[simplicial skeleton]]_.

# Contents
* table of contents
{:toc}

## Definition
 {#Definition}

A [[strict category]] $\mathcal{C}$ is called __skeletal__ if any two [[objects]] that are [[isomorphism|isomorphic]] are actually already [[equality|equal]] (in the fixed [[set]] of [[objects]] $Obj(\mathcal{C})$), hence if all its [[isomorphisms]] are [[automorphisms]].

(If in addition all isomorphisms are in fact [[identity morphisms]], then one speaks of a *[[gaunt category]]*. In the other extreme, if all morphisms of a skeletal category are isomorphisms, then one has a [[skeletal groupoid]]).

A __skeleton__ of a category $C$ is defined to be a skeletal [[subcategory]] of $C$ whose [[inclusion functor]] exhibits it as [[equivalence of categories|equivalent]] to $C$. A __weak skeleton__ of $C$ is any skeletal category which is [[equivalence of categories|weakly equivalent]] to $C$.

In the absence of the [[axiom of choice]], it is more appropriate to define a __skeleton__ of $C$ to be a weak skeleton as defined above.

## Constructions

In this section, we work in [[set-level foundations]] to construct skeleta of categories and more general objects.

### Existence of Skeletons of Categories

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

If the [[axiom of choice]] holds, then every category $C$ has a skeleton (in the strongest sense).
=--

+-- {: .proof}
###### Proof
Simply choose one object in each isomorphism class and one isomorphism to that object from each other object in that class.

In more detail, generate the full subcategory $sk(C)$ containing just the chosen objects. Denote by $in\colon sk(C)\to C$ the inclusion. We exhibit a weak inverse of $in$ as a functor
$-':x\mapsto x'$ constructed as follows. For every object $x$ one has chosen already the unique object $x'$ in $sk(X)$ isomorphic to $x$, but one also needs to make a choice of isomorphism $i_x\colon x\to x'$ for every $x$. This enables to conjugate between $C(x,y)$ and $C(x',y')$ by

$$
(x\stackrel{f}\to y)\mapsto (x'\stackrel{i_x^{-1}}\to x\stackrel{f}\to y\stackrel{i_y}\to y').
$$

The rule for *morphisms* $-': f\mapsto f' := i_y\circ f\circ i_{x}^{-1}$ is clearly functorial. Let us show that $-'$ is a weak inverse of $in$. In one direction, given $y\in sk(C)$ we compute $(in_{y})' = y$ (strict equality); in another direction, given $x\in C$, notice that $i^{-1}_{in_{x'}}:in_{x'}\cong x$  for $x\in C$ is an isomorphism. It suffices to show that these isomorphisms for all $x\in C$ together form a *natural* isomorphism $i^{-1}_{in}:in_{-'}\to id_C$; the naturality diagram is commutative precisely because of the conjugation formula for the functor $-'$ for morphisms. This completes the proof that $-'$ is indeed a weak inverse of $in$.
=--

In fact, the statement that every (possibly [[small category|small]]) category has a skeleton is _equivalent_ to the axiom of choice if "subcategory" and "equivalence" have their naive ('strong') meanings.  For given a [[surjection]] $p:A\to B$ in $Set$, make $A$ into a category with a unique isomorphism $a\cong a'$ iff $p(a)=p(a')$; then a skeleton of $A$ supplies a splitting of $p$.

Even with the more general notion of weak or ana-[[equivalence of categories]], some amount of choice is required to show that every category has a skeleton.  It would be interesting to know the precise strength of the statement "every category is weakly equivalent to a skeletal one."

### Skeletons as an Endo-Pseudofunctor on $\mathfrak{Cat}$

In the presence of choice we can thusly turn the process of taking the skeleton of a category into an endo-pseudofunctor on $\mathfrak{Cat}$, the $2$-category of categories.

For each category $\mathcal{C}$ let $sk_\mathcal{C}$ denote a skeleton of $\mathcal{C}$ and let $E_\mathcal{C}:sk_\mathcal{C}\simeq\mathcal{C}:E_\mathcal{C}^{\sim1}$ denote the skeletal equivalence at $\mathcal{C}$, with $\epsilon^{sk}_\mathcal{C}:E_\mathcal{C}\circ E^{\sim1}_\mathcal{C}\Rightarrow1_\mathcal{C}$ the skeletal transformation at $\mathcal{C}$. We define an endo-pseudofunctor

$$
sk:\mathfrak{Cat}\to\mathfrak{Cat}
$$

as follows:

1. $sk(\mathcal{C})=sk_\mathcal{C},$
1. $sk(F:\mathcal{C}\to\mathcal{D})=E^{\sim1}_\mathcal{D}\circ F\circ E_\mathcal{C}:sk_\mathcal{C}\to\mathcal{C}\to\mathcal{D}\to sk_\mathcal{D},$
1. $sk(\alpha:F\Rightarrow G)=i\circ\alpha\circ i^{-1}:sk(F)\Rightarrow sk(G),$ where
$$
i\circ\alpha\circ i^{-1}:sk(F)\Rightarrow sk(G)
$$
is defined by
$$
(i\circ\alpha\circ i^{-1})_X=i_{G(X)}\circ\alpha_X\circ i_{F(X)}^{-1}:F(X)'\to F(X)\to G(X)\to G(X)'.
$$
1. For each category $\mathcal{C}$, the unitor $\iota^{sk}_\mathcal{C}:1_{sk(\mathcal{C})}\Rightarrow sk(1_\mathcal{C})$ is defined by
$$
\iota^{sk}_\mathcal{C}=1_{1_{sk(\mathcal{C})}},
$$
since
$$sk(1_\mathcal{C})=E^{\sim1}_\mathcal{C}\circ1_\mathcal{C}\circ E_\mathcal{C}=E^{\sim1}_\mathcal{C}\circ E_\mathcal{C}=1_{sk(\mathcal{C})}.$$
1. For each pair of composable functors $F:\mathcal{C}\to\mathcal{D}$, $G:\mathcal{B}\to\mathcal{C}$, the associator $\alpha^{sk}_{F,G}:sk(F)\circ sk(G)\Rightarrow sk(F\circ G)$ is defined by
$$
\alpha^{sk}_{F,G}=1_{E^{\sim1}_\mathcal{D}\circ F}\star\epsilon^{sk}_\mathcal{C}\star1_{G\circ E_\mathcal{B}}:E^{\sim1}_\mathcal{D}\circ F\circ E_\mathcal{C}\circ E_\mathcal{C}^{\sim1}\circ G\circ E_\mathcal{B}\Rightarrow E^{\sim1}_\mathcal{D}\circ F\circ G\circ E_\mathcal{B}.
$$

### Skeleton of an Indexed Category

Using the above definition, we can canonically define the skeleton of an indexed category.

Let $\psi:\mathcal{B}^{op}\to\mathfrak{Cat}$ be an indexed category. We define the **indexed skeleton of $\psi$**, denoted

$$
sk(\psi):\mathcal{B}^{op}\to\mathfrak{Cat},
$$

to be the indexed category given by postcomposing $\psi$ with $sk$, so $sk(\psi)=sk\circ\psi.$ That is, for each $I\in\mathcal{B}$ we take a skeleton $sk(\psi(I))$ of the category $\psi(I)$ indexed by $I$ with equivalence $E_I:\psi(I)\simeq sk(\psi(I)):E_I^{\sim1}$, we extend functors using the equivalences at various skeletons, and we extend natural transformations using skeletal isomorphisms in the codomain categories (although $\mathcal{B}^{op}$ has only identity $2$-cells since it's a promoted $1$-category, so the action on $2$-cells is trivial).

### Skeleton of a Fibered Category

Using the above definition and the [[Grothendieck construction]], we can define the skeleton of a fibration using the skeleton of an indexed category.

Let $p:\mathcal{E}\to\mathcal{B}$ be a fibration. We define the **fibered skeleton of $p$**, denoted

$$
sk(p):sk^p_\mathcal{E}\to\mathcal{B}
$$

to be the Grothendieck completion of the indexed skeleton of the indexing by $p$. That is, $sk(p)$ is the fibration obtained by turning $p$ into an indexed category using the Grothendieck construction, taking the indexed skeleton of this indexed category, then turning the resulting skeletal indexed category back into a fibration (via the Grothendieck construction again). We refer to the total category $sk^p_\mathcal{E}$ in the resulting fibration as the **$p$-skeleton** of $\mathcal{E}$, which is different than $sk_\mathcal{E}$ since we have only skeletalized the fiber categories of our original fibration and not the total category of the original fibration. In general, we will have that
$$
sk_\mathcal{E}\ncong sk^p_\mathcal{E},
$$
although they are trivially equivalent. As an example of this construction in action, we can pass from the monomorphism fibration on a category with pullbacks to the subobject fibration by taking the fibered skeleton. For details, see the subobject fibration section of [[codomain fibration]].

