Grothendieck topos

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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# Contents
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## Definition

Classically, we have:

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A __Grothendieck topos__ $\mathcal{T}$ is a category that admits a [[geometric embedding]]

$$
  \mathcal{T} \stackrel{\stackrel{lex}{\leftarrow}}{\hookrightarrow}
  PSh(C)
$$

in a [[presheaf]] category, i.e., a full and faithful functor that has a left exact left adjoint.

This is equivalently the [[category of sheaves]] ([[Set]]-valued [[presheaves]] satisfying the sheaf condition) over a [[small category|small]] [[site]].

=--

See at *[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]* for discussion of this characterization.

Since smallness can be relative, we also have:

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For a given fixed [[ETCS|category of sets]] $S$, a __Grothendieck topos__ over $S$ is a [[category of sheaves]] ($S$-valued [[presheaves]] satisfying the sheaf condition) over a [[site]] which is [[small category|small]] relative to $S$, that is a site [[internal category|internal]] to $S$.
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Note that a Grothendieck topos is a [[topos]] because (or if) $S$ is.

The [[site]] is not considered part of the structure; different sites may give rise to [[equivalence of categories|equivalent]] category of sheaves.

By the general theory of [[geometric morphisms]], every Grothendieck topos sits inside a category of [[presheaf|presheaves]] by a [[geometric embedding]] $Sh(S) \hookrightarrow PSh(S)$.

* This may be taken as an alternative definition of [[sheaf]]: since [[Lawvere-Tierney topology|Lawvere-Tierney topologies]] are bijectively given by [[geometric embeddings]],  instead of explicitly defining a sheaf as a presheaf satisfying [[descent]], one may define categories of sheaves as geometric embeddings into presheaf categories.

* For details on the relation between the two perspectives see [[geometric embedding]].

* This perspective is useful for defining the [[vertical categorification]] of sheaves: [[stacks]] and [[∞-stacks]]: the higher categories of these may be defined as geometric embeddings into higher categories of presheaves. This has been worked out in detail for [[(∞,1)-categories]]. See [[(∞,1)-category of (∞,1)-sheaves]].

* Sometimes it is useful to distinguish between [[petit topos]] and [[gros topos]].

## Properties

### General

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

Every Grothendieck topos is a [[locally presentable category]].

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([Borceux 94, vol 3, prop. 3.4.16](#Borceux94))

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###### Proposition

Every Grothendieck topos is a [[total category]] and a [[cototal category]].

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###### Proof

From the page [[total category]], totality follows from the fact that a Grothendieck topos is

* [[cocomplete category|Cocomplete]],
* [[well-powered category|Well-copowered]] (because [[quotient]] objects of an [[object]] $X$ are in [[bijection]] with [[equivalence relations]] on $X$, and there is a [[small set]] of these because a Grothendieck topos is well-powered), and
* Has a [[generator]] (e.g., the [[coproduct]] of all [[representable functor|representables]] for a [[small site]] presentation).

Dually, a Grothendieck topos is

* [[complete category|Complete]],
* [[well-powered category|Well-powered]] (because [[subobjects]] of an [[object]] $X$ are in [[bijection]] with elements of a [[small set|small]] [[hom-set]] $\hom(X, \Omega)$), and
* Has a [[cogenerator]] (e.g., the [[product]] of [[power objects]] $\Omega^c$ where $c$ ranges over representables for a small site presentation).

Therefore a Grothendieck topos is also cototal.

=--

### Giraud\'s axiomatic characterization
{#Giraud}

Giraud characterized Grothendieck toposes as categories satisfying certain [[exactness property|exactness]] and [[small set|small]] [[complete category|completeness]] properties (where "small" is again relative to the given category of sets $S$). The exactness properties are elementary (not depending on $S$), and are satisfied in any elementary [[topos]], or even a [[pretopos]].

[[Giraud's theorem]] characterises a Grothendieck topos as follows:

1. a [[locally small]] [[category]] with a small [[generating set]],
2. with all finite [[limits]],
3. with all small [[coproducts]], which are [[disjoint coproduct|disjoint]], and [[pullback stability|pullback-stable]],
4. where all [[congruences]] have effective [[quotient objects]], which are also pullback-stable.

These conditions are equivalent to

* a [[locally small]] [[infinitary pretopos]] with a small generating set.

See the [[Elephant]], theorem C.2.2.8.  (There, the assumption of local smallness is not stated explicitly, but it is included in the definition of $\infty$-pretopos by way of [[well-powered category|well-poweredness]]; on the nLab it is not so included, so we have to state it explicitly.  To see that it is necessary, note that if $U$ and $V$ are [[Grothendieck universes]] with $U\in V$, then $Set_V$ satisfies all the other conditions relative to $Set_U$, but is not locally small and is not a Grothendieck topos.)  See also [Wikipedia](https://secure.wikimedia.org/wikipedia/en/wiki/Topos#Giraud.27s_axioms).

Sometimes (3,4) are combined and strengthened to the statement that the category has all small [[colimits]], which are effective and pullback-stable.  However, this is a mistake for two reasons: it is a significantly stronger axiomatisation (since without the small generating set, not every infinitary pretopos has this property), and it is not valid in weak foundations (while the definition given above is).

### Street's axiomatic characterization
{#Streetcharacterization}

Augmenting the aforementioned Proposition \ref{total} that Grothendieck toposes are total categories, [Street](#Street) more or less characterizes Grothendieck toposes as *lex total* categories having the same "size" as $Set$.

In more detail: we take as background set theory ZFC + "there is a strongly inaccessible cardinal" $\kappa$; equivalently, the existence of a single [[Grothendieck universe]] $U$ (a set of "small sets"). Supposing given a model $V$ of $ZFC+universe$, "category" shall then refer to category theory interpreted in $V$. Let $Set$ be the category of small sets; note that the set of morphisms of $Set$ has size $\kappa$.

Recall that locally small category $E$ is [[lex total category|lex total]] if the [[Yoneda embedding]] $y: E \to [E^{op}, Set]$ has a [[left exact functor|left exact]] [[left adjoint]].

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###### Theorem
**(Street)**
A category $E$ is a Grothendieck topos iff it is lex total and the size of the set of isomorphism classes of objects is $\kappa$ or less.
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###### Remark
This result is in the spirit of saying "every Grothendieck topos is the category of [[sheaves]] with respect to the [[canonical topology]] on itself". Putting aside set-theoretic issues, it suggests that Grothendieck toposes be seen as analogous to [[frames]], which may be defined as lex total objects in $\mathbf{2}$-$Cat$. In this setting, the appropriate morphisms are left exact left adjoints, so that Grothendieck toposes and [[geometric morphisms]] between them would be analogous to [[locales]] and [[continuous maps]] between them.

One can deduce formally that lex total categories are [[locally cartesian closed category|locally cartesian closed]] [[Heyting pretoposes]].
=--

### As lex reflections of categories of presheaves

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

### As localic groupoids
 {#AsLocalicGroupoids}

Every Grothendieck topos is equivalent to the [[classifying topos of a localic groupoid]], every Grothendieck topos [[point of a topos|with enough points]] is equivalent to the classifying topos of a [[topological groupoid]], and in fact the 2-category [[Topos]] of Grothendieck toposes is equivalent to a localization of that of [[localic groupoids]].

See at _[[classifying topos of a localic groupoid]]_ for more.

### Internal logic

Being an [[elementary topos]], a Grothendieck topos has an [[internal logic]] that can be taken to be [[higher-order logic]] or a form of [[dependent type theory]].  It is also useful to take its internal [[geometric logic]], in particular because that is preserved by [[geometric morphisms]] and because every geometric theory has a Grothendieck [[classifying topos]].

Although the internal logic of a Grothendieck topos is [[constructive logic]], there are a number of principles that are true in every Grothendieck topos (at least, assuming the base topos [[Set]] is [[classical mathematics|classical]]) but are not constructively provable (and in particular can fail in other toposes).  Roughly speaking, many of these axioms assert that classicality fails "in only a [[small category|small]] way".  These include:

* [[WISC]]: every set has a weakly initial set of covers.
* The (constructive) [[axiom of multiple choice]] (AMC).
* The [[axiom of stack completions]].
* (allegedly) the local [[small cardinality selection axiom]] (SCSA).
* Every [[complete small category]] is a preorder.

### 2-monadicity over locally presentable categories

The [[2-category]] of Grothendieck toposes and [[logical morphisms]] is [[2-monad|2-monadic]] over the 2-category of [[locally presentable categories]] and [[cocontinuous functors]]. This is proven in [Bunge and Carboni 1995](#BungeCarboni1995). See [[symmetric topos]] for more details.

### Enrichment over accessible categories with directed colimits

The [[2-category]] of Grothendieck toposes and [[geometric morphisms]] is [[enriched bicategory|enriched as a bicategory]] over the 2-category of [[accessible categories]] with [[directed colimits]] (note that accessible categories always have $\kappa$-filtered colimits for some cardinal $\kappa$, but not necessarily directed colimits unless they are $\omega$-accessible) and functors preserving [[directed colimits]]. This is proven in [Di Liberti 2022](#DiLiberti22).

## In weak foundations

We have two definitions of a Grothendieck topos:

*  the category of sheaves on some small site,
*  a category that satisfies Giraud\'s axioms (as listed above).

The theorem that these are equivalent can be proved in quite weak foundations, whether [[finitist mathematics|finitist]], [[predicative mathematics|predicative]], or [[constructive mathematics|constructive]] (or all three at once), as long as we axiomatize correctly given the caveats listed in the previous section.  Some hard-nosed predicativists (and even hard-nosed ZFC fundamentalists) may object to the language (on the ground that large categories such as $Set$ and other nontrivial Grothendieck toposes don\'t really exist), but they should accept the theorems when suitably phrased.

In predicative mathematics, however, we cannot prove that every Grothendieck topos is in fact a [[topos]]!  In fact, it is immediate that [[the category of sets]] is a Grothendieck topos, but $Set$ is an elementary topos if and only if [[power sets]] are small, which is precisely what predicativists doubt.  One can use the term __Grothendieck pretopos__ to avoid implying that we have an elementary topos.  On the other hand, since Grothendeick toposes came first, perhaps it is the definition of 'elementary topos' that is too strong.

