general covariance

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## Idea
 {#Idea}

In [[physics]], the term _general covariance_ is meant to indicate the property of a [[physical system]] or [[model (in theoretical physics)]] whose [[configuration space|configurations]], [[action functional]] and [[equations of motion]] are all [[equivariance|equivariant]] under the [[action]] of the [[diffeomorphism group]] on the [[smooth manifold]] underlying the [[spacetime]] or the [[worldvolume]] of the system. Here "covariance" is as in "[[covariant tensor]]" another term for behaviour under diffeomorphisms.

The term _[[general relativity]]_ for [[Einstein]]-[[gravity]] originates in at least closely related ideas (see _[History and original formulation](#HistoryAndOriginalFormulation)_ below), and Einstein-gravity is the archtypical example of a generally covariant physical system:

here, the [[configuration space]] of physical fields over a [[smooth manifold]] $\Sigma$ is not quite the space of [[Riemannian metrics]] on $\Sigma$ itself, but is the [[quotient]] $[\Sigma,\mathbf{Fields}]/\mathbf{Diff}(\Sigma)$ of this space by the [[action]] of the [[diffeomorphism group]] $\mathbf{Diff}(\Sigma)$: two Riemannian metrics $g_1$ and $g_2$ on $\Sigma$ represent the same field of [[gravity]] on $\Sigma$ if there is a [[diffeomorphism]] $f : \Sigma \stackrel{\simeq}{\to} \Sigma$ such that $g_2 = f^* g_1$.

Or rather, such a diffeomorphism is a [[gauge transformation]] between the two field configurations. The configuration space is not the naive [[quotient]] of fields by diffeomorphisms as above, but is the [[homotopy quotient]], or _[[action groupoid]]_, denoted $[\Sigma,\mathbf{Fields}]\sslash \mathbf{Diff}(\Sigma)$. In the physics literature this action groupoid is most familiar in its [[infinitesimal cohesion|infinitesimal approximation]], the corresponding [[Lie algebroid]], whose formal dual is a [[BRST complex]] whose degree-1 elements are accordingly called the _diffeomorphism ghosts_ (see there).

As with all [[gauge transformations]], they relate physical configurations which may be nominally different, but [[equivalence|equivalent]].
Therefore _general covariance_ is an instance of the general _[[principle of equivalence]]_ in mathematics which says that sensible statements about [[objects]] must respect the [[isomorphisms]] and more general [[equivalences]] between these objects.

A physical system which is not generally covariant in this sense is hence one where the [[smooth manifold]] $\Sigma$ as above, underlying [[spacetime]]/[[worldvolume]] is not regarded as modelling an absolute physical system (such as the [[observable universe]] in gravity), but a [[subsystem]] that is equipped with ambient [[structure]] that [[spontaneous symmetry breaking|breaks]] the diffeomorphism symmetry. Notably systems like [[electromagnetism]] or [[Yang-Mills theory]] have traditionally been written in a non-generally covariant form describing gauge fields on a fixed gravitational background, as for instance the space inhabited by a particle accelerator. This ambient structure on the spacetime $\Sigma$ breaks its general diffeomorphism invariance and hence the effective resulting theory on this background is not generally covariant (a special case of the general phenomenon of [[spontaneous symmetry breaking]]).

On the other hand, such a model consisting of background (e.g. the particle accelerator) and quantum fields propagating in it is ultimately to be understood as an approximation to a more encompassing model in which also the background is dynamical, and which is again generally covariant. Specific for electromagnetism and Yang-Mills theory this refined generally covariant model is known as _[[Einstein-Maxwell theory]]_ or more generally _[[Einstein-Yang-Mills theory]]_.

The idea of general covariance has a long and convoluted history and the literature witnesses plenty of disagreement about how to interpret and formalize it in technical detail ([Norton](#Norton)). Already early arguments by [[Einstein]] himself (e.g. the "[[hole paradox]]" ([Einstein-Grossmann](#EinsteinGrossmann))) show that the discussion has suffered from the beginning from lack of the basic [[category theory|category theoretic]] concept  of [[isomorphism]]  in the [[category]] [[Diff]] of [[smooth manifolds]]. Below in _[Formalization in homotopy type theory](#InHomotopyTypeTheory)_ we indicate a formalization of general covariance that is general, fundamental, and accurately reflects the role of the term in theoretical physics.