## Properties

In this section we collect some properties of skeleta in [[set-level foundations]].

### Non-invariance

Of course, two categories can be equivalent even if one is skeletal and the other is not.  Thus, the notion of skeletal category violates the [[principle of equivalence]].  (It doesn't make sense to ask whether a *particular* category, skeletal or not, "violates the principle of equivalence".)

### Skeletons without Choice

Without any choice, we have the following theorems.

+-- {: .num_theorem}
###### Theorem
Any [[thin category]] (i.e. any [[preordered set]]) has a weak skeleton.
=--
+-- {: .proof}
###### Proof
In this case, we can take the objects of the skeleton of $C$ to be the isomorphism classes of $C$.  If $C$ is thin, then we can define a partial ordering on its set of isomorphism classes, making them into a skeleton of $C$.
=--

+-- {: .num_theorem}
###### Theorem
Any two skeletons (in the strong sense) of a category are isomorphic.
=--
+-- {: .proof}
###### Proof
Let $\mathcal{C}$ be a category, with $sk (\mathcal{C})$ and $sk (\mathcal{C})'$ two skeletons of $\mathcal{C}$, and for each object $Y\in\mathcal{C}$ denote by $X_Y$ and $X_Y'$ respectively the unique objects in $sk (\mathcal{C})$ and $sk (\mathcal{C})'$ which are isomorphic to Y, and denote their respective isomorphisms by $i_Y:Y\to X_Y$ and $i_Y':Y\to X_Y'$. We define a functor $F:sk (\mathcal{C})\to sk (\mathcal{C})'$ by
$$
F(X_Y)=X_Y',
$$
$$
F(f:X_Y\to X_Z)=i_Z'\circ i_Z^{-1}\circ f\circ i_Y\circ i_Y'^{-1}:X_Y'\to Y\to X_Y\to X_Z\to Z\to X_Z'.
$$
We then have
$$
F(1_{X_Y})=i_Y'\circ i_Y^{-1}\circ 1_{X_Y}\circ i_Y\circ i_Y'^{-1}=i_Y'\circ i_Y^{-1}\circ i_Y\circ i_Y'^{-1}=i_Y'\circ1_Y\circ i_Y'^{-1}=i_Y'\circ i_Y'^{-1}=1_{X_Y'}=1_{F(X_Y)},
$$
$$
F(f:X_B\to X_C)\circ F(g:X_A\to X_B)=i_C'\circ i_C^{-1}\circ f\circ i_B\circ i_B'^{-1}\circ i_B'\circ i_B^{-1}\circ g\circ i_A\circ i_A'^{-1}
$$
$$
=i_C'\circ i_C^{-1}\circ f\circ i_B\circ 1_{B}\circ i_B^{-1}\circ g\circ i_A\circ i_A'^{-1}=i_C'\circ i_C^{-1}\circ f\circ i_B\circ i_B^{-1}\circ g\circ i_A\circ i_A'^{-1}
$$
$$
=i_C'\circ i_C^{-1}\circ f\circ1_{X_B}\circ g\circ i_A\circ i_A'^{-1}=i_C'\circ i_C^{-1}\circ f\circ g\circ i_A\circ i_A'^{-1}=F(f\circ g),
$$
so $F$ is a functor. The inverse $F^{-1}: sk (\mathcal{C})'\to sk (\mathcal{C})$ is defined in the obvious way, and is functorial by similar computations.
=--

Notice that the [[axiom of choice]] fails in general when one considers [[internal category|internal categories]].  Hence not every [[internal category]] has a skeleton. A necessary condition for an internal category $X_1 \rightrightarrows X_0$ to have a skeleton is the existence quotient $X_0/X_1$ - the object of orbits under the action of the core of $X$. If the quotient map $X_0 \to X_0/X_1$ has a section, then one could consider $X$ to have a skeleton, but this condition isn't sufficient for the induced inclusion functor to be a weak equivalence of internal categories when this makes sense (i.e. if the category is internal to a site).

+--{: .query}
[[David Roberts]]: The claim above about the necessity of the existence of the quotient needs to be checked.
=--

### Equivalents of Choice

Define a *coskeleton* of a category $C$ to be a skeletal category $S$ with a surjective equivalence $C\to S$.  In [[Categories, Allegories]] it is shown that the following are equivalent.

1. The axiom of choice holds.
1. Any two ana-equivalent categories are strongly equivalent.
1. Any two weakly equivalent categories are strongly equivalent.
1. Every small category has a weak skeleton.
1. Every small category has a coskeleton.
1. Any two weak skeletons of a given small category are isomorphic.
1. Any two coskeletons of a given small category are isomorphic.

For convenience we add:

1. Given an inhabited family $\{S_i\}_I$ of equinumerous sets there exists $0 \in I$  and a family of isomorphisms of the permutation groups $\{Aut(S_0) \to Aut(S_i)\}_I$.
1. Given a family $\{S_i\}_I$ of inhabited equinumerous sets, there exists a family $(x_i)_I$ such that $x_i \in S_i$ for all $i \in I$.

### Uniqueness of Constructions

It is well-known that objects defined by [[universal property|universal properties]] in a category, such as [[limits]] and [[colimits]], are not unique on the nose, but only unique up to unique canonical isomorphism.  It can be tempting to suppose that in a *skeletal* category, where any two isomorphic objects are equal, such objects will in fact be unique on the nose.  However, under the most appropriate definition of "unique," this is *not* true (in general), because of the presence of automorphisms.

More explicitly, consider the notion of [[cartesian product]] in a category.  Although we colloquially speak of "a product" of objects $A$ and $B$ as being the *object* $A\times B$, strictly speaking a product consists of the object $A\times B$ *together with* the projections $A\times B\to A$ and $A\times B\to B$ which exhibit its universal property.  Thus, even if the category in question is skeletal, so that there can be only one object $A\times B$ that is a product of $A$ and $B$, in general this object can still "be the product of $A$ and $B$" in many different ways.

For example, given any projections $A\times B\to A$ and $A\times B\to B$ that exhibit $A\times B$ as a product of $A$ and $B$, we can compose them both with any automorphism of $A\times B$ to get a new, different, pair of projections that also exhibit $A\times B$ as a product of $A$ and $B$.  In fact, the universal property of a product implies that any two pairs of projections are related by an automorphism of $A\times B$, so this example is generic.  Thus, even in a skeletal category, we cannot speak of "the" product of $A$ and $B$, except in the same [[generalized the|generalized sense]] that makes sense in any category.  A formal way to say this is that the "category of products of $A$ and $B$," while still *equivalent* to the [[trivial category]], as it is in any category with products, will not be *isomorphic* to the trivial category even when the ambient category is skeletal.

(It is true in a few cases, though, that skeletality implies uniqueness on the nose.  For instance, a [[terminal object]] can have no nonidentity automorphisms, so in a skeletal category, a terminal object (if one exists) really is unique on the nose.)

## In intensional/homotopy type theory

All the discussion above pertains to [[set-level foundations]].  In [[intensional type theory]] such as [[homotopy type theory]], additional care is needed.

A first guess is to define a [[precategory]] $C$ to be skeletal if for any objects $a,b$ there is a function $\Vert a\cong b\Vert \to (a=b)$, i.e. if $a$ and $b$ are merely isomorphic then they are equal.  (Here $\Vert-\Vert$ denotes the [[propositional truncation]].)  However, if the [[type]] $ob(C)$ of objects is not a set, then $a=b$ is not a [[proposition]], and so under this definition a category could "be skeletal" in more than one way.

Arguably a better definition is to say that a precategory $C$ is **skeletal** if for any objects $a,b$ the canonical function
$$ (a=b) \to \Vert a\cong b\Vert $$
is an equivalence.  Since $\Vert a\cong b\Vert$ is always a proposition, this entails that $a=b$ is also a proposition.  Thus, this implies that $ob(C)$ *is* a set, i.e. that $C$ is a [[strict category]].

Note the analogy between this condition and the notion of [[univalent category]]: the only difference is the propositional truncation.  It follows that if a precategory is both skeletal and univalent, then $a\cong b$ is always a proposition (since it is equivalent to its propositional truncation), and thus the category is [[gaunt category|gaunt]].  Indeed, the gaunt categories are precisely those that are both skeletal and univalent.