Similarly, in finitist mathematics, we cannot prove that every Grothendieck topos has a [[natural numbers object]]; while in strongly predicative mathematics, we cannot prove that every Grothendieck topos is [[cartesian closed category|cartesian closed]].  In each case, once a property is accepted of $Set$ (the [[axiom of infinity]] and small [[function sets]], in these examples), it can be proved for all Grothendieck toposes.

Constructivism as such is irrelevant; even in [[classical mathematics]], most Grothendieck toposes are not [[boolean topos|boolean]].  However, for an analogous result, try the theorem that the [[category of presheaves]] on a [[groupoid]] (and hence any category of sheaves contained within it) is boolean.  (Again, $Set$ itself is an example of this.)

The theorem that every Grothendieck topos is [[cocomplete category|cocomplete]] is a subtle point; it fails only in finitist predicative mathematics.  (The key point in the proof is to generate the [[transitive relation|transitive closure]] $\sim^*$ of a binary relation $\sim$.  One proof defines $a \sim^* b$ to mean that $a \sim x_0 \sim \cdots \sim x_{n-1} \sim b$ for some $n$, which is predicative but infinitary; another defines $a \sim^* b$ to mean that $a \sim' b$ for every transitive relation $\sim'$ that contains $\sim$, which is finitary but impredicative.)

## Examples and non-examples

By definition, every [[category of sheaves]] is a Grothendieck topos.  Examples include:

* Every [[presheaf]] category $PSh(C) = Set^{C^{op}}$ is a Grothendieck topos.

* In particular [[Set]] is a Grothendieck topos, because it is equivalent to $PSh(\ast)$.

* Broadly, most of the [[toposes]] that people have worked with are Grothendieck.

On the other hand, there are [[topos#ElementaryTopos|(elementary) toposes]] that fail to be Grothendieck toposes:

* Because a Grothendieck topos [[complete category|has all small limits]], $FinSet$ is not Grothendieck.

* More generally, the category $Set_{\lt \pi}$ of sets with [[cardinality]] below any bound $\pi$ is not a Grothendieck topos, even when $\pi$ is a [[strong limit cardinal]] so that $Set_{\lt \pi}$ is an (elementary) topos.

* Similarly, any category $FinSet^{C^{op}}$ of $FinSet$-valued presheaves is a topos but not a Grothendieck topos, and the same goes more generally for $Set_{\lt \pi}^{C^{op}}$ for $\pi$ a [[strong limit cardinal]].

* The [[effective topos]], the maps in which can be thought of as computable functions, is not Grothendieck.

* Other non-Grothendieck toposes occur in the topos-theoretic approach to [[nonstandard analysis]].

## Related concepts

* [[topos]]

  * **Grothendieck topos**,  [[category of sheaves]]

    * [[geometric logic]], [[geometric type theory]]

  * [[2-topos]]

  * [[(∞,1)-sheaf (∞,1)-topos]]

* [[lex total category]]

[[!include locally presentable categories - table]]

## History
  {#History}

From [[Colin McLarty]] (sent to the [Categories mailing list](https://www.mta.ca/~cat-dist/) on Apr 14, 2018):

> At the start of 1958 [[Grothendieck]] believed the correct [[Weil cohomology]] of a [[scheme]] $S$ would be the [[abelian sheaf cohomology|derived functor cohomology]] of some [[category]] $Ab(S)$
of [[sheaf of abelian groups|sheaves of Abelian groups]] on $S$ ---and $Ab(S)$ was likely to be the category of all Abelian [[group objects]] in some [[category of sheaves]] of [[sets]] on $S$.  He had no name yet for such conjectural categories of sheaves of sets but for later reference I will call this category $T(S)$.  It would require some new notion of sheaves of sets more general than the existing notion using topological spaces.  He had no concrete idea what this new notion of sheaf might be.  His experience in Kansas in 1955 suggested it  was likely to come from some new notion of espace etale over a scheme. and it should have some good exactness properties.

> On April 21, 1958 Grothendieck went to the Seminaire Chevalley to hear [[Serre]] describe isotrivial fiber bundles on a variety V, which are bundles that become trivial when pulled back to some surjective family of finite, unramified maps to V.  Serre showed this gave the expected one-dimensional Weil cohomology groups of V.  Grothendieck immediately told Serre it would work in all dimensions--which Serre found "very optimistic"" at the time.

> All of this is well documented in familiar sources.

> Grothendieck's 1973 topos lectures in Buffalo show that during Serre's talk Grothendieck saw the Weil espaces etales over a scheme $S$ should be patched together from Serre's unramified maps to $S$.  That is, the sought category $T(S)$ of sheaves of sets on S should be generated by the unramified maps to S, and should have arbitrary colimits of these, and finite limits.  He first sought to axiomatize this roughly the way his Tohoku paper had axiomatized sheaves of groups.

> A few changes came soon: He realized finite unramified maps should be replaced by flat unramified maps (just as topological espaces etales are trivial locally in the fiber and not on the base).  He abandoned the axiomatic approach to $T(S)$ as too vague and shifted to construction by sites.  And for sites he began treating sheaves as a kind of functor, rather than as espaces etales.  Only after the notion ot site developed would Giraud give his topos axioms, and no one has yet really taken a precise notion of espace etale much  beyond the topological case.

> But April 21 1958 was the birth of [[topos theory]]. The term topos came
later.  (Lots of people are not named on the day they are born.)

> As of the summer of 1973 Grothendieck's stated preferred definition of topos was still: a category with arbitrary [[colimits]], [[finite limits]], and a small generating set. He says over and over this is not quite adequate for proofs.  He says proofs require the notion of [[site]], or else the [[Giraud axioms]], but he calls the vaguer idea more intuitive and says that is the way to think about a topos.

> Also in 1973 Grothendieck says the objects in any topos should be seen as [[espaces etales]] over the [[terminal object]] of the topos, in a generalized sense that includes saying any orbit of a group action lies "etale" over a [[fixed point]].  Today, it is not obvious that this can work well in general.  In [[SGA4]] Grothendieck had already given evidence that it would work for [[petit topos]] but not [[gros topos]].  But still in 1973 he did say it. And it works perfectly for petit etale toposes---as long as we generalize the notion  of espace etale to include a Galois orbit (in an extension field) lying over a single point (in the ground field).

## References

Basic exposition:

* [[Michael Barr]], [[Colin McLarty]], [[Charles Wells]],  *Variable set theory*, prepared for *Scientific American* but unpublished (~1985) [[pdf](https://www.math.mcgill.ca/barr/papers/vst.pdf), [[BarrMcLartyWells-VariableSetTheory.pdf:file]]]

A quick introduction of the basic facts of sheaf-topos theory is chapter I, "Background in topos theory" in

* [[Ieke Moerdijk]], _Classifying Spaces and Classifying Topoi_ Lecture Notes in Mathematics 1616, Springer (1995)

Textbook accounts include

* {#Borceux94} [[Francis Borceux]], vol 3 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

Grothendieck topoi appear around section III,4 there.
A proof of Giraud's theorem is in appendix A.

The proof of Giraud's theorem for [[(∞,1)-topoi]] is section 6.1.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

Street's characterization of Grothendieck toposes is given in

* {#Street}  [[Ross Street]], _Notions of topos_, Bull. Australian Math. Soc. 23 (1981), 199-208; MR83a:18014. ([pdf link](http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ23_02%2FS000497270000705Xa.pdf&code=c6248e4fa6a517c9bb451e4f5abcf862))

These are also called Giraud frames in

* {#Vickers22} [[Steven Vickers]], *Generalized point-free spaces, pointwise* [[arXiv:2206.01113](https://arxiv.org/abs/2206.01113)]

On 2-monadicity of the opposite of the 2-category of Grothendieck toposes:

* {BungeCarboni1995} [[Marta Bunge]], [[Aurelio Carboni]], _The Symmetric Topos_  ,  JPAA **105** (1995) pp.233-249.

On enrichment of the 2-category of Grothendieck toposes:

* {#DiLiberti2022} [[Ivan Di Liberti]], _General facts on the Scott Adjunction_, Applied Categorical Structures 30.3 (2022): 569-591.

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Grothendieck universe

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### Context
#### Foundations
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[[!include foundations - contents]]
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#### Universes
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[[!include universe - contents]]
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# Contents
* table of contents
{:toc}

## Idea

Problems of set theory arise by the unjustified recursion of the naive notion of a 'collection of things.' If 'Col' is one notion of collections (such as 'Set' or 'Class'), then the notion 'Col of all Cols' is in general problematic, as it is subject to the construction of Russell-style paradoxes (although it is not the only source of such paradoxes).

One way out is to consider a hierarchy of notions of collections: Postulate that the collection of all 'Col's is not a 'Col' itself but instead is another notion of collection, 'Col+'. We may thus speak of the 'Col+ of all Cols.' Similarly, the collection of all 'Col+'-type collections may be taken to be 'Col++,' and so on.

One formalization of this idea is that of a _Grothendieck universe_. This is defined to be a set $ U $ that behaves like a 'set+ of all sets' in that all the standard operations of set theory (union, power set, etc.) can be performed on its elements.

Although developed for application to [[category theory]], the definition is usually given in a form that only makes sense in a membership-based [[set theory]]. On this page, we consider only that version; for a form that makes sense in structural set theory, please see [[universe in a topos]].

## Definition

A **Grothendieck universe** is a [[pure set]] $ U $ such that:

1. for all $ u \in U $ and $t\in u$, we have $ t \in U $ (i.e., $ U $ is _transitive_);
1. for all $ u \in U $, we have $ \mathcal{P}(u) \in U $;
1. $ \varnothing \in U $;
1. for all $ I \in U $ and functions $ u: I \to U $, we have $ \displaystyle \bigcup_{i \in I} u_{i} \in U $.

Some authors leave out (3), which allows the empty set $ \varnothing $ itself to be a Grothendieck universe.  Other authors use the set $ \mathbb{N} $ of [[natural numbers]] in place of $ \empty $ in (3), which prevents the countable set $ V_{\omega} $ of [[hereditarily finite sets]] from being a Grothendieck universe.