## History and original formulation
 {#HistoryAndOriginalFormulation}

> The question of general covariance of physical theories in space and time can be traced back to the famous debate between [[Gottfried Wilhelm Leibniz]] and [[Samuel Clarke]] (the latter assisted by Sir [[Isaac Newton]]) on the [[ontology|ontological]] status of space in the years 1715–1716 ([Alexander](#Alexander)), the central question being if space exists as a substance or as an absolute being and absolute motion is present (Clarke) or if it is constituted only in relation to co-existent things allowing for relativism in motions only (Leibniz). This kind of problems also played an important role when the general theory of relativity was being developed in the years around 1910. While [[Albert Einstein]] first characterized generally covariant field equations as inadmissible since they did not determine the [[Riemannian metric|metric field]] uniquely as shown in the hole argument ( _Lochbetrachtung_ ) in the appendix of ([Einstein-Grossmann](#EinsteinGrossmann)), he later accepted ([Einstein 1916](#Einstein1916)) that all [[physical laws]] had to be expressed by [[equations]] that are valid in all [[coordinate systems]], i. e., which are covariant (generally covariant) under arbitrary [[substitution|substitutions]].

>> _Die allgemeinen Naturgesetze sind durch Gleichungen auszudrücken, die für alle Koordinatensysteme gelten, d. h. die beliebigen Substitutionen gegenüber kovariant (allgemein kovariant) sind._ ([Einstein 1916 p. 776](#Einstein1916))

> The hole argument was dismissed by the reasoning that it is not the [[spacetime]] [[Riemannian metric|metric]] that has to be fixed uniquely by the [[equations of motion|field equations]], but only the physical phenomena that occur in spacetime need to be given a unique expression with reference to any description of spacetime. All physical statements are given in terms of spacetime coincidences; [[measurement|measurements]] result in statements on meetings of material points of the measuring rods with other material points or in coincidences between watch hands and points on the clockface. The introduction of a reference system merely serves the easy description of the totality of all these coincidences (point-coincidence argument) ([Einstein 1916 p. 776f](#Einstein1916)).

> from ([Brunetti-Pormann-Ruzzi](#BrunettiPorrmannRuzzi))

## Modern formulation in differential geometry
 {#ModernFormulationInDifferentialGeometry}

We discuss the modern formulation of general covariance in [[differential geometry]].

### In gravity

Let $\Sigma \in $ [[SmoothMfd]] be a [[smooth manifold]]. Write $Riem(\Sigma)$ for the space of ([[pseudo-Riemannian metric|pseudo]]-)[[Riemannian metrics]] on $\Sigma$. For $f : \Sigma \to \Sigma$ a [[diffeomorphism]], there is a [[function]] $f^* : Riem(\Sigma) \to Riem(\Sigma)$ which sends a Riemannian metric to its pullback:

$$
  (f^*g)(v,w) \coloneqq g(f_* v, f_* w)
  \,,
$$

where $f_* : T\Sigma \to T\Sigma$ is the canonical map induced on the [[tangent bundle]] (see at _[[derivative]]_).

Say that two metrics $g_1, g_2$ are [[gauge transformation|gauge equivalent]] if there is a diffeomorphism $f$ such that $g_2 = f^* g_1$.  This is an [[equivalence relation]]. Write $Riem(\Sigma)/Diff(\Sigma)$ for the corresponding set of [[equivalence classes]].

The statement of _general covariance_ is that the distinct configurations of the gravitational field form the set $Riem(\Sigma)/Diff(\Sigma)$. In particular, if $\Sigma$ is [[compact topological space|compact]], then the [[Einstein-Hilbert action]] functional which *a priori* is defined on $Riem(\Sigma)$ descends to $Riem(\Sigma)/Diff(\Sigma)$

$$
  S_{EH} : Riem(\Sigma)/Diff(\Sigma) \to \mathbb{R}
  \,.
$$

While this captures the idea of general covariance accurately, for further development of the theory of [[gravity]], however, the set $Riem(\Sigma)/Diff(\Sigma)$ needs to be refined. It is really equipped with the structure of a [[smooth space]] $\mathbf{Riem}(\Sigma)/\mathbf{Diff}(\Sigma)$ (in order to perform [[variational calculus]] and hence derive the [[equations of motion]] of the theory), and second it is to be refined to a [[smooth infinity-groupoid|smooth groupoid]] $\mathbf{Riem}(\Sigma)\sslash\mathbf{Diff}(\Sigma)$.

