generalized smooth space > relationships

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     sodipodi:linespacing="125%"><tspan
       sodipodi:role="line"
       id="tspan8563"
       x="-21.7328552"
       y="10.212161">Smith</tspan></text>
  <text
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     style="font-size:9.60000038px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-align:center;line-height:125%;writing-mode:lr-tb;text-anchor:middle;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans;-inkscape-font-specification:Bitstream Vera Sans"
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  <text
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     style="font-size:9.60000038px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-align:center;line-height:125%;writing-mode:lr-tb;text-anchor:middle;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans;-inkscape-font-specification:Bitstream Vera Sans"
     x="-87.340805"
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  <text
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     style="font-size:9.60000038px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-align:center;line-height:125%;writing-mode:lr-tb;text-anchor:middle;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans;-inkscape-font-specification:Bitstream Vera Sans"
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       y="67.652336">Frölicher</tspan></text>
  <text
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     style="font-size:9.60000038px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-align:center;line-height:125%;writing-mode:lr-tb;text-anchor:middle;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans;-inkscape-font-specification:Bitstream Vera Sans"
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     sodipodi:linespacing="125%"><tspan
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</svg>


generalized smooth spaces > history

< [[generalized smooth spaces]]
[[!redirects generalized smooth spaces -- history]]


generalized Spin(7)-manifold

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###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _generalized Spin(7)-structure_ on an [[8-manifold]] $X$ is a [[reduction of the structure group]] of the [[generalized tangent bundle]] $T X \oplus T^\ast X$ along the inclusion of the [[direct product group]] of two copies of [[Spin(7)]] into the [[spin group|spin]]-[[Narain group]], along

$$
  Spin(7) \times Spin(7) \longrightarrow Spin(8,8)
  \,.
$$

This generalizes the [[reduction of the structure group|reduction]] of the plain [[tangent bundle]] along the inclusion of [[Spin(7)]] into [[Spin(8)]]

$$
  Spin(7) \hookrightarrow Spin(8)
$$

which goes with [[Spin(7)-manifolds]], whence the name.

## Properties

### As exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

## Related concepts

* [[generalized Calabi-Yau manifold]]

* [[generalized G2-manifold]]

## References

The concept was maybe first considered in

* {#Witt04} Frederik Witt, section 5 of _Generalised $G_2$-manifolds_, _Commun.Math.Phys. 265 (2006) 275-303_ ([arXiv:math/0411642](https://arxiv.org/abs/math/0411642))

Further discussion includes

* [[Dimitrios Tsimpis]], section 4.1. of _M-theory on eight-manifolds revisited: $N=1$ supersymmetry and generalized $Spin(7)$ structures_, JHEP 0604 (2006) 027 ([arXiv:hep-th/0511047](https://arxiv.org/abs/hep-th/0511047))

* [[Mariana Graña]], [[Carlos S. Shahbazi]], [[Marco Zambon]], _Spin(7)-manifolds in compactifications to four dimensions_, High Energ. Phys. (2014) 2014: 46 ([arXiv:1405.3698](https://arxiv.org/abs/1405.3698))

[[!redirects generalized Spin(7)-manifolds]]

[[!redirects generalised Spin(7)-manifold]]
[[!redirects generalised Spin(7)-manifolds]]

[[!redirects generalized Spin(7)-structure]]
[[!redirects generalized Spin(7)-structure]]

[[!redirects generalised Spin(7)-structure]]
[[!redirects generalised Spin(7)-structure]]


generalized tangent bundle

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###Context###
#### Differential geometry
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#Contents#
* table of contents
{:toc}

## Definition

In the context of [[generalized complex geometry]] one says for $X$ a [[manifold]], $T X$ its [[tangent bundle]] and $T^* X$ the [[cotangent bundle]] that the fiberwise [[direct sum]]-bundle $T X \oplus T^* X$ is the _generalized tangent bundle_.

More generally, a [[vector bundle]] $E \to X$ that sits in an [[exact sequence]] $T^* X \to E \to T X$ is called a _generalized tangent bundle_, such as notably those underlying a [[Courant Lie 2-algebroid]] over $X$.

## Properties

### As an associated bundle

The ordinary [[tangent bundle]] is the canonical [[associated bundle]] to the [[general linear group]]-[[principal bundle]] classified by the morphism

$$
  g_{T X} : X \to \mathbf{B} GL(n)
$$

to the [[smooth infinity-groupoid|smooth]] [[moduli stack]] of $GL(n)$.

Similarly there is a canonical morphism

$$
  (g_{T X}, g^{-T}_{T X}) : X \to \mathbf{B} O(n,n)
$$

to the [[moduli stack]] which is the [[delooping]] of the [[Narain group]] $O(n,n)$. This classifies the $O(n,n)$-principal bundle to which $T X \oplus T^* X$ is associated.

### Reduction of structure group

Where a [[reduction of structure groups|reduction of the structure group]] of the tangent bundle along $\mathbf{B} O(n) \hookrightarrow \mathbf{B} GL(n)$ is equivalently a [[vielbein]]/[[orthogonal structure]]/[[Riemannian metric]] on $X$, so a reduction of the structure group of the generalized tangent bundle along $\mathbf{B} (O(n) \times O(n)) \to \mathbf{B}O(n,n)$ is a _[[generalized vielbein]]_, defining a _[[type II geometry]]_ on $X$.

Other [[reduction of the structure group|reductions]] yield other geometric notions, for instance:

* reduction along $U(n,n) \to O(2n,2n)$ is a [[generalized complex structure]];

* further reduction along $SU(n,n)  \to U(n,n) \to O(2n,2n)$ is a [[generalized Calabi-Yau manifold]] structure.