## Consequences

From the definition above, one can prove additional closure properties of a universe $ U $, including the usual codings in pure set theory of [[function sets]] and [[cartesian products]] and [[disjoint unions]] of sets, using the following lemmata:

+-- {: .num_lemma}
###### Lemma
If $ t $ is a [[subset]] of $ u $ and $ u \in U $, then $ t \in U $.
=--
+-- {: .proof}
###### Proof
By (2), $ \mathcal{P}(u) \in U $. Then as $ t \in \mathcal{P}(u) $, we have $ t \in U $ by (1).
=--

+-- {: .num_lemma}
###### Lemma
If $ u, v \in U $, then $ u \cup v \in U $.
=--
+-- {: .proof}
###### Proof
As $ \empty \in U $ by (3), so are $ \star \stackrel{\text{df}}{=} \mathcal{P}(\empty) $ and $ TV \stackrel{\text{df}}{=} \mathcal{P}(\star) $ by (2). Even in [[constructive mathematics]], $ 2 = \{ \bot,\top \} $ is a subset of $ TV $, so $ 2 \in U $ by Lemma 1. Then $ (\bot \mapsto u,\top \mapsto v) $ is a function from $ 2 \to U $, so the union $ u \cup v $ in $ U $ by (4).
=--

Then using their usual encodings in set theory:

* the nullary cartesian product $ \star $ is $ \mathcal{P}(\varnothing) $ as in the previous proof;
* the binary cartesian product $ u \times v $ is a subset of $ \mathcal{P}(\mathcal{P}(u \cup v)) $;
* the general cartesian product $ \displaystyle \prod_{i \in I} u_{i} $ is a subset of $ \displaystyle \mathcal{P} \left( I \times \bigcup_{i \in I} u_{i} \right) $;
* the nullary disjoint union is $ \varnothing $;
* the binary disjoint union $ u \uplus v $ is a subset of $ 2 \times (u \cup v) $;
* the general disjoint union $ \displaystyle \biguplus_{i \in I} u_{i} $ is a subset of $ \displaystyle I \times \bigcup_{i \in I} u_{i} $;
* the set of functions $ u \to v $ is a subset of $ \mathcal{P}(u \times v) $.

## Terminology: Small/Large

Given a universe $ U $, an element of $ U $ is called a **$ U $-small set**, while a subset of $ U $ is called **$ U $-moderate**. Every $ U $-small set is $ U $-moderate by requirement (1) of the definition. If the universe $ U $ is understood, we may simply say **small** and **moderate**.

The term _$ U $-large_ is ambiguous; it sometimes means 'not small' but sometimes means the same as 'moderate' (or 'moderate but not small'). The reason is that language that distinguishes 'small' from 'large' in terms of sets and [[proper class|proper classes]] translates fairly directly into terms of $ U $-small and $ U $-moderate sets. To be precise, if we redefine 'set' to mean '$ U $-small set,' then every proper class in this new world of sets will be represented by a $ U $-moderate set (a subset of $ U $). Those sets that are not even $ U $-moderate are 'too large' to be translated into language of proper classes.

(Note, though, that not all $ U $-moderate sets represent proper classes in the language of set theory relative to the world of $ U $-small sets, only those that are _first-order definable_ from $ U $-small sets. In fact, if $ \kappa $ is the [[cardinal number|cardinality]] of the universe $ U $, then there are only $ \kappa $ proper classes relative to $ U $, but there are $ 2^{\kappa} $-many $ U $-moderate sets.)

As defined above, these concepts violate the [[principle of equivalence]], as two sets may be [[isomorphism|isomorphic]] yet have different properties with respect to $ U $. However, a set which is isomorphic to a $ U $-small or $ U $-moderate set is called **essentially** $ U $-small or $ U $-moderate; these respect the [[principle of equivalence]].

## Axiom of Universes

If $ U $ is a Grothendieck universe, then it is easy to show that $ U $ is itself a model of [[ZFC]] (minus the axiom of infinity unless you modify (3) to rule out countable universes). Therefore, one cannot prove in ZFC the existence of a Grothendieck universe containing $ \mathbb{N} $, and so we need extra set-theoretic axioms to ensure that uncountable universes exist. Grothendieck's original proposal was to add the following **axiom of universes** to the usual axioms of set theory:

* For every set $ s $, there exists a universe $ U $ that contains $ s $, i.e., $ s \in U $.

In this way, whenever any operation leads one outside of a given Grothendieck universe (see applications below), there is guaranteed to be a bigger Grothendieck universe in which one lands. In other words, every set is small if your universe is large enough!

Later, Mac Lane pointed out that often, it suffices to assume the existence of *one* uncountable universe. In particular, any discussion of 'small' and 'large' that can be stated in terms of sets and proper classes can also be stated in terms of a single universe $ U $ (with 'large' meaning '$ U $-moderate but not $ U $-small').

## Large cardinals

If $ U $ is a Grothendieck universe, then one can prove in ZFC that it must be of the form $ V_{\kappa} $, where $ \kappa $ is a (strongly) [[inaccessible cardinal]] ([Williams](#Williams)). Here, $ V_{\kappa} $ is the $ \kappa $-th set in the [[von Neumann hierarchy]] of pure sets. Conversely, every such $ V_{\kappa} $ is a Grothendieck universe. Thus, the existence of Grothendieck universes is equivalent to the existence of inaccessible cardinals, and so the axiom of universes is equivalent to the 'large cardinal axiom' that 'there exist arbitrarily large inaccessible cardinals.'

It is worth noting, for those with foundational worries, that the axiom of universes is much, much weaker than many large cardinal axioms which are routinely used, and believed to be consistent, by modern set theorists. Of course, one cannot _prove_ the consistency of any large cardinal axiom (if it really is consistent) except by invoking a stronger one.

## Structural Version

An equivalent concept (at least for the purposes of category theory) can also be defined in structural set theories (like [[ETCS]]). Please see [[universe in a topos]].

## Examples

The set $ V_{\omega} $ of [[hereditarily finite sets]] (finite sets of finite sets of...) is a Grothendieck universe, unless you phrase axiom (3) in the definition to specifically rule it out. In this way, the axiom of infinity can be seen as a simple universe axiom (stating that at least one universe exists), and Mac Lane's axiom that an uncountable universe exists is merely one step further.

If you refrain from using the axiom of universes (except perhaps once, to get $ \mathbb{N} $ as above), then the set of all sets (or cardinal numbers) that you can actually construct is a Grothendieck universe. Of course, you cannot possibly have proved that this universe exists, but the intuition that you ought be able to form the collection of 'everything that we've used so far' is the justification for the axiom of universes.

Similarly, if you use the axiom of universes at most $ n $ times, then the set of all sets that you can construct with this restriction is a Grothendieck universe. Thus, we can find a sequence $ U_{1} \in U_{2} \in U_{3} \in \ldots $ of universes. The [[axiom of replacement]] then allows us to form the union (a directed [[colimit]]) $ \bigcup_{n \lt \omega} U_{n} $. This will not be a universe (it violates (4), by definition), but we can use the axiom of universes again to show that it is in some universe $ U_{\omega} $. Proceeding in this way, we can construct a tower of universes indexed by the [[ordinal numbers]].

The set of all sets that can be constructed using the axioms of ZFC together with the axiom of universes is, if it exists, again a universe which contains all the $ U_{\alpha} $ constructed above. Of course, it cannot be shown to exist using only ZFC and the axiom of universes; the axiom of universes is not the final word on large cardinal axioms by any means.

## Applications

Let $ U Set $ be the [[category]] of $ U $-small sets, a [[full subcategory]] of [[Set]]. It is common, especially when $ U $ is understood, to redefine $ Set $ to mean $ U Set $; here we keep the distinction for clarity. However, when $ Set $ means $ U Set $, sometimes $ SET $ is used to mean the category of _all_ sets.

A category whose set of morphisms is (essentially) $ U $-small may be called a $ U $-[[small category]]; it can also be thought of as an [[internal category]] in $ U Set $. A category whose [[hom-sets]] are all (essentially) $ U $-small may be called [[locally small category|locally]] $ U $-small; it can also be thought of as an [[enriched category]] over $ U Set $. Every $ U $-small category is locally $ U $-small.

A category whose set of morphisms is $ U $-moderate may be called a $ U $-[[large category|moderate category]]; again '$ U $-large' may mean 'not $ U $-small,' '$ U $-moderate,' or both. In practice, most $ U $-moderate categories are locally $ U $-small and vice versa, but there is no theorem that this must be true. Note that $ U Set $ itself is $ U $-moderate and locally $ U $-small but not $ U $-small.

All notions of category theory that reference size, such as [[complete category|completeness]] and [[locally presentable category|local presentability]], must then be relativized to $ U $. In order to move from a category defined in one universe to another, we need a procedure of [[universe enlargement]].

### Presheaf Categories

Let $ C $ be a $ U $-small category. Then the category of $ U $-[[presheaf|presheaves]] on $ C $ (the [[functor category]] $ [C^{op},U Set] $) is also $ U $-moderate and locally $ U $-small but not $ U $-small unless $ C $ is empty. ($ U Set $ itself is the special case of this where $ C $ is the [[point]].) These arguments go as follows:

* $ U PSh(C) $ is $ U $-moderate: An upper bound for the size of $ [C^{op},U Set] $, hence of the set $ Obj([C^{op},U Set]) $ is the size of $ \{ F: Obj(C) \times Mor(C) \to U \} $, where both $ Obj(C) $ and $ Mor(C) $ are in $ U Set $. Hence, we are looking at the cardinal number $ |U|^{|u| \times |v|} $, where $ u = Obj(C) $ and $ v = Mor(C) $. Use the fact that any Grothendieck universe must be infinite (since it has $ \varnothing $, $ \mathcal{P}(\varnothing) $, etc.), and the result follows from cardinal arithmetic that $ \kappa^{\lambda} = \kappa $ if $ \lambda \lt \kappa $ and $ \kappa $ is infinite.

* $ U PSh(C) $ is locally $ U $-small: An upper bound for the size of the set of morphisms between two functors $ F,G: C^{op} \to U Set $ is the disjoint union indexed by the objects $ c $ of $ C $ over the $ U $-sets $ G(c)^{F(c)} $. Now $ G(c)^{F(c)} \in U $ as it is a function set and $ \displaystyle \bigcup_{c \in Obj(C)} G(c)^{F(c)} $ by the assumption that unions stay in $ U $.