Unlike in set-level foundations, in intensional type theory not every precategory (and not even every univalent category) has a skeleton.  Indeed, if the axiom [[sets cover]] fails, then there can be univalent categories that are not weakly equivalent to any strict category.  For this reason, the notion of skeleton tends to be less useful outside of set-level foundations, although particular concrete examples may turn out to be skeletal.

## Examples

* Every [[poset]] is skeletal.

* The [[walking parallel pair]] is a skeletal category which is not a poset.

* The [[free object|free]] [[category]] on a [[directed acyclic graph]] is skeletal.

* The [[category]] of [[Archimedean ordered fields]] and [[monotonic]] [[field homomorphisms]] is skeletal.

All of the above examples are in fact [[gaunt categories|gaunt]]: not only are any two isomorphic objects equal, but any isomorphism is an identity morphism.

* The [[delooping]] of a [[group]] is a skeletal category that is not gaunt (unless the group is [[trivial group|trivial]]).  In intensional type theory, this is a non-univalent precategory; its [[Rezk completion]] is no longer skeletal according to the above improved definition.

### Posets as skeleta of prosets
 {#PosetsAsSkeletaOfProsets}

In terms of [[(n,r)-category]]-theory one may essentially identify [[preordered sets]] with *[[thin categories]]* or *[[(0,1)-categories]]*. Under this identification, the passage of skeleta of categories corresponds to choosing an [[equivalence of categories|equivalent]] [[partial order]] inside a [[preorder]].

For more details (and more precise statements, see at *[[relation between preorders and (0,1)-categories]]*.

## Related concepts

* [[skeletal groupoid]]

## References

* {#MacLane71} [[Saunders MacLane]], p. 91 of: _[[Categories for the Working Mathematician]]_, Graduate texts in mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#RiehlCTInContext} [[Emily Riehl]], p. 34 in: *[[Category Theory in Context]]*, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf)]

* {#Richter19} [[Birgit Richter]], §2.6 in: *From categories to homotopy theory*, Cambridge Studies in Advanced Mathematics **188**, Cambridge University Press (2020) [[doi:10.1017/9781108855891](https://doi.org/10.1017/9781108855891), [book webpage](https://www.math.uni-hamburg.de/home/richter/catbook.html), [pdf](https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf)]

See also:

* Wikipedia, *Skeleton_(category_theory)*

Presenting a [[pretorsion theory]] on [[Cat]] whose torsion(-free) objects are the [[groupoids]] ([[skeletal categories]], respectively), hence whose "trivial objects" are the [[skeletal groupoids]]:

* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]

[[!redirects skeleton of a fibration]]
[[!redirects fibered skeleton]]
[[!redirects skeleta]]
[[!redirects skeletons]]
[[!redirects skeletal]]
[[!redirects skeletal category]]
[[!redirects skeletal categories]]


sketch

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

\tableofcontents

\section{Idea}

The notion of a *sketch* ([Bastiani & Ehresmann 1972](#BastianiEhresmann72)) is one formalisation of the notion of a *[[theory]]*. It is [[diagram|diagrammatic]], and has the advantage of being very close to [[category theory]], allowing it to very naturally express the category theoretic structure which is required to construct a [[model]] of the theory ([[finite products]], say). On the other hand, it is not a very concise notion: as Example \ref{ExampleSketchUnitalMagmas} illustrates, writing down the full details of a sketch even in the simplest examples takes time!

There is a precise correspondence between categories of models of sketches and [[accessible category|accessible categories]] and [[locally presentable category|locally presentable categories]], discussed below.

The notion of a sketch generalises that of a [[Lawvere theory]]. See Example \ref{ExampleLawvereTheory}.

\section{Details}

\subsection{Sketches}

\begin{defn} A _sketch_ is a [[small category|small]] [[category]] $T$ equipped with a set $L$ of [[cones]] and a set $C$ of [[cocones]]. Alternatively, it is a [[directed graph]] equipped with a set $D$ of [[diagram|diagrams]], a set $L$ of [[cones]], and a set $C$ of [[cocones]]. \end{defn}

\begin{defn} A _realized sketch_ is one where all the cones in $L$ are [[limit]] cones and all the cocones in $C$ are [[colimit]] cocones. \end{defn}

\begin{defn} A _limit sketch_ is a sketch with $C = \emptyset$. \end{defn}

\begin{defn} A _colimit sketch_ is a sketch with $L = \emptyset$. \end{defn}

\begin{defn} A _finite product sketch_ is a limit sketch in which the only cones are those of [[finite product]] diagrams. \end{defn}

\begin{defn} A _finite limit sketch_ is a limit sketch in which the only cones are those of [[finite limit]] diagrams. \end{defn}

\subsection{Models of sketches}

\begin{defn} A _model of a sketch_ in a category $\mathcal{C}$ is a [[functor]] $T\to \mathcal{C}$ taking each cone in $L$ to a limit cone and each cocone in $C$ to a colimit cocone. \end{defn}

In particular, $T$ is realized if and only if its identity functor is a model.

\begin{defn} If one takes the definition of a sketch to be that involving directed graphs, a _model of a sketch_ in a category $\mathcal{C}$ is a morphism of directed graphs from the directed graph of the sketch to the underlying directed graph of $\mathcal{C}$, so that diagrams are taken to commutative diagrams, cones are taken to limit cones, and co-cones are taken to colimit cones.\end{defn}

Frequently the notion of model is restricted to the case $\mathcal{C}=Set$.

\begin{defn} A [[category]] is _sketchable_ if it is the category of models (in $Set$) of a sketch. \end{defn}

\section{Examples}

\begin{example} A sketch, more precisely a finite product sketch, for the theory of pointed sets can be constructed as follows. The directed graph can be taken to be the following.

\begin{centre}
  \begin{tikzpicture}
    \fill (0,0) circle[radius=0.1];
    \node at (0,-0.3) {$v_{1}$};
    \fill (3,0) circle[radius=0.1];
    \node at (3,-0.3) {$v_{2}$};
    \draw[shorten <= 0.5em, shorten >= 0.5em, ->] (0,0) -- (3,0);
  \end{tikzpicture}
\end{centre}

The set of diagrams can be taken to be empty. The set of cones can be taken to be the set with the single cone given by the vertex $v_{1}$, i.e. a cone of the empty diagram. The set of co-cones can be taken to be empty.

A model of this sketch necessarily sends the vertex $v_{1}$ to a product of the empty diagram, hence to a one element set $1$; sends the vertex $v_{2}$ to any set $X$; and sends the arrow from $v_{1}$ to $v_{2}$ to an arrow from $1$ to $X$, that is, to an element of $X$, as required.
\end{example}

\begin{example} \label{ExampleSketchUnitalMagmas} A sketch, more precisely a finite product sketch, for the theory of unital magmas (sets with a binary operation which has a two sided unit) can be constructed as follows. The directed graph can be taken to be the following.

\begin{centre}
  \begin{tikzpicture}
    \fill (0,0) circle[radius=0.1];
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,3) circle[radius=0.1];
    \fill (3,-3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >= 0.5em, ->] (0,0) to node[auto] {$e$} (3,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, bend right=60] (3, 0) to node[auto, swap] {$p_{1}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, transform canvas={yshift=-0.1em}, bend left=60] (3, 0) to node[auto] {$p_{2}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, <-] (3, 0) to node[auto] {$m$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=0.1em}, bend right] (3, 3) to node[auto, swap] {$p_{1}$} (0,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, 3) to node[auto] {$p_{2}$} (3,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend left] (3, 3) to node[auto] {$e, -$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=0.1em}, bend left] (3, -3) to node[auto] {$p_{2}$} (0,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, -3) to node[auto, swap] {$p_{1}$} (3,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend right] (3, -3) to node[auto, swap] {$-, e$} (6,0);
  \end{tikzpicture}
\end{centre}