[[!include Spin(8)-subgroups and reductions -- table]]

## Related concepts

* [[tangent bundle]]

* [[Courant Lie 2-algebroid]]

* [[exceptional tangent bundle]]

[[!redirects generalized tangent bundles]]


generalized the

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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In natural language, the definite article ('the' in English) is generally used only for nouns which are uniquely characterized by context.  Hence we have "the Moon" and "the book I was just reading," but only "a car" or "a wild late-night party." (Sometimes "the book I was just reading" is abbreviated to "the book", but it should be clear from context that only one book could be meant.) Philosophers call this use of the definite article to pick out an individual [[definite description]].

In [[mathematics]], and especially in [[category theory]], [[homotopy theory]]/[[homotopy type theory]] and generally in [[higher category theory]], it is common to use "the" more generally for [[objects]]/[[types]] which are characterized only up to [[coherence|coherent]] [[isomorphism]]/[[equivalence in an (infinity,1)-category|equivalence]]: Such need not actually be unique, but any two of them are [[isomorphism|isomorphic]] or more generally [[coherence|coherently]] [[equivalence in homotopy type theory|equivalent]] in an [[essentially unique]] way (meaning that the [[∞-groupoid]] of all such objects is [[contractible space|contractible]]).

For example, one speaks (assuming that any exists) of "the" [[terminal object]] of a category, "the" [[Cartesian product]] of two objects, "the" [[adjoint functor|left adjoint]] of a functor, and so on.  Outside of pure [[category theory]] we have examples such as "the" Dedekind-complete ordered field (the field of [[real numbers]]).

## Formalization

The notion of a "generalized the" can be formalized and treated uniformly in [[homotopy type theory]]. Here one may define an [[introduction rule]] for *the* as follows:

$$
  (A:Type),((a,p):IsContr(A)) \vdash (the(A,a,p):A).
$$

Here the [[term]] $(a,p)$ is one witness for the [[contractible type|contractibility]] of the [[type]] $A$. Then we have $the(A,a,p)=a:A$.

Since $IsContr(A)$ is itself [contractible](contractible+type#properties_18), we could say that $(a,p)$ is *the* witness for the contractibility of the type $A$, which may explain why we do not generally mention it.

If we wish to extend this treatment to types which are [[propositions]], we might call such types $FactThat(P)$, for some statement $P$. Then if $P$ holds, we can introduce a term $the(FactThat(P))$.

## References

The term "generalized the" was introduced by [[James Dolan]] and may first appear in print here:

* [[John Baez]], *Higher-dimensional algebra II: 2-Hilbert spaces*, Adv. Math. 127 (1997), 125-189.  ([arXiv:q-alg/9609018](http://arxiv.org/abs/q-alg/9609018))

Quoting:

> Given a tensor product of the 2-Hilbert spaces $H$ and $H'$, we often write its underlying 2-Hilbert space as $H \otimes H'$. This notation may tempt one to speak of ‘the’ tensor product of $H$ and $H'$, which is is legitimate if one uses the generalized ‘the’ as advocated by Dolan [7]. In a set, when we speak of ‘the’ element with a given property, we implicitly mean that this element is unique.  In a category, when we speak of ‘the’ object with a given property, we merely mean that this object is unique up to isomorphism — typically a specified isomorphism. Similarly, in a 2-category, when we speak of ‘the’ object with a given property, we mean that this object is unique up to equivalence — typically an equivalence that is specified up to a specified isomorphism. This is the sense in which we may refer to ‘the’ tensor product of $H$ and $H'$. The generalized ‘the’ may be extended in an obvious recursive fashion to $n$-categories.

Discussion in [[homotopy type theory]]:

* [[David Corfield]], _Expressing 'The Structure of' in Homotopy Type Theory_, Synthese 197, 681–700 ([doi:10.1007/s11229-017-1569-7](https://doi.org/10.1007/s11229-017-1569-7), [pdf](https://ncatlab.org/davidcorfield/show/Expressing+%27The+Structure+of%27+in+Homotopy+Type+Theory)), revised and expanded as Chapter 3 of [Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy](https://ncatlab.org/davidcorfield/show/Modal+Homotopy+Type+Theory).

[[!redirects the]]
[[!redirects generalised the]]
[[!redirects generalized the]]

[[!redirects unique]]
[[!redirects unique up to]]
[[!redirects unique up to unique isomorphism]]
[[!redirects unique up to unique coherent isomorphism]]
[[!redirects unique up to unique equivalence]]
[[!redirects unique up to unique coherent equivalence]]

[Spanish Version](https://chicksgold.com/blog/generalized-the) by [ChicksGold](https://chicksgold.com).


generalized uniform structures - table

**Generalized uniform structures**

| [[proarrow]] | [[monad]] | [[Rezk completion|Rezk-complete]] version | [[pro-object|pro]]-monad | symmetric versions | | | |
| - | - | - | - | - | - | - | - |
| [[binary relation]] | [[preorder]] | [[partial order]] | [[quasiuniformity]] | [[symmetric relation]] | [[equivalence relation]] | [[equality]] | [[uniformity]] |
| [[binary function]] to $[0,\infty)$ | [[quasipseudometric]] | [[quasimetric]] | [[quasiprometric]] | [[symmetric binary function]] | [[pseudometric]] | [[metric]] | [[prometric]] |
| [[topogeny]] | [[quasiproximity]] | | [[syntopogeny]] | [[symmetric topogeny]] | [[proximity]] | | [[symmetric syntopogeny]] |


generalized universal bundle

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###Context###
#### Homotopy
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[[!include homotopy - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Universal bundles -- or [[mapping cocylinder]]s -- are intermediate steps in the computation of [[homotopy fibers]], dual to the that way [[mapping cone]] are intermediate steps in the computation of [[homotopy cofibers]].

It is familiar from topology that one can form the path fibration $P X \to X$ of a [[topological space]]. This can be understood as an example of a general construction where one computes [[homotopy pullbacks]] of the [[point]] -- or, if things are not [[groupoid]]al, [[comma objects]].

Since universal bundles are examples of this construction, we here speak of _generalized universal bundles_. Another appropriate term might be _generalized path fibrations_.