Now let $ C $ be a $ U $-moderate category (and not small). Then the category of $ U $-presheaves on $ C $ is not even locally $ U $-small, nor is it even $ U $-moderate (it is 'too large'). However, it is locally $ U $-moderate. Also, it is quite possible, if $ C $ is a $ U $-[[large site|moderate site]], that the category of $ U $-[[sheaf|sheaves]] on $ C $ is $ U $-moderate and locally $ U $-small.

+-- {: .standout}
*Note:* Here we are considering presheaves on $ C $ with values in $ U $-small sets. In many cases, a more appropriate notion of '$ U $-small presheaf' is that discussed at [[small presheaf]], namely a presheaf that is a $ U $-small colimit of representables.
=--

## Alternative Approaches

* A different, potentially much more elegant and natural proposal for solving the problem to be solved by Grothendieck universes is that described at [[michaelshulman:category of all sets|category of all sets]]. Don't get your hopes up too high, though; even if it works, it isn't quite the category theory you're used to.

## Related concepts

* [[Tarski-Grothendieck set theory]]

## References
 {#References}

The original account:

* [[Nicolas Bourbaki]], _Univers_, Appendix to Expose I in [[SGA 4]] (1963-64)

Further early discussion:

* [[Saunders MacLane]], *One universe as a foundation for category theory*,  In: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) 192-200 [[doi:10.1007/BFb0059147](https://doi.org/10.1007/BFb0059147)]

Comprehensive historical review with further references:

* [[Ralf Krömer]], §6.4.4 in: *Tool and object: A history and philosophy of category theory*, Science Networks. Historical Studies **32**, Springer (2007) [[doi:10.1007/978-3-7643-7524-9](https://doi.org/10.1007/978-3-7643-7524-9)]

Most texts on [[category theory]] and related topics mention the topic of Grothendieck universes without providing details. Exceptions are:

* {#Isbell66} [[John Isbell]], *Structure of categories*, Bulletin of the American Mathematical Society **72** (1966), 619-655. [[euclid:1183528163](http://projecteuclid.org/euclid.bams/1183528163), [ams:1966-72-04/S0002-9904-1966-11541-0](https://www.ams.org/journals/bull/1966-72-04/S0002-9904-1966-11541-0/)]

* [[Johann Sonner]], *On the formal definition of categories*, Math. Zeitschr. **80** (1962) 163-176 [[eudml:170093](https://eudml.org/doc/170093), [[Sonner-FormalDefCategories.pdf:file]]]

* [[Horst Schubert]], §3.2 in: *Categories*, Springer (1972) [[doi:10.1007/978-3-642-65364-3](https://doi.org/10.1007/978-3-642-65364-3)]

Discussions spelling out more details:

* [[Mike Shulman]], _Set theory for category theory_ [[arXiv:0810.1279](http://arxiv.org/abs/0810.1279)]

* {#Murfet06} [[Daniel Murfet]], _Foundations for category theory_ (2006) [[pdf](http://therisingsea.org/notes/FoundationsForCategoryTheory.pdf), [[Murfet-GrothendieckUniverses.pdf:file]]]

* [[Zhen Lin Low]], _Universes for category theory_ [[arxiv/1304.5227](http://arxiv.org/abs/1304.5227)]

* [[Paul Blain Levy]], _Formulating Categorical Concepts using Classes_ [[arXiv:1801.08528](https://arxiv.org/abs/1801.08528)]

The proof that a Grothendieck universe is equivalently a set of $ \kappa $-small sets for $ \kappa $ an [[inaccessible cardinal]] is in

* {#Williams} N. H. Williams, _On Grothendieck universes_, Compositio Mathematica, tome 21 no 1 (1969) ([numdam]( http://www.numdam.org/item?id=CM_1969__21_1_1_0))

SGA uses universes and much of modern results in algebraic geometry use general results from SGA, including Wiles proof of Fermat's theorem. Colin McLarty discusses how to remove the need for universes in Wiles' proof in

* [[Colin McLarty]], _What does it take to prove Fermat's last theorem? -- Grothendieck and the logic of number theory_, [pdf](http://www.cwru.edu/artsci/phil/Proving_FLT.pdf)

category: foundational axiom

[[!redirects Grothendieck universe]]
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Grothendieck's Galois theory

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

## Idea

In [[topology]], the [[fundamental theorem of covering spaces]] says that for a sufficiently nice [[topological space]], the [[fundamental group]] at a point can be reconstructed as a group of [[deck transformations]] of the [[universal covering space]], which is the same as the [[automorphism]]s of the [[fiber]] over that point of the projection map. The deck transformations are [[monodromies]] induced by loops at the base point.

The functor which assigns to a point the fiber over it, generalizes to [[fiber functors]] in the [[Tannaka duality|Tannakian formalism]] of Grothendieck which defines in more general setups the [[fundamental groupoid]] as the group of automorphisms of the appropriate fiber functor. See also at *[[fundamental group of a topos]]*.

Now, Grothendieck's [[Galois theory]] means to generalize to the context of [[schemes]] this familiar correspondence between

[[covering spaces]] of $X$ : $\pi_1(X)$[[action|-sets]]

for  [[locally path-connected space|locally path connected]], [[semi-locally simply connected space|semilocally simply connected]] [[topological spaces]] $X$.

The objects on the left are not difficult to define for schemes (at least naively -- one really needs trivialisations over étale [[coverage|covers]]), but it may not be entirely immediate what the [[fundamental group]] defined in terms of loops should be.

The reason Galois's name is attached to this theory is that in the case of the scheme $Spec(k)$ associated to a field $k$, the objects corresponding to the covering spaces are simply the schemes $Spec(k')$ for [[field extension]]s $k'$, and the fundamental group of $Spec(k)$ is the [[Galois group]] of $k$.  More generally Grothendieck's Galois theory assigns to any scheme $X$ an [[algebraic fundamental group]] $\pi_1(X)$, which is a [[profinite group]].

### Generalizations

The basic idea of Grothendieck's Galois theory may be extended to objects in an [[topos]] -- leading to a notion of [[fundamental group of a topos]] --  and then further to objects in any [[(∞,1)-topos]]. For more on this see [[homotopy group of an ∞-stack]].

## Details

### Preliminaries

+-- {: .un_defn}
###### Definition
Given an arrow $f:x \to y$ in a category $C$ the _category of arrows compatible with $f$_, denoted $Comp(f)$ is the full [[subcategory]] of the [[undercategory]] $x \downarrow C$ on the arrows that [[coequalizer|coequalize]] the same pairs $g,h:w\rightrightarrows x$ that $f$ does.
=--

+-- {: .un_defn}
###### Definition
An arrow $f:x\to y$ in a category $C$ is a _[[strict epimorphism]]_ if it is [[initial object|initial]] in $Comp(f)$.
=--

It is not obvious, but a strict epimorphism is an epimorphism.

### Grothendieck's axioms

In what follows, Let $C$ be a [[category]] and $F:C \to Set$ a [[functor]]. The axioms presented here are as in

J. P. Murre, _Lectures on an introduction to Grothendieck's theory of the fundamental group_, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp.

and copied also in

* [[Eduardo Dubuc]], C. S. de la Vega _On the Galois theory of Grothendieck_, Bol. Acad. Nac. Cienc. (Cordoba) **65** (2000) 111--136. [arXiv](http://arxiv.org/abs/math.CT/0009145)

Some terminology: $X\in C$ is called _finite_ if $F(X)$ is a finite [[set]]. Let $\int_F C$ denote the [[category of elements]] of $F$, in which an object $(X,a)$ is called finite if $X$ is finite.

* **G0)** The full subcategory of $\int_F C$ on the finite objects is [[cofinal subcategory|cofinal]].

* **G1)** $C$ has all finite [[limit]]s

* **G2)** $C$ has an [[initial object]], finite [[coproduct]]s and [[quotient object|quotients]] by finite [[group]]s

* **G3)** Given $f:x\to z$ in $C$ there is a factorisation $x \stackrel{e}{\to} y \stackrel{i}{\to} z$ where $e$ is a strict epimorphism and $i$ is a mono. Also, $y$ is assumed to be a [[coproduct|direct sum]]mand of $z$.

* **G4)** $F$ preserves finite limits

* **G5)** $F$ preserves initial object, finite sums, quotients by finite group actions and sends strict epimorphisms to surjections

* **G6)** $F$ reflects isomorphisms

***

The functor $F$ is called the *fibre functor*, and the pair $(C,F)$ is sometimes called a *Galois category*.

It follows from the axioms that $F$ is a [[pro-representable functor]]. The [[automorphism group]] of the [[pro-object]] $P$ representing $F$ is (should be. I'm not familiar enough with pro-objects) a [[profinite group]] $\pi$. This acts on $F(X) = [P,-]$ by precomposition (talking out of my depth here -- it's getting a bit vague) and so $F$ lifts to a functor to $\pi-Set$, and Grothendieck's result is that this functor is an [[equivalence of categories]].

There are several modifications one can make the above. In the case that $C$ is the category of covering spaces of a nice enough space, the functor $F$ is [[representable functor|representable]] by the [[universal covering space]], and so there is a 'representable' version of the above, not needing to utilise profinite groups. One can also consider just the connected-objects version, and end up with an equivalence to the category of _[[transitive action|transitive]]_ $\pi$-sets.

## The classical case of fields

Even for the classical case of the inclusion of [[field]]s, Grothendieck's Galois theorem gives more general statement than the previously known. This is the Grothendieck's version of the Galois correspondence theorem for fields:

Let $K \subset L$ be a finite dimensional [[Galois extension]] of fields. Let $Gal[L : K]$ denote the group of $K$-automorphisms of $L$ and $Gal [L : K]-finSet$ the category of finite $Gal[L : K]$-sets. By $SplitfinAlg_K(L)$ denote the finite dimensional $K$-algebras which are split over $L$; here $L$ itself is an object. Consider the [[representable functor]] $h_L = Hom_K(-,L):SplitfinAlg_K(L)\to Set$. It takes values in the [[subcategory]] of finite sets and it comes with a canonical $Gal[L : K]$-action. In other words, this functor factors through $Gal [L : K]-finSet$. Moreover, the corresponding functor

$$
SplitfinAlg_K(L)\to Gal [L : K]-finSet
$$

is an [[equivalence of categories]].