Finally, for setups that admit to introduce [[fermions]]/[[spinors]] into the model one needs to refine Riemannian metrics to [[vielbein]] fields/[[orthogonal structures]]. The fully refined generally covariance smooth configuration groupoid is then $[\Sigma, \mathbf{orth}]\sslash \mathbf{Diff}(\Sigma)$, discussed in more detail [below](#GravityInHoTT).

### Relation to the "principle of equivalence" in gravity
 {#RelationToPrincipleOfEquivalenceInGravity}

The _[[equivalence principle (physics)|principle of equivalence]]_ in [[general relativity]] is formally the statement that around every point in a ([[pseudo-Riemannian metric|pseudo]]-)[[Riemannian manifold]] one can find a [[coordinate system]] such that the [[Levi-Civita connection]] vanishes ("[[Riemann normal coordinates]]"), which means that to **first [[infinitesimal space|infinitesimal]] order** around that point particle dynamics subject to the force of gravity is equivalent to dynamics in [[Minkowski spacetime]] with vanishing field of gravity.

By the above, this is a special case of the principle of general covariance: for every field configuration $g$ and every given point there is a gauge equivalent field configuration $f^* g$ such that the "force of gravity" (the Levi-Civita connection) vanishes at that point.

It is via this relation that the physical "[[equivalence principle (physics)|principle of equivalence]]" relates to the mathematical _[[principle of equivalence]]_: this says that formulations need to respect the given notion of [[equivalence]]/[[gauge transformation]], and so

[[principle of equivalence]] in mathematics $\Rightarrow$ principle of general covariance $\Rightarrow$ principle of equivalence in physics .

### In other topological field theories

## Formalization in homotopy type theory
 {#InHomotopyTypeTheory}

We discuss here that general covariance in field theory has a natural formalization in [[homotopy type theory]], hence internal to any [[(∞,1)-topos]]. For exposition, background and further details on the discussion of [[classical field theory|classical]]/[[quantum field theory]] in this fashion see ([Schreiber, ESI lectures](#SchreiberLectures)) and ([Schreiber-Shulman](#SchreiberShulman)).

The same kind of construction yields important [[moduli stacks]], for instance the [[moduli stack of Riemann surfaces]], see [this remark](moduli+space+of+curves#InTermsOfTheHigherToposOfSmoothStacks) there.

### Informal introduction
 {#InformalIntroduction}

Let $\Sigma$ be a [[smooth manifold]] to be thought of as [[spacetime]].

Then the central idea of _general covariance_ is that for

$$
  i_1 \;\colon\; U \hookrightarrow \Sigma
$$

and

$$
  i_2 \;\colon\; U \hookrightarrow \Sigma
$$

two [[subsets]]/[[submanifolds]], they should be regarded as [[equivalence|equivalent]] if there is a [[diffeomorphism]] $\phi \;\colon\; \Sigma \stackrel{}{\longrightarrow} \Sigma$ which takes one to the other, hence such that
$i_2 = \phi^\ast i_1 \coloneqq\phi \circ i_1 $.

That this statement can be puzzling if one thinks of the case $U = \ast$ as being just a single point is the content of the historical "[[hole argument]] [[paradox]]".

But with just a little bit of formalization the apparent paradox is resolved, because the above evidently just says that the "[[moduli space]]" for "subsets of spacetime" is not the manifold $\Sigma$ itself, but is rather a "[[moduli stack]]" namely the [[quotient stack]] $\Sigma //Diff(\Sigma)$ of $\Sigma$ by the [[action]] of the [[diffeomorphism group]].

Indeed, the technical term "[[quotient stack]]" is precisely defined by the condition that for $U$ of the shape of a [[disk]]/[[coordinate chart]], then two maps

$$
  i \;\colon\; U \hookrightarrow \Sigma \longrightarrow \Sigma//Diff(\Sigma)
$$

to it are equivalent if (and only if) there is a diffeomorphism relating them, as above.

So if in a generally covariant field theory spacetime is not actually the manifold $\Sigma$, but rather the quotient stack $\Sigma//Diff(\Sigma)$, then also a [[field (physics)|field]] in this generally covariant field theory should be a field on that quotient stack, not on $\Sigma$ itself.

