classifying topos

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

#Contents#
* table of contents
{:toc}

## Idea

The _classifying topos_ for a given type of mathematical [[structure]] $T$ --- for example the structures: "[[group]]", "[[torsor]]", "[[ring]]", "[[category]]"  etc. --- is a ([[Grothendieck topos|Grothendieck]]) [[topos]] $S[T]$ such that [[geometric morphisms]] $f: E \to S[T]$ are the same as structures of this sort in the topos $E$, i.e. groups [[internalization|internal to]] $E$, torsors internal to $E$, etc.  In other words, a classifying topos is a [[representing object]] for the functor which sends a topos $E$ to the category of structures of the desired sort in $E$.

In particular for $E$ a [[sheaf topos]] on a [[topological space]] $X$ and $G$ a (bare, i.e. discrete) [[group]], a $G$-torsor in $E$ is a $G$-[[principal bundle]] over $X$. There is a classifying topos denoted $B G$, such that the [[groupoid]] $G Bund(X)$ of $G$-[[principal bundle]]s over $X$ is equivalent to geometric morphims $Sh(X) \to B G$:

$$
  G Bund(X) \simeq Topos(Sh(X), B G)
  \,.
$$

This is evidently analogous to the notion of [[classifying space]] in [[topology]], which for the discrete group $G$ is a [[topological space]] $\mathcal{B} G$ such that

$$
  \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G)
  \,.
$$

Hence one can think of classifying topoi as a grand generalization of the notion of [[classifying space]] in topology.

## Definition

In a tautological way, every [[topos]] $F$ is the classifying topos for something, namely for the categories of [[geometric morphisms]] $E \to F$ into it. The concept of [[geometric theory]] allows one to usefully interpret these categories as _categories of certain structures in $E$_ :

as decribed in _[Geometric theories -- In terms of sheaf topoi](geometric+theory#InTermsOfSheafTopoi)_, every [[sheaf topos]] $F$ is a completion $S[T]$ of the [[syntactic category]] $C_T$ of _some_ [[geometric theory]] $T$

$$
  F \simeq S[T]
  \,.
$$

And structures of type $T$ in $E$ is what geometric morphisms $E \to F$ classify.

So the **classifying [[topos]]** for the [[geometric theory]] $T$ is a [[Grothendieck topos]] $S[T]$ equipped with a "universal model $U$ of $T$".  This means that for any Grothendieck topos $E$ together with a model $X$ of $T$ in $E$, there exists a unique (up to isomorphism) [[geometric morphism]] $f: E \to S[T]$ such that $f^*$ maps the $T$-model $U$ to the model $X$.  More precisely, for any Grothendieck topos $E$, the category of $T$-models in $E$ is equivalent to the category of geometric morphisms $E \to S[T]$.

The fact that a classifying topos is like the ambient [[set theory]] but equipped with that universal model is essentially the notion of _[[forcing]]_ in [[logic]]: the passage to the [[internal logic]] of the classifying topos _forces_ the universal model to exist.

If $C_T$ is the [[syntactic category]] of $T$, so that $T$-models are the same as [[geometric category|geometric functors]] out of $C_T$, then this universal model can be identified with a certain geometric functor

$$
  U : C_T \to S[T]
  \,.
$$

Its universality property means that any geometric functor

$$
  X : C_T \to E
$$

factors essentially uniquely as

$$
  X : C_T \stackrel{U}{\to} S[T] \stackrel{f^*}{\to} E
$$

for $U$ the universal model and $f^*$ the [[left adjoint]] part of a [[geometric morphism]].  More precisely, composition with $U$ defines an equivalence between the category of geometric morphisms $E\to S[T]$ and the category of geometric functors $C_T\to E$.

More specifically, for any [[cartesian theory]], [[regular theory]] or [[coherent theory]] $\mathbb{T}$ (which in ascending order are special cases of each other and all of geometric theories), the corresponding [[syntactic category]] $\mathcal{C}_{\mathbb{T}}$ comes equipped with the structure of a [[syntactic site]] $(\mathcal{C},\mathbb{T}, J)$ (see there) and the classifying topos for $\mathbb{T}$ is the [[sheaf topos]] $Sh(\mathcal{C}_{\mathbb{T}}, J)$.

Classifying toposes can also be defined over any [[base topos]] $S$ instead of [[Set]].  In that case "Grothendieck topos" is replaced by "[[bounded topos|bounded]] $S$-topos". The _general existence of classifying toposes_ for geometric theories for bounded $S$-toposes is then intimately connected to the existence of the [[classifying topos for the theory of objects]] which in turn hinges on the existence of a [[natural number object]] in $S$. See [below](#ForObjects) and, for further details, [[classifying topos for the theory of objects]] or Blass ([1989](#blass)).

If the classifying topos of a geometric theory $T$ is a [[presheaf topos]], one calls $T$ a _[[theory of presheaf type]]_.

## Background on the theory of theories

The notion of _classifying topos_ is part of a trend, begun by [[Bill Lawvere|Lawvere]], of viewing a mathematical [[theory]] in [[logic]] as a [[category]] with suitable [[exact functor|exactness]] properties and which contains a "generic model", and a [[model]] of the theory as a functor which preserves those properties.  This is described in more detail at [[internal logic]] and [[type theory]], but here are some simple examples to give the flavor.  The original example is that of a 'finite products theory':

* **Finite products theory.**  Roughly speaking, a 'finite products theory', '[[Lawvere theory]]', or '[[algebraic theory]]' is a [[theory]] describing some mathematical structure that can be defined in an arbitrary category with finite [[product]]s.  An example would be the theory of [[groups]]. As explained in the entry for [[Lawvere theory]], for each such theory $T$ there is a category with finite products $C_{fp}[T]$ -- the [[syntactic category]], which serves as a "classifying category" for $T$, in that models of $T$ in the category of sets correspond to product-preserving [[functor]]s $f : C_{fp}[T] \to Set$.   More generally, for any category with finite products, say $E$, models of $T$ in $E$ correspond to product-preserving functors $f : C_{fp}[T] \to E$.

* **Finite limits theory.** Next up the line is the notion of  'finite limits theory', sometimes called an [[essentially algebraic theory]].  This is roughly a theory describing some structure that can be defined in an arbitrary category with [[finite limit]]s (also called a [[finitely complete category]]).  An example of a finite limits theory would be the theory of categories.   (The notion of 'category' requires finite limits, while the notion of 'group' does not, because categories but not groups involve a _partially defined_ operation, namely composition of morphisms.)    Every finite limits theory $T$ admits a [[syntactic category|classifying category]] $C_{fl}(T)$: a finitely complete category such that models of $T$ in a category $E$ with finite limits correspond to functors $f: C_{fl}(T) \to E$ that preserve finite limits.  (Such functors are called [[left exact]], or 'lex' for short.)

* **Geometric theory.** Further up the line, a [[geometric theory]] is roughly a theory which can be formulated in that fragment of first-order logic that deals in finite limits and arbitrary (small) colimits, plus certain [[familial regularity and exactness|exactness]] properties the details of which need not concern us.  The point is that a category with finite limits, small colimits, and appropriate exactness is just a Grothendieck topos, and a functor preserving finite limits and small colimits is just the inverse image part of a geometric morphism.  Just as in the previous two cases, any 'geometric theory' has a classifying category $S[T]$ (which is now a Grothendieck topos) which possesses a "generic object" for that theory, and T-models in any other Grothendieck topos E can be identified with geometric morphisms $f\colon E\to S[T]$, or specifically with their inverse image parts.

Each type of theory may be considered a $2$-theory, or [[doctrine]].  Furthermore, each type of theory can be promoted to a theory "further up the line", by [[completion|freely adding]] the missing structure to the classifying category.  This can always be done purely formally, but in a few cases this promotion also has other, more explicit descriptions.

For instance, to go from a finite products theory $T$ to the corresponding finite limits theory, we can take the opposite of the category of [[finitely presentable object|finitely presentable models]] of $T$ in $Set$, thanks to [[Gabriel-Ulmer duality]].  Similarly, to go from a finite limits theory to the classifying topos of the corresponding geometric theory, we can take the category of presheaves on the classifying category of the finite limits theory.

## Properties

### Geometric morphisms equivalent to morphisms of sites
 {#GeometricMorphismsAndMorphismsOfSites}

The fact that classifying toposes are what they are all comes down,
if spelled out explicitly, to the fact that a [[geometric morphism]]
$f : \mathcal{E} \to \mathcal{F}$ of toposes can be identified with a certain
morphism of [[site]]s $C_{\mathcal{E}}$, $C_{\mathcal{F}}$  for these toposes, going the other way round, $C_\mathcal{E} \leftarrow C_{\mathcal{F}}$, and having certain properties. If here $C_\mathcal{F}$ is the [[syntactic site]] of some [[theory]] $\mathbb{T}$ and we choose $C_{\mathcal{E}}$ to be the [[canonical site]] of $\mathcal{E}$ (itself equipped with the [[canonical coverage]]) this makes manifest why the geometric morphisms in $\mathcal{F}$ correspond to [[model]]s of $\mathbb{T}$ in $\mathcal{E}$.

We now say this in precise manner. In the following a _[[cartesian site]]_ means a [[site]] whose underlying category is [[finitely complete category|finitely complete]].

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

Let $(\mathcal{C}, J)$ and $(\mathcal{D}, K)$ be [[cartesian site]]s such that $\mathcal{C}$ is a [[small category]], $\mathcal{D}$ is an [[essentially small category]] and the [[coverage]] $K$ is [[subcanonical coverage|subcanonical]].

Then a [[geometric morphism]] of the corresponding [[sheaf toposes]]

$$
  f : Sh(\mathcal{D}, K) \to Sh(\mathcal{C}, J)
$$

is induced by a [[morphism of sites]]

$$
  (\mathcal{D}, K) \leftarrow (\mathcal{C}, J)
$$

precisely if the [[inverse image]] of $f$ respects the [[Yoneda embedding]]s $j$ as

$$
  \array{
    \mathcal{D } &\leftarrow& \mathcal{C}
    \\
    {}^{\mathllap{j_{\mathcal{D}}}}\downarrow && \downarrow^{\mathrlap{j_{\mathcal{C}}}}
    \\
    Sh(\mathcal{D}, K) &\stackrel{f^*}{\leftarrow}&
    Sh(\mathcal{C}, J)
  }
  \,.
$$

=--

This appears as ([Johnstone, lemma C2.3.8](#Johnstone)).

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

It suffices to observe that the factorization, if it exists, is a morphism of sites.

=--

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

Let $(\mathcal{C},J)$ be a [[small category|small]] [[cartesian site]] and let $\mathcal{E}$ be any [[sheaf topos]]. Then we have an [[equivalence of categories]]

$$
  Topos(\mathcal{E}, Sh(\mathcal{C}, J))
  \simeq
  Site((\mathcal{C}, J), (\mathcal{E}, C))
$$

between the [[geometric morphism]]s from $\mathcal{E}$ to $Sh(\mathcal{C}, J)$ and the morphisms of [[sites]] from $(\mathcal{C}, J)$ to the [[big site]] $(\mathcal{E}, C)$ for $C$ the [[canonical coverage]] on $\mathcal{E}$.

=--

This appears as ([Johnstone, cor. C2.3.9](#Johnstone)).

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

This means that a sheaf topos $Sh(\mathcal{C},J)$ is the classifying topos for the theory of [[local algebras]] determined by the [[site]] $(\mathcal{C},J)$.

=--

## Examples

We list and discuss explicit examples of classifying toposes.