There is an infinitary version as well, generalizing the classical Galois theorem on infinitary Galois extensions.

Thus let $K\subset L$ be an arbitrary Galois extension. Now $Gal[L:K]$ denotes the profinite Galois group and $Gal[L:K]-profinSpace$ the category or profinite $Gal[L:K]$-spaces. $SplitAlg_K(L)$ denotes the category of $K$-algebras split over $L$ (possible infinite-dimensional). Then there is a canonical anti-equivalence of categories

$$
SplitAlg_K(L)\to Gal [L : K]-profinSpace
$$

(factorizing a profinite-space version of the representable functor $Hom_K(-,L)$).

A special case of this is the following: the category of [[étale scheme|étale k-schemes]] reps. [[étale group scheme|étale group schemes]] for a field $k$ is equivalent to the category of sets equipped with an [[action]] of the [[absolute Galois group]] reps. to the category of [[Galois module|Galois modules]] of the absolute Galois group.

## Galois theorem for locales and topoi

Let $E$ be a [[Grothendieck topos]]. Then there exist an open [[localic groupoid]] $G$ such that $E$ is equivalent to the category of étale presheaves over $G$. ([Joyal & Tierney 1984](#JoyalTierney84), see also at *[[classifying topos of a localic groupoid]]*).

## Related entries

* [[classifying topos of a localic groupoid]]

## References

The original development of the theory by [[Grothendieck]]  is in

* [[Alexander Grothendieck]], (1971), *[[SGA1]] -- Revetements  étales et groupe fondamental*, Lecture Notes in Maths. 224, Springer-Verlag.

See also:

* J. P. Murre, _Lectures on an introduction to Grothendieck's theory of the fundamental group_, Tata Inst. of Fund. Res. Lectures on Mathematics _40_, Bombay, 1967. iv+176+iv pp.

* {#JoyalTierney84} [[André Joyal]], [[Myles Tierney]], *An extension of the Galois theory of Grothendieck*,  Mem. Amer. Math. Soc. **309** (1984) [[ISBN:978-1-4704-0722-3](https://bookstore.ams.org/memo-51-309)]

A more recent treatment can be found in

* [[Eduardo Dubuc]], C. S. de la Vega, _On the Galois theory of Grothendieck_, Bol. Acad. Nac. Cienc. (Cordoba) **65** (2000) 111--136. [arXiv:math.CT/0009145](http://arxiv.org/abs/math.CT/0009145)

and more related categorical and topos theoretic aspects in

* [[Eduardo Dubuc]], _Localic Galois theory_, Adv. Math. __175__:1 (2003), 144–167 doi; _On the representation theory of Galois and atomic topoi_, JPAA __186__:3 (2004) 233–275 doi

A very approachable account is given in

* Marco Antònio Delgado Robalo, _Galois theory towards dessins d'enfants_, masters thesis, Lisboa 2009, [pdf](https://fenix.tecnico.ulisboa.pt/downloadFile/395139415315/dissertacao.pdf)

(This has the advantage of looking towards Grothendieck's [[children's drawing|dessins d'enfants]].)

Some basic intuitions are explained in

* Pierre Cartier, _A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry_, Bull. Amer. Math. Soc. __38__ (2001), 389-408, [pdf](http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf)

The construction for general [[topos]]es is described in section 8.4 of

* [[Peter Johnstone]], _Topos theory_, Academic Press (1977)

and, a current state-of-the-art description is in

* [[Marta Bunge]], _Galois groupoids and covering morphisms in topos theory_,  Galois theory, Hopf algebras, and semiabelian categories,  131--161, Fields Inst. Commun. __43__, Amer. Math. Soc., Providence, RI, 2004, [links](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.7071).

A modern approach from classical via Grothendieck up to [[categorical Galois theory]] based on precategories and adjunctions is in

* [[Francis Borceux]], [[George Janelidze]], _[[Galois Theories]]_, Cambridge Studies in Adv. Math. __72__, 2001. xiv+341 pp.

The application of a general Tannakian theorem of Saavaedra Rivano, as corrected by Deligne, to the "[[differential Galois theory]]" for differential instead of algebraic equations is in the last chapter of Deligne's [[Catégories Tannakiennes]].

* [[George Janelidze]], Dietmar Schumacher, [[Ross Street]], _Galois theory in variable categories_, Applied Categorical Structures __1__, Number 1, 103-110, DOI:
* Federico G. Lastaria, _On separable algebras in Grothendieck Galois theory_, Le Matematiche 51:3, 1996, [link](http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/464/0)

[[!redirects Grothendieck Galois theory]]
[[!redirects Grothendieck's Galois theory]]
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category: Galois theory


Grothendieck-Lefschetz trace formula

## Related concepts

* [[Lefschetz trace formula]]

## References

* Wikipedia, _[Grothendieck trace formula](http://en.wikipedia.org/wiki/Grothendieck_trace_formula)_

* [[Akhil Mathew]], _[Grothendieck trace formula](https://amathew.wordpress.com/tag/grothendieck-trace-formula/)_

[[!redirects Grothendieck trace formula]]


Grothendieck-Maltsiniotis infinity-category

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## Idea

An [[algebraic definition of higher categories|algebraic definition]] of [[∞-groupoid]]s and [[∞-categories]] similar to (but older than) the definition [[Batanin ∞-categories]].

## References

The original definition is indicated in words in sections 1-13 of

* [[Alexander Grothendieck]], _[[Pursuing Stacks]]_

This has been extracted and formalized in

* [[Georges Maltsiniotis]], _Infini groupoïdes d'après Grothendieck_ ([web](http://www.math.jussieu.fr/~maltsin/textes.html))

A comprehensive account of a simplified version of the definition and its generalization from $\infty$-groupoids to $\infty$-categories is given in

* [[Georges Maltsiniotis]], _Grothendieck $\infty$-groupoids and still another definition of $\infty$-categories_ ([arXiv:1009.2331](http://arxiv.org/abs/1009.2331))

A definition as [[models]] of a [[dependent type theory]] is given in

* [[Eric Finster]], [[Samuel Mimram]], _A Type-Theoretical Definition of Weak ω-Categories_ ([arxiv:1706.02866](https://arxiv.org/abs/1706.02866))

The relation to [[Batanin ∞-categories]] is discussed in

* {#Ara10} [[Dimitri Ara]], _Sur les $\infty$-groupoïdes de Grothendieck_ Thesis, under the supervision of G. Maltsiniotis, (2010) ([pdf](http://www.normalesup.org/~ara/files/these.pdf))

A brief survey is provided in

* [[Mike Shulman]], ([blog entry](http://golem.ph.utexas.edu/category/2010/09/grothendieckmaltsiniotis_categ.html))

On the [[homotopy hypothesis]] for Grothendieck 3-groupoids:

* {#HenryLanari} [[Simon Henry]], [[Edoardo Lanari]], *On the homotopy hypothesis in dimension 3*, Theory and Applications of Categories **39** 26 (2023) 735-768 [[arxiv/1905.05625](https://arxiv.org/abs/1905.05625), [tac:39-26](http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/39/26/39-26.pdf)].

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Grothendieck-Riemann-Roch theorem

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## Idea

The _Grothendieck-Riemann-Roch theorem_ describes  (the failure of) the [[natural transformation|naturality]] of the behaviour of a _[[Chern character]]_ under [[fiber integration|push forward]] along [[proper maps]].

It says, in the formulation of ([Atiyah-Hirzebruch 61](#AtiyahHirzebruch61)), that for $f \colon X \longrightarrow Y$ a [[flat morphism]] of [[schemes]] which are [[flat morphism|flat]] and [[regular scheme|regular]] [[quasi-projective varieties]] over the [[spectrum of a commutative ring|spectrum]] of a [[Dedekind domain]], then the [[Chern character]] of the [[fiber integration|push-forward]] of some $E$ is the push-forward of the [[cup product]] of the Chern-character of $E$ with the [[Todd class]]. Hence it says that "Chern-cup-Todd is natural under pushforward" along proper maps, and generally along [[K-orientation|K-oriented]] maps.

If $f \colon X \to Y$ is a [[proper map]], then there is a [[commuting diagram]]

$$
  \array{
    K_0(X)
    &\overset{ch(-)\cup Td(X)}{\longrightarrow}&
    H_\bullet(X; \mathbb{Q})
    \\
    \Big\downarrow\mathrlap{{}^{\int_f}}
    &&
    \Big\downarrow\mathrlap{{}^{\int_f}}
    \\
    K_0(Y)
    &\underset{ch(-)\cup Td(Y)}{\longrightarrow}&
    H_\bullet(Y;\mathbb{Q})
    \mathrlap{\,,}
  }
$$

where...

If $X$ is an [[algebraic curve]], then the Riemann-Roch theorem reduces to a statement about the [[Euler characteristic]]/[[genus of a curve|curve]]. This generalizes to [[arithmetic geometry]] with the notion of [[genus of a number field]].

There are various extensions of the Grothendieck-Riemann-Roch theorem,
such as the [[Atiyah-Singer index theorem]] (for elliptic operators and elliptic
complexes), the Connes-Moscovici local index formula (a non-commutative/equivariant version of Atiyah-Singer), and Bismut's hypoelliptic index formula in Bott-Chern cohomology (for non-K\"aeler complex manifolds).

Another approach due to Kashiwara and Schapira relies on the use of methods of Hochschild cohomology and its microlocalized version (which gives a refined index theorem, that takes care of the information related to the propagation of singularities).
It is essentially divided in two parts: the functorial one, which is based on Hochschild and cyclic homology (very similar to Toen and Vezzosi's approach to the construction of the Chern character), and the computational one (due to Bresler-Nest-Tsygan and others), that explicitly describes the relation between the functorial construction of the Chern character and its more classical construction. It is this last comparison statement that introduces the Todd class and makes the Riemann-Roch/index formula look complicated, despite the fact that it simply expresses the functoriality of Hochshild and cyclic homology classes with respect to the push-forward, that is evident. The microlocalized version of the index/RRG theorem is necessary to have a better understanding of the "deformation of the Laplacian" methods (introduced by Witten in his paper on the Morse inequalities, and used by Bismut in his work on the hypoelliptic Laplacian, by Laumon in his paper on local constants of functional equations, and by Kedlaya in his paper on p-adic Weil II) that have become a central tool in the study of cohomology theories and of (additive) index-type formulas and (multiplicative) product-type formulas.