The set of diagrams can be taken to have six elements, the first consisting of

\begin{centre}
  \begin{tikzpicture}
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >=0.5em, <-] (3, 0) to node[auto] {$m$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, 3) to node[auto] {$p_{2}$} (3,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend left] (3, 3) to node[auto] {$e, -$} (6,0);
  \end{tikzpicture}
\end{centre}

the second consisting of the following

\begin{centre}
  \begin{tikzpicture}
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,-3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >=0.5em, <-] (3, 0) to node[auto] {$m$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, -3) to node[auto, swap] {$p_{1}$} (3,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend right] (3, -3) to node[auto, swap] {$-, e$} (6,0);
  \end{tikzpicture}
\end{centre}

the third consisting of the following

\begin{centre}
  \begin{tikzpicture}
    \fill (0,0) circle[radius=0.1];
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >= 0.5em, ->] (0,0) to node[auto] {$e$} (3,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, bend right=60] (3, 0) to node[auto, swap] {$p_{1}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=0.1em}, bend right] (3, 3) to node[auto, swap] {$p_{1}$} (0,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend left] (3, 3) to node[auto] {$e, -$} (6,0);
  \end{tikzpicture}
\end{centre}

the fourth consisting of the following

\begin{centre}
  \begin{tikzpicture}
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, bend left=60] (3, 0) to node[auto] {$p_{2}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, 3) to node[auto] {$p_{2}$} (3,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend left] (3, 3) to node[auto] {$e, -$} (6,0);
  \end{tikzpicture}
\end{centre}

the fifth consisting of the following

\begin{centre}
  \begin{tikzpicture}
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,-3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, bend right=60] (3, 0) to node[auto, swap] {$p_{1}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, -3) to node[auto, swap] {$p_{1}$} (3,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend right] (3, -3) to node[auto, swap] {$-, e$} (6,0);
  \end{tikzpicture}
\end{centre}

and the sixth consisting of the following.

\begin{centre}
  \begin{tikzpicture}
    \fill (0,0) circle[radius=0.1];
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \fill (3,-3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >= 0.5em, ->] (0,0) to node[auto] {$e$} (3,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, bend left=60] (3, 0) to node[auto, swap] {$p_{2}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=0.1em}, bend left] (3, -3) to node[auto] {$p_{2}$} (0,0);
   \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=-0.1em}, bend right] (3, -3) to node[auto, swap] {$-, e$} (6,0);
  \end{tikzpicture}
\end{centre}

The set of cones can be taken to have four elements, the first being the leftmost vertex (cone of the empty set), the second consisting of

\begin{centre}
  \begin{tikzpicture}
    \fill (0,0) circle[radius=0.1];
    \fill (3,0) circle[radius=0.1];
    \fill (3,3) circle[radius=0.1];

    \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=0.1em}, bend right] (3, 3) to node[auto, swap] {$p_{1}$} (0,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, 3) to node[auto] {$p_{2}$} (3,0);
  \end{tikzpicture}
\end{centre}

the third consisting of

\begin{centre}
  \begin{tikzpicture}
    \fill (0,0) circle[radius=0.1];
    \fill (3,0) circle[radius=0.1];
    \fill (3,-3) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >=0.5em, ->, transform canvas={yshift=0.1em}, bend left] (3, -3) to node[auto] {$p_{2}$} (0,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, ->] (3, -3) to node[auto, swap] {$p_{1}$} (3,0);
  \end{tikzpicture}
\end{centre}

and the fourth consisting of the following.

\begin{centre}
  \begin{tikzpicture}
    \fill (3,0) circle[radius=0.1];
    \fill (6,0) circle[radius=0.1];
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, bend right=60] (3, 0) to node[auto, swap] {$p_{1}$} (6,0);
    \draw[shorten <= 0.5em, shorten >=0.5em, <-, transform canvas={yshift=-0.1em}, bend left=60] (3, 0) to node[auto] {$p_{2}$} (6,0);
  \end{tikzpicture}
\end{centre}

The set of co-cones can be taken to be empty.

In a model of this sketch, the leftmost vertex is sent to a one element set $1$, the middle vertex is sent to an arbitrary set $X$, the top vertex is sent to the product $1 \times X$, the bottom vertex is sent to the product $X \times 1$, and the right vertex is sent to the product $X \times X$. The arrow $e$ picks out an element $e_{X}$ of $X$.

The arrow $e,-$ is sent to an arrow $e_{X} \times id: 1 \times X \rightarrow X \times X$, which is forced by the universal property of $X \times X$ and the fact that the diagrams

\begin{centre}
  \begin{tikzcd}
    1 \times X \ar[r, "e_{X} \times id"] \ar[d, "p_{1}", swap] & X \times X \ar[d, "p_{1}"] \\
    1 \ar[r, "e_{X}", swap] & X
  \end{tikzcd}
\end{centre}

and

\begin{centre}
  \begin{tikzcd}
    1 \times X \ar[r, "e_{X} \times id"] \ar[dr, "p_{2}", swap] & X \times X \ar[d, "p_{2}"] \\
      & X
  \end{tikzcd}
\end{centre}

commute to really be the product of the arrows $e_{X}$ and $id$. Similarly, the arrow $-,e$ is sent to an arrow $id \times e_{X}: X \times 1 \rightarrow X \times X$, which is forced by the fact that the diagrams

\begin{centre}
  \begin{tikzcd}
    X \times 1 \ar[r, "id \times e_{X}"] \ar[dr, "p_{1}", swap] & X \times X \ar[d, "p_{1}"] \\
      & X
  \end{tikzcd}
\end{centre}

and

\begin{centre}
  \begin{tikzcd}
    X \times 1 \ar[r, "id \times e_{X}"] \ar[d, "p_{2}", swap] & X \times X \ar[d, "p_{2}"] \\
    1 \ar[r, "e_{X}", swap] & X
  \end{tikzcd}
\end{centre}

commute to really be the product of the arrows $e_{X}$ and $id$.

The arrow $m$ is sent to a map $m: X \times X \rightarrow X$ which is arbitrary except that the diagrams

\begin{centre}
  \begin{tikzcd}
    1 \times X \ar[r, "e_{X} \times id"] \ar[dr, "p_{2}", swap] & X \times X \ar[d, "m"] \\
      & X
  \end{tikzcd}
\end{centre}

and

\begin{centre}
  \begin{tikzcd}
    X \times 1 \ar[r, "id \times e_{X}"] \ar[dr, "p_{1}", swap] & X \times X \ar[d, "m"] \\
      & X
  \end{tikzcd}
\end{centre}

are forced to commute.

Putting all of this together, we see that we exactly have a unital magma.

\end{example}

\begin{example} \label{ExampleLawvereTheory} A [[Lawvere theory]] is a special case of a (limit) sketch, where the category is one with a distinguished object $X$ such that all objects are ([[isomorphic]] to) powers of $X$, and $C = \emptyset$ and $L$ is the set of all product cones. \end{example}

## Properties

### Relation to accessible and locally representable categories
 {#RelationToLocallyRepresentableCategories}

\begin{proposition}
\label{LimitSketchableCategoriesAreAccessible}
The categories of models of sketches are equivalently the [[accessible categories]].
\end{proposition}

\begin{proposition}
\label{LimitSketchableCategoriesAreLocallyPresentable}
The categories of models of limit-sketches are the [[locally presentable categories]].
\end{proposition}
([Adámek & Rosický 1994, Cor. 1.52](#AdámekRosický49))

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

From the discussion there we have that

* an [[accessible category]] is equivalently:
  * a [[full subcategory]] of a [[presheaf category]] that's closed under $\kappa$-filtered colimits for some $\kappa$
  * the category of models of a sketch

* a [[locally presentable category]] is equivalently:
  * a *reflective* full subcategory of a presheaf category that's closed under $\kappa$-filtered colimits for some $\kappa$
  * the category of models of a *limit* sketch
  * an accessible category with all small limits
  * an accessible category with all small colimits

We can "break in half" the difference between the two and define

* a [[locally multipresentable category]] to be equivalently:
  * a *multireflective* full subcategory of a presheaf category that's closed under $\kappa$-filtered colimits for some $\kappa$
  * the category of models of a *limit and coproduct* sketch
  * an accessible category with all small *connected* limits
  * an accessible category with all small multicolimits

and

* a *weakly locally presentable category* to be equivalently:
  * a *weakly reflective* full subcategory of a presheaf category that's closed under $\kappa$-filtered colimits for some $\kappa$
  * the category of models of a *limit and epi* sketch
  * an accessible category with all small products
  * an accessible category with all small weak colimits

=--

### Monoidal Structures on the Category of Sketches

The category of sketches is well behaved: it is complete, cocomplete, cartesian closed and has a second symmetric monoidal closed structure.

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

The category of sketches is [[topological category|topological]] over the category of directed [pseudographs](https://ncatlab.org/nlab/show/pseudograph).