One generalizaton of "generalized universal bundles" is that the objects in question need not be groupoidal, i.e. they behave like [[directed spaces]]. In this case the [[homotopy pullbacks]] familiar from topology are replaced by [[comma object]] constructions. This is useful in various applications. For instance the constructions [[category of elements]] and [[Grothendieck construction]] can be understood as such directed homotopy pullbacks of the point.

See also

* [[fibration sequence]]

  * [[homotopy limit]]

  * [[homotopy pullback]]

* [[principal bundle]]

* [[principal 2-bundle]]

* [[principal ∞-bundle]]

and in particular

* [[universal principal ∞-bundle]].

## Definition

Let $C$ be a [[closed monoidal category]] with [[interval object]] $I$. Then for any [[pointed object]] $pt \stackrel{pt_B}{\to}B$ in $C$ the **generalized universal $B$-bundle** is (if it exists) the morphism

$$
  p : \mathbf{E}_{pt} B \to B
$$

which is the total composite vertical morphism of the [[pullback]] diagram

$$
  \array{
    \mathbf{E}_{\mathrm{pt}}B
    &\longrightarrow&
    pt
    \\
    \big\downarrow
      &&
    \big\downarrow{}^{\mathrlap{pt_B}}
    \\
    [I,B]
    &\stackrel{d_1}{\longrightarrow}&
    B
    \\
    \big\downarrow{}^{\mathrlap{d_0}}
    \\
    B
  }
  \,.
$$

So the object $\mathbf{E}_{pt}B := [I,B]\times_{B} pt $ is defined to be the [[pullback]] of the diagram
$ [I,B] \stackrel{d_1}{\to} B \stackrel{pt_B}{\leftarrow} pt$ and the morphism $\mathbf{E}_{pt}B \to B$ is the composite of the left vertical morphism in the above diagram which comes from the definition of [[pullback]] and $d_0$.

Then a (generalized) "$B$-bundle" on some object $X$ is a morphism $P \to X$ which is the [[pullback]] of the generalized universal $B$-bundle $\mathbf{E}_{pt}$ along a "classifying morphism" $g : X \to B$

$$
  \array{
    P &\longrightarrow& \mathbf{E}_{pt}
    \\
    \big\downarrow && \big\downarrow
    \\
    X &\stackrel{g}{\longrightarrow}& B
  }
$$

This can be understood as a "(directed) [[homotopy pullback]]" of the point:

If one defines, as one does, a (possiby [[directed object|directed]]) homotopy between two morphisms $f,g : A \to B$ to be a morphism $\eta : A \to [I,B]$ such that $d_0^* \eta = f $ and $d_1^* \eta = g$, then $P$ is the "lax pullback" (really [[comma object]]) of the point along $g$

$$
  \array{
    P &\to& *
    \\
    \big\downarrow &\swArrow& \big\downarrow{}^{\mathrlap{pt_B}}
    \\
    X &\stackrel{g}{\longrightarrow}& B
  }
  \,.
$$

The generalized universal bundle can be constructed in this way if we take $X = B$:

$$
  \array{
    \mathbf{E}_{pt} &\longrightarrow& *
    \\
    \big\downarrow{}^{\mathrlap{p}}
    &\swArrow&
    \big\downarrow{}^{\mathrlap{pt_B}}
    \\
    B &\stackrel{id}{\longrightarrow}& B
  }
  \,.
$$

The [[fiber]] of the generalized universal bundle is the [[loop space object|loop monoid]] $\Omega_{pt} B$:

$$
  \array{
    \Omega_{pt} B &\longrightarrow& \mathbf{E}_{pt} &\longrightarrow& *
    \\
    \big\downarrow & & \big\downarrow^{p}
    &\swArrow& \big\downarrow{}^{\mathrlap{pt_B}}
    \\
    *
    &\stackrel{pt_B}{\longrightarrow}
    &
    B
      &\stackrel{id}{\longrightarrow}&
    B
  }
  \,.
$$

the sequence

$$
  \Omega_{pt}B \stackrel{i}{\to} \mathbf{E}_{pt} B \stackrel{p}{\to} B
$$

is exact in that $i$ is the kernel of $p$ in the sense of kernels of morphisms of [[pointed objects]] (see there).

## Examples

### Groupoid incarnations of universal principal bundles

In (higher) categorical contexts, take the interval object to the the interval category $I := \{a \to b\}$. Then

#### Ordinary $G$-principal bundles

For $C =$ [[Cat]], $B := \mathbf{B}G$ a one-object groupoid corresponding to a group $G$ with the unique point, $\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G = G//G$ is the [[action groupoid]] of $G$ acting on itself. The sequence of groupoids is
$$
  G \to G \sslash G \to \mathbf{B}G
  \,.
$$

This is the universal $G$-bundle in its groupoid incarnation. It is a theorem by Segal from the 1960s that indeed this maps, under [[geometric realization]] to the familiar universal $G$-bundle in $Top$. Moreover, it can be seen that every $G$-principal bundle $P \to X$ in the ordinary sense is the pullback of $\mathbf{E} G$ in the following sense:

the $G$-bundle $P \to X$ is classified by a nonabelian $G$-valued 1-cocycle (the transition function of any of its local trivializations), which is an [[anafunctor]]

$$
  \array{
    \hat X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
    \\
    \big\downarrow{}^{\mathrlap{\pi}}
    \\
    X
  }
  \,.
$$

(For instance $\hat X$ could be the Čech groupoid of a [[cover]] of $X$.)