One may hope for an extension of the Riemann-Roch-Grothendieck theorem to the
setting of a general proper morphism in non-strict [[global analytic geometry]] using
an extension of Bismut's approach to a setting of exotic [[global Hodge theory]]. However, it seems that for arithmetic applications (e.g., the study of special values of arithmetic L-functions), one will clearly have to prove a refined RRG theorem for strict [[global analytic geometry|global analytic spaces]], using semistable compactifications (à la Deligne "théorie de Hodge II, III") and logarithmic methods: the (purely analytic) non-strict theorem will not suffice.

## Related concepts

* [[rational Todd class is Chern character of Thom class]]

* [[Hirzebruch-Riemann-Roch theorem]]

* [[arithmetic Riemann-Roch theorem]]

* [[push-forward in generalized cohomology]]

* [[transfer index conjecture]] for [[Becker-Gottlieb transfer]]

## References

The formulation in terms of [[topological K-theory]] is due to:

* {#AtiyahHirzebruch61} [[Michael Atiyah]], [[Friedrich Hirzebruch]], _Vector bundles and homogeneous spaces_, Proc. Symp. Pure Math. 3, 7–38 (1961)

Early textbook accounts:

* {#Karoubi78} [[Max Karoubi]], Chapter V.4 of: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 ([pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3))

See also:

* Wikipedia, _[Grothendieck-Riemann-Roch theorem](http://en.wikipedia.org/wiki/Grothendieck%E2%80%93Riemann%E2%80%93Roch_theorem)_

* Wikipedia, _[Riemann-Roch theorem for surfaces](http://en.wikipedia.org/wiki/Riemann–Roch_theorem_for_surfaces)_

Discussion of Riemann-Roch over [[arithmetic curves]] is in

* {#Neukirch92} [[Jürgen Neukirch]], chapter III, section 3 of _Algebraische Zahlentheorie_ (1992), English translation _Algebraic Number Theory_,  Grundlehren der Mathematischen Wissenschaften 322, 1999 ([pdf](http://www.plouffe.fr/simon/math/Algebraic%20Number%20Theory.pdf))

The refinement to [[differential cohomology]], hence [[differential K-theory]],  is discussed in section 6.2 of

* [[Ulrich Bunke]], [[Thomas Schick]], _Differential K-theory. A survey_ ([arXiv:1011.6663](http://arxiv.org/abs/1011.6663))

Over [[algebraic stacks]]/[[Deligne-Mumford stacks]] the GRR theorem is discussed in

* Roy Joshua, _Riemann-Roch for algebraic stacks_ ([pdf I](http://www.math.osu.edu/~joshua.1/rr1.pdf), [pdf II](http://www.math.osu.edu/~joshua.1/rr2.pdf), [pdf III](http://www.math.osu.edu/~joshua.1/rr3.pdf))

* [[Bertrand Toën]], _Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford_, K-Theory 18
(1999), no. 1, 33–76. 1, 23

* Dan Edidin, _Riemann-Roch for Deligne-Mumford stacks_ ([arXiv:1205.4742v1](http://arxiv.org/abs/1205.4742v1))

For a [[categorification]] of GRR, see

* Marc Hoyois, Pavel Safronov, Sarah Scherotzke, Nicolò Sibilla, _The categorified Grothendieck--Riemann--Roch theorem_, ([arXiv:1804.00879](https://arxiv.org/abs/1804.00879))

[[!redirects Riemann-Roch theorem]]


Grothendieck-Teichmüller tower

#Contents#
* table of contents
{:toc}

## Idea

The _Grothendieck-Teichmüller tower_ construction involves [[moduli spaces]] outlined in [[Grothendieck]]'s _[[Esquisse d'un programme]]_ and then developed by [[Vladimir Drinfel'd]] and others. It involves the [[Teichmüller groupoids]] which are the [[fundamental groupoids]] of [[moduli stacks]] of [[genus]] $g$ [[curves]] with $n$ points removed. It is a basis for the definition of the _Grothendieck-Teichmüller group_ which is by the definition inertia-preserving [[automorphism group]] of the Grothendieck-Teichmüller tower.

## Definitions

...

### Grothendieck-Teichmüller Lie algebra

GT Lie algebra appears as the tangent Lie algebra to the GT group.

Drinfel'd defined it more explicitly as follows.

Let $\mathfrak{lie}_n$ be the degree completion of the [[free Lie algebra]] on $n$-generators over a fixed ground field $K$ of characteristic $0$.
Let $\mathfrak{der}_n$ be the space of $K$-linear derivations $\mathfrak{lie}_n\to
\mathfrak{lie}_n$. A derivation $u\in\mathfrak{der}_n$ is __tangential__ if there exist
$a_i\in\mathfrak{lie}_n$ for $i=1,\ldots,n$, such that $u(x_i)=[x_i,a_i]$. In particular, $u$ is determined by elements $a_1,\ldots,a_n$ and is below denoted
by $(a_1,\ldots,a_n)$. The tangential derivations form a Lie subalgebra
$\mathfrak{tder}_n\subset\mathfrak{der}_n$. The __GT Lie algebra__ is the subspace
$\mathfrak{grt}\subset\mathfrak{tder}_2$ consisting of the tangential derivations
of the form $(0,\psi)$, where the identities
$$
\psi(x,y)=-\psi(y,x),\,\,\,for\,\,\,all\,\,\,x,y,
$$
$$
\psi(x,y)+\psi(y,z)+\psi(z,x) = 0,\,\,\,whenever\,\,\,\,x+y+z=0,
$$
and the pentagon-type identity
$$
\psi(t^{1,2},t^{2,34})+\psi(t^{12,3},t^{3,4})=
\psi(t^{2,3},t^{3,4})+\psi(t^{1,23},t^{23,4})+\psi(t^{1,2},t^{2,3}),
$$
hold, and which is equipped with the __Ihara Lie bracket__
$$
[\psi_1,\psi_2]_{Ihara} = (0,\psi_1)(\psi_2)-(0,\psi_2)(\psi_1)+[\psi_1,\psi_2].
$$

## Properties

### Relation to Drinfeld associators

The GT group acts freely on the set of [[Drinfeld associators]].

### Relation to the absolute Galois group of the rational numbers

+-- {: .num_theorem}
###### Theorem
**(Drinfeld, Ihara, Deligne)**

There is an inclusion of the [[absolute Galois group]] of the [[rational numbers]] into the Grothendieck-Teichmüller group (recalled e.g. [Stix 04, theorem 6](#Stix04)).

=--

### Relation to the motivic Galois group

The Grothendieck-Teichmüller group is supposed to be a [[quotient]] of the [[motivic Galois group]]. This is a conjecture due to ([Drinfeld 91](#Drinfeld91)).

### Relation to the graph complex
 {#RelationToTheGraphComplex}

The Grothendieck-Teichmüller Lie algebra is isomorphic to the 0th cohomology of [[Kontsevich]]'s [[graph complex]] ([Willwacher 10](#Willwacher10)).

### Relation to deformation quantization and the cosmic Galois group

[[Grothendieck]] predicted that the GT group is closely related to the absolute Galois group. [[Maxim Kontsevich]] later conjectured its [[action]] on certain spaces of [[quantum field theories]] and outlined its [[motivic cohomology|motivic]] aspects.

This was later proven by [[Vasily Dolgushev]], see at _[formal deformation quantization -- Motivic Galois group action on the space of quantizations](deformation+quantization#MotivicGaloisGroup)_ for details and pointers.

For more see also at _[[cosmic Galois group]]_ for more on this.

## Related concepts

* [[stable symplectic category]],

* [[motivic Galois group]], [[cosmic Galois group]]

## References

### General

The Grothendieck-Teichmüller group $GRT$ was originally introduced in

* [[Vladimir Drinfel'd]], _On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$_, [abs](http://mi.mathnet.ru/aa199), Rossiĭskaya Akademiya Nauk. Algebra i Analiz (in Russian) 2 (4): 149–181, ISSN 0234-0852, MR1080203 translation in Leningrad Math. J. 2 (1991), no. 4, 829–860

inspired by

* [[Alexander Grothendieck]], _Sketch of a program_, London Math. Soc. Lect. Note Ser., 242, Geometric Galois actions, 1, 5–48, Cambridge Univ. Press, Cambridge, 1997.

Review:

* {#Fresse18} [[Benoit Fresse]], _Little discs operads, graph complexes and Grothendieck--Teichmüller groups_, in [[Haynes Miller]] (ed.) _[[Handbook of Homotopy Theory]]_ ([arXiv:1811.12536](https://arxiv.org/abs/1811.12536))

See also:

* [[Leila Schneps]], [[Pierre Lochak]], _Geometric Galois actions_, 1, London Math. Soc. Lecture Note Ser. 242 ([doi](http://dx.doi.org/10.1017%2FCBO9780511666124))-- it contains a reprint of Groethendieck's Esquisse and some surveys by contributors, including Leila Schneps, _The Grothendieck-Teichmueller group GT: a survey_, [pdf](http://www.math.jussieu.fr/~leila/SchnepsGT.pdf)

* [[Maxim Kontsevich]], _Operads and motives in deformation quantization_, Lett. Math. Phys. 48 (1999), 35-72, [math.QA/9904055](http://arxiv.org/abs/math/9904055) [doi](https://doi.org/10.1023/A:1007555725247)

* {#FresseHOGTG} [[Benoit Fresse]], *Homotopy of Operads and Grothendieck-Teichmüller Groups (Volumes 1 & 2)*, Mathematical Surveys and Monographs **217** (2017) [[ISBN:978-1-4704-3480-9](https://bookstore.ams.org/surv-217/), [website](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/), [pdf (draft of Part I)](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.648.9110&rep=rep1&type=pdf)]

> The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, we set up the background required by an in-depth study of this subject, and we make precise the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.