=--

The above proposition gives the category of sketches Cartesian products - however these are often not the sketches one would expect when thinking of the product of two theories. Instead consider the tensor product:
+-- {: .num_prop }
###### Proposition

Let $S,T$ be sketches. We define the sketch $S \otimes T$ to be:

The vertices of $S \otimes T$ are the product of the set of vertices from $S$, $T$. The set of arrows is given as
$$
\{ (\alpha, b) | \alpha \in \mathsf{Edge}(S), b \in \mathsf{Vertex}(T)\}
\cup \{ (a, \beta) | a \in \mathsf{Vertex}(S), \beta \in \mathsf{Edge}(T)\}
$$
where the source of $(\alpha,b)$ is $(s_S(\alpha), b)$, and vice versa. $S$ is often called the _horizontal_ structure and $T$ as the _vertical_ structure.
The set of diagrams is the union of the following three sets:

* The horizontal diagrams are constant in the second parameter: $H = \{ (D, b) | D \in \mathsf{Diagrams}(S), b \in \mathsf{Vertex}(T) \}$

* The vertical diagrams are constant in the first parameter: $V =  \{ (a, D) |  a \in \mathsf{Vertex}(S), D \in \mathsf{Diagrams}(T) \}$

* Also add every square diagram: $C$ is the set of squares for each edge $\alpha$ in $S$, $\beta \in T$

\begin{center}
    \begin{tikzcd}
        (a,b) \ar[r, "(\alpha{,}b)"] \ar[d, "(a{,}\beta)"] & (a',b) \ar[d, "(a'{,} \beta)"] \\
        (a, b') \ar[r, "(\alpha{,} b')"] & (a',b')
     \end{tikzcd}
\end{center}

* The set of cones and cocones are define analogously to the set of commuting diagrams, except _only_ the vertical and horizontal cones are taken.

This tensor product, along with the unit $(\ast, \emptyset, \emptyset, \emptyset)$, gives the category of sketches a monoidal structure.
=--

This monoidal structure is useful for considering structures like double categories (i.e. categories in the category of categories).
+-- {: .num_prop }

###### Proposition

Let $S,T$ be sketches, and $X$ some category. Then the category of models of $S$ in the category of models of $T$ in $X$ is equivalent to the category of models of $S \otimes T$ in $X$.

=--

One can ask when $S \otimes T$ has the same models as $T \otimes S$, i.e. when $S$-models in the category of $T$-models are the same as $T$-models in the category of $S$-models.
This is the case, roughly, when the colimits and limits specified in the sketches commute. For example, since limits always commute, you can swap the sketches if both are limit sketches. And you can swap a finite product sketch with a sifted colimit sketch etc. For more precise statements see David Bensons article and the references therein.

In general one can not swap the order in the monoidal product.
For example take the (terminal object + coproduct)-sketch $S$ whose models are maps $X \to X \coprod 1$ and the finite product sketch $M$ whose models are monoids. Look at models of these in $Set$:
Since in the category of monoids the terminal object is also initial, the coproduct of a monoid with the terminal object is isomorphic to that monoid again. Therefore $S$-models in $M$-Mod are monoids with an endomorphism.
On the other hand $M$-models in $S$-Mod are pairs of monoids one of which has one element more, plus a homomorphism between them. These categories do not seem to be equivalent.

## Related concepts

* [[internalization]]

## References

### General

Original articles:

* {#Ehresmann68} [[Charles Ehresmann]], _Esquisses et types de structures algébriques_, Bul. Inst. Polit. Iasi **14** 1–2 (1968) 1-14 [[[Ehresmann-EsquisseEtTypes.pdf:file]]]

* {#BastianiEhresmann72} [[Andrée Bastiani]], [[Charles Ehresmann]], _Categories of sketched structures_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome **13** no 2 (1972)  104-214 ([numdam:CTGDC_1972__13_2_104_0](http://www.numdam.org/item/?id=CTGDC_1972__13_2_104_0))

Overview in:

* [[Charles Wells]], *Sketches: Outline with references*, 1993 ([citeseer:10.1.1.217.3531](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.217.3531), [[Wells_Sketches.pdf:file]])

See also:

* [[Michael Barr]], *Notes on Sketches*, Technical report of the Electrotechnical Laboratory (computer language section), Tsukuba, Japan (1997) [[pdf](https://www.math.mcgill.ca/barr/papers/sketch.pdf), [[Barr-NotesOnSketches.pdf:file]]]

Textbook accounts:

* [[Michael Barr]], [[Charles Wells]], Section 4 of: *[[Toposes, Triples, and Theories]]*, Originally published by: Springer-Verlag, New York, 1985, republished in:
Reprints in [Theory and Applications of Categories, No. 12 (2005) pp. 1-287](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)

and with emphasis on the relation to [[locally presentable category|locally presentable]] and [[accessible categories]]:

* {#MakkaiPare} [[Michael Makkai]], [[Robert Paré]], Chapter 3 onwards in: _Accessible categories: The foundations of categorical model theory_,  Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989 ([ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104))

* {#AdámekRosický49} [[Jiří Adámek]], [[Jiří Rosický]], Def. 1.49 onwards in: _[[Locally presentable and accessible categories]]_, Cambridge University Press (1994)

Proof that not only every sketchable category is [[accessible category|accessible]] but that conversely every [[accessible category]] is sketchable:

* [[Christian Lair]], *Catégories modelables et catégories esquissables*, Diagrammes **6** 5 (1981) [[numdam:DIA_1981__6__A5_0](http://www.numdam.org/item/?id=DIA_1981__6__A5_0), [pdf](http://www.numdam.org/article/DIA_1981__6__A5_0.pdf)]

The tensor product of sketches is investigated here:

* David D. Benson, Multilinearity of sketches, [TAC, vol.11, nr. 3, pp.269-277](http://www.tac.mta.ca/tac/volumes/1997/n11/n11.pdf)

The category of sketches itself was studied as a [[categorical semantics]] for [[type theory]] in:

* {#Gray} [[John W. Gray]], _The Category of Sketches as a Model for Algebraic Semantics_, Categories in Computer Science and Logic: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference Held June 14-20, 1987 with Support from the National Science Foundation. Vol. 92. American Mathematical Soc., 1989.

See also

* [[Martin Brandenburg]], *Large limit sketches and topological space objects* ([arXiv:2106.11115](https://arxiv.org/abs/2106.11115))

### In enriched category theory

Discussion of sketches the generality of [[enriched category theory]]:

* {#Kelly82} [[G. Max Kelly]], *Structures defined by finite limits in the enriched context, I*, Cahiers de topologie et géométrie différentielle **23** 1 (1982)  3-42 [[numdam:CTGDC_1982__23_1_3_0](http://archive.numdam.org/item/CTGDC_1982__23_1_3_0)]

* [[Francis Borceux]], [[Carmen Quinteriro]], [[Jiří Rosický]], *A theory of enriched sketches*, Theory and Applications of Categories, **4** 3 (1998) 47-72 [[tac:4-03](http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf)]

### Generalisations

A generalisation internal to a [[2-category]] is discussed in:

* Yoshiki Kinoshita, [[John Power]], and Makoto Takeyama. _Sketches_, Journal of Pure and Applied Algebra 143.1-3 (1999): 275-291.

* [[Ross Street]]. _Pointwise extensions and sketches in bicategories_ (2014), ([arXiv:1409.6427](https://arxiv.org/abs/1409.6427)).

* [[Ross Street]]. _Complete objects with respect to a theory_.

[[!redirects sketches]]

[[!redirects finite product sketch]]
[[!redirects finite product sketches]]

[[!redirects finite limit sketch]]
[[!redirects finite limit sketches]]

[[!redirects limit sketch]]
[[!redirects limit sketches]]


Sketches of an Elephant

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

This page collects material related to the book

* [[Peter Johnstone]]

  **Sketches of an Elephant -- A Topos Theory Compendium**

  Oxford University Press 2002

  Volume 1 ([ISBN:9780198534259](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198534259))

  Volume 2 ([ISBN:9780198515982](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198515982))

on [[topos theory]].

Like _Gravitation_, the title can be taken to refer not only to the subject matter but also to the immense size and scope of the book itself. Like _The Lord of the Rings_, it consists of 6 parts arranged evenly into 3 volumes (but without appendices). Actually, Volume 3 has not yet been published (so who knows? it may have appendices after all!).

The _Elephant_ is a good reference for anything related to topos theory, and we may often cite it here. However, it introduced many terminological changes, some of which may not be widely accepted or even known. (Fortunately, it will tell you about these in the text.)