The universal groupoid bundle $\mathbf{E}G \to \mathbf{B}G$ may now be pulled back along this anafunctor to yield the groupoid bundle $g^* \mathbf{E}G \to X$ given by the total left vertical morphism in

$$
  \array{
    g^* \mathbf{E}G &\longrightarrow& \mathbf{E}G
    \\
    \big\downarrow && \big\downarrow
    \\
    \hat X &\stackrel{g}{\longrightarrow}& \mathbf{B}G
    \\
    \big\downarrow{}^{\mathrlap{\pi}}
    \\
    X
  }
  \,.
$$

This bundle of groupoids is weakly equivalent to the $G$-principal bundle we started with in that there is a morphism of bundles of groupoids (with $P$ regarded as a bundle of discrete groupoids)

$$
  \array{
     g^* \mathbf{E}G &&\stackrel{\simeq}{\longrightarrow}&& P
     \\
     & \searrow && \swarrow
     \\
     &&
     X
  }
  \,.
$$

In fact that horizontal morphism is an acyclic fibration in the [[folk model structure]], i.e. a  [[k-surjective functor]] for all $k$.

This is recalled in the following reference.

#### $G$-principal 2-bundles

For $C = 2Cat$, strict 2-categories , $B := \mathbf{B}G$ a strict one-object 2-groupoid corresponding to a strict [[2-group]] $G$ with the unique point, $\mathbf{E}_{pt} \mathbf{B}G = \mathbf{E}G$ was described under the name $INN(G)$ in

* Urs Schreiber, David Roberts, _The inner automorphism 3-group of a strict 2-group_, Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 193-244
([arXiv](http://arxiv.org/abs/0708.1741v2))

This was shown to be _action bigroupoid_ of $G$ acting on itself in

* Igor Bakovič, _Bigroupoid 2-torsors _ PhD thesis, Munich (2008) ([pdf](http://edoc.ub.uni-muenchen.de/9209/)).

One can show that every $G$-principal 2-bundle as described in

* Toby Bartels, _2-Bundles_ ([arXiv](http://arxiv.org/abs/math.CT/0410328))

* Christoph Wockel, _..._

* Igor Bakovič, _Bigroupoid 2-torsors _ PhD thesis, Munich (2008) ([pdf](http://edoc.ub.uni-muenchen.de/9209/)).

indeed is recovered as the pullback of $\mathbf{E} G \to \mathbf{B}G$ along the corresponding cocycle, along the lines described above.

The way this works is indicated briefly in the last section of Roberts-Schreiber above. A more detailed description for the moment is in the notes

* Urs Schreiber: [[2bundrecon.pdf:file]]

### Universal $n$-category bundles: $n$-subobject classifiers
 {#UniversanNCategoryBundles}

One can take $B$ to be something very different from the familiar classifying groupoids. Taking it to be $n Cat$ yields the [[subobject classifiers]] of higher [[topos|toposes]]:

* $\mathbf{E}_{pt} (-1)Cat \to (-1)Cat$ is $\{\top\} \to \{\top, \bottom\}$, the [[subobject classifier]] in [[Set]].

* $\mathbf{E}_{pt} 0Cat \to 0Cat$ is $Set_* \to Set$, the forgetful functor from [[pointed sets]], which is the  2-[[subobject classifier]] in [[Cat]]. Pullback of this creates the [[category of elements]] of a [[presheaf]].

* $\mathbf{E}_{pt} Cat \to Cat$ is $Cat_* \to Cat$. Pullback of this is the [[Grothendieck construction]].

It was David Roberts in the blog comment

* David Roberts, [comment in: 101 things to do with a 2-classifier](http://golem.ph.utexas.edu/category/2008/01/101_things_to_do_with_a_2class.html#c014559)

who first pointed out that these (higher) subobject classifiers are just generalized universal bundles in the above sense.

These cases for $n= 0$ and $n=1$ have been considered in the context of universal category bundles in

* Kathryn Hess, _[[HessLackBundCat.pdf:file]]_ .

The discussion there becomes more manifestly one of bundles if one regards all morphisms $C \to Set$ appearing there as being the right legs of [[anafunctors]].

There is a well-understood version of this for $n = (\infty,1)$, i.e. for [[(infinity,1)-category|(∞,1)-categories]]. This is described at [[universal fibration of (∞,1)-categories]].

### Action groupoids as generalized bundles

A morphism $\rho : B \to F$
to a  [[pointed object]] $F$ (needs not be a basepoint preserving morphism!) can be regarded as a _representation_ of $B$ on the point of $F$. The pullback of the universal $F$-bundle along this morphism

$$ \rho^* \mathbf{E}_{pt} F \to B$$

can be addressed as the _$F$-bundle **$\rho$-associated** to the universal $B$-bundle $\mathbf{E}_{pt}B$.

If $B$ is a groupoid, then $\rho^* \mathbf{E}_{pt} F$ is the [[action groupoid]] of $B$ acting on the point of $F$.

Further pulling this back along a cocycle $g : \hat X \to B$
of a $B$-principal bundle yields the $\rho$-accociated bundle of that.

For instance for $B = \mathbf{B}G$ and $F = Vect$
with $\rho : \mathbf{B}G \to Vect$ a [[representation]] of the group $G$ on a vector space $V$, the $\rho$-associated $\mathrm{Vect}$-bundle on $\mathbf{B}G$ is

$$
  V \to V \sslash G \to \mathbf{B}G
  \,.
$$

Pulling that further back along the cocycle $g : \hat X \to \mathbf{B}G$ classifying a $G$-principal bundle $P \to X$, one obtains the familiar vector bundle $P \times_G V \to X$ which is $\rho$-associated to $P$, along the lines described above:

$$
  \array{
     g^* \rho^*\mathbf{E}_{pt}F
     &&\stackrel{\simeq}{\longrightarrow}&& P\times_G V
     \\
     & \searrow && \swarrow
     \\
     &&
     X
  }
  \,.
$$

For more on this see at *[[infinity-action|$\infty$-action]]*

[[!redirects generalized universal bundles]]


generalized vielbein

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An ordinary [[vielbein]]/[[orthogonal structure]] is a [[reduction of structure groups|reduction of the structure group]] of the [[tangent bundle]] of a [[smooth manifold]] from the [[general linear group]] $GL_n$ to its [[maximal compact subgroup]], the [[orthogonal group]].