For $\mathbf{Fields}$ a [[moduli space]]/[[moduli stack]] of [[field (physics)|fields]], in a non-generally covariant field theory a field configuration is simply a map

$$
  \Phi \;\colon\; \Sigma \longrightarrow \mathbf{Fields}
$$

and accordingly the [[configuration space|space of all field configurations]] is the [[mapping space]] $[\Sigma, \mathbf{Fields}]$.

From this it is clear that for a generally covariant field theory we are instead to declare that the space of field configurations is

$$
  [\Sigma//Diff(\Sigma), \;\mathbf{Fields}]
  \,.
$$

And it is at this point that the formalism of [[homotopy type theory]]/[[(infinity,1)-topos theory|higher topos theory]] works a little wonder for us. Namely the formalism allows us to prove, and this is what is discussed below, that

$$
  [\Sigma//Diff(\Sigma),\; \mathbf{Fields}]
  \simeq
  [\Sigma,\; \mathbf{Fields}]//Diff(\Sigma)
  \,.
$$

In words, the right hand side is the time-honored answer: two fields _on_ a spacetime manifold $\Sigma$ which are such that one goes over into the other when pulled back along a diffeomorphism are [[gauge equivalence|gauge equivalent]]. This is the statement of general covariance, derived here, formally, from just the condition that any two shapes _in_ spacetime are to be equivalent if related by a diffeomorphism.

Here to read the above equivalence as a theorem, we have to read the left hand side, as it should, be "in the context of $Diff(\Sigma)$-actions". Such [[context]]-dependence is precisely what _[[dependent type theory|dependent]]_ [[homotopy type theory]] takes care of, and this is what the following technical statement deals with.

### Introduction in terms of type theory

A basic idea of traditional [[dependent type theory]] is of course that [[types]] $A$ may appear _in [[context]]_ of other types $\Gamma$. The traditional interpretation of such a dependent type

$$
  x : \Gamma \vdash A(x) : Type
$$

is that of a $\Gamma$-parameterized family or _[[bundle]]_ of types, whose [[fiber]] over $x \in \Gamma$ is $A(x)$.

But in [[homotopy type theory]], types are [[homotopy types]] and, hence so are the contexts. A [[type in context]] is now in general something more refined than just a family of types. It is really a family of types equipped with _[[equivariance]]_ structure with respect to the [[homotopy groups]] of the context type.

Hence in homotopy theory [[types in context]] generically satisfy an [[equivariance]]-principle.

Specifically, if the context type is [[n-connected object in an (infinity,1)-topos|connected]] and [[pointed object|pointed]], then it is equivalent to the [[delooping]] $\Gamma \simeq \mathbf{B}G$ of an [[∞-group]] $G$. One finds -- this is discussed at _[[∞-action]]_ -- that the [[context]] defined by the type $\mathbf{B}G$ is that of $G$-[[equivariance]]: every type in the context is a type in the original context, but now equipped with a $G$-[[∞-action]]. In a precise sense, the homotopy type theory of $G$-$\infty$-actions is equivalent to plain homotopy type theory _in [[context]]_ $\mathbf{B}G$.

In the following we discuss this for the case that $G$ is an [[automorphism ∞-group]] of a type $\Sigma$ which is regarded as representing [[spacetime]] or a [[worldvolume]]. We show that in this context the rules of homotopy type theory automatically induce the principle of general covariance and naturally produce configurations spaces of generally covariant field theories: for $\mathbf{Fields}$ a moduli object for the given fields, so that a field configuration is a function $\phi : \Sigma \to \mathbf{Fields}$, the configuration space of _covariant_ fields is the [[function type]] $(\Sigma \to \mathbf{Fields})$ but formed in the "general covariance context" $\mathbf{B}\mathbf{Aut}(\Sigma)$. When [[categorical semantics|interpreted]] in [[smooth infinity-groupoid|smooth models]], $\mathbf{Conf}$ is the smooth groupoid of field configurations and diffeomorphism gauge transformations acting on them, the [[Lie integration|Lie integrations]] of the [[BRST complex]] whose degree-1 elements are the _diffeomorphism ghosts_.

More precisely, we show the following.