### For nothing

Since the [[empty theory|empty geometric theory]] has a unique model in any Grothendieck topos, its classifying topos is the terminal Grothendieck topos, namely $Set$.

Note that $Set$ has no non-trivial [[subtopos|subtoposes]].  Thus relative to the empty signature, the empty theory is complete: either a sequent $\sigma$ follows from $\mathbb{T}_{\Sigma_\emptyset}$ or $\{\sigma\}$ is inconsistent.  In other words, the only toposes classifying theories over the empty signature are $Set$ and the inconsistent topos $\mathbf{1}$.

The empty theory is not the only theory classified by $Set$: any theory that has a unique model in any Grothendieck topos will do.  For instance, the theories of initial objects, of terminal objects, and of natural numbers objects are all classified by $Set$.  Note that these theories have nonempty signatures, e.g. to axiomatize initial objects one has to add the sequent $\top\vdash_x\bot$ to the [[theory of objects]] below, where $x$ is the unique sort.

### For contradictions

The contradictory theory $\{\top\vdash\bot\}$ has no models in any nontrivial Grothendieck topos.  Thus its classifying topos is the initial Grothendieck topos $\mathbf{1}$ (which is a [[strict initial object]]).

More generally, any theory that has no models in any nontrivial Grothendieck topos is classified by $\mathbf{1}$, such as the theory of [[zero objects]].

### For objects
 {#ForObjects}

The [[presheaf topos]] $[FinSet, Set]$ on the [[opposite category]] of [[FinSet]] is the classifying topos for the [[theory of objects]], sometimes called the "object classifier". This is not to be confused with the notion of an [[object classifier]] in an [[(∞,1)-topos]]  and maybe better called in full the _[[classifying topos for the theory of objects]]_.

For $E$ any [[topos]], a [[geometric morphism]] $E \to [FinSet,Set]$ is equivalently just an [[object]] of $E$, given by the inverse image of $FinSet(\{ * \}, -)$.

### For pointed objects
 {#ForPointedObjects}

Similarly, the presheaf topos $[FinSet_*, Set]$ (where $FinSet_*$ is the category of finite [[pointed sets]]) classifies [[pointed objects]]; cf. [this question](http://mathoverflow.net/questions/85600/what-do-gamma-sets-classify) and answer.  This is the topos of "$\Gamma$-sets"; see [[Gamma-space]].

### For groups

We discuss the [[finite product theory]] of [[group]]s. This theory has a classifying category $C_{fp}(Grp)$.  $C_{fp}(Grp)$ is a category with finite products equipped with an object $G$, the "[[walking]] group", a morphism $m: G \times G \to G$ describing multiplication, a morphism $inv : G \to G$ describing inverses, and a morphism $i: 1 \to G$ describing the identity element of $G$, obeying the usual group axioms.   For any category with finite products, say $E$, a finite-product-preserving functor $f: C_{fp}(Grp) \to E$ is the same as a [[group object]] in $E$.   For more details, see [[Lawvere theory]].

We can promote $C_{fp}(Grp)$ to a category with finite limits, $C_{fl}(Grp)$, by adjoining all finite limits.   As mentioned above, one way to do this is to take the category of models of $C_{fp}(Grp)$ in Set, which is simply $Grp$, and then take the full subcategory of [[finitely presentable object|finitely presentable]] groups.  By [[Gabriel-Ulmer duality]], the opposite of this is $C_{fl}(Grp)$.
For any category with finite products, say $E$, a left exact functor $f: C_{fl}(Grp) \to E$ is the same as a [[group object]] in $E$.

We can further promote $C_{fl}(Grp)$ to a [[Grothendieck topos]] by taking the category of [[presheaves]].  This gives the classifying topos for groups:
$$       S[Grp] = Set^{C_{fl}(Grp)^{op}}  \, .
$$
By invoking [[Diaconescu's theorem]], for any Grothendieck topos, say $E$, a left exact left adjoint functor $ f^*: S[Grp] \to E $ is the same as a [[group object]] in $E$.

### For rings

The discussion above for groups can be repeated verbatim for rings, since they too are described by a finite products theory.

### For (inhabited) linear orders
 {#ForLinearOrders}

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

The category of [[cosimplicial sets]] $[\Delta, Set]$ -- hence the [[presheaf topos]] over the [[opposite category]] $\Delta^{op}$ of the [[simplex category]] --  is the classifying topos for [[inhabited set|inhabited]] [[linear orders]].

=--

This appears as ([Moerdijk 95, prop. 5.4](#Moerdijk95)).

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

For ease of notation we discuss this in [[Set]], hence we show that [[geometric morphisms]] $Set \to PSh(\Delta^{op})$ are equivalently [[linear orders]]. Or, by [[Diaconescu's theorem]], that [[flat functors]]

$$
  X : \Delta^{op} \to Set
$$

are equivalently linear orders. Evidently, such a functor is in particular a [[simplicial set]] and we will show that $X$ being flat is equivalent to this simplicial set being the [[nerve]] of an [[inhabited set|inhabited]] linear order regarded as a [[category]] (a [[(0,1)-category]]).

First assume that $X$ is a [[flat functor]]. Since (by the discussion there) this preserves all [[finite limits]] that exist in $\Delta^{op}$, equivalently that it sends the finite [[colimits]] that exist in $\Delta$ to limits in $Set$, it in particular sends the gluings of intervals

$$
  \begin{aligned}
    [n] & \simeq [k] \coprod_{[0]} [l] \;\;\;\; (n = k + l)
    \\
    & \simeq [1] \coprod_{[0]} [1] \coprod_{[0]} \cdots \coprod_{[0]} [1]
  \end{aligned}
$$

in $\Delta$ to [[isomorphisms]]

$$
  \begin{aligned}
    X_n & \simeq X_k \times_{X_0} X_l
    \\
    & \simeq
      X_1 \times_{X_0} \cdots \times_{X_0} X_1
  \end{aligned}
  \,.
$$

This are the [[Segal space|Segal relations]] that say that $X$ is the [[nerve]] of a [[category]].

Moreover, since [[monomorphisms]] are characterized by [[pullbacks]], $F$ being flat means that it sends jointly epimorphic families of morphisms in $\Delta$ to monomorphisms in $Set$. In particular, the epimorphic family $\{\partial_0 : [0] \to [1], \partial_1 : [0] \to [1]\}$ is sent to an injection

$$
  (d_0, d_1) : X_1 \hookrightarrow X_0 \times X_0
  \,.
$$

Since $X_1$ is the set of [[morphisms]] of the category that $X$ is the nerve of, this means that there is at most one morphism in this category from any one object to any other. Hence this category is a [[poset]].

Finally to show that this poset is an inhabited linear order, we use the fact that a functor is flat precisely if its [[category of elements]] [[filtered category|cofiltered]].

This means

1. The category of elements is inhabited, hence the poset of which $X$ is the nerve is inhabited.

1. For every two elements $y, z \in X_0$ there exist morphisms $\alpha, \beta : [0] \to [k]$ in $\Delta$ and $w \in X_k$ such that $X(\alpha) : w \mapsto y$ and $X(\beta) : w \mapsto z$. Since $X$ is the nerve of a poset, this means that there is a totally ordered set $w  = (w_0 \leq  \cdots \leq w_k)$ and $y$ and $z$ are among its elements $y = w_{\alpha(0)}$, $z = w_{\beta(0)}$. Accordingly we have either $y \leq z$ or $z \leq y$ and hence $X$ is in fact the nerve of a [[total order]].

1. If $y,z$ are elements in the total order  with  $y \leq z$ and $z \leq y$, this means that in the nerve we have elements $(y,z) \in X_1$ and $(z,y) \in X_1$ with $d_0(y,z) = d_1(z,y)$ and $d_1(y,z) = d_1(z,y)$.

   By co-filtering, there exists a [[cone]] over this diagram in the category of elements, hence morphisms $\alpha, \beta : [1] \to [k]$ in $\Delta$ and $w \in X_k$ such that

   1. $X(\alpha) : w \mapsto (y,z)$ and $X(\beta) : w \mapsto (z,y)$;

   1. $\partial_0 \circ \alpha = \partial_1 \circ \beta$ and $\partial_1 \circ \alpha = \partial_0 \circ \beta$.

   Here the last condition in $\Delta$ can only hold if $\alpha = \beta = const_{i}$, hence if $y = z$.

Conversely, assume that $X$ is the nerve of a linear order. We show that then it is a flat functor $X : \Delta^{op} \to Set$.

(...)

=--

### For intervals
 {#ForIntervals}

[[Andre Joyal]] showed that $Set^{{\Delta}^{op}}$, the category of [[simplicial sets]], is the classifying topos for [[linear intervals]].

Specifically a [[geometric morphism]] from $Set$ to $Set^{{\Delta}^{op}}$ is an [[linear interval]] in [[Set]], meaning a [[totally ordered set]] with distinct [[top]] and [[bottom]] elements.  In general, a linear interval is a model for the one-sorted [[geometric theory]] whose [[signature]] consists of a binary [[relation]] $\leq$ and two [[constant|constants]] $0$, $1$, subject to the following [[axiom|axioms]]:

* $\vdash (x \leq x)$
* $\exists_y (x \leq y) \wedge (y \leq z) \vdash (x \leq z)$
* $(x \leq y) \wedge (y \leq x) \vdash (x = y)$
* $\vdash (x \leq y) \vee (y \leq x)$
* $\vdash (0 \leq x) \wedge (x \leq 1)$
* $(0 = 1) \vdash false$

(Joyal calls this a **strict** linear interval; by removing the hypothesis of distinct top and bottom, one arrives at a weaker notion he calls "linear interval". Linear intervals in this sense are classified by the topos $Set^{\Delta_{a}^{op}}$, where $\Delta_a$, sometimes called the algebraist's Delta or the augmented [[simplex category]], is the category of all finite ordinals including the empty one.)

The generic such interval is $\Delta^1 \in Set^{{\Delta}^{op}}$; see [[generic interval]] for more details and references.

### For abstract circles

The category of [[cyclic sets]] is the classifying topos for [[abstract circles]] ([Moerdijk 96](#Moerdijk96)).

### For local rings

#### Local rings

The classifying topos for [[local ring|local rings]] is the [[big Zariski topos]] of the [[scheme]] $Spec(\mathbb{Z})$.  A **local ring** is a model of the geometric theory of commutative unital rings subject to the extra axioms

* $(0 = 1) \vdash false$
* $x + y = 1 \vdash \exists_z (x z = 1) \vee \exists_z (y z = 1)$

In a topos of sheaves over a [[sober space]], a local ring is precisely what algebraic geometers usually call a "sheaf of local rings": namely, a sheaf of rings all of whose [[stalk|stalks]] are local. See [[locally ringed topos]]. This is a special case of the case of [Cover-preserving flat functors](#CoverPreservingFLatFunctors) below.

#### Strict local rings
 {#StrictLocalRings}

For $Spec R$ an [[affine scheme]], the [[étale topos]] $Sh(X_{et})$ classifies "[[strict local ring|strict local R-algebras]]". The [[point of a topos|points of this topos]] are _[[strict Henselian ring|strict Henselian R-algebras]]_ ([Hakim, III.2-4](#Hakim)) and ([Wraith](#Wraith)).

See also [this MO discussion](http://mathoverflow.net/questions/48690/what-does-an-etale-topos-classify)

### For principal bundles {#PrincipalBund}

Essentially every topos may be regarded as a classifying topos for certain [[torsor]]s/[[principal bundle]]s.