Accordingly, whenever we have a [[reduction of structure groups]] along the inclusion $H \hookrightarrow G$ of a [[maximal compact subgroup]], we may speak of a _generalized vielbein_.

## Definition

Let $G$ be a [[Lie group]] and let $H \hookrightarrow G$ be the inclusion of a [[maximal compact subgroup]]. Write

$$
  i : \mathbf{B}H \to \mathbf{B}G
$$

for the induced morphism of [[smooth infinity-groupoid|smooth]] [[moduli stacks]] of [[principal bundles]].

Notice that

1. these form a bundle

  $$
    \array{
      G/H &\to& \mathbf{B}H
      \\
      && \downarrow^{\mathrlap{i}}
      \\
      && \mathbf{B}G
    }
  $$

  exhibiting the [[coset]] $G/H$ as the [[homotopy fiber]] of $i$;

1. under [[geometric realization]] $i$ becomes an [[weak homotopy equivalence|equivalence]]

   $$
     {\vert i\vert}  : {\vert \mathbf{B} H\vert} = B H \simeq B G = {\vert \mathbf{B}G\vert}
   $$

Then for $X$ a [[smooth manifold]] or more generally a [[smooth infinity-groupoid]] equiped with a map $g : X \to \mathbf{B}G$ an _$i$-generalized vielbein_ is a lift $e$ in

$$
  \array{
     && \mathbf{B}H
     \\
     & {}^{\mathllap{e}}\nearrow & \downarrow^{\mathrlap{i}}
     \\
     X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,.
$$

The [[moduli space]] of $i$-generalized vielbeing relative $g$ is the [[twisted cohomology]]

$$
  \mathbf{H}_{/\mathbf{B}G}(g,i)
  \,.
$$

## Properties

* Locally on $X$ the moduli space of generalized vielbeins is the [[coset]] $G/H$.

## Examples

* The ordinary notion of [[vielbein]] is obtained for

  $$
    \array{
       GL_n/O(n) &\to& \mathbf{B}O(n)
       \\
       && \downarrow
       \\
       && \mathbf{B}GL_n
    }
    \,.
  $$

* in the context of [[generalized complex geometry]] one considers generalized vielbeins arising from reduction along $O(n)\times O(n) \to O(n,n)$ of the [[generalized tangent bundle]]

  $$
    \array{
       O(n)\backslash O(n,n)/O(n) &\to& \mathbf{B}(O(n) \times O(n))
       \\
       && \downarrow
       \\
       && \mathbf{B}O(n,n)
    }
    \,.
  $$

* in the context of [[exceptional generalized geometry]] one considers vielbeins arising from reduction along $H_n \to E_n$ for $E_n$ an [[exceptional Lie group]].

## Related concepts

* [[super-vielbein]]

## References

Section _[Fields](geometry+of+physics#Fields)_ at

* _[[geometry of physics]]_

[[!redirects generalized vielbeins]]


generating function

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

> This entry is about the generating functions in the sense of algebraic combinatorics. For another notion see [[generating function in classical mechanics]].

#Contents#
* table of contents
{:toc}

## Idea

A _generating function_ is an element of $R[\![z]\!]$, the [[rig]] of formal [[power series]] over the rig $R$ (which is often taken to be the [[natural numbers]] or the [[rational number]]s), used for purposes of [[combinatorics]].  A general element takes the form
\[f(z) = \sum_{n=0}^{\infty}f_n z^n.\]
If $R$ is taken to be the [[real numbers]] or the [[complex numbers]] (or any subrig), then we can ask whether the power series has a radius of convergence $r \ge 0$ and if there's an [[analytic continuation]]; if so, then we also say that the continuation is the generating function.

When $R = \mathbb{N},$ we can think of the coefficients on $z^n$ as counting the number of ways to put a particular structure on the [[finite set]] $n$.  (You get [[structure types]] if you take this literally.)

Multiplying generating functions in the same variable gives
\[ \left(\sum_{n=0}^{\infty}f_n z^n\right)\left(\sum_{n=0}^{\infty}g_n z^n\right) = \sum_{n=0}^{\infty}\sum_{k=0}^n f_k g_{n-k} z^n,\]
which effectively says to split up the set into two parts, put the $f$ structure on the first part and the $g$ structure on the second part.

_Ordinary_ generating functions (OGFs) describe structures on [[totally ordered sets]], while _exponential_ generating functions (EGFs) apply to unordered [[sets]] (whose elements may be distinguished by having different _labels_).  For example, the generating function for being an unordered finite set is
\[ e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots ,\]
while the generating function for being a finite ordered set is
\[ \frac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots .\]

If we assume that the zeroth term is $1$, then multiplying generating functions in different variables gives
\[ \left(\sum_{n=0}^{\infty}f_n x^n\right)\left(\sum_{n=0}^{\infty}g_n z^n\right) = 1 + (f_1 x + g_1 z) + (f_2 x^2 + f_1 g_1 x z + g_2 z^2) + \cdots\]
If we take the product of countably many generating functions $f^i(x_i)$ and then set $x_i = p_i^{-s},$ where $p_i$ is the $i$th prime, then we get
\[ 1^{-s} + f^1_1 2^{-s} + f^2_1 3^{-s} + f^1_2 4^{-s} + f^3_1 5^{-s} + f^1_1 f^2_1 6^{-s} + \cdots + \prod_{i} f^i_{e_i} (p_i^{e_i})^{-s} + \cdots, \]
which is called the _Dirichlet_ generating function for the family $f^i.$

The product of two Dirichlet generating functions gives
\[ \left(\sum_{n=1}^{\infty}f_n n^{-s}\right)\left(\sum_{n=1}^{\infty}g_n n^{-s}\right) = \sum_{n=1}^{\infty} \sum_{d|n} f_d g_{n/d} n^{-s},\]
which effectively says to _factor_ the term into two dimensions, apply $f$ to the first and $g$ to the second.