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

Consider in homotopy type theory two types
$\vdash \Sigma, \mathbf{Fields} : Type$, to be called _spacetime_ and _field moduli_. Let

$$
  \vdash \mathbf{B}\mathbf{Aut}(\Sigma) : Type
$$

be the image of the name of $\Sigma$, with essentially unique term

$$
  \vdash \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)
  \,.
$$

Perform the canonical context extension of $\Sigma$ and trivial context extension of $\mathbf{Fields}$ to get types in context

$$
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)
  \vdash
  \Sigma : Type
$$

and

$$
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)
  \vdash
  \mathbf{Fields} : Type
  \,.
$$

Form then the type of field moduli "$\mathbf{Conf} \coloneqq (\Sigma \to \mathbf{Fields})$" in this context:

$$
  \mathbf{Conf} \coloneqq
  \;\;\;\;\;
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)
  \vdash (\Sigma \to \mathbf{Fields}) : Type
  \,.
$$

=--

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

The [[categorical semantics]] of $\mathbf{Conf}$ in the model [[Smooth∞Grpd]] of the homotopy type theory and for $\Sigma \in $ [[SmoothMfd]] $\hookrightarrow Smooth \infty Grpd$ is given by the [[diffeological space|diffeological groupoid]]

$$
  \mathbf{Conf} = [\Sigma, \mathbf{Fields}]\sslash \mathbf{Diff}(\Sigma)
$$

whose objects are field configurations on $\Sigma$ and whose morphisms are diffeomorphism gauge transformations between them. In particular the corresponding [[Lie algebroid]] is dual to the [[BRST complex]] of fields with diffeomorphism ghosts in degree 1.

=--

### The ambient theory

Write $\mathbf{H}$ for the ambient [[homotopy type theory]], or rather an [[categorical semantics|interpretation]] given by an [[(∞,1)-topos]].
For standard applications in [[physics]] we have $\mathbf{H} = $ [[Smooth∞Grpd]] or [[SmoothSuper∞Grpd]] or similar, but none of the following general discussion depends on a concrete choice for $\mathbf{H}$.

Pick once and for all an [[object]]

$$
  \Sigma \in \mathbf{H}
$$

supposed to represent the [[space]] underlying [[spacetime]] or the [[worldvolume]] of a [[model (in theoretical physics)]],

### The diffeomorphism group

Write

$$
  \mathbf{Aut}(\Sigma) \in Grp(\mathbf{H})
$$

for the [[automorphism ∞-group]] of $\Sigma$. As discussed there, this is the [[loop space object]] of the  [[homotopy image]]-factorization of

$$
  * \stackrel{\vdash \Sigma}{\to} Type
  \,,
$$

hence the factorization by an  [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] followed by a [[monomorphism in an (infinity,1)-category|monomorphism]]:

$$
  * \stackrel{}{\to} \mathbf{B}\mathbf{Aut}(\Sigma)
    \stackrel{}{\hookrightarrow} Type
  \,.
$$

In the standard [[categorical semantics|interpretation]] of the homotopy type theory in $\mathbf{H} = $ [[Smooth∞Grpd]] $\Sigma$ could be an ordinary [[smooth manifold]] or [[orbifold]], in particular, and then $\mathbf{Aut}(\Sigma) = \mathbf{\Diff}(\Sigma)$ is the [[diffeomorphism group]] of $\Sigma$, regarded as a [[diffeological space|diffeological]] [[group object]]. In view of this archetypical example we will in the following often say _[[diffeomorphism]]_ for short instead of _auto-equivalence in $\mathbf{H}$_ and similarly refer to $\mathbf{Aut}(\Sigma)$ loosely as the _diffeomorphism group_ of $\Sigma$. But even in the specifical model $\mathbf{H} = $ [[Smooth∞Grpd]]/[[SmoothSuper∞Grpd]], $\Sigma$ can be much more general than a [[smooth manifold]] or [[supermanifold]] or [[orbifold]].

### The context of general covariance

Write then $\mathbf{B} \mathbf{Aut}(\Sigma) \in \mathbf{H}$ for the [[delooping]] of the diffeomorphism group. The essentially unique [[term]] of this type is to be thought of as being $\Sigma$ itself, and so we
write it as

$$
  \vdash \Sigma : \mathbf{B} \mathbf{Aut}(\Sigma)
  \,.
$$

By the discussion at _[[∞-action]]_, a [[type in context]] of $\mathbf{B}\mathbf{Aut}(\Sigma)$

$$
  \Sigma : \mathbf{B} \mathbf{Aut}(\Sigma) \vdash V(\Sigma)
$$

is a type $\vdash V : Type$ in the absolute context equipped with an $\mathbf{Aut}(\Sigma)$-[[∞-action]] $\rho : V \times \mathbf{Aut}(\Sigma) \to V$ on it. The [[dependent sum]]