#### Over bare groups {#BareGTor}

For any (bare / discrete) [[group]] $G$, write $\mathbf{B}G$ for its [[delooping]] groupoid, the groupoid with a single object and $G$ as its endomorphisms. The presheaf topos

$$
  G Set := PSh(\mathbf{B}G)
$$

of [[permutation representation]]s (objects are [[set]]s equipped with a $G$-[[action]], morphisms are $G$-equivariant maps between these) is the classifying topos for $G$-[[torsor]]s.

For example, if $X$ is a [[topological space]], geometric morphisms from the [[sheaf topos]] $Sh(X)$ of [[sheaves]] on (the [[category of open subsets]] of) $X$ to $G Set$ are the same as $G$-[[principal bundle]]s over $X$

$$
  G Bund(X) \simeq Topos(Sh(X), G Set)
  \,.
$$

This follows via [[Diaconescu's theorem]], which asserts that [[geometric morphism]]s $Sh(X) \to Sh(\mathbf{B}G)$ are equivalent to [[flat functor]]s

$$
  \mathbf{B}G \to Sh(X)
  \,.
$$

Such a flat functor picks a single sheaf on $X$ and encodes a $G$-action on this sheaf such that this sheaf is the sheaf of [[section]]s of a $G$-[[principal bundle]] on $X$.

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

Let $G$ be a (bare, discrete) group, write $\mathcal{B}G \in $ [[Top]] for the ordinary [[classifying space]] and $\mathbf{B}G \in $ [[Grpd]] the one-object groupoid version of $G$. There is a canonical [[geometric morphism]]s

$$
  PSh(\mathbf{B}G)
  \to
  Sh(\mathcal{B}G)
  \,.
$$

This is a _weak homotopy equivalence_ of toposes, in that it induces isomorphisms on [[geometric homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of the terminal object.

=--

This is ([Moerdijk 95, theorem 1.1, proven in chapter IV](#Moerdijk95)).

#### In terms of geometric theories {#BareGTorGeomTheo}

A [[geometric theory]] $T$ whose [[model]]s are $G$-torsors can be described as follows.  It has one sort, $X$, and one unary operation $g:X\to X$ for every element $g\in G$.  It has algebraic axioms $\top\vdash_x \;1(x) = x$ and $\top\vdash_x \;g(h(x)) = (g h)(x)$, which make $X$ into a $G$-set, and geometric axioms $\top \vdash\; \exists x \in X$ (inhabited-ness), $g(x) = x \;\vdash_x \;\bot$ for all $g\neq 1$ (freeness), and $\top\vdash_{x,y}\; \bigvee_{g\in G}\; g(x) = y$ (transitivity).

#### Over topological groups {#TopGTor}

If $G$ is a general [[topological group]] we have a [[simplicial object|simplicial]] [[topological space]] $G^{\times \bullet}$. The category $Sh(G^{\times \bullet})$ of [[sheaves on a simplicial topological space|sheaves on this simplicial space]] is a topos.

This is such that for $X$ a topological space, geometric morphisms $Sh(X) \to Sh(G^{\times \bullet})$ classifies topological $G$-principal bundles on $X$.

This idea admits generalizations to [[localic groups]] --- and even to [[localic groupoids]].  For more details, see [[classifying topos of a localic groupoid]] .

#### The universal $G$-bundle topos {#UniversalBundle}

At [[generalized universal bundle]] and [[principal ∞-bundle]] it is discussed that the principal bundle classified by a morphims into a classifying object is its [[homotopy fiber]], and how the universal bundle is a replacement of the point such that its ordinary pullback models that [[homotopy pullback]].

Concretely, for $G$ a [[group]] and $\mathbf{B}G = \{\bullet \stackrel{g \in G}{\to} \bullet\}$ in [[∞Grpd]] its [[delooping]] [[groupoid]], the universal $G$-bundle is really just the point inclusion

$$
  \array{
    *
    \\
    \downarrow
    \\
    \mathbf{B}G
  }
$$

in that for $X \to \mathbf{B}G$ a morphism, the corresponding $G$-[[principal ∞-bundle]] in [[∞Grpd]] is the [[homotopy pullback]]

$$
  \array{
    P &\to& *
    \\
    \downarrow &{}^{\simeq}\swArrow& \downarrow
    \\
    X &\to& \mathbf{B}G
  }
  \,.
$$

We can send this morphism $(* \to \mathbf{B}G)$ in [[Grpd]] with

$$
  PSh(-) : Grpd \to Toposes
$$

to the [[2-category of toposes]] to get a [[geometric morphism]]

$$
  \array{
    PSh(*) = Set
    \\
    \downarrow^{\mathrlap{p}}
    \\
    PSh(\mathbf{B}G) = Set^G
  }
  \,.
$$

By the rules of morphisms of [[site]]s we have that the [[inverse image]] $p^* : PSh(\mathbf{B}G) \to Set$ is precomposition with $p : * \to \mathbf{B}G$, i.e. the functor that just forgets the $G$-action on a set.

Its [[left adjoint]] $p_! : Set \to PSh(\mathbf{B}G)$ is the functor

$$
  p_! : S \mapsto S \times G
$$

which sends a set $S$ to the $G$-set $S \times G$ equipped with the evident $G$-action induced by that of $G$ on itself.

Because for $(V,\rho)$ any set with $G$-action $\rho$ we have naturally

$$
  Hom_{Set}(S,V) \simeq Hom_{Set^G}(S \times G, (V,\rho))
  \,.
$$

The object

$$
  p_!(*) = G \in PSh(\mathbf{B}G)
$$

singled out in this way is the universal object in $Set^G$, namely $G$ equipped with the canonical $G$-action on itself.

It ought to be true that the topos-incarnation of the $G$-principal bundle on a topological space $X$ classified by a [[geometric morphism]] $Sh(X) \to PSh(\mathbf{B}G)$ is the $(2,1)$-pullback

$$
  \array{
    \mathcal{P} &\to& Set
    \\
    \downarrow &{}^{\simeq}\swArrow& \downarrow
    \\
    Sh(X) &\to& PSh(\mathbf{B}G)
  }
  \,.
$$

> needs more discussion...

### For general localic groupoids
  {#ForLocalicGroupoids}

In fact, _any_ [[Grothendieck topos]] can be thought of as a classifying topos for some [[localic groupoid]].  This is related to the discussion above, since Joyal and Tierney showed that any Grothendieck topos is equivalent to the $B G$ for some [[localic groupoid]] $G$.  A useful discussion of this idea starts
[here](http://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_2.html#c012724).

### For flat functors

As a special case of the above, any [[presheaf topos]], i.e. any topos of the form $Set^{C^{op}}$, is the classifying topos for [[flat functors]] from $C$ (sometimes also called "$C$-[[torsor]]s").  In other words, geometric morphisms $E \to Set^{C^{op}}$ are the same as [[flat functor]]s $C \to E$.  This is [[Diaconescu's theorem]].  If $C$ has finite limits, then a flat functor $C \to E$ is the same as a functor that preserves finite limits.

### For geometric theories / cover-preserving flat functors on a site
 {#CoverPreservingFLatFunctors}

Another way, apart from that [above](#ForLocalicGroupoids), of viewing any [[Grothendieck topos]] $E$ as a classifying topos is to start with a small [[site]] of definition for it.  Any such site gives rise to a [[geometric theory]] called the theory of [[cover-preserving functor|cover-preserving]] flat functors on that site (also called the [[theory of flat functors| theory of J-continuous flat functors]], for syntactic details see there!).  The classifying topos of this theory is again $E$.

Moreover, for any object $X$ of $E$, there is a small site of definition
for $E$ which includes $X$, and thus for which $X$ is (part of) the universal object.

We have:

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

Every [[sheaf topos]] has a [[cartesian site]] $(\mathcal{C}, J)$ of definition.

This $Sh(\mathcal{C}, J)$ is the classifying topos for cover-preserving [[flat functor]]s out of $\mathcal{C}$.

Every category of such functors is the category of [[model]]s of some geometric theory, and for every geometric theory there is such a cartesian site.

=--

This appears as ([Johnstone, remark D3.1.13](#Johnstone)).

### For local algebras
 {#LocalAlgebras}

As a special case or rather re-interpretation of the [above](#CoverPreservingFLatFunctors), let $\mathcal{T}$ be any
[[essentially algebraic theory]] and equip its [[syntactic category]] $\mathcal{C}_{\mathbb{T}}$ with some [[coverage]] $J$. Then the [[sheaf topos]] $Sh(\mathcal{C}_{\mathbb{T}}, J)$ is the classifying topos for _local $\mathbb{T}$-algebras_ :

for $Sh(X)$ any [[sheaf topos]] a [[geometric morphism]]

$$
  \mathcal{O} : Sh(X) \to Sh(\mathcal{C}_{\mathbb{T}}, J)
$$

is

1. a $\mathbb{T}$-[[algebra over an algebraic theory|algebra]] in $Sh(X)$, hence a [[sheaf]] of $\mathbb{T}$-algebras over the site $X$;

1. such that this sheaf of algebras is local as seen by the respective topologies.

See [[locally algebra-ed  topos]] for more on this.

By prop. \ref{EveryToposIsAClassifyingToposForLocalAlgebras} we have that every [[sheaf topos]] is the classifying topos of _some_ theory of local algebras.

The [[vertical categorification]] of this situation to the context of [[(∞,1)-category]] theory is the notion of [[structured (∞,1)-topos]] and of [[geometry (for structured (∞,1)-toposes)]]:

The geometry $\mathcal{G}$ is the [[(∞,1)-category]] that plays role of the syntactic theory. For $\mathcal{X}$ an [[(∞,1)-topos]], a model of this theory is a limits and covering-preserving [[(∞,1)-functor]]

$$
  \mathcal{G} \to \mathcal{X}
  \,.
$$

The [[Yoneda embedding]] followed by [[∞-stackification]]

$$
  \mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G})
  \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G})
$$

constitutes a model of $\mathcal{G}$ in the (Cech) [[∞-stack]] [[(∞,1)-topos]] $Sh_{(\infty,1)}(\mathcal{G})$ and exhibits it as the classifying topos for such models (geometries):

This is _[[Structured Spaces]]_ [prop 1.4.2](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0459v1.pdf#page=26).

## As a generalization of the notion of classifying space in topology

In view of the analogy between the classifying topos denoted $B G$, such that the [[groupoid]] $G Bund(X)$ of $G$-[[principal bundle]]s over $X$ is equivalent to geometric morphims $Sh(X) \to B G$:

$$
  G Bund(X) \simeq Topos(Sh(X), B G)
  \,
$$

and the notion of [[classifying space]] in [[topology]], which for the discrete group $G$ is a [[topological space]] $\mathcal{B} G$ such that

$$
  \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G)
  \,
$$

we should expect there to be a topos analog of the total space, $E G$, for the classifying space. This analog is the *generic G-torsor*, which is an internal $G$-torsor in the topos $Set^G$.  The important aspect of the space $E G$ is that as a principal $G$-bundle over $\mathcal{B} G$, it is a *universal element*, i.e. the natural transformation $Hom(X, \mathcal{B}G) \to G Bdl(X)$ that it induces (by the [[Yoneda lemma]]) is the isomorphism which exhibits $\mathcal{B}G$ as the object representing the functor $X \mapsto G Bdl(X)$.  For the same Yoneda reasons, the classifying topos $Sh(C_T)$ of any geometric theory $T$ comes with a *generic $T$-model*, which is a $T$-model in $Sh(C_T)$ which represents the functor $E \mapsto T Mod(E)$ in the same way.  For $T$ = the theory of $G$-torsors, this generic model is the generic $G$-torsor.