#Contents#
* automatic table of contents goes here
{:toc}

## A Toposes as Categories

### A1 Regular and cartesian closed categories

#### A1.1 Preliminary assumptions

* [[category theory]]

* [[reflective subcategory]]

* [[monadicity theorem]]

* [[adjoint lifting theorem]]

* [[Grothendieck construction]]

#### A1.2 Cartesian categories

* [[finitely complete category]] / [[cartesian category]]

* [[cartesian functor]]

#### A1.3 Regular categories

* [[strong epimorphism]]

* [[conservative functor]]

* [[regular category]]

  * [[regular epimorphism]]

* [[exact category]]
* [[regular and exact completion]]

#### A1.4 Coherent categories

* [[coherent category]]

* [[disjoint coproduct]]

* [[extensive category]]
* [[pretopos]]
* [[Heyting category]]
* [[Boolean category]]

#### A1.5 Cartesian closed categories

* [[cartesian closed category]]
* [[locally cartesian closed category]]

* [[cartesian closed functor]]
* [[Frobenius reciprocity]]

* [[dependent product]], [[dependent sum]]

#### A1.6 Subobject classifiers

* [[subobject classifier]]

### A2 Toposes - basic theory

#### A2.1 Definition and examples

* [[topos]]

* Examples

  * [[Set]], [[FinSet]]

  * [[category of sheaves]], [[presheaf]]

#### A2.2 The monadicity theorem

* [[monadicity theorem]]

#### A2.3 The Fundamental Theorem

#### A2.4 Effectiveness, positivity and partial maps

#### A2.5 Natural number objects

* [[natural number object]]

#### A2.6 Quasitoposes

* [[quasitopos]]

### A3 Allegories

#### A3.1 Relations in regular categories

* [[relation]]
* [[Rel]]

#### A3.2 Allegories and tabulations

* [[allegory]]
* [[bicategory of relations]]
* [[1-category equipped with relations]]

#### A3.3 Splitting symmetric idempotents

* [[idempotent]]

#### A3.4 Division allegories and power allegories

### A4 Geometric morphisms - basic theory

#### A4.1 Definition and examples

* [[geometric morphism]]

  * [[direct image]]

  * [[inverse image]]

  * [[geometric transformation]]

* [[essential geometric morphism]]

## B 2-Categorical Aspects of Topos Theory

### B1 Indexed categories and fibrations

#### B1.1 Review of 2-categories

* [[2-category]]
* [[2-limit]]

  * [[inverter]]

#### B1.2 Indexed categories

* [[indexed category]]

* [[indexed functor]]

* [[pseudofunctor]]

#### B1.3 Fibrations

* [[Grothendieck fibration]]

* [[bifibration]]

* [[Grothendieck construction]]

#### B1.4 Limits and colimits

* [[limit]]

* [[dependent product]], [[dependent sum]]

#### B1.5 Descent conditions and stacks

* [[descent]]

* [[stack]]

### B2 Internal and locally internal categories

#### B2.1 Review of enriched categories

* [[enriched category]]

* [[enriched category theory]]

#### B2.2 Locally internal categories

* [[locally internal category]]

#### B2.3 Internal categories and diagram categories

* [[internal category]], [[internal diagram]]

* [[internal (co-)limit]]

#### B2.4 The indexed adjoint functor theorem

* [[adjoint functor theorem]]

* [[indexed adjoint functor theorem]]

#### B2.5 Discrete opfibrations

* [[discrete fibration|discrete opfibration]]

#### B2.6 Filtered colimits

* [[filtered colimit]]

#### B2.7 Internal profunctors

* [[profunctor]]

* [[internal profunctor]]

### B3 Toposes over a base

#### B3.1 $\mathcal{S}$-Toposes as $\mathcal{S}$-indexed categories

* [[indexed topos]]

* [[bounded geometric morphism]]

* [[bounded topos]]

#### B3.2 Diaconescu's theorem

* [[torsor]], [[principal bundle]]

* [[Diaconescu's theorem]]

* [[Sierpinski topos]]

#### B3.3 Giraud's theorem

* [[Giraud's theorem]]

#### B3.4 Colimits in Top

* [[Topos]]

* The paragraph before B3.4.8 refers to A4.1.13, but should probably refer instead to A4.1.15.

## C Toposes as Spaces

### C1 Sheaves on a locale

#### C1.1 Frames and nuclei

* [[frame]]
* [[Heyting algebra]]

#### C1.2 Locales and spaces

* [[locale]]

#### C1.3 Sheaves, local homeomorphisms and frame-valued sets

* [[sheaf]]
* [[local homeomorphism]]

#### C1.4 Continuous maps

(...)

#### C1.5 Some topological properties of toposes

* [[locally connected topos]]

### C2 Sheaves on a site

#### C2.1 Sites and coverages

* [[coverage]]

  * [[regular coverage]]

* [[Grothendieck topology]]

  * [[Grothendieck pretopology]]

* [[site]]

  * [[standard site]]

* Lemma 2.1.7 is incorrect as stated: one should assume that $A$ is a sheaf for a (sifted) coverage. See [here](http://math.stackexchange.com/a/358709) for details.

#### C2.2 The topos of sheaves

* [[sheaf]]

* [[category of sheaves]],

* [[Grothendieck topos]], [[sheaf topos]]

  * [[Giraud's theorem]]

  * [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]

* [[subcanonical topology]], [[canonical topology]]

* [[dense sub-site]]

* The statement that the induced coverage inherits closure properties requires assumptions on either the coverage or the subcategory. See the discussion [here](http://nforum.mathforge.org/discussion/3693/induced-coverage/).

#### C2.3 Morphisms of sites

* [morphism of sites](http://ncatlab.org/nlab/show/site#Morphisms)

#### C2.4 Internal sites and pullbacks

...

#### C2.5 Fibrations of sites

...

### C3 Classes of geometric morphisms

#### C3.1 Open maps

#### C3.2 Proper maps

* [[proper geometric morphism]]

* [[separated geometric morphism]]

* [[compact topos]], [[Hausdorff topos]]

#### C3.3 Locally connected morphisms

* [[locally connected topos]]

* [[locally connected geometric morphism]]

#### C3.4 Tidy morphisms

#### C3.5 Atomic morphisms

* [[atomic geometric morphism]]

#### C3.6 Local maps

* [[local geometric morphism]]

* [[local topos]]

* Example C3.6.15(e) says that $(E/l(M))_{ce}$ is equivalent to $Set$, but the referred-to paper "Local maps of toposes" seems to say that it should be equivalent to $E$ instead.

## D Toposes as theories

### D1 First-order categorical logic

#### D1.1 First-order languages

* [[signature]]

* [[term]]

* [[formula]]

* [[context]]

* [[sequent]]

* [[axiom]]

* [[theory]]

  * [[algebraic theory]]

  * [[propositional theory]]

  * [[Horn theory]]

  * [[regular theory]]

  * [[coherent theory]]

  * [[first-order theory]]

  * [[geometric theory]]

  * [[infinitary first-order theory]]

#### D1.2 Categorical semantics

* [[categorical semantics]]

* [[internal logic]]

* [[models in presheaf toposes]]

#### D1.3 First-order logic

#### D1.4 Syntactic categories

* [[syntactic category]]

#### D1.5 Classical completeness

### D2 Sketches

#### D2.1 The concept of sketch

* [[sketch]]

#### D2.2 Sketches and theories

* [[theory]]

* [[Lawvere theory]]

#### D2.3 Sketchable and accessible categories

* [[locally presentable category]]

* [[accessible category]]

#### D2.4 Properties of model categories

### D3 Classifying toposes

#### D3.1 Classifying toposes via syntactic sites

* [[syntactic site]]

  * [[regular coverage]]

  * [[coherent coverage]]

* [[classifying topos]]

#### D3.2 The object classifier

* [[theory of objects]]

* [[classifying topos for the theory of objects]]

#### D3.3 Coherent toposes

* [[coherent topos]]

#### D3.4 Boolean classifying toposes

#### D3.5 Conceptual completeness

* [[conceptual completeness]]

### D4 Higher-order logic

#### D4.1 Interpreting higher-order logic in a topos

#### D4.2 $\lambda$-Calculus and cartesian closed categories

* [[cartesian closed category]]

* [[lambda-calculus]]

#### D4.3 Toposes as type theories

#### D4.4 Predicative type theories

* [[predicative mathematics]]

#### D4.5 Axioms of choice and Booleanness

* [[axiom of choice]]
* [[internally projective object]] --- note that Lemma 4.5.3(iii) in the Elephant is not quite strong enough to imply (i) and (ii).
* [[excluded middle]]

#### D4.6 De Morgan's law and the Gleason cover

#### D4.7 Real numbers in a topos

* [[real numbers object]]

### D5 Aspects of finiteness

#### D5.1 Natural number objects revisited

#### D5.2 Finite cardinals

* [[finite set]]
* [[finite object]]