$$
  \vdash \sum_{\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)} V(\Sigma) : Type
  \,,
$$

which we also write

$$
  V \sslash \mathbf{Aut}(\Sigma)
  \coloneqq
  \sum_{\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)}
   V(\Sigma)
  \,,
$$

is the total space of the universal [[associated ∞-bundle|associated]] $V$-[[fiber ∞-bundle]]:

$$
  \array{
     V &\to& \sum_{\mathbf{B}\mathbf{Aut}(\Sigma)} V(\Sigma)
     \\
      && \downarrow^{\overline{\rho}}
     \\
      && \mathbf{B}\mathbf{Aut}(\Sigma)
  }
  \,.
$$

Hence the [[categorical semantics|interpretation]] of the context $\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)$  is the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}\mathbf{Aut}(\Sigma)}$ and equivalently is the [[(∞,1)-topos]] of objects in $\mathbf{H}$ equipped with $G$-[[∞-actions]] and of $G$-[[equivariance|equivariant]] [[morphisms]] between them:

$$
  \mathbf{H}_{/\mathbf{B}\mathbf{Aut}(\Sigma)}
  \simeq
  Act_{\mathbf{H}}(\mathbf{Aut}(\Sigma))
  \,.
$$

Hence a [[type in context]] $\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)$ is a "generally covariant type" with respect to $\Sigma$ in the sense that it transforms [[covariant functor|covariantly]] by equivalences under diffeomorphisms of $\Sigma$. In summary then

**Fact.** $\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)$ _is the **[[context]] of general covariance** with respect to $\Sigma$.

In the precise formal sense.

In particular, $\Sigma$ itself is canonically equipped with the defining action of $\mathbf{Aut}(\Sigma)$ on it, which [[syntax|syntactically]] we may write

$$
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma) \vdash \Sigma : Type
$$

and which [[categorical semantics|semantically]] is exhibited by the universal  [[associated ∞-bundle|associated]] $\Sigma$-[[fiber ∞-bundle]]

$$
  \array{
    \Sigma &\to& \sum_{\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)} \Sigma
    \\
    && \downarrow^{\overline{\rho_{\Sigma}}}
    \\
    && \mathbf{B} \mathbf{Aut}(\Sigma)
  }
  \,,
$$

given by the [[pullback]] of the [[universe]] $\widetilde Type \to Type$ along the defining [[monomorphism in an (infinity,1)-category|inclusion]] $\mathbf{B}\mathbf{Aut} \hookrightarrow Type$.

Here the total space

$$
  \Sigma \sslash \mathbf{Aut}(\Sigma)
  \coloneqq
  \sum_{\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma) } \Sigma
$$

is the [[homotopy quotient]] or [[action groupoid]] of $\Sigma$ by $\mathbf{Aut}(\Sigma)$. This is the type characterized by the fact that a [[function]] $f : U \to \Sigma \sslash \mathbf{Aut}(\Sigma)$ is a function to $\Sigma$ which is regarded as ([[gauge equivalence|gauge]]) [[equivalence|equivalent]] to another function to $\Sigma$ if both differ by postcomposition with a diffeomorphism of $\Sigma$.

### Generally covariant field configuration spaces

Let now

$$
  \mathbf{Fields} \in \mathbf{H}
$$

be an object that represents the [[moduli ∞-stack]] of field configurations on $\Sigma$ for some [[model (in theoretical physics)]] to be described. For instance for $G \in Grp(\mathbf{H})$ an [[∞-group]] and $\mathbf{H}$ a [[cohesive homotopy type theory]], we could have $\mathbf{Fields} = \mathbf{B}G_{conn}$ the moduli for a choice of $G$-[[connection on an ∞-bundle|principal ∞-connection]], being the moduli for $G$-([[higher gauge theory|higher]])[[gauge fields]]. For general $\mathbf{Fields} = X \in \mathbf{H}$ we may always regard $X$ as the [[target space]] of a [[sigma-model]].

Then the [[internal hom]]

$$
  \mathbf{Fields}(\Sigma) \coloneqq [\Sigma, \mathbf{Fields}] \in \mathbf{H}
$$

hence the [[function type]]

$$
  \vdash \Sigma \to \mathbf{Fields} : Type
$$

is the _naive_ [[configuration space]] of the model. This is not generally covariant, precisely so by the above definition: it is not in the generally covariant context $\mathbf{H}_{/\mathbf{B} \mathbf{Aut}(\Sigma)}$.