Videos from a seminar at the Newton Institute:

* Newton Institute Programme Jan-April 2013: Grothendieck-Teichmüller Groups, Deformation and Operads, [seminars](https://www.newton.ac.uk/event/gdo/seminars) (some with videos)

Relation to [[automorphism groups]] of the [[profinite completion]] of [[braid groups]]:

* [[Pierre Lochak]], [[Leila Schneps]], *The Grothendieck-Teichmüller group and automorphisms of braid groups*, in: *The Grothendieck Theory of Dessins d’Enfant*, Cambridge University Press (1994, 2011) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/Fls.pdf), [doi:10.1017/CBO9780511569302](https://doi.org/10.1017/CBO9780511569302)]

The Drinfeld conjecture is stated in

* {#Drinfeld91} [[Vladimir Drinfel'd]], _On quasi-triangular quasi-Hopf algebras and a group closely related with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$, Leningrad Math. J. 2 (1991) 829--860.

See also:

* {#Stix04} [[Jakob Stix]], _The Grothendieck-Teichmüller group and Galois theory of the rational numbers_, 2004 ([[StiXGaloisAndGT.pdf:file]])

* [[Anton Alekseev]], Charles Torossian, _The Kashiwara-Vergne conjecture and Drinfeld's associators_, [arxiv/0802.4300](http://arxiv.org/abs/0802.4300)

### Relation to graph complexes

Relation to [[graph complexes]]:

$H^0\big( Graphs_0(\mathbb{R}^2) \big)$ is the [[Lie algebra]] of the [[Grothendieck-Teichmüller group]]:

* {#Willwacher10} [[Thomas Willwacher]], _M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra_, Invent. math. (2015) 200: 671 ([arxiv:1009.1654](http://arxiv.org/abs/1009.1654))
* [[M. Kontsevich]], _Derived Grothendieck–Teichmüller group and graph complexes [after T. Willwacher]_, Seminaire Bourbaki 1126 (2016-2017) [pdf](https://www.bourbaki.fr/TEXTES/1126.pdf)

* {#DolgushevRogers12} [[Vasily Dolgushev]], [[Christopher Rogers]], _Notes on algebraic operads, graph complexes, and Willwacher's construction_, In: Mathematical aspects of quantization 583 (2012): 25--145. ([arXiv:1202.2937](https://arxiv.org/abs/1202.2937))
* V. A. Dolgushev, C. L. Rogers, T. H. Willwacher, _Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields_, Ann. Math. __182__:3 (2015) 855--943 [doi](https://doi.org/10.4007/annals.2015.182.3.2)

* [[Vasily Dolgushev]], _A manifestation of the Grothendieck-Teichmueller group in geometry_ ([slides pdf](https://math.temple.edu/~vald/AMSslides.pdf))

See also:

* Kevin Morand, _A note on multi-oriented graph complexes and deformation quantization of Lie bialgebroids_ ([arXiv:2102.07593](https://arxiv.org/abs/2102.07593))

[[!redirects Grothendieck-Teichmueller tower]]
[[!redirects Grothendieck-Teichmüller group]]
[[!redirects Grothendieck-Teichmuller tower]]
[[!redirects Grothendieck-Teichmuller group]]
[[!redirects Grothendieck-Teichmüller groups]]

[[!redirects Grothendieck-Teichmueller group]]

[[!redirects groupe de Grothendieck-Teichmüller]]


ground ring

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#### Algebra
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[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A __ground ring__ (or __base ring__, __field of [[scalars]]__, etc) is a [[ring]] $k$ (usually a [[commutative ring]] or even a [[field]]) which is fixed in a given situation, such that everything takes place 'over' $k$.  There is no technical definition here; rather, it is the meaning of 'over' that must be defined in any particular case.

The elements of the ground ring are often called __[[scalar|scalars]]__.  Note that a '[[scalar field]]' in the sense of [[physics]] does not refer to $k$, although its meaning does depend on $k$.

There are analogies between the ground ring and the base space of a [[bundle]].  There are also generalisations in which $k$ might be, for example, a [[monad]].  It is also important to consider [[base change]] from one ground ring to another, mediated by a ring homomorphism or even a [[bimodule]].

## Examples

Perhaps most fundamentally, the categories [[Mod]] and [[Vect]] depend on (respectively) a ground ring and a ground field.  That is, a [[module]] is not just a module but a $k$-module (either left or right when $k$ might not be commutative), and a [[vector space]] is not just a vector space but a vector space over $k$.  Conversely, the ground ring/field itself appears as the unit of the [[tensor product]] in these categories.

Other terms which depend on a ground ring/field include:

*  [[graded module]]
*  [[graded vector space]]
*  [[algebraic variety]]
*  [[relative scheme]]
*  [[2-vector space]]
*  [[category algebra]]
*  [[groupoidification]]
*  [[Hopf algebra]]
*  [[symmetric function]]
*  [[coalgebra]]
*  [[associative algebra]]
*  [[nonassociative algebra]]
*  [[fusion category]]
*  [[vector bundle]]
*  [[almost scheme]]
*  [[tangent vector]]
*  [[cotangent vector]]
*  [[exterior algebra]]
*  [[Hilbert space]]
*  [[Banach space]]
*  [[topological vector space]]
*  [[Legendre polynomial]]
*  [[differential form]]
*  [[inner product space]]
*  [[affine space]]

This list is very incomplete, made mostly by searching for 'ground ring', 'base field', etc.

## Related concepts

* [[positive characteristic]]

[[!redirects ground rings]]

[[!redirects ground field]]
[[!redirects ground fields]]

[[!redirects field of scalars]]
[[!redirects fields of scalars]]

[[!redirects base ring]]
[[!redirects base field]]


ground state

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#### Quantum systems
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[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For a [[quantum system]], a _ground state_ is a [[quantum state]] of minimal [[energy]].

## Related concept

* [[quantum adiabatic theorem]]

* [[adiabatic quantum computation]]

* [[topological quantum computation]], [[anyons]]

[[!include states and observables -- content]]

## References

See also:

* Wikipedia, *[Ground state](https://en.wikipedia.org/wiki/Ground_state)*

[[!redirects ground states]]


group

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#### Category theory
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=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

Classically, a **group** is a [[monoid]] in which every element has an [[inverse]] (necessarily unique). When written with a view toward [[group objects]] (see Internalization below), one should rather say that a group is a monoid together with an inversion operation.

An **[[abelian group]]** is a group in which moreover the order in which two elements are multiplied is irrelevant.

## Delooping ##

To some extent, a group "is" a [[groupoid]] with a single object, or more precisely a [[pointed object|pointed]] groupoid with a single object.

The [[delooping]] of a group $G$ is a [[groupoid]] $\mathbf{B} G$ with

* $Obj(\mathbf{B}G) = \{\bullet\}$

* $Hom_{\mathbf{B}G}(\bullet, \bullet) = G$.

Since for $G, H$ two groups, [[functors]] $\mathbf{B}G \to \mathbf{B}H$ are canonically in bijection with group homomorphisms $G \to H$, this gives rise to the following statement:

Let [[Grpd]] be the 1-[[category]] whose objects are [[groupoids]] and whose [[morphisms]] are [[functors]] (discarding the [[natural transformations]]). Let [[Grp]] be the category of groups. Then the [[delooping]] functor

$$
  \mathbf{B} : Grp \to Grpd
$$

is a [[full and faithful functor]]. In terms of this functor we may regard groups as the full [[subcategory]] of groupoids on groupoids with a single object.

It is in this sense that a group really is a groupoid with a single object.

But notice that it is unnatural to think of [[Grpd]] as a 1-category. It is really a [[2-category]], namely the sub-2-category of [[Cat]] on groupoids.

And the category of groups is _not_ equivalent to the full sub-2-category of the 2-category of groupoids on one-object groupoids.

The reason is that two functors:

$$
  \mathbf{B}f_1, \mathbf{B}f_2 : \mathbf{B}G \to \mathbf{B}H
$$

coming from two group homomorphisms $f_1, f_2 : G \to H$ are related by a [[natural transformation]] $\eta_h : \mathbf{B}f_1 \to \mathbf{B}f_2$ with single component $\eta_h : \bullet \mapsto h \in Mor(\mathbf{B} H)$ for each element $h \in H$ such that the homomorphisms $f_1$ and $f_2$ differ by the [[inner automorphism]] $Ad_h : H \to H$

$$
  (\eta_h : \mathbf{B}f_1 \to \mathbf{B}f_2)
  \Leftrightarrow
  (f_2 = Ad_h \circ f_1)
  \,.
$$

To fix this, look at the category of [[pointed object|pointed]] groupoids with [[pointed functor|pointed functors]] and pointed natural transformations.  Between group homomorphisms as above, only identity transformations are pointed, so $Grp$ becomes a full sub-$2$-category of $Grpd_*$ (one that happens to be a $1$-[[1-category|category]]).  (Details may be found in the appendix to [[Lectures on n-Categories and Cohomology]] and should probably be added to [[pointed functor]] and maybe also [[k-tuply monoidal n-category]].)

## Generalizations

### Internalization

A **[[group object]]** [[internalization|internal to]] a [[category]] $C$ with finite [[product|products]] is an object $G$ together with maps $mult:G\times G\to G$, $id:1\to G$, and $inv:G\to G$ such that various diagrams expressing associativity, unitality, and inverses commute.

Equivalently, it is a functor $C^{op}\to Grp$ whose underlying functor $C^{op} \to Set$ is [[representable functor|representable]].

For example, a group object in [[Diff]] is a [[Lie group]].  A group object in [[Top]] is a [[topological group]].  A group object in [[Sch/S]] (the category or [[relative schemes]]) is an $S$-[[group scheme]].  And a group object in $CAlg^{op}$, where [[CAlg]] is the category of [[commutative algebras]], is a (commutative) [[Hopf algebra]].

A group object in [[Grp]] is the same thing as an abelian group (see [[Eckmann-Hilton argument]]), and a group object in [[Cat]] is the same thing as an [[internal category]] in [[Grp]], both being equivalent to the notion of [[crossed module]].

### In higher categorical and homotopical contexts

Internalizing the notion of _group_ in [[higher category theory|higher categorical]] and [[homotopy theory|homotopical]] contexts yields various generalized notions. For instance

* a [[2-group]] is a group object in [[Grpd]]

* an [[n-group]] is a group object internal to [[n-groupoid]]s

* an [[∞-group]] is a [[group object in an (∞,1)-category]].