#### D5.3 Finitary algebraic theories

* [[algebraic theory]]

* [[Lawvere theory]]

#### D5.4 Kuratowski-finiteness

* [[finite set]]
* [[finite object]]

#### D5.5 Orbitals and numerals

## E Homotopy and Cohomology

### E1 Homotopy theory for toposes

* [[homotopy groups in an (∞,1)-topos]]

#### E1.1 Path-connectedness for locales

* [[locale]]

#### E1.2 The fundamental groupoid via paths

#### E1.3 The fundamental groupoid via coverings

#### E1.4 Natural homotopy

### E2 Algebraic homotopy theory

#### E2.1 Quillen model structures

* [[model category]]

#### E2.2 Model structure for simplicial sets

* [[model structure on simplicial sets]]

#### E2.3 Model structures for sheaves

* [[model structure on simplicial presheaves|model structure on simplicial sheaves]]

#### E2.4 Axiomatic theory of open maps

### E3 Cohomology theory

* [[cohomology]]

* [[abelian sheaf cohomology]]

#### E3.1 Abelian groups and modules in a topos

* [[abelian group]], [[abelian sheaf]]

* [[module]]

#### E3.2 Cech cohomology

* [[Cech cohomology]]

#### E3.3 Torsors and non-abelian cohomology

* [[nonabelian cohomology]]

* [[principal bundle]]

* [[principal ∞-bundle]]

#### E3.4 Classifying toposes and classifying spaces

* [[classifying topos]]

#### E3.5 Cohomological applications of descent theory

* [[descent]]

## F Toposes as Mathematical Universes

### F1 Synthetic differential geometry

* [[synthetic differential geometry]]

#### F1.1 Properties of the generic ring

* [[classifying topos]]

#### F1.2 Rings of line type

* [[line object]]

#### F1.3 Well-adapted models

* [[Models for Smooth Infinitesimal Analysis]]

#### F1.4 Tiny objects

* [[tiny object]]

#### F1.5 Synthetic integration theory

* [[integration axiom]]

#### F1.6 Intrinsic infinitesimal

* [[infinitesimal object]]

* [[amazing right adjoint]]

### F2 Realizability toposes

#### F2.1 Schönfinkel algebras and assemblies

#### F2.2 Realizability toposes

* [[realizability topos]]

#### F2.3 Modified Realizability

#### F2.4 Synthetic domain theory

* [[synthetic domain theory]]

### F3 The free topos

#### F3.1 The free topos as a mathematical universe

#### F3.2 Disjunction and existence properties

#### F3.3 Doing without the natural numbers

#### F3.4 Recursive functions in the free topos

### F4 Topos theory and set theory

#### F4.1 Internal sets in a topos

#### F4.2 Algebraic set theory

* [[algebraic set theory]]

#### F4.3 Independence proofs via classifying toposes

#### F4.4 Independence of the axiom of choice

* [[axiom of choice]]

category: reference

[[!redirects Sketches of an elephant]]

[[!redirects Sketches of an Elephant -- A Topos Theory Compendium]]

[[!redirects Elephant]]


skew-commutative algebra

A __skew-commutative algebra__ is a [[graded object|graded]] [[associative algebra|algebra]] $A$ (with the product often written as $\wedge$) such that

*  $x \wedge y = (-1)^{pq}\, y \wedge x$ whenever $x$ is in grade $p$ and $y$ is in grade $q$.

Actually, there is a _twisted [[tensor product]]_ on the category $Gr Mod$ of graded modules such that a skew-commutative algebra is really a [[commutative algebra]] (that is an [[abelian monoid|abelian]] [[monoid object]]) in $Gr Mod$, but I forget how that goes.


skew-simplicial set

#Contents#
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{:toc}

## Idea

One can possibly enlarge the [[simplex category]] to a one of the several interesting categories in homological algebra like [[cycle category]] of Connes and the skeletal category of finite sets.
Such examples can be founded axiomatically as kind of extensions of the simplex category by a group with special properties. The resulting notion is a __skew-simplicial group__ (synonym: crossed simplicial group): the presheaves on a skew-simplicial group are skew-simplicial sets of the corresponding kind. In addition to cyclic and symmetric simplicial sets other important applications are [[dihedral set|dihedral]], [[quaternionic  set|quaternionic]] and hyperoctahedral sets/homology (cf. Loday [MR89e:18024](http://www.ams.org/mathscinet-getitem?mr=89e:18024)). The formalism has been found by Krasauskas and a bit later and independently also by Fiedorowicz and Loday.

##Related Lab entries

* [[crossed group]]

* [[cyclic set]]

* [[dihedral homology]]

## References

The important special cases are [[symmetric set]], [[cyclic set]], [[simplicial set]], [[dihedral set]].

Original references are

* Z. Fiedorowicz, Jean-Louis Loday, _Crossed simplicial groups and their associated homology_,  Trans. Amer. Math. Soc. __326__ (1991),  no. 1, 57--87, [MR91j:18018](http://www.ams.org/mathscinet-getitem?mr=91j:18018), [doi](http://dx.doi.org/10.2307/2001855)
* R. Krasauskas, _Skew-simplicial groups_ (Russian) Litovsk. Mat. Sb.  __27__ (1987),  no. 1, 89--99, [MR88m:18022](http://www.ams.org/mathscinet-getitem?mr=88m:18022) (English transl. [[krasauskas.pdf:file]])

A connection to geometry of surfaces is sketched in

* R. Krasauskas, _Crossed simplicial groups of framed braids and mapping class groups of surfaces_, Lith. Math. J. __36__, 263--281 (1996) [doi](https://doi.org/10.1007/BF02986853)

A recent treatise on connection to geometry is

* Tobias Dyckerhoff, [[Mikhail Kapranov]], _Crossed simplicial groups and structured surfaces_, [arxiv/1403.5799](http://arxiv.org/abs/1403.5799) In: Stacks and Categories in Geometry, Topology, and Algebra. Contemporary Mathematics __643__, 37--110, Amer. Math. Soc. (2015) [doi](https://doi.org/10.1090/conm/643/12896)

They also play a major role in

* S. Balchin, _Homotopy of planar Lie group equivariant presheaves_, J. Homotopy Relat. Struct. 13, 555--580 (2018) [doi](https://doi.org/10.1007/s40062-017-0193-z)

[[!redirects skew-simplicial set]]
[[!redirects skew-simplicial sets]]
[[!redirects crossed simplicial set]]
[[!redirects crossed simplicial sets]]
[[!redirects skew-simplicial group]]
[[!redirects skew-simplicial groups]]
[[!redirects crossed simplicial group]]
[[!redirects crossed simplicial groups]]


skew-symmetric matrix

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## Definition

A square [[matrix]] $A$ is called _skew-symmetric_ if it is [[equality|equal]] to minus own [[transpose matrix]]: $A = - A^T$.

## Related concepts

* [[symmetric matrix]]

* [[hermitean matrix]]

* [[triangular matrix]]

## References

See also

* Wikipedia, _[Skew-symmetric matrix](https://en.wikipedia.org/wiki/Skew-symmetric_matrix)_

[[!redirects skew-symmetric matrices]]


skewfield

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## Definition

A __skewfield__ (also spelled skew-field), or __reciprocal ring__ is a [[unitality|unital]] [[ring]] where each non-[[zero]] [[element]] has a two-sided [[inverse]] (but [[zero]] does not), or equivalently, an [[associative unital algebra|associative unital]] $\mathbb{Z}$-[[reciprocal algebra]] where the multiplicative identity element $1$ is not zero.

A __division ring__ is a [[unitality|unital]] [[ring]] where each non-[[zero]] [[element]] has a left inverse and a right inverse, or equivalently, an [[associative unital algebra|associative unital]] $\mathbb{Z}$-[[division algebra]] where the multiplicative identity element $1$ is not zero. However in all associative unital division algebras, left and right inverses are the same, for the same reason that associative unital [[quasigroups]] and associative unital [[invertible magmas]] are both [[groups]]. As a result, division rings are the same as skewfields.

A [[commutative ring|commutative]] skewfield is called a _[[field]]_, but sometimes in specialized works on skewfields one often says simply _field_ for _skewfield_.

In [[constructive mathematics]] and [[internalization|internally]], the same issues appear for skewfields as for [[fields]], and may be dealt with in the same way.

[[linear algebra|Linear algebra]] is often understood in the generality of division rings, namely the usual notions of _[[linear basis]]_, _[[dimension]]_, _[[linear map]]_, [[matrix]] of a linear map with respect to two bases and so on, and even [[Gauss elimination]] procedure, hold without changes for left or right [[vector spaces]] over a division rings.