## Related concepts

* [[theory of flat functors|theory of cover-preserving flat functors]]

* [[representable functor]]

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

* [[classifying space]], [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]]

* [[universal principal bundle]], [[universal principal infinity-bundle]]

* [[classifying morphism]]

* [[classifying (infinity,1)-topos]]

* [[natural number object]]

## References

### General

Early references containing some remarks on the formation of the concept are

* [[Myles Tierney]], _Forcing Topologies and Classifying Toposes_ , pp.211-219 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory_  , Academic Press New York 1976.

* {#J77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Also available as Dover reprint Mineola 2014)

Standard textbook references for classifying topoi of theories

* {#Johnstone}[[Peter Johnstone]], _[[Sketches of an Elephant]]_ , Oxford UP 2002. (In particular, sections B4.2 pp.424-432, D3.2 pp.901-910)

* [[Francis Borceux]], _Handbook of categorical algebra_, (in series Enc. Math. Appl.) vol. 3, Categories and sheaves, Cambridge Univ. Press 1994, Ch.4 Classifying toposes

* [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, Springer Heidelberg 1994. (chap. VIII)

A more advanced reference containing several developments of the general theory, especially in relation with the view of toposes as ‘bridges’, is the monograph

* [[Olivia Caramello]], [_Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic ‘bridges’_](https://global.oup.com/academic/product/theories-sites-toposes-9780198758914), Oxford University Press, 2017.

The relation between the existence of natural number objects and classifying toposes is discussed in

* {#blass} [[Andreas Blass]], *Classifying topoi and the axiom of infinity*, Algebra Universalis **26** (1989) 341-345 [[doi:10.1007/BF01211840](https://doi.org/10.1007/BF01211840)]

  > (in relation to the [[axiom of infinity]])

The study of classifying spaces of topological categories is described in the monograph

* [[Ieke Moerdijk]], _Classifying spaces, classifying topoi_,
Lec. Notes Math. 1616, Springer Verlag 1995

The original theory for a general algebraic theory is developed in

* M. Makkai, G. Reyes, _First-order categorical logic_, Lecture Notes in Mathematics 611, Springer 1977.

The results for the continuous groupoids include

* {#Moerdijk88} [[Ieke Moerdijk]],  The classifying topos of a continuous groupoid I, _Trans. A.M.S._ **310** (1988), 629-668.

* {#Moerdijk90} [[Ieke Moerdijk]], _The classifying topos of a continuous groupoid  II_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, **31** no. 2 (1990), 137-168.  ([web](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1990__31_2_137_0))

* {#Moerdijk95} [[Ieke Moerdijk]], _Classifying spaces and classifying topoi_,  Lecture Notes in Math. **1616**, Springer-Verlag, New York, 1995.

* {#Moerdijk96} [[Ieke Moerdijk]], _Cyclic sets as a classifying topos_, 1996 ([[MoerdijkCyclic.pdf:file]])

Classifying toposes as [[locally algebra-ed (infinity,1)-toposes]] are discussed in section 1.4 of

* [[Jacob Lurie]], _[[Structured Spaces]]_
 {#Lurie}

The [[étale topos]] as a classifying topos for [[strict local rings]] is discussed in

* {#Hakim} [[Monique Hakim]], _Topos annelés et schémas relatifs_, Sec. III.2-4
* {#Wraith} [[Gavin Wraith]], _Generic Galois theory of local rings_

[[Nikolai Durov]] has introduced somewhat a generalization of topos called [[vectoid]] and quite flexible notion of a classifying vectoid in

* [[Nikolai Durov]], _Classifying vectoids and generalisations of operads_, [arxiv/1105.3114](http://arxiv.org/abs/1105.3114), the translation of "Классифицирующие вектоиды и классы операд", Trudy MIAN, vol. 273

### Relation to forcing
 {#ReferencesRelationToForcing}

Reviews of the interpretation of _[[forcing]]_ as the passge to classifying toposes include

* [[Andreas Blass]], [[Andrej Ščedrov]], *Classifying topoi and finite forcing*, Journal of Pure and Applied Algebra **28** (1983) 111-140 [<a href="https://doi.org/10.1016/0022-4049(83)90085-3">doi:10.1016/0022-4049(83)90085-3</a>]

* [[Andrej Ščedrov]], *Forcing and classifying topoi*, Memoirs of the American Mathematical Society (1984) [[AMS:memo-48-295](https://bookstore.ams.org/memo-48-295/5)]

For more see

* {#Roberts15} [[David Roberts]], _Class forcing and topos theory_, talk at _[Topos at l'IHES](https://indico.math.cnrs.fr/event/747/)_ 2015 ([talk notes pdf](https://www.dropbox.com/s/vk1efw7lsvta80p/Roberts_IHES.pdf?dl=0),  [video recording](https://youtu.be/4AaSySq8-GQ))

[[!redirects classifying toposes]]
[[!redirects classifying topoi]]
[[!redirects classifying locale]]
[[!redirects classifying locales]]


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

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

The [[classifying topos]] $\mathcal{S}[\mathbb{O}]$ for the [[theory of objects]] $\mathbb{O}$, or the **object classifier**, as it is also called[^termin], is  the [[presheaf topos]] $[FinSet, Set]$ on the [[opposite category]] of [[FinSet]].

What motivates the terminology, is that for any [[topos]] $E$, [[geometric morphisms]] $E \to \mathcal{S}[\mathbb{O}]$ correspond to [[objects]] of $E$.

This is because by [a standard fact](morphism+of+sites#LemmaForClassifyingToposes) [[geometric morphisms]]

$$
  E \to PSh(FinSet^{op})
$$

are equivalent to [[morphisms of sites]]

$$
  E \leftarrow FinSet^{op}
$$

hence to [[finite limit]]-preserving such [[functors]]. Since finite limits in $FinSet^{op}$ are [[finite colimits]] in [[FinSet]] and since $FinSet$ is generated under finite colimits from the singleton set $\ast$, such functors are uniquely determined by their image of $\ast$, hence by a choice of object in $E$.

[^termin]: As this is not to be confused with the notion of an [[object classifier]] in an [[(∞,1)-topos]], we prefer to call it in full the _classifying topos for the theory of objects_.

[^ultra]: For another remarkable property of this inclusion functor see at [[ultrafilter monad]].

The construction is due to [[Gavin Wraith]] and constituted an important step towards the general theorem on the existence of [[classifying topos|classifying toposes]] for geometric theories in the early development of [[topos theory]].

Similarly

* $PSh(FinSet_\ast^{op})$ is the classifying topos for [[pointed objects]].

* write $Fin\infty Grpd$ for the [[full sub-(∞,1)-category]] on [[∞Grpd]] which is generated under [[finite (∞,1)-colimits]] from the point $\ast$ ([[Higher Algebra|HA, def. 1.4.2.8]]), then the [[(∞,1)-presheaf (∞,1)-topos]] $PSh_\infty(Fin\infty Grpd^{op})$ is the [[classifying (∞,1)-topos]] for objects;

* write $Fin\infty Grpd_\ast$ for pointed finite $\infty$-groupoids in this sense, then $PSh_\infty((Fin\infty Grpd_\ast)^{op})$ is the classifying $(\infty,1)$-topos for pointed objects. See also at _[[spectrum object]]_ _[via excisive functors](spectrum%20object#ViaExcisiveFunctors)_.

## Properties

### Generic object

The _generic_ or **universal object** $U$ is the inclusion $FinSet\hookrightarrow Set$: every object $X$ of $E$ arises from some geometric morphism $f$ as $X\cong f^\ast(U)$.[^ultra]

### The symbolic link to algebra

As the role of the object classifier bears some resemblance to the role of the [[polynomial ring]] $k[x]$ over a ground ring $k$ that 'classifies' elements of k-algebras $A$ via $A\cong Hom_k(k[x], A)$, it is traditionally denoted $\mathcal{S}[U]$, the 'adjunction' of the free (=generic) object to the base topos $\mathcal{S}$, in our case $Set$.

The analogy between topos theory and algebra is pursued further in Bunge&Funk (2006) where, in the context of [[Lawvere distribution|topos distributions]] and the '[[symmetric algebra]]' of a topos (aka the [[symmetric topos]]), $\mathcal{S}[U]$ is shown to play the role of the [[real line]] $\mathbf R$ in [[functional analysis]].

### The relative case

What concerns base toposes $\mathcal{S}$ other than $Set$, it is a theorem due to [[Andreas Blass]] ([Blass 1989](#blass)) that _$\mathcal{S}$ has an object classifier $\mathcal{S}[\mathbb{O}]$ precisely if $\mathcal{S}$ has a [[natural number object]]_.

A consequence of this, discussed in sec. B4.2 of (Johnstone 2002,I p.431), is that classifying toposes for [[geometric theories]] over $\mathcal{S}$ exist precisely if the object classifier $\mathcal{S}[\mathbb{O}]$ exists.

### An alternative characterization

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

The classifying topos $[FinSet, Set]$ is [[equivalence of categories|equivalent]] to the category of finitary [[endofunctors]] on [[Set]] (those that commute with [[filtered colimits]]):

$$
  [FinSet, Set] \simeq End_f(Set)
  \,.
$$

=--

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

Because every [[set]] is the [[filtered colimit]] over its finite [[subsets]].

Constructively we should take a little care over what is meant by "[[finite set]]". A set is the filtered colimit of its _Kuratowski_ finite subsets, but a Kuratowski finite set is the image of some finite cardinal $\{0,\ldots, n-1\}$. The issue is that, unless the superset has decidable equality, we are not necessarily able to eliminate duplicates from the list of elements in the Kuratowski finite subset.
We find that the set is still a filtered colimit of finite cardinals, though not necessarily subsets. The category $FinSet$ can be taken to have the natural numbers as its objects, with morphisms being functions between the corresponding finite cardinals.

=--

### The monoidal point of view

Since the category $End_f(Set) \hookrightarrow End(Set)$ of [[finitary functors]] is naturally a [[monoidal category]] under composition, this induces the structure of a (non-cartesian) [[monoidal category]] also on the [[classifying topos]] $[FinSet, Set]$ and hence makes it a [[monoidal topos]]. A [[monoid]] with respect to this monoidal structure is equivalently a [[finitary monad]].

A related point of view is that $FinSet^{op}$ is the [[free construction|free]] [[cartesian monoidal category]] generated by an object (or by the terminal category), and its [[free cocompletion]] $[FinSet, Set]$ is the free [[cartesian monoidal category|cartesian]] [[monoidally cocomplete category]] generated by an object. Thus $[FinSet, Set]$ plays the role of a cartesian analogue to $[\mathbb{P}^{op}, Set]$, the free [[symmetric monoidal category|symmetric]] [[monoidally cocomplete category]] on an object, where $\mathbb{P} = Core(FinSet)$ is the [[core]] of [[FinSet]], the [[permutation groupoid]]. And in the same way that $[\mathbb{P}^{op}, Set]$ is a [[monoidal topos]] whose monoids are symmetric or permutative [[operads]] (as discussed at _[operad -- a detailed conceptual treatment](operad#Conceptual)_), so $[FinSet, Set]$ is seen as a monoidal topos whose monoids are cartesian operads, aka [[Lawvere theories]]. Some material on this can be found at _[[toddtrimble:Towards a doctrine of operads]]_.

### The functorial point of view

Consider the forgetful [[2-functor]] $U:GrTop^{op}\to Cat$ from the opposite of the 2-category $GrTop$ of Grothendieck toposes and geometric morphisms that maps a topos to its underlying category and a geometric morphism to its [[inverse image functor]]. The object classifier $\mathcal{S}[\mathbb{O}]$ is a [[representing object]] for $U$.