* a [[loop space]] is a group object in [[Top]]

* generally there is a notion of [[groupoid object in an (infinity,1)-category|group object in an (infinity,1)-category]].

And the notion of [[loop space object]] and [[delooping]] makes sense (at least) in any [[(infinity,1)-category]].

Notice that the relation between group objects and deloopable objects becomes more subtle as one generalizes this way. For instance not every [[groupoid object in an (infinity,1)-category|group object in an (infinity,1)-category]] is [[delooping|deloopable]]. But every group object in an [[(infinity,1)-topos]] is.

### Weakened axioms

Following the practice of [[centipede mathematics]], we can remove certain properties from the definition of group and see what we get:

* remove inverses to get [[monoids]], then remove the identity to get [[semigroups]];
* or remove associativity to get [[loop (algebra)|loops]], then remove the identity to get [[quasigroups]];
* or remove all of the above to get [[magma|magmas]];
* or instead allow (in a certain way) for the binary operation to be partial to get [[groupoids]], then remove inverses to get [[categories]], and then remove identities to get [[semicategory|semicategories]]
* etc.

## Examples

### Special types and classes

* [[simple group]]

* [[finite group]], [[progroup]]

  * [[classification of finite simple groups]]

  * [[sporadic finite simple groups]]

* [[abelian group]]

  * [[finite abelian group]]

* [[divisible group]]

* [[acyclic group]]

* [[topological group]]

  * [[discrete group]]

  * [[Kac-Moody group]]

* [[Lie group]]

* [[group of Lie type]]

### Concrete examples

Standard examples of [[finite group]]s include

* [[group of order 2]] $\mathbb{Z}_2$

* [[symmetric group]] $\Sigma_n$

* [[cyclic group]]

* [[braid group]] $Br_n$

* [[Monster group]]

Standard examples of non-finite groups include

* group of [[integer]]s $\mathbb{Z}$ (under [[addition]]);

* group of [[real number]]s without 0 $\mathbb{R}\setminus \{0\}$ under [[multiplication]].

* [[Prüfer group]]

Standard examples of [[Lie groups]] include

* [[orthogonal group]]

* [[unitary group]]

* [[Spin group]], [[spin^c group]]

Standard examples of [[topological group]]s include

* [[string group]]

### Counterexamples

For more see [[counterexamples in algebra]].

1. A non-[[abelian group|abelian]] [[group]], all of whose [[subgroup]]s are [[normal subgroup|normal]]:

   $$
   Q \coloneqq \langle a, b | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
   $$

1. A [[finitely presented group|finitely presented]], infinite, [[simple group]]

   [[Thomson's group]] T.

1. A [[group]] that is not the [[fundamental group]] of any [[3-manifold]].

   $$
   \mathbb{Z}^4
   $$

1. Two [[finite group|finite]] non-[[isomorphism|isomorphic]] groups with the same [[order profile]].

   $$
   C_4 \times C_4, \qquad C_2 \times \langle a, b, | a^4 = 1, a^2 = b^2, a b = b a^3 \rangle
   $$

1. A counterexample to the converse of [[Lagrange's theorem]].

   The [[alternating group]] $A_4$ has order $12$ but no [[subgroup]] of order $6$.

1. A [[finite group]] in which the product of two [[commutator]]s is not a commutator.

   $$
   G = \langle (a c)(b d), (e g)(f h), (i k)(j l), (m o)(n p), (a c)(e g)(i k), (a b)(c d)(m o), (e f)(g h)(m n)(o p), (i j)(k l)\rangle \subseteq S_{16}
   $$

## Related concepts

* [[2-group]]

* [[n-group]]

* [[∞-group]]

* [[monoid]], [[monoid object]],

* **group**, [[group object]]

  * [[discrete group]]

    * [[order of a group]]

      * [[p-primary group]]

    * [[finite group]], [[profinite group]]

    * [[finitely generated group]]

  * [[subgroup]]

    * [[torsion subgroup]]

    * [[stabilizer]],  [[centralizer]], [[normalizer]]

  * [[isogeny]]

  * [[coset]], [[coset space]]

  * [[abelian group]], [[anabelian group]],

    * [[group completion]]

  * [[group commutator]], [[commutator subgroup]], [[abelianization]]

  * [[group character]]

  * [[group cohomology]]

  * [[group extension]]

  * [[normed group]], [[bornological group]]

  * [[topological group]], [[Lie group]],

  * [[loop group]]

  * [[cogroup]]

  * [[multivalued group]]

  * is a commutative pregroup as mentioned in [[pregroup grammar]]

* [[ring]], [[ring object]]

* [[automorphism group]], [[automorphism 2-group]], [[automorphism ∞-group]],

  * [[group of bisections]]

* [[center]], [[center of an ∞-group]],

* [[inner automorphism group]]

* [[outer automorphism group]], [[outer automorphism ∞-group]]

* [[group presentation]]

[[!include oidification - table]]

## Literature

For more see also the references at *[[group theory]]*.

The terminology "group" was introduced (for what today would more specifically be called *[[permutation groups]]*) in

* [[Évariste Galois]], *[[Galois' last letter|letter to Auguste Chevallier]]*, (May 1832)

The original article that gives a definition equivalent to the modern definition of a group:

* [[Heinrich Weber]], *Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist*, Mathematische Annalen 20:3 (1882), 301–329 ([doi:10.1007/bf01443599](http://dx.doi.org/10.1007/bf01443599))

Introduction of group theory into ([[quantum physics|quantum]]) [[physics]] (cf. *[[Gruppenpest]]*):

* [[Hermann Weyl]], §III in: *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics*, Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]

Textbook account in relation to applications in [[physics]]:

* {#Sternberg94} [[Shlomo Sternberg]], *Group Theory and Physics*, Cambridge University Press 1994 ([ISBN:9780521558853](https://www.cambridge.org/gb/academic/subjects/mathematics/algebra/group-theory-and-physics?format=PB&isbn=9780521558853))

See also:

* Wikipedia, *Group_(mathematics)*

* [[bananaspace]], *[群](https://www.bananaspace.org/wiki/%E7%BE%A4)* (Chinese)

{#TTFormalizations} Formalization of group structure in [[dependent type theory]]:

in [[Coq]]:

* Farida Kachapova, *Formalizing groups in type theory* [[arXiv:2102.09125](https://arxiv.org/abs/2102.09125)]

and with the [[univalence axiom]]

* [[unimath]] -> [UniMath.Algebra.Groups](https://unimath.github.io/doc/UniMath/d4de26f//UniMath.Algebra.Groups.html)

in [[Agda]]:

* [agda-unimath](https://unimath.github.io/agda-unimath/) -> [group-theory.groups](https://unimath.github.io/agda-unimath/group-theory.groups.html)

* [[Martín Escardó]], *[Groups](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#groups-sip)*, §3.33.10 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

in [[cubical Agda]]:

* [[1lab]]: *[Algebra.Group](https://1lab.dev/Algebra.Group.html)*

in [[Lean]]:

* [Lean Community](https://leanprover-community.github.io/) --> [mathlib](https://leanprover-community.github.io/mathlib-overview.html) --> [algebra.group.defs](https://leanprover-community.github.io/mathlib_docs/algebra/group/defs.html#top) --> [group](https://leanprover-community.github.io/mathlib_docs/algebra/group/defs.html#group)

Exposition in a context of [[homotopy type theory]]:

* [[Egbert Rijke]], Section 19 in: *Introduction to Homotopy Type Theory*, Cambridge Studies in Advanced Mathematics, Cambridge University Press [[arXiv:2212.11082](https://arxiv.org/abs/2212.11082)]

Alternative discussion (under [[looping and delooping]]) of groups in [[homotopy type theory]] as pointed connected [[homotopy 1-types]]:

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 4 of: *[[Symmetry]]* (2021) [[pdf](https://unimath.github.io/SymmetryBook/book.pdf)]

[[!redirects groups]]

category: group theory


group action on an sphere

[[!redirects group actions on spheres]]
[[!redirects group actions on spheres]]

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

#Contents#
* table of contents
{:toc}

## Idea

The possible [[actions]] of well-behaved [[topological groups]] (such as [[compact Lie groups]]) on [[topological space|topological]] [[n-spheres]] display various interesting patterns in their classification.

This entry is meant to eventually list and discuss some of these. For the moment it mainly just collects some references.

## Properties

(...)

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

Given an continuous [[action]] of the [[circle group]] on the [[topological space|topological]] [[4-sphere]], its [[fixed point]] space is of one of two types:

1. either it is the [[0-sphere]] $S^0 \hookrightarrow S^4$

1. or it has the [[rational homotopy theory|rational homotopy type]] of an even-dimensional sphere.

=--

([Félix-Oprea-Tanré 08, Example 7.39](#FelixOpreaTanre08))

(...)

## References

Discussion of [[free actions|free]] [[group actions on spheres]] by [[finite groups]] includes

* {#Wall78} [[C. T. C. Wall]], _Free actions of finite groups on spheres_, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/wall7.pdf))

* [[Alejandro Adem]], _Constructing and deconstructing group actions_ ([arXiv:0212280](http://arxiv.org/abs/math/0212280))

Discussion of [[circle group]]-actions on spheres includes

* {#FelixOpreaTanre08} [[Yves Félix]], John Oprea, [[Daniel Tanré]], _Algebraic Models in Geometry_, Oxford University Press 2008

The subgroups of [[special orthogonal group|SO(8)]] which act freely on $S^7$ have been classified in

* J. A. Wolf, _Spaces of constant curvature_, Publish or Perish, Boston, Third ed., 1974

and lifted to actions of [[Spin group|Spin(8)]] in

* S. Gadhia, _Supersymmetric quotients of M-theory and supergravity backgrounds_, PhD thesis, School of Mathematics, University of Edinburgh, 2007

Further discussion of these actions of $Spin(8)$ on the [[7-sphere]] is in

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], Sunil Gadhia, [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], _Half-BPS M2-brane orbifolds_, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. ([arXiv:1007.4761](http://arxiv.org/abs/1007.4761), [Euclid](https://projecteuclid.org/euclid.atmp/1408561553))

where they are related to the [[black brane|black]] [[M2-brane]] [[BPS state|BPS]]-solutions of [[11-dimensional supergravity]] at [[ADE-singularities]].