But by the [[inverse image]] of the $\mathbf{B}\mathbf{Aut}(\Sigma)$-[[dependent product]] [[étale geometric morphism]]

$$
  \mathbf{H}_{/\mathbf{B}G}
  \stackrel{\overset{\sum_{\mathbf{B}\mathbf{Aut}(\Sigma)}}{\to}}{
  \stackrel{\overset{}{\leftarrow}}{\underset{\prod_{\mathbf{B}\mathbf{Aut}(\Sigma)}}{\to}}}
  \mathbf{H}
$$

which is [[context enlargement]] by $\mathbf{B}\mathbf{Aut}(\Sigma)$,
the moduli type $\mathbf{Fields}$ is freely moved to the general covariant context, where it is regarded as equipped with the [trivial ∞-action](infinity-action#TrivialAction). Accordingly we will write just $\mathbf{Fields} \in \mathbf{H}_{/\mathbf{B}\mathbf{Aut}(\Sigma)}$ with that trivial action understood, which is justified by the precise syntactic expression for it:

$$
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma) \vdash \mathbf{Fields} : Type
$$

We may then form the _configuration space of fields in the generally covariant context_ . As before, a field $\phi$ should be a function on $\Sigma$ with values in the moduli type of field configurations, but now we interpret this statement in the generally covariant context. Syntactically this simply means that a field is now a term in $\mathbf{B}\mathbf{Aut}(\Sigma)$-context

$$
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)
  \vdash
  \phi : \Sigma \to \mathbf{Fields}
$$

and that accordingly the configuration space of fields is

$$
  \Sigma : \mathbf{B}\mathbf{Aut}(\Sigma)
  \vdash
  \Sigma \to \mathbf{Fields} : Type
  \,.
$$

The [[categorical semantics|semantics]] of this is the
[[internal hom]] in the [[slice (∞,1)-topos]], the _[Internal hom of ∞-actions](infinity-action#CartesianClosedMonoidalStructure)_

$$
  \mathbf{Conf}
    \coloneqq
  [\Sigma \sslash \mathbf{Aut}(\Sigma), \mathbf{Fields}]
  \in
  \mathbf{H}_{/\mathbf{Aut}(\Sigma)}
  \,,
$$

The central observation now is that discussed at _[∞-action -- Examples -- General covariance](infinity-action#GeneralCovariance)_:

$$
  \mathbf{Conf} \simeq [\Sigma,\mathbf{Fields}] \sslash \mathbf{Aut}(\Sigma)
$$

is the [[homotopy quotient]] of the naive fields $\in \mathbf{Fields}(\Sigma)$ by the action of the [[diffeomorphism group]], exhibiting a [[gauge equivalence]] between any two field configurations that differ after pullback along a diffeomorphism.

This is precisely as it should be for configuration space of generally covariant theories. We have found:

**Fact**. In terms of homotopy type theory, configuration spaces of a generally covariant theory over $\Sigma$ are precisely the ordinary configuration spaces of fields, but formed in the context $\mathbf{B}\mathbf{Aut}(\Sigma)$:

$$
  \mathbf{Conf} =_{def}
  \;\;\;\;\;\;
\Sigma : \mathbf{B}\mathbf{Aut}(\Sigma) \vdash (\Sigma \to \mathbf{Field}) : Type
  \,.
$$

### Generally covariant field of gravity
 {#GravityInHoTT}

We now spell out the example of ordinary [[Einstein]]-[[gravity]] in this language. Plenty of further examples work analogously.

For pure [[gravity]], we choose $\mathbf{H} = $ [[Smooth∞Grpd]] or $=$[[SynthDiff∞Grpd]].