For more on this functorial approach to geometric theories see at [[geometric theory#FunctorialDefinition]] or Johnstone ([2002, vol.1 pp.424ff](#JT02)).

### The generalized space point of view

The universal property says that maps (geometric morphisms) from a Grothendieck topos $\mathcal{E}$ to $\mathcal{S}[\mathbb{O}]$ are equivalent to the objects of $\mathcal{E}$, the sheaves over the generalized space of whatever $\mathcal{E}$ classifies. But $\mathcal{S}[\mathbb{O}]$ is the generalized space of "sets", so sheaves should be understood as continuous set-valued maps.
This is seen most clearly when $\mathcal{E}$ is sheaves over an [[point-free topology|ungeneralized space]] $X$. The sheaf is equivalent to a local homeomorphism to $X$, and the set-valued map takes points of $X$ to the stalks, the fibres of the local homeomorphism.

So what are the sheaves over $\mathcal{S}[\mathbb{O}]$, i.e. its objects? They are continuous maps $F$ from the space of sets to itself. All continuous maps (geometric morphisms) preserve filtered colimits of points, and it follows that $F$ is determined by its action on finite cardinals. Hence $\mathcal{S}[\mathbb{O}]$ is - at least - a subcategory of $[FinSet, Set]$.

### The object classifier and exponentiability

Recall that a [[locale]] $X$ is  called _exponentiable_ (in the category of locales) if the [[exponential object|exponential]] $Y^X$ exists for all locales $Y$. Interestingly, exponentiability of $X$ hinges on the existence of the single exponential $S^X$ where $S$ is the [[Sierpinski space]]: $Y^X$ exists for all $Y$ iff $S^X$ exists.

In the 2-category $GrTop$ the object classifier $\mathcal{S}[\mathbb{O}]$ takes over the role of the Sierpinski space and we have the following

+-- {: .num_prop}
###### Proposition
A [[Grothendieck topos]] $\mathcal{E}$ is [[exponentiable topos|exponentiable]] iff the exponential $\mathcal{S}[\mathbb{O}]^\mathcal{E}$ exists.

=--

This result is due to Johnstone-Joyal ([1982, p.257](#JJ82)) and occurs as theorem 4.3.1 of Johnstone ([2002, vol.1 p.433](#J02)).

## The pointed object classifier

[[pointed object|Pointed objects]] are important in [[homotopy theory]]. Similarly to objects, they arise as models of a geometric theory and this [[theory of objects|theory of pointed objects]] $\mathbb{O}_\ast$  is, of course, classified by a topos, namely, the presheaf topos $[FinSet_\ast, Set]$ on the opposite $FinSet_\ast^{op}$ of the category of finite [[pointed sets]] whose skeleton is [[Segal's category]], hence $[FinSet_\ast, Set]$ is equivalent to the topos of "$\Gamma$-sets": cf. [[Gamma-space]] and for the role as a classifying space the following MO-discussion: [link](http://mathoverflow.net/questions/85600/what-do-gamma-sets-classify) .[^classi]

[^classi]: More generally, classifying toposes for [[Horn theory|universal Horn theories]] $\mathbb{T}$ correspond to the respective toposes of covariant set-valued functors on the category of finitely presentable models of $\mathbb{T}$ ([Blass & Scedrov (1983)](#BlassScedrov83)).

Since the theory $\mathbb{O}_\ast$ is an extension of $\mathbb{O}$, it can be treated relatively as an internal theory in $\mathcal{S}[\mathbb{O}]$. There it is just the theory of elements of the generic set $G$, in other words the propositional theory for the discrete space of $G$. Discrete spaces are always got as slice toposes, so $\mathcal{S}[\mathbb{O}_\ast]$ is equivalent to

$$[FinSet, Set]/G\simeq [(\underset{FinSet^{op}}{\int} G)^{op},Set]\;.$$

$G$ is the inclusion functor $FinSet \to Set$, mapping each natural number $n$ to its finite cardinal $\{0,\ldots,n-1\}$.

The forgetful map (geometric morphism) from $\mathcal{S}[\mathbb{O}_\ast]$ to $\mathcal{S}[\mathbb{O}]$, forgetting the point, is the generic local homeomorphism. Every local homeomorphism is a bipullback of it.

## Related entries

* [[theory of objects]]

* [[classifying topos]]

* [[geometric theory]]

* [[exponentiable topos]]

* [[Sierpinski space]]

## References

* [[Peter Johnstone|P. T. Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). (sec. 6.3)

* {#JT02}[[Peter Johnstone]], _[[Sketches of an Elephant]]_ , Oxford UP 2002. (sections B4.2 pp.424-432, D3.2 pp.901-910)

* {#JJ82}[[Peter Johnstone]], [[André Joyal]], _Continuous categories and exponentiable toposes_ ,  JPAA **25** (1982) pp.255-296.

* [[Peter Johnstone]], [[Gavin Wraith]], _Algebraic Theories in a Topos_ , pp.141-242 in LNM **661** Springer Heidelberg 1978. (ch. IV, pp.175ff)

* [[Saunders Mac Lane]], [[Ieke Moerdijk]]: _[[Sheaves in Geometry and Logic]]_, Springer Heidelberg 1994. (sec.VIII.4)

* {#blass} [[Andreas Blass]], _Classifying topoi and the axiom of infinity_ , Algebra Universalis **26** (1989) 341-345 [[doi:10.1007/BF01211840](https://doi.org/10.1007/BF01211840)]

* {#BlassScedrov83} [[Andreas Blass]], [[Andrej Scedrov]], *Classifying topoi and finite forcing*, Journal of Pure and Applied Algebra **28** (1983) 111-140 [doi;10.1016/0022-4049(83)90085-3]

* [[Marta Bunge]], [[Jonathon Funk]], _Singular Coverings of Toposes_ , Springer LNM **1890** Heidelberg 2006.

* [[Richard Garner]], _Lawvere theories, finitary monads and Cauchy-completion_ , JPAA **218** no.11 (2014) pp.1973–1988. ([preprint](http://comp.mq.edu.au/~rgarner/Papers/FSet.pdf))
 {#Garner13}

* [[Gavin Wraith]], _Topos Elémentaire et Arithmétique_ , Cah. Top. Géom. Diff. Cat. **XIV-2** (1973) pp.217-218. ([proceedings colloque Amiens 1973](http://archive.numdam.org/article/CTGDC_1973__14_2_153_0.pdf), 3.92 MB)

[[!redirects classifying topos for the theory of objects]]
[[!redirects classifying topos for the theory of pointed objects]]


classifying topos of a localic groupoid

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

#Contents#
* table of contents
{:toc}

## Idea

The _classifying topos of a localic groupoid_ $\mathcal{G}$ is an incarnation of a [[localic groupoid]] (possibly a [[topological group]]) in the category of [[toposes]]. At least in good cases, [[geometric morphisms]] into it classify $\mathcal{G}$-[[groupoid principal bundles]], whence the name.

## Definition

A [[localic groupoid]] is a [[groupoid object]] $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ [[internalization|internal to]] [[locales]] (which we can think of as Grothendieck-[[(0,1)-toposes]]). If both $\mathcal{G}_0$ and $\mathcal{G}_1$ happen to be [[spatial locales]], hence [[topological spaces]], then this is a _[[topological groupoid]]_.

Let $N_\bullet \mathcal{G} : \Delta^{op} \to Locales$ be the [[simplicial object]] in locales given by the [[nerve]] of $\mathcal{G}$. By applying the [[sheaf topos]] functor  $Sh : Locale \to Topos$ to this, we obtain  a [[simplicial object|simplicial]] [[topos]] $Sh(N \mathcal{G}) : [n] \mapsto Sh(N_n \mathcal{G})$. Let $tr_2 Sh(N \mathcal{G})$ be its [[simplicial skeleton|2-truncation]], then the [[2-limit|2-colimit]]

$$
  Sh(\mathcal{G})
   \coloneqq
  \underset{\longrightarrow}{\lim}_n tr_2 Sh(N_\bullet \mathcal{G})
$$

in the 2-category [[Topos]] is called the [[classifying topos]] of $\mathcal{G}$.

This has a more explicit description along the lines discussed at [[sheaves on a simplicial topological space]]:

For $E \in Sh(\mathcal{G}_0)$ a [[sheaf]] on the [[topological space]] of its [[objects]], say that a _$\mathcal{G}_1$-action_ on $E$ is a continuous [[groupoid action]] of $\mathcal{G}_\bullet$ on the [[etale space]] $Sp(E) \to \mathcal{G}_0$ over $\mathcal{G}_0$ that corresponds to the sheaf $E$, hence for each morphisms $f \colon x \to y$ in $\mathcal{G}$ a continuous function $\rho(f) \colon Sp(E)_x \to Sp(E)_y$ that satisfies the usual [[action]] property. These sheaves with $\mathcal{G}_1$-action and with the evident [[homomorphisms]] between them form a [[category]], and this is $Sh(\mathcal{G})$.

## Properties
 {#Properties}

### Exhaustion of the category of all toposes
 {#ExhaustionOfTheCategoryOfToposes}

\begin{proposition}\label{EveryGrToposIsSheavesOnLocalicGroupoid}
([Joyal-Tierney 84](#JoyalTierney84))
\linebreak
For every [[Grothendieck topos]] $\mathcal{E}$ there is a [[localic groupoid]] $\mathcal{G}$ such that $\mathcal{E} \simeq Sh(\mathcal{G})$.
\end{proposition}

\begin{proposition}
([Moerdijk 88, theorem 5](#Moerdijk88))
\linebreak
The construction in Prop. \ref{EveryGrToposIsSheavesOnLocalicGroupoid} extends to an [[equivalence of 2-categories]] between that of Grothendieck toposes [[Topos]] and a [[localization]] of that of localic groupoids.
\end{proposition}

\begin{proposition}
([Butz-Moerdijk 98](#ButzMoerdijk98))
\linebreak
If $\mathcal{E}$ has [[point of a topos|enough points]], then the [[localic groupoid]] in Prop. \ref{EveryGrToposIsSheavesOnLocalicGroupoid} may in fact be taken to be a [[topological groupoid]] $\mathcal{G}$ such that $\mathcal{E} \simeq Sh(\mathcal{G})$.
\end{proposition}

## Related entries

* [[Grothendieck's Galois theory]]

## References

The original result for localic groupoids and arbitrary Grothendieck toposes is due to:

* {#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)]

The above equivalence of categories can in fact be lifted
to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in

* {#Moerdijk88} [[Ieke Moerdijk]], _The classifying topos of a continuous groupoid I_ , Trans. Amer. Math. Soc. Volume 310, Number 2, (1988) ([pdf](http://www.ams.org/journals/tran/1988-310-02/S0002-9947-1988-0973173-9/S0002-9947-1988-0973173-9.pdf))

* [[Ieke Moerdijk]], _The classifying topos of a continuous groupoid II_ , Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168. ([pdf](http://www.numdam.org/article/CTGDC_1990__31_2_137_0.pdf))

The restriction to topological groupoids and Grothendieck toposes with enough points is due to

* {#ButzMoerdijk98} [[Carsten Butz]], [[Ieke Moerdijk]], _Representing topoi by topological groupoids_, Journal of pure and applied algebra 130 (1998) 223-235 ([pdf](http://ns.synrc.com/publications/cat/Topology/Representing%20Topoi%20by%20Topological%20Groupoids.pdf))

[[!redirects classifying topos of a topological groupoid]]


classifying topos of a topological groupoid > history

> see at _[[classifying topos of a localic groupoid]]_


Claude Chevalley

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

* [St Andrews history entry](http://www-history.mcs.st-andrews.ac.uk/Biographies/Chevalley.html)

* related $n$Lab entries: [[Chevalley's theorem on constructible sets]], [[valuative criterion of properness]], [[valuative criterion of separatedness]]

category:people
[[!redirects C. Chevalley]]


Claude Crépeau

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Claude_Cr%C3%A9peau)

## Selected writings

Introducing the notion of [[quantum teleportation]]:

* {#BBCJPW93} [[Charles H. Bennett]], [[Gilles Brassard]], [[Claude Crépeau]], [[Richard Jozsa]], [[Asher Peres]], [[William K. Wootters]]: *Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels*, Phys. Rev. Lett. **70** 1895 (1993) [[doi:10.1103/PhysRevLett.70.1895](https://doi.org/10.1103/PhysRevLett.70.1895)]

category: people

[[!redirects Claude Crepeau]]


Claude Lévi-Strauss

**Claude Lévi-Strauss** (1908-2009) was a French anthropologist and a towering figure of 20th century social sciences.

After meeting the linguist [[Roman Jakobson]] during WW2 in New York, Lévi-Strauss extended the structuralist outlook from [[linguistics]] to anthropology. By conceiving of marriage systems and myths as abstract networks of relations he opened up new avenues for research and paved the way for the application of mathematical methods in anthropology. His views as exposed in the monumental _Structures élémentaires de la parenté_ (1949/1967), the four volume _Mythologiques I-IV_ (1964,1967,1968,1971), and his other writings, were as influential as controversial.