Sometimes we take the *exponents* on $z$ to be in a rig other than the natural numbers.  For example, we might set $z^a \cdot z^b = z^{\max(a,b)}$ and $(z^a)^b = z^{a+b};$ such a system lets us talk about the [[rig of costs|cost]] of operations done in parallel (max) or sequentially (+).  Similarly, we could take the exponents to be [[binary string]]s when considering instantaneous codes, or [[finite field]] elements when considering structures on a finite collection of objects.

## Examples

The OGF $T(z)$ for binary [[trees]] (rooted in the plane, counted by number of internal nodes) satisfies $T(z) = 1 + z T(z)^2$, which can be solved as $T(z) = \frac{1-\sqrt{1-4z}}{2z}$.  The coefficient of $z^n$ in $T(z)$ is the $n$th _Catalan number_ $\binom{2n}{n}/(n+1)$.

Fixed-point free [[involutions]] on a set have the EGF $e^{z^2/2}$, while arbitrary involutions have EGF $e^{z+z^2/2}$, and  [[partition#of_sets|partitions]] have EGF $e^{e^z-1}$.

In [[quantum field theory]] generating functions often appear as [[partition functions]] and [[vacuum amplitudes]] as functions of [[source fields]].

## Related concepts

* [[Bogoliubov's formula]] for [[quantum observables]] in [[perturbative quantum field theory]] generated from an [[S-matrix]] functional

* [[species]]

* [[Witten conjecture]]

## References

Early articles exploring the foundations of generating functions include:

* [[Peter Doubilet]], [[Gian-Carlo Rota]], and [[Richard Stanley]], "On the foundations of combinatorial theory (VI): The idea of generating function", in _Sixth Berkeley Symposium on Mathematical Statistics and Probability_, Vol. II: _Probability Theory_, University of California, 1972, pp. 267–318. ([euclid](https://projecteuclid.org/euclid.bsmsp/1200514223))

* {#Joyal81} [[André Joyal]], _Une théorie combinatoire des séries formelles_ , Advances in Mathematics 42:1-82 (1981) doi, [MR84d:05025](http://www.ams.org/mathscinet-getitem?mr=633783)

Textbook accounts include:

* [[Herbert Wilf]], _generatingfunctionology_ (Third Edition), A K Peters, 2006. ([book webpage](https://www.math.upenn.edu/~wilf/DownldGF.html)) ([author pdf](https://www.math.upenn.edu/~wilf/DownldGF.html))

* [[Philippe Flajolet]] and [[Robert Sedgewick]], _Analytic Combinatorics_, CUP, 2009. ([author pdf](http://algo.inria.fr/flajolet/Publications/book.pdf))

[[!redirects generating functions]]
[[!redirects ordinary generating function]]
[[!redirects exponential generating function]]

[[!redirects generating functional]]
[[!redirects generating functionals]]


generating function in classical mechanics

> to be merged with [[canonical transformation]]

## Local picture

Local picture is explained in

* [[Landau-Lifschitz]], Mechanics, vol I. of Course of theoretical physics, chapter 23, Canonical transformations

Suppose we choose a Lagrangean submanifold, what gives the splitting between submanifold coordinates $q_i$, $i = 1,\ldots, n$ and the corresponding moment $p_i$ where the Hamiltonian is $H(p,q,t)$ and $p = (p_1,\ldots, p_n)$, $ = (q_1,\ldots, q_n)$. The canonical transformation by definition preserves the equations of motion; let the new coordinates be $Q_j$ and momenta $P_j$ and the new Hamiltonian is $K$. Thus both variations

$$
\delta \int p d q - H d t = \delta \int P d Q - K d t  = 0
$$

vanish. Here we of course write $p d q := \sum_{i = 1}^n p_i d q_i$. Subtracting the left and right hand side variations we get that the difference must be a (variation of the integral of the) total differential of a function, which can be chosen as a function of some set of old and new coordinates in a consistent way. For example in 1 dimension, we have 4 possibilities of one new and one old generalized coordinate.

$$
p d q - H d t = P d Q - K d t + d F
$$

$$
d F = p d q - P d Q + (K - H) d t
$$

therefore if $F = F (q,Q, t)$ then $\frac{\partial F}{\partial q} = p$, $\frac{\partial F}{\partial Q} = -P$, $K = \frac{\partial F}{\partial t} + H$, what gives the relation between the old and new coordinates and momenta and the new Hamiltonian $K$, which must be expressed in terms of $P, Q, t$.

If $F = F(q,P,t)$ then we add $P Q$, use the Leibniz rule $d (P Q) = P d Q + Q d P$ and we see that for the generating function of the "second kind", $\Phi(q,P,t) = F(q,P,t) + Q P$ the total differential

$$
d \Phi = d (F + P Q) = p d q + Q d P + (K - H) d t
$$
and $\frac{\partial \Phi}{\partial q} = p$, $\frac{\partial F}{\partial P} = Q$, $K = \frac{\partial \Phi}{\partial t} + H$. A similar analysis can be done straightfowardly for other possibilities of the choice of arguments.

## Global picture

An invariant picture of generating functions on symplectic manifolds is in

* C. Viterbo, _Symplectic topology as the geometry of generating functions_, Math. Ann. __292__ (1992), 685–710, [MR1157321](http://www.ams.org/mathscinet-getitem?mr=1157321), [doi](http://dx.doi.org/10.1007/BF01444643)

* C. Viterbo, _Generating functions, symplectic geometry, and applications_, Proc. ICM  537-547 [doi](https://doi.org/10.1007/978-3-0348-9078-6_47)

* L. Traynor, _Symplectic homology via generating function_, Geom. Funct. Anal. 4 (1994) 718-748, [MR1302337](http://www.ams.org/mathscinet-getitem?mr=1302337), [doi](http://dx.doi.org/10.1007/BF01896659)

* David Théret, _A complete proof of Viterbo’s uniqueness theorem on generating functions_, Topology and its Applications __96__ (1999) 249–266 [pdf](https://core.ac.uk/download/pdf/82459899.pdf)

An adaption of generating functions to the setup of symplectic micromorphisms is in

*  Alberto S. Cattaneo, Benoit Dherin, Alan Weinstein, _Symplectic microgeometry II: generating functions_, [arxiv/1103.0672](http://arxiv.org/abs/1103.0672)

[[!redirects generating functions in classical mechanics]]


generation of fundamental particles

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[standard model of particle physics]] has the curious property that its content of [[fermions]] appears in three sets of particles that share all properties except that their [[rest mass]] increases drastically from one set to the next. These are called the three "generations" "families" of fermionic particles.