## Examples

The most famous noncommutative example is the skewfield of [[quaternions]].

The [[Frobenius theorem]] states that apart from the fields of [[real number|real]] and [[complex numbers]] and quaternions, there are no associative finite-dimensional [[division algebras]] over the real numbers; and even if one includes nonassociative finite-dimensional division algebras one obtains only one more example (the [[octonions]]). See at _[[normed division algebra]]_ for more on this.

## Related concepts

* [[division algebra]]

* [[reciprocal algebra]]

* [[Dieudonné determinant]]

* [[division rig]]

## References

* [[Peter Draxl]], _Skew fields_, London Math. Soc. Lecture notes __81__, Cambridge University Press 1983 ([doi:10.1017/CBO9780511661907](https://doi.org/10.1017/CBO9780511661907), ISBN:9780511661907)

[[!redirects division ring]]
[[!redirects division rings]]
[[!redirects reciprocal ring]]
[[!redirects reciprocal rings]]
[[!redirects skew field]]
[[!redirects skew fields]]
[[!redirects skew-field]]
[[!redirects skew-fields]]
[[!redirects skewfield]]
[[!redirects skewfields]]


Skip Garibaldi

* [webpage](http://www.mathcs.emory.edu/~skip/home/Skip_Garibaldi.html)

category: people


Skolem's paradox

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* table of contents
{: toc}

## Idea

According to the [[Löwenheim-Skolem theorem]], for a [[first-order logic|first-order]] [[theory]] with a [[countable set|countable]] alphabet if there is an [[infinite set|infinite]] [[model theory|model]], then there is a countable model. Let us consider the language of some form of [[set theory]] and a model satisfying the [[axiom of infinity]]. Then Cantor's [[diagonal argument]] can be carried out internally within the model and provides internally _uncountable_ "sets" in the _countable_ model.

The resolution of this apparent [[paradox]] is that, while this conclusion is true _internally_, it is not true _externally_: namely any two infinite sets are countable externally in that model, hence there is a $1$--$1$ function between any two of them including for a model of some uncountable set $X$ and of its power set $P(X)$. However, that function (or its graph) is not in the model! One can enlarge the model by adding the function (and more). But this extended model will necessary have $P(X)$ uncountable externally and there is no $1$--$1$ function from $X$ to $P(X)$ externally any more.

## Remark

The paradox highlights a general feature of first-order logic namely that it cannot discern between countable and uncountable models. In fact, Skolem cast set theory as a _first-order theory_ precisely to display the deficiency of axiomatizations of the uncountable, a move that was not accepted by [[Ernst Zermelo|Zermelo]].

It is somewhat ironic that Skolem effectively helped consolidating what is today called [[Zermelo-Fraenkel set theory]] and inaugurating first-order model theory in what was basically an attempt to discredit set-theoretic foundations and the  axiomatic method in general.

## Related entries

* [[Lindström's theorem]]

* [[second-order logic]]

## References

* [[Thoralf Skolem]], *[[Some remarks on axiomatized set theory|Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre]]*, Mathematikerkongressen i Helsingfor 4-7 Juli 1922. English transl. pp.290-301 of van Heijenoort (ed.), _From Frege to Gödel_ , Harvard UP 1967.

* D. van Dalen, H.-D. Ebbinghaus, _Zermelo and the Skolem paradox_ , Bull. Symbolic Logic **6**  (2000) pp.145–161.

* C. McCarty, N. Tennant, _Skolem's Paradox and Constructivism_ , J. Phil. Logic **16** (1987) pp.165-202. ([draft](https://u.osu.edu/tennant.9/files/2014/09/mccarty_tennant_jpl1987-1ncyai0.pdf))

[[!redirects Skolem paradox]]
[[!redirects Skolem Paradox]]
[[!redirects Skolem's paradox]]
[[!redirects Skolem's Paradox]]
[[!redirects Skolem's paradox]]
[[!redirects Skolem's Paradox]]
[[!redirects Skolem\'s paradox]]
[[!redirects Skolem\'s Paradox]]


Skyrme hadrodynamics with heavy mesons -- references

### Skyrme hadrodynamics with heavy quarks/mesons
 {#SkyrmeHadrodynamicsWithHeavyMesonsReferences}

Inclusion of [[heavy flavor|heavy]] [[flavor (particle physics)|flavors]] into the [[Skyrme model]] for [[quantum hadrodynamics]]:

#### Inclusion of strange quarks/kaons

Inclusion of [[strange quarks]]/[[kaons]] into the [[Skyrme model]]:

* [[Curtis Callan]], [[Igor Klebanov]], _Bound-state approach to strangeness in the Skyrme model_, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5)

* [[Curtis Callan]], K. Hornbostel, [[Igor Klebanov]], _Baryon masses in the bound state approach to strangeness in the skyrme model_, Physics Letters B Volume 202, Issue 2, 3 March 1988, Pages 269-275 (doi10.1016/0370-2693(88)90022-6)

* [[Norberto Scoccola]], D. P. Min, H. Nadeau, [[Mannque Rho]], _The strangeness problem: An $SU(3)$ skyrmion with vector mesons_, Nuclear Physics A Volume 505, Issues 3–4, 25 December 1989, Pages 497-524 (doi:10.1016/0375-9474(89)90029-8)

Review:

* [[Igor Klebanov]], section 6 of: _Strangeness in the Skyrme model_, in: D. Vauthrin, F. Lenz, J. W. Negele,  _Hadrons and Hadronic Matter_, Plenum Press 1989 ([doi:10.1007/978-1-4684-1336-6](https://link.springer.com/book/10.1007/978-1-4684-1336-6))

* [[Mannque Rho]], Section 2.2 of: _Cheshire Cat Hadrons_, Phys. Rept. 240 (1994) 1-142 ([arXiv:hep-ph/9310300](https://arxiv.org/abs/hep-ph/9310300), doi:10.1016/0370-1573(94)90002-7)

#### Inclusion of charm quarks/D-mesons

Inclusion of [[charm quarks]]/[[D-mesons]] into the [[Skyrme model]]:

* [[Mannque Rho]], D. O. Riska, [[Norberto Scoccola]], _Charmed baryons as soliton - D meson bound states_, Phys. Lett.B 251 (1990) 597-602 ([spire:297771](https://inspirehep.net/literature/297771), doi:10.1016/0370-2693(90)90802-D)

* Yongseok Oh, Dong-Pil Min, [[Mannque Rho]], [[Norberto Scoccola]], _Massive-quark baryons as skyrmions: Magnetic moments_, Nuclear Physics A
Volume 534, Issues 3–4 (1991) Pages 493-512 (doi:10.1016/0375-9474(91)90458-I)

#### Inclusion of bottom quarks/B-mesons

Inclusion of further [[heavy flavor|heavy]] [[flavor (particle physics)|flavors]] beyond [[strange quark]]/[[kaons]], namely [[charm quarks]]/[[D-mesons]] and [[bottom quarks]]/[[B-mesons]], into the [[Skyrme model]]:

* [[Mannque Rho]], D. O. Riska, [[Norberto Scoccola]], _The energy levels of the heavy flavour baryons in the topological soliton model_, Zeitschrift für Physik A Hadrons and Nuclei volume 341, pages 343–352 (1992) ([doi:10.1007/BF01283544](https://doi.org/10.1007/BF01283544))

* Arshad Momen, Joseph Schechter, Anand Subbaraman, _Heavy Quark Solitons: Strangeness and Symmetry Breaking_, Phys. Rev. D49:5970-5978, 1994 ([arXiv:hep-ph/9401209](https://arxiv.org/abs/hep-ph/9401209))

* Yongseok Oh, Byung-Yoon Park, Dong-Pil Min, _Heavy Baryons as Skyrmion with $1/m_Q$ Corrections_, Phys. Rev. D49 (1994) 4649-4658 ([arXiv:hep-ph/9402205](https://arxiv.org/abs/hep-ph/9402205))

Review:

* [[Mannque Rho]], _Massive-quark baryons as Skyrmions_,  Modern Physics Letters A, Vol. 06, No. 23 (1991) ([doi:10.1142/S0217732391002268](https://doi.org/10.1142/S0217732391002268))

* [[Norberto Scoccola]], _Heavy quark skyrmions_, ([arXiv:0905.2722](https://arxiv.org/abs/0905.2722), [doi:10.1142/9789814280709_0004](https://doi.org/10.1142/9789814280709_0004)), Chapter 4 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))