## On category theory

_La définition des catégories, comme systèmes formés à la fois d'un ensemble de termes et de l'ensemble des relations entre ces termes, correspond bien à celle qu'on peut donner du mythe, et la notion de morphisme, qui n'exprime rien d'autre que l'existence d'une relation entre deux termes sans se préoccuper d'en préciser la nature logique, semble trancher le même genre de dilemme que celui dont la solution me fut révélée il y a vingt-cinq ans quand, après qu'un mathématicien illustre mais de la vieille école m'eut dit qu'il ne pouvait m'aider à débrouiller les problèmes de parenté parce qu'il ne connaissait que l'addition, la soustraction, la multiplication et la division, et que le marriage n'était assimilable à aucune de ces opérations, un autre plus jeune et déjà nommé m'assura qu'il lui était indifférent de savoir ce qu'était mathématiquement parlant le marriage, pourvu qu'entre les différents types de marriage on puisse définir des relations_.  Lévi-Strauss  ([1971](#LS71), p.568)

The two mathematicians are Jacques Hadamard, during WW2 a French expatriate in New York and [[André Weil]] who provided a mathematical appendix to _Les structures élémentaires de la parenté_. Lévi-Strauss gives another account of his encounters with them in Lévi-Strauss ([1990](#LSE90), p.79).

## Related entries

* [[canonical formula of myth]]

* [[kinship]]

* [[structuralism]]

## Related pages

* wikipedia entry: [link](https://en.wikipedia.org/wiki/Claude_Levi-Strauss)

* online publications by Lévi-Strauss on persée: [link](http://www.persee.fr/author/persee_29817)

## References

* Mauro W. Barbosa De Almeida, _Symmetry and Entropy: Mathematical Metaphors in the Work of Lévi-Strauss_ , Current Anthropology **31** no.4 (1990) pp.367-385.

* Maurice Godelier, _Lévi-Strauss_ , Seuil Paris 2013.

* Claude Lévi-Strauss, _The Structural Study of Myth_ , J. American Folklore **78** no. 278 (1955) pp.428-444.

* Claude Lévi-Strauss, _Les mathématiques de l'homme_ , Bull. int. sci. soc.  Unesco **6** no.4 (1954) pp.643-653.  (available online)

* {#LS71} Claude Lévi-Strauss, _Mythologiques IV: L'Homme nu_ , Plon Paris 1971.

* Claude Lévi-Strauss, _Réflexions sur l'atome de parenté_ , L'Homme **13** no.3 (1973) pp.5-30. ([link](http://www.persee.fr/doc/hom_0439-4216_1973_num_13_3_367355))

* {#LSE90}Claude Lévi-Strauss, Didier Eribon, _De près et de loin_ , Odile Jacob Paris 1990.

* Dan Sperber, _On Anthropological Knowledge_ , chapter 3, Cambridge UP 1985. ([pdf](http://www.dan.sperber.fr/wp-content/uploads/On-Anthropological-Knowledge.pdf))

* Terence S. Turner, _The crisis of late structuralism_ , pp.205-243 in Turner, _The Fire of the Jaguar_ , Hau books Chicago 2017. ([link](https://haubooks.org/the-fire-of-the-jaguar/))

[[!redirects Levi-Strauss]]
[[!redirects Claude Lévi-Strauss]]
[[!redirects Lévi-Strauss]]

category: people


Claude Quitté

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=140888)

## Selected writings

On [[commutative algebra]] via [[constructive mathematics]]:

* [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

category: people


Claude Roger

Claude Roger is a French mathematician based at Lyon 1. There is a summary of his contribution to research at the [website](http://math.univ-lyon1.fr/~okra/Claude2009/Claude-resume.html) of a colloquium (*ALGEBRES DE LIE DE DIMENSION INFINIE : GEOMETRIE ET COHOMOLOGIE*) held in his honour.


Claude Sabbah

* [homepage](http://www.math.polytechnique.fr/~sabbah), [articles](http://www.math.polytechnique.fr/~sabbah/articles.html)

Related $n$lab articles: [[meromorphic connection]], [[Riemann-Hilbert problem]], [[Fuchsian equation]], [[Stokes phenomenon]], [[Hodge theory]], [[D-geometry]], [[D-module]]

category: people


Claude Schochet

* [webpage](http://www.math.wayne.edu/~claude/)

## related pages

* [[homotopical structure on C*-algebras]]

category: people


Claude Shannon

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

## related $n$Lab entries

* [[entropy]]

* [[sampling theory]]

category: people


Claude-Alain Pillet

* [personal page](http://pillet.univ-tln.fr/)

## Selected writings

On [[open quantum systems|open]] [[quantum systems]] ([[quantum probability]], [[quantum noise]], [[quantum channels]], ...):

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems I -- The Hamiltonian Approach*, Lecture Notes in Mathematics **1880**, Springer (2006) [[doi:10.1007/b128449](https://doi.org/10.1007/b128449)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems II -- The Markovian approach*,  Lecture Notes in Mathematics **1881**, Springer (2006) [[doi:10.1007/b128451](https://doi.org/10.1007/b128451)]

* [[Stéphane Attal]], [[Alain Joye]], [[Claude-Alain Pillet]] (eds.), *Open Quantum Systems III -- Recent Developments*, Lecture Notes in Mathematics **1882**, Springer (2006) [[doi:10.1007/b128453](https://doi.org/10.1007/b128453)]

category: people


Claudia de Rham

* [institute page](https://www.imperial.ac.uk/people/c.de-rham)

* [inspire page](https://inspirehep.net/authors/1030566)

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

## Selected writings

On [[defect branes]]:

* [[Claudia de Rham]], *The Effective Field Theory of Codimension-two Branes*, JHEP 0801:060 (2008) [[arXiv:0707.0884](https://arxiv.org/abs/0707.0884), [doi:10.1088/1126-6708/2008/01/060](https://doi.org/10.1088/1126-6708/2008/01/060)]

category: people


Claudia Faggian

* [personal page](https://www.irif.fr/~faggian/)

## Selected writings

On [[linear logic]] and [[quantum circuit diagrams]]:

* {#LagoFaffian12} [[Ugo Dal Lago]], [[Claudia Faggian]], _On Multiplicative Linear Logic, Modality and Quantum Circuits_, EPTCS **95** (2012) 55-66 [[arXiv:1210.0613](http://arxiv.org/abs/1210.0613), [doi:10.4204/EPTCS.95.6](https://doi.org/10.4204/EPTCS.95.6)]

category: people


Claudia Scheimbauer

* [webpage](https://people.maths.ox.ac.uk/scheimbauer/)

## related $n$Lab entries

* [[factorization homology]]

category: people


Claudio Carmeli

* [webpage](http://www.dime.unige.it/en/users/claudio-carmeli)

## Selected writings

On [[supergeometry]] ([[supermanifolds]], [[super Lie groups]]) from the perspective of [[functorial geometry]]:

* [[Claudio Carmeli]], Lauren Caston, [[Rita Fioresi]], _Mathematical Foundations of Supersymmetry_, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( [ISBN:978-3-03719-097-5](https://bookstore.ams.org/emsserlec-15/), [arXiv:0710.5742](https://arxiv.org/abs/0710.5742))

category: people


Claudio Dappiaggi

* [webpage](http://wwwiexp.desy.de/sfb676/researchers/dappiaggi/)

## related entries

* [[AQFT on curved spacetime]]

* [[field bundle]]

category: people


Claudio Hermida

[[Claudio Hermida]] is a Portuguese-Canadian [[category theory|category theorist]] and theoretical [[computer science|computer scientist]]. His works include the study of [[higher category theory|higher categorical]] [[descent theory]], higher categorical [[fibrations]] (including ongoing work with [[Marta Bunge]]), [[multicategories]], the relationship between [[extra structure|structure]] and [[extra property|property]] in terms of [[2-algebra|2-categorical algebra]] and [[fibered categories]], etc.

* [homepage](http://maggie.cs.queensu.ca/chermida) at Queen's University

## Selected writings

On [[fibered 2-categories]] (and discussing [[fibered limits]]):

* {#Hermida} [[Claudio Hermida]], *Some properties of $Fib$ as a fibred 2-category*, Journal of Pure and Applied Algebra **134** (1999) 83-109 [doi:10.1016/S0022-4049(97)00129-1, [ps.gz](http://maggie.cs.queensu.ca/chermida/papers/jpaa.ps.gz)]

* _Descent on 2-fibrations and strongly 2-regular 2-categories_, Appl. Categ. Structures __12__ (2004), no. 5--6, 427--459.

On [[higher monadic descent]]:

* [[Claudio Hermida]], *Descent on 2-fibrations and strongly 2-regular 2-categories*, Appl. Categ. Structures **12** 5-6 (2004)  427-459 [[doi:10.1023/B:APCS.0000049311.17100.da](https://doi.org/10.1023/B:APCS.0000049311.17100.da)]

category: people


Claudio Pisani

* [arXiv page](https://arxiv.org/search/math?searchtype=author&query=Pisani%2C+C)

## Selected writings

On [[multicategories]]:

* [[Claudio Pisani]], _Sequential multicategories_, ([arXiv:1402.0253](https://arxiv.org/abs/1402.0253))

On [[operads]] as [[double functors]] between certain [[double categories]]:

* [[Claudio Pisani]], _Operads as double functors_ ([arXiv:2208.07028](https://arxiv.org/abs/2208.07028))

category: people


Claudio Procesi

* [webpage](http://www.mat.uniroma1.it/~procesi),

__Claudio Procesi__ is an Italian mathematician. Most of his work is in [[algebra]], [[representation theory]], [[Lie theory]], [[combinatorics]] and study of [[moduli spaces]] in [[algebraic geometry]], [[configuration spaces]] and arrangements of hyperplanes.

He was a teacher of [[Corrado de Concini]] ([wikipedia](http://it.wikipedia.org/wiki/Corrado_De_Concini)) with whom he often collaborates. They together invented so-called [[wonderful compactifications]] of [[symmetric spaces]] and of [[moduli spaces]].