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

Any reason for this striking pattern presently remains mysterious.

While the particle content of any one single generation would naturally be explained by [[GUT]] models, in terms of a single [[irreducible representation]] of a [[simple group|simple]] [[gauge group]] containing that of the [[standard model of particle physics]], this does not explain why there are three almost identical copies of this.

One suggestion to explain this phenomenon appears in [[intersecting D-brane models]]. See there at _[Generations of fermions](intersecting+D-brane+model#GenerationsOfFermions)_.

## Related concepts

* [[flavour anomaly]]

* [[fourth generation of fundamental particles]]

## References

### General

See also

* Wikipedia, _Generation (particle physics)_

### In string theory

On [[model (in theoretical physics)|model building]] with realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in

* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))

* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys.Rev.D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))

* {#Mayes19} [[Van Eric Mayes]], _All Fermion Masses and Mixings in an Intersecting D-brane World_ ([arXiv:1902.00983](https://arxiv.org/abs/1902.00983))

* {#GemillHowingtonMayes19} Jordan Gemmill, Evan Howington, Van E. Mayes, _One String to Rule Them All: Neutrino Masses and Mixing Angles_ ([arXiv:1907.07106](https://arxiv.org/abs/1907.07106))

[[!redirects generations of fundamental particles]]

[[!redirects generation of fermions]]
[[!redirects generations of fermions]]

[[!redirects family of fundamental particles]]
[[!redirects families of fundamental particles]]

[[!redirects family of fermions]]
[[!redirects families of fermions]]

[[!redirects generation of fundamental fermions]]
[[!redirects generations of fundamental fermions]]

[[!redirects first generation of fundamental particles]]
[[!redirects second generation of fundamental particles]]

[[!redirects first generation of fermions]]
[[!redirects second generation of fermions]]


generator

The term _generator_ can refer to several distinct but related concepts.

See:

* [[generators and relations]]

* [[separator]]

[[!redirects generator: disambiguation]]

[[!redirects generators]]


generator > history

< [[generator]]
[[!redirects generator>History]]


generator > history II

< [[generator]]


generator > history III

< [[generator]]


generators and relations

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

# Generators and relations
* table of contents
{: toc}

## Idea

A mathematical structure is said to be _presented by generators and relations_ if it is obtained from a structure _[[free construction|freely]] generated_ by a [[set]] of _generators_ by imposing _relations_ among these generators.

For instance an [[associative algebra]] $A$ may be obtained by imposing on the [[tensor algebra]] of the underlying vector space (which is the [[free monoid|free algebra]] on that vector space) the product relations, which identify the formal juxtaposition of two elements with their intended product. This is for instance how _[[Clifford algebras]]_ or _[[universal enveloping algebras]]_ are traditionally defined.

Similarly a [[group presentation]] specifies a group in terms of a set of generators and relations.  The study of group presentations form and important part of [[combinatorial group theory]].

But the concept of _generators and relations_ is very general and applies to all kinds of structures. Below we discuss a general formalization using concepts from [[category theory]] and [[universal algebra]].

## Definition

Let $C$ be a [[category]], $T$ a [[monad]] on $C$, and $A$ a $T$-[[algebra for a monad|algebra]].  A **presentation of $A$ by generators and relations** is a [[coequalizer]] diagram

$$ T R \; \rightrightarrows \; T G \to A $$

in the category $C^T$ of $T$-algebras.  Here $G\in C$ is the *object of generators* and $R\in C$ is the *object of relations*.

The $T$-algebra map $T G \to A$ corresponds to a unique morphism $G\to A$ in $C$, which we think of as the "inclusion" of the generators into $A$ (though in general it need not be any kind of [[monomorphism]]).  Similarly, the $T$-algebra maps $T R \;\rightrightarrows\; T G $ correspond to unique morphisms $R \;\rightrightarrows\; T G$ in $C$, which we think of as a [[binary relation]] on $T G$ (though in general, the maps need not be jointly monic either).

This definition can be generalised to any [[concrete category]] with [[free objects]], not necessarily monadic.

## Examples

### Monads on Set

The classical case is when $C=$[[Set]].  Then $G$ and $R$ are sets, and a presentation of a $T$-algebra by generators and relations amounts to first constructing the free $T$-algebra $T G$ on the generators, then specifying (via the maps $R \;\rightrightarrows\; T G$) a set of pairs of elements of $T G$ to be identified (the relations).  The presented $T$-algebra $A$ is the [[quotient set]] by these relations.

Probably the *most* classical case is when $T$ is the free [[group]] monad on $Set$.

### Categories

[[Cat]] (to be precise, [[Str Cat]]) is the category of algebras for a monad on the category of [[quivers]].  Therefore, we have a notion of a [[presentation of a category by generators and relations]].

### Theories and Monads

[[Lawvere theories]], [[operads]], and even ([[accessible functor|accessible]]) [[monads]] can be realized as the algebras for monads on some categories.  Thus, we can talk about presentations of these sorts of objects by generators and relations.  In this context, each "generator" is an *operation*, and each "relation" is an *axiom* that the operations must satisfy.  For instance, the usual definition of a group in terms of binary, nullary, and unary operations satisfying associativity, unitality, and inversion axioms can be regarded as a presentation of the Lawvere theory (or monad) of groups by generators and relations.