## writings

* [publications](http://www.mat.uniroma1.it/~procesi/publi.html)

* _Lie groups, an approach through invariants and representations_, Universitext, Springer 2006, [gBooks](http://books.google.co.in/books?id=Sl8OAGYRz_AC&printsec=frontcover&hl=hr&source=gbs_atb)

* _On the $n!$ conjecture_, [ps](http://www.mat.uniroma1.it/~procesi/lavori/nfatt-b.ps), Séminaire Bourbaki 2001/2002. (Astérisque __290__, Exposé 898) (2002)

* {#deConciniProcesi95} [[Corrado de Concini]], [[Claudio Procesi]], _Wonderful models of subspace arrangements_, Selecta Mathematica, New Series Vol. 1, No. 3, 1995 ([pdf](https://link.springer.com/content/pdf/10.1007/BF01589496.pdf))

* Corrado De Concini, Claudio Procesi, _Topics in hyperplane arrangements, polytopes and box-splines_, Universitext 223, Springer 2010.

## related $n$Lab entries

* [[wonderful compactification]]

* [[Fulton-MacPherson compactification]]

category: people
[[!redirects C. Procesi]]


Claudio Rebbi

* [InSpire page](https://inspirehep.net/authors/992139)

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

## Selected writings

Early discussion of [[quantization]] of the [[string]] [[sigma-model]] in [[light-cone gauge]]:

* [[Peter Goddard]], [[Jeffrey Goldstone]], [[Claudio Rebbi]], [[Charles B. Thorn]], *Quantum dynamics of a massless relativistic string*, Nuclear Physics B **56** 1 (1973) 109-135 [doi:10.1016/0550-3213(73)90223-X]

category: people


Claudio Teitelboim

* [website](http://www.uchile.cl/?_nfpb=true&_pageLabel=conUrl&url=6568)

* [Wikipedia entry](http://de.wikipedia.org/wiki/Claudio_Bunster)

## Selected writings

On [[BV-formalism|BV-]][[BRST complex|BRST]]-[[quantization]] of [[gauge theories]]:

* [[Marc Henneaux]], [[Claudio Teitelboim]], *[[Quantization of Gauge Systems]]*, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [doi:10.2307/j.ctv10crg0r](https://doi.org/10.2307/j.ctv10crg0r), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

Introducing [[BTZ black holes]]:

* [[Máximo Bañados]], [[Claudio Teitelboim]], [[Jorge Zanelli]], _The Black Hole in Three Dimensional Space Time_, Phys. Rev. Lett. 69 (1992) 1849-1851 ([arXiv:hep-th/9204099](https://arxiv.org/abs/hep-th/9204099))

* [[Máximo Bañados]], [[Marc Henneaux]], [[Claudio Teitelboim]], [[Jorge Zanelli]], _Geometry of the 2+1 Black Hole_, Phys. Rev. D **48** (1993) 1506-1525 [[arXiv:gr-qc/9302012](https://arxiv.org/abs/gr-qc/9302012)]

On [[self-dual higher gauge fields]]:

* {#HenneauxTeitelboim88} [[Marc Henneaux]], [[Claudio Teitelboim]], _Dynamics of chiral (self-dual) $p$-forms_, Physics Letters B Volume 206, Issue 4, 2 June 1988, Pages 650-654 (doi:10.1016/0370-2693(88)90712-5)

On [[Jackiw-Teitelboim gravity]]:

* {#Teitelboim83} [[Claudio Teitelboim]], _Gravitation and hamiltonian structure in two spacetime dimensions_, Physics Letters B Volume 126, Issues 1–2, 23 June 1983, Pages 41-45 (doi:10.1016/0370-2693(83)90012-6

## Related $n$Lab entries

* [[gauge theory]]

* [[BV-BRST formalism]]

category: people

[[!redirects Claudio Bunster]]


Claudio Ternullo

* [personal page](https://www.claudioternullo.com/)

## Selected publications

* [[Claudio Ternullo]], *Maddy on the Multiverse*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_3](https://doi.org/10.1007/978-3-030-15655-8_3), [pdf](https://www.claudioternullo.com/_files/ugd/8787da_7474d586e9034716b2eea201b3e00191.pdf)]

category: people


Claus Hertling

__Claus Hertling__ is a German mathematician, a specialist in singularity theory who studied under Brieskorn and Manin.

## Selected works

He studied Frobenius manifolds and a generalization F-manifolds with examples from [[integrable system]]s, nonabelian [[Hodge theory]] and singularities.

* Claus Hertling, _Frobenius manifolds and moduli spaces for singularities_, Cambridge University Press 2002.

* [[Claus Hertling]], Christian Sevenheck, _Twistor structures, $tt^*$-geometry and singularity theory_, pp. 49--74 in: "From tQFT to tt* and integrability", Proceedings of Symposia in Pure Mathematics __78__ (2007, Augsburg, Germany), AMS 2008; [arxiv:0807.2199](https://arxiv.org/abs/0807.2199)

* [[Claus Hertling]], [[Claude Sabbah]], _Examples of non-commutative Hodge structure_ (v1 title: Fourier-Laplace transform of flat unitary connections and TERP structures), [arxiv:0912.2754](https://arxiv.org/abs/0912.2754)

category: people


Claus Michael Ringel

Claus Michael Ringel is a German algebraist.

* personal [page at Bielefeld](https://www.math.uni-bielefeld.de/~ringel)

## Selected work

On [[Hall algebras]]:

* [[Claus M. Ringel]], *The Hall multiplication]*, lecture notes  (2006) [[web](https://www.math.uni-bielefeld.de/~ringel/lectures/green/Welcome.html)]

Ringel realized that [[Hall algebra|Hall's construction]] of an associative algebra structure from the combinatorics of flags of [[p-group|p-abelian groups]] (defined in terms of numbers of extensions) can be applied to the category of representations of a quiver over a finite field $\mathbf{F}_q$ and related it to quantum groups in

* C. M. Ringel, _Hall algebras and quantum groups_, Invent. Math. __101__ (3): 583--591, [doi](https://doi.org/10.1007/BF01231516) [MR1062796](https://mathscinet.ams.org/mathscinet-getitem?mr=1062796)
* wikipedia: [Ringel-Hall algebra](https://en.wikipedia.org/wiki/Ringel%E2%80%93Hall_algebra)

This work influenced [[George Lusztig]]'s work on quantum groups where Lusztig has constructed "the upper half" of quantum group as a Hall algebra adapted to perverse sheaves over flag variety of a quiver.

Other works include

* C. M. Ringel, _The simple Artinian spectrum of a finite dimensional algebra_ (in Dekker Lect. Notes 51, 1979)

category: people
[[!redirects C. M. Ringel]]
[[!redirects Claus Ringel]]
[[!redirects Claus M. Ringel]]


Claus Montonen

* [Mathematics Genealogy Project page](http://genealogy.math.ndsu.nodak.edu/id.php?id=52885)

On [[electric-magnetic duality]]:

* [[Claus Montonen]], [[David Olive]], _Magnetic Monopoles As Gauge Particles?_ Phys. Lett. B72 (1977) 117-120 ([spire:121372](https://inspirehep.net/literature/121372), doi:10.1016/0370-2693(77)90076-4)

The [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of ([[fuzzy 3-sphere|fuzzy]] [[3-sphere]]-shape:

* Masato Arai, [[Claus Montonen]], Shin Sasaki, _Vortices, Q-balls and Domain Walls on Dielectric M2-branes_, JHEP 0903:119, 2009 ([arXiv:0812.4437](https://arxiv.org/abs/0812.4437))

## Related $n$Lab entries

* [[Montonen-Olive duality]] ([[S-duality]])

category: people


Clay Cordova

* [webpage](https://www.ias.edu/scholars/clay-cordova)

## Selected writings

On the [[3d-3d correspondence]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Complex Chern-Simons from M5-branes on the Squashed Three-Sphere_ ([arXiv:1305.2891](https://arxiv.org/abs/1305.2891))

Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]] and the [[3d-3d correspondence]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Toda Theory From Six Dimensions_, J. High Energ. Phys. (2017) 2017: 106 ([arxiv:1605.03997](https://arxiv.org/abs/1605.03997))

On [[D=5 super Yang-Mills theory]]:

* [[Clay Cordova]], [[Daniel Jafferis]], _Five-Dimensional Maximally Supersymmetric Yang-Mills in Supergravity Backgrounds_ ([arXiv:1305.2886](https://arxiv.org/abs/1305.2886))

On [[anomaly cancellation]] via the [[Green-Schwarz mechanism]] from a [[higher gauge theory]]-perspective

* [[Francesco Benini]], [[Clay Cordova]], [[Po-Shen Hsin]], *On 2-Group Global Symmetries and Their Anomalies*,  J. High Energ. Phys. **2019** 118 (2019) ([arXiv:1803.09336](https://arxiv.org/abs/1803.09336), doi:10.1007/JHEP03(2019)118)

and specifically so in [[D=6 N=(2,0) SCFT]] and [[D=6 N=(1,0) SCFT]]:

* [[Clay Cordova]], [[Thomas Dumitrescu]], [[Kenneth Intriligator]], from p. 18 in: _2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories_ ([arXiv:2009.00138](https://arxiv.org/abs/2009.00138))

On [[generalized global symmetries]]:

* [[Clay Cordova]], [[Thomas T. Dumitrescu]], [[Kenneth Intriligator]], [[Shu-Heng Shao]], *Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond* [[arXiv:2205.09545](https://arxiv.org/abs/2205.09545)]

## Related entries

* [[higher gauge theory]]

* [[Green-Schwarz mechanism]]

* [[de Sitter spacetime]]

category: people


cleavage

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

#Contents#
* table of contents
{:toc}

## Idea

Being a [[Grothendieck fibration]] is a [[property-like structure]] on a [[functor]], like the existence of [[limits]] in a [[category]]: it is defined by the existence of certain [[objects]] (in this case, [[cartesian morphisms]]) which, when they exist, are unique up to unique [[isomorphism]].  Any [[property]]-like [[structure]] can be "algebraicized" by requiring a specific *choice* of the objects that are required to exist; a *cleavage* is this "algebraicization" for the property of being a [[Grothendieck fibration]].

## Definition

Let $p\colon E\to B$ be a functor.  A **cleavage** of $p$ is a choice, for each $e\in E$ and $u\colon b\to p(e)$ in $B$, of a single [[cartesian arrow]] $f:e'\to e$ such that $p(f) = u$.  Evidently if there exists a cleavage for $p$, then $p$ is a [[Grothendieck fibration]].  If $p$ is equipped with a cleavage, it said to be **cloven**.  Conversely, if we assume the [[axiom of choice]], then every fibered category has a cleavage.

If the cartesian arrow $f$ is an identity whenever $u$ is, the cleavage is said to be **normal**, and if each composite $f_2 f_1$ is the specified lifting of $u_2 u_1$, the cleavage is said to be a **splitting**.  Any cleavage can be modified to become normal, but not necessarily to become split.  (Any fibration is, however, *equivalent* to a split fibration.)

## Cloven fibrations vs pseudofunctors

Given a cleavage of $p$ and an arrow $u:b'\to b$ in $B$, there is a functor $u^*\colon p^{-1}(b)\to p^{-1}(b')$ which to every object $e\in p^{-1}(b)$ assigns the domain of the specified arrow in the cleavage which is above $u$ and whose codomain is $e$.  This correspondence extends to a functor, thanks to the universal property of the cartesian arrows.

The functor $u^*$ may be called either the *inverse image functor* along $u$, or the *direct image functor* of $u$, depending on the context; see the remarks on notation at [domain opfibration](http://ncatlab.org/nlab/show/domain+opfibration#notation).  It depends on the choice of cleavage as well as on $u$, although different cleavages produce canonically naturally isomorphic functors.  If one doesn't choose a cleavage, then $u^*$ can still be defined as an [[anafunctor]].

The direct image functors corresponding to varying morphisms in $B$ together form a [[pseudofunctor]] $B^{op}\to Cat$.  The inverse [[Grothendieck construction]] produces a cloven fibration from a pseudofunctor, setting up an equivalence of 2-categories (in fact, a [[strict 2-equivalence of strict 2-categories]]) between cloven fibrations and pseudofunctors $B^{op} \to Cat$.  If we either assume the axiom of choice, or allow the morphisms in [[Cat]] to be anafunctors, then this extends to an equivalence between pseudofunctors and not-necessarily-cloven fibrations.

## Non-strict cleavages

There is a corresponding notion of cleavage for a [[Street fibration]], namely a choice of, for each $e\in E$ and $u\colon b\to p(e)$ in $B$, a cartesian arrow $f:e'\to e$ in $E$ and an isomorphism $v\colon p(e') \xrightarrow{\cong} b$ such that $u\circ v = p(f)$.  Such a cleavage induces, for every $u\colon b'\to b$, a functor between the [[essential fibers]] of $b$ and $b'$, and thereby a pseudofunctor and another equivalence of 2-categories (though not a strict 2-equivalence).

## References

The original reference:

* [[Alexander Grothendieck]], §VI.7 of: _Revêtements Étales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61_ ([[SGA 1]]) , LNM **224** Springer (1971) [updated version with comments by M. Raynaud: [arxiv.0206203](http://arxiv.org/abs/math/0206203)]

Review:

* [[Angelo Vistoli]], Def. 3.9 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

[[!redirects pseudo-splitting]]
[[!redirects split fibration]]
[[!redirects cloven fibration]]
[[!redirects split opfibration]]
[[!redirects cloven opfibration]]
[[!redirects pseudo-splittings]]
[[!redirects split fibrations]]
[[!redirects cloven fibrations]]
[[!redirects split opfibrations]]
[[!redirects cloven opfibrations]]
[[!redirects cleavages]]


Clebsch-Gordan coefficient

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[tensor product of representations|tensor product]] $\rho_{j_1} \otimes \rho_{j_2}$ of two [[representations]] of some [[Lie group]] $G$ (by default often the [[special orthogonal group]] $SO(3)$) and given its decomposition into a [[direct sum]] of [[irreducible representations]] $\{\rho_{t_{tot}}\}$ by an [[isomorphism]]

$$
  \rho_{j_1} \otimes \rho_{j_2} \stackrel{\simeq}{\longrightarrow} \underset{j_{tot}}{\oplus} C_{j_1 j_2}{}^{j_{tot}} \rho_{j_{tot}}
$$

then the [[matrix]] elements of this [[linear map]] in some standard [[basis]] are called -- in the [[physics]] literature -- the _Clebsch-Gordan coefficients_ or equivalently (up to a constant) the _Wigner 3j symbols_ .

Specifically for $G = SO(3)$ the [[rotation group]] in 3-dimensional [[Cartesian space]], then the standard [[basis]] elements of the [[representation]] $\rho_{j}$ of total [[angular momentum]] $j \in \mathbb{N}$ are traditionally denoted

$$
  |j,m\rangle \in \rho_{j}
  \;\;\,,\;\;
  -j \leq m \leq j
$$

and their [[inner product]] is traditionally denoted by

$$
  \langle j_1, m_1 | j_2, m_2\rangle
  \in
  \mathbb{C}
  \,.
$$

In these basis elements that above [[matrix]] then has components given by

$$
  \langle (j_1, m_1)\otimes(j_2,m_2) | j_{tot}, m_{tot}\rangle
  \in
  \mathbb{C}
  \,.
$$

These expressions are specifically the _Clebsch-Gordan coefficients_ as they appear in the physics literature.

## Related concepts

* [[Wigner 6-j symbol]]

* [[Wigner 9-j symbol]]

* [[Fierz identity]]

* [[fusion ring]]

## References

Lecture notes include for instance

* M. Tuckerman, _Quantum mechanics and dynamics -- Addition of angular momenta -- [The general problem](http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_8/node2.html))_

[[!redirects Clebsch-Gordan coefficients]]

[[!redirects 3j symbol]]
[[!redirects 3j symbols]]
[[!redirects 3-j symbol]]
[[!redirects 3-j symbols]]

[[!redirects Wigner 3j symbol]]
[[!redirects Wigner 3j symbols]]
[[!redirects Wigner 3-j symbol]]
[[!redirects Wigner 3-j symbols]]


cleft extension

Let $H$ be a Hopf algebra, a right $H$-[[comodule algebra]] $E$ is an __$H$-extension__ of a subalgebra $U\subset E$ if $U=E^{co H}$ is precisely  the subalgebra of $H$-coinvariants. The $H$-extension $U\subset E$ is **cleft** if there exist a __convolution-invertible $H$-comodule map__ $\gamma:H\to E$.

If $U\hookrightarrow E$ is a cleft $H$-extension, then the cleavage $\gamma$ can always be chosen normalized in the sense that $\gamma(1)=1$; because if it is not normalized we can rescale $\gamma$ to form a normalized cleavage $\gamma'=\gamma^{-1}(1)\gamma$ (indeed, $1$ is [[grouplike element|group-like]], hence $\gamma(1)$ is invertible with inverse $(\gamma(1))^{-1}=\gamma^{-1}(1)$).

It is easy to show that the rule

$$h\triangleright u := \sum \gamma(h_{(1)})u\gamma^{-1}(h_{(2)})$$

defines a [[measuring]] $\triangleright:H\otimes U\to U$ i.e. $h\triangleright(uv)=\sum (h_{(1)}\triangleright u)(h_{(2)}\triangleright v)$ and if $\gamma$ is chosen normalized, then $h\triangleright 1 = \epsilon(h) 1$. Define a convolution invertible map $\sigma\in Hom_k(H\otimes H,U)$ by

$$\sigma(h,k) = \sum \gamma(h_{(1)})\gamma(k_{(1)})\gamma^{-1}(h_{(2)}k_{(2)}),\,\,\,\,\,h,k\in H.$$

Then the pair $(\triangleright,\sigma)$ defines the data for the __cocycled [[crossed product algebra]]__ $U\sharp_\sigma H$ which is canonically isomorphic to $B$ as an $H$-extension of $U\cong U\otimes 1\hookrightarrow U\sharp_\sigma H$, and i.e. as a right $H$-comodule algebra with the isomorphism fixing $U$ as given.

Conversely, every cocycled product $U\sharp_\sigma H$ is cleft via $\gamma:
h\mapsto 1\sharp h$ and the cocycle $\sigma$ built out of $\gamma$ is the same one, which helped build the cocycled crossed product.

Every cleft extension is a particular case of a [[Hopf-Galois extension]].

* Y. Doi, [[Mitsuhiro Takeuchi]], _Cleft comodule algebras for a bialgebra_, Comm. Alg. __14__ (1986) 801--818 [doi](https://doi.org/10.1080/00927878608823337)
* [[Shahn Majid]], _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

* [[Susan Montgomery]], _Hopf algebras and their actions on rings_, CBMS 82, AMS 1993.

There are generalizations for Hopf algebroids:

* [[Gabriella Böhm]], [[Tomasz Brzeziński]], _Cleft extensions of Hopf algebroids_, Appl. Cat. Str. 14 (2006) 431-469; [math.QA/0510253](http://arxiv.org/abs/math.QA/0510253)

There are some globalizations of cleft extensions. For the smash product case of the globalization some details are written in

* [[Zoran Škoda]], _Čech cocycles for quantum principal bundles_, [arxiv/1111.5316](http://arxiv.org/abs/1111.5316)
* Z. Škoda, [[zoranskoda:cleft extension of a space cover]]

[[!redirects cleft extensions]]
[[!redirects cleft comodule algebra]]
[[!redirects cleft Hopf-Galois extension]]


Clemens Berger

* [website](http://math.unice.fr/~cberger/)

## Selected writings

(...)

On [[generalized Reedy categories]]:

* {#BergerMoerdijk} [[Clemens Berger]], [[Ieke Moerdijk]], *On an extension of the notion of Reedy category*, Mathematische Zeitschrift **269** (2011) 977–1004 [[arXiv:0809.3341](http://arxiv.org/abs/0809.3341), [doi:10.1007/s00209-010-0770-x](https://doi.org/10.1007/s00209-010-0770-x)]

On [[model structures on algebras over an operad]]:

* {#BergerMoerdijkAlgebras} [[Clemens Berger]], [[Ieke Moerdijk]], _Resolution of coloured operads and rectification of homotopy algebras_, in *Categories in Algebra, Geometry and Mathematical Physics*, Contemp. Math. **431** (2007) 31-58 [[arXiv:math/0512576](http://arxiv.org/abs/math/0512576), [ams:conm/431](www.ams.org/books/conm/431)]

On [[Morita equivalence|Morita theory]] for/via [[strong monads]]:

* {#BergerRatkovic19} [[Clemens Berger]], [[Kruna Ratkovic]], *Gabriel-Morita theory for excisive model categories*, Appl Categor Struct **27** (2019) 23–54 [[arXiv:1705.03863](https://arxiv.org/abs/1705.03863), [doi;10.1007/s10485-018-9539-x](https://doi.org/10.1007/s10485-018-9539-x)]

category: people


Clement Delcamp

* [personal webpage](http://www2.mpq.mpg.de/Theorygroup/CIRAC/people/staff_and_contacts/44.1.html)

## Selected writings

On [[topological phases of matter]] via [[4d TQFT]] and [[higher gauge theory]]/[[higher parallel transport]]:

* [[Clement Delcamp]], *Excitation basis for (3+1)d topological phases*, Journal of High Energy Physics **2017** (2017) 128 ([arXiv:1709.04924](https://arxiv.org/abs/1709.04924) doi:10.1007/JHEP12(2017)128)

* [[Clement Delcamp]], Apoorv Tiwari, *On 2-form gauge models of topological phases*, JHEP 05 (2019) 064 ([arXiv:1901.02249](https://arxiv.org/abs/1901.02249))

category: people


Cliff P. Burgess

* [personal page](https://physics.mcmaster.ca/~cburgess/cburgess/)

## Selected writings

On [[partial differential equations]] appearing in [[mathematical physics]] and their solution by [[special functions]] ([[hypergeometric functions]]):

* [[Cliff P. Burgess]], *Primer on Partial Differential Equations for Physicists*, lecture notes (1990) [[pdf](https://physics.mcmaster.ca/~cburgess/Notes/mathphys.pdf), [[Burgess-PDEs.pdf:file]]]

On [[quantum gravity]] as an [[effective field theory|effective]] [[perturbative quantum field theory]]:

* [[Cliff Burgess]], *Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory*, Living Rev. Rel. **7** 5 (2004) [[arXiv:gr-qc/0311082](https://arxiv.org/abs/gr-qc/0311082), [doi:10.12942/lrr-2004-5](https://doi.org/10.12942/lrr-2004-5)]

On [[supergravity]] coupled to the (non-supersymmetric) [[standard model of particle physics]]:

* [[Cliff P. Burgess]], [[Fernando Quevedo]], *Who's Afraid of the Supersymmetric Dark? The Standard Model vs Low-Energy Supergravity* ([arXiv:2110.13275](https://arxiv.org/abs/2110.13275))

category: people

[[!redirects Cliff Burgess]]