## Existence

+-- {: .num_theorem}
###### Theorem
Every $T$-algebra has a presentation by generators and relations.
=--
+-- {: .proof}
###### Proof
For any $T$-algebra $A$, the following diagram is a coequalizer (also called the [[Beck coequalizer]]):
$$ T^2 A \; \underoverset{\mu}{\T a}{\rightrightarrows}\; T A \xrightarrow{a} A$$
where $a\colon T A \to A$ is the structure map of the $T$-algebra $A$ and $\mu$ is the multiplication of the monad $T$.
=--

This particular presentation is of importance in the [[monadicity theorem]].

However, in practice one is usually interested in more "economical" presentations where $G$ and $R$ are smaller than $A$.  For example, for monads over [[Set]], when $G$ and $R$ are [[finite sets]], then we say that $A$ is __[[finitely presented object|finitely presented]]__.

## Remarks

Note that knowing only the category of $T$-algebras does not suffice to determine the notion of "presentation by generators and relations"; we also have to exhibit it as the category of algebras for a monad $T$ on some other category $C$.  Often the choice of $C$ is obvious, but there are cases in which the same category can be realized as the category of algebras for multiple monads on multiple different categories, yielding different notions of "generators and relations".  For example, the category of finitary monads on $Set$ (= Lawvere theories) can be realized as monadic over the category of finitary endofunctors of Set, or over the category of $\mathbb{N}$-indexed sets.

## Categorifications

If $C$ is an [[n-category]] and $T$ a suitable sort of monad, then we have a similar notion except that coequalizers must be replaced by ($n$-truncated) [[codescent object|codescent objects]], i.e. [[geometric realization]].

## Related concepts

* [[finitely generated object]]

[[!redirects generators and relations]]
[[!redirects presentation by generators and relations]]
[[!redirects presentations by generators and relations]]

[[!redirects presentation]]
[[!redirects presentations]]


generic interval

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

#Contents#
* table of contents
{:toc}

## Idea

The _generic interval_ is the standard [[1-simplex]] $\Delta_1$ in [[sSet]].

## Properties

What makes this interval 'generic' is the following result:

\begin{theorem}

_The [[topos]] [[sSet]] of [[simplicial sets]] is the [[classifying topos]] for [[linear orders]] with distinct [[top]] and [[bottom]] elements, i.e. intervals (aka nontrivial totally [[distributive lattices]]) with generic interval $\Delta_1=Hom_{\Delta}(\quad,[1])$.

If $I$ is an interval in a topos $\mathcal{E}$, then it corresponds to a [[geometric morphism]] $S_I \,\colon\, \mathcal{E}\rightarrow sSet$ with [[direct image]] part $S_I_* \,\colon\, \mathcal{E}\rightarrow sSet$ the [[singular simplicial set|singular functor]] and [[inverse image]] part the [[geometric realization]] functor $|\quad|_I \,\colon\, sSet\rightarrow \mathcal{E}$.

\end{theorem}

This result was announced by [[André Joyal]] in 1974 at the [[Isle of Thorns]], the first published proof appears in [Johnstone 1979](#Johnstone79) and in book form in [MacLane-Moerdijk 1994](#MacLaneMoerdijk94), sec. VIII.8).

\begin{remark}

The classifying topos point of view is helpful, because it is the choice of an algebraic structure, of the kind classified, in a [[topological topos]] that gives rise to an exact singular/realization pair.  ([Lawvere 2013](#GrandisLawvere13))

\end{remark}

\begin{remark}

The generic interval and the results of [Wraith 1993](#Wraith93) play a role in Lawvere's attempt to view classical combinatorial topology as part of greater landscape situated in the [[presheaf topos]] on the category of nonempty finite sets, which is the classifying topos for nontrivial [[Boolean algebras]] and contains [[simplicial complexes]] and the category of [[groupoids]] [[Grpd]] as [[reflective subcategories]]  (cf. [Lawvere 1989](#Lawvere89), pp.273-275).

\end{remark}

## Related Concepts

* [[topological topos]]

* [[Aufhebung]]

* [[interval object]]

* [[sSet]]

## References ##

* {#Johnstone79} [[Peter Johnstone]], _On a topological topos_ , Proc. London Math. Soc. **3** 38 (1979) pp.237-271. (cf. this n-café blog ([link](http://golem.ph.utexas.edu/category/2014/04/on_a_topological_topos.html)))

* {#Lawvere89} [[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_, Contemporary Mathematics **92** (1989) pp.261-299 ([doi:10.1090/conm/092](http://dx.doi.org/10.1090/conm/092)).

* [[F. William Lawvere]], _Categories of Spaces may not be Generalized Spaces as Exemplified by Directed Graphs_, Revista Colombiana de Matemáticas **XX** (1986) pp.179-186. Reprinted with commentary in TAC **9** (2005) pp.1-7. ([pdf](ftp://ftp.tac.mta.ca/tac/html/tac/reprints/articles/9/tr9.pdf))

* {#Wraith93} [[Gavin C. Wraith]], _Using the Generic Interval_, Cah. Top. Géom. Diff. Cat. **XXXIV** 4 (1993) pp.259-266. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1993__34_4/CTGDC_1993__34_4_259_0/CTGDC_1993__34_4_259_0.pdf))

* {#MacLaneMoerdijk94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994.

* [[F. William Lawvere]], _Core Varieties, Extensivity, and Rig Geometry_, TAC  **20** no.14 (2008) pp.497-503. ([pdf](ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/14/20-14.pdf))

* [[F. William Lawvere]], [[Ross Street]], [[Todd Trimble]], _catlist exchange October 2011_ , ([link](http://comments.gmane.org/gmane.science.mathematics.categories/6980)) .

* {#GrandisLawvere13} [[Marco Grandis]], [[F. William Lawvere]], _catlist exchange November 2013_ , ([link](http://comments.gmane.org/gmane.science.mathematics.categories/7920)).