If we denote by $D^n \in $ [[SynthDiff∞Grpd]] the first order [[infinitesimal object|infinitesimal neighbourhood]] of the origin in the [[Cartesian space]] $\mathbb{R}^n$, then

$$
  GL(n) = \mathbf{Aut}(D^n)
$$

is the [[automorphism ∞-group]] of $D^n$. Accordingly we write the unique term of the [[delooping]] $\mathbf{B}GL(n)$ as

$$
  \vdash D^n : \mathbf{B}GL(n)
  \,.
$$

The fields of Einstein gravity are [[orthogonal structures]] ([[Riemannian metrics]]) on a [[smooth manifold]] $\Sigma \in $ [[SmoothMfd]] $\hookrightarrow \mathbf{H}$ of [[dimension]] $n$.
As discussed at _[[orthogonal structure]]_ and _[[vielbein]]_, we are to regard $\Sigma$ in the [[context]] of the [[delooping]] of the [[general linear group]] $GL(n) \in Grp(\mathbf{H})$ via its [[tangent bundle]] $T \Sigma \to \Sigma$, by which we always mean here the $GL(n)$-[[principal bundle]] to which the tangent bundle is [[associated bundle|associated]].

By the discussion at _[[principal ∞-bundle]]_ this is modulated by a morphism

$$
  (\Sigma \stackrel{\vdash T \Sigma}{\to} \mathbf{B} GL(n))
  \in
  \mathbf{H}_{/\mathbf{B} GL(n)}
$$

to the [[delooping]] $\mathbf{B}GL(n)$ of $GL(n)$ (the [[moduli stack]] of $GL(n)$-[[principal bundles]]) in that we have a [[fiber sequence]]

$$
  \array{
    T \Sigma &\to& \Sigma
    \\
    && \downarrow^{\mathrlap{\vdash T \Sigma}}
    \\
    && \mathbf{B}GL(n)
  }
$$

in $\mathbf{H}$. (A detailed exposition of this and the following, with further pointers, is in ([Schreiber, ESI lectures](#SchreiberLectures)).)

Therefore the [[syntax]] of the tangent bundle as a [[dependent type]] is

$$
  D^n : \mathbf{B}GL(n) \vdash T \Sigma(D^n) : Type
$$

and since $D^n$ is essentially unique we will notationally suppress it in the [[succedent]] on the right and just write

$$
  D^n : \mathbf{B}GL(n) \vdash T \Sigma : Type
  \,.
$$

Let then

$$
  (\mathbf{B} O(n) \stackrel{\mathbf{orth}}{\to} \mathbf{B} GL(n))
  \in
  \mathbf{H}_{/\mathbf{B}GL(n)}
$$

be the [[delooping]] of the inclusion $O(n) \to GL(n)$ of the [[maximal compact subgroup]] of $GL(n)$, the [[orthogonal group]] $O(n)$, regarded as an [[object]] in the [[slice-(∞,1)-topos]] over $\mathbf{B}GL(n)$. Since this sits in the [[homotopy fiber sequence]]

$$
  \array{
    GL(n)/O(n) &\to& \mathbf{B}O(n)
    \\
    && \downarrow^{\mathrlap{orth}}
    \\
    && \mathbf{B}GL(n)
  }
$$

with the [[coset]] [[smooth space]] $GL(n)/O(n)$, the [[syntax]] of this object is the [[dependent type]]

$$
  D^n : \mathbf{B}GL(n) \vdash GL(n)/O(n) : Type
  \,.
$$

In view of the [[equivalence of (∞,1)-categories]]

$$
  \mathbf{H}_{/\mathbf{B}GL(n)} \simeq Act_{\mathbf{H}}(GL(n))
$$

this expresses the canonical $GL(n)$-[[∞-action|action]] on the coset $GL(n)/O(n)$ (by mutliplication from the "other side").

The [[syntax]] of the [[moduli stack]] of [[vielbein]] fields /  [[Riemannian metrics]] on $\Sigma$ is

$$
  \vdash \prod_{D^n : \mathbf{B}GL(n)} T \Sigma \to GL(n)/O(n) : Type
  \,.
$$

This almost verbatim expresses the familiar statement:

> A [[vielbein]] on $\Sigma$ is a $GL(n)$-equivariant map from $T \Sigma$ to the coset $GL(n)/O(n)$.

The [[categorical semantics]] of such a [[vielbein]] $e$ is as a diagram

$$
  \array{
    \Sigma && \stackrel{}{\to}&& \mathbf{B}O(n)
    \\
    & \searrow &\swArrow_{e}& \swarrow_{\mathrlap{orth}}
    \\
    && \mathbf{B}GL(n)
  }
  \,.
$$

This in turn almost verbatim expresses the familar equivalent statement

> A vielbein is a [[reduction of the structure group]] of the $GL(n)$-[[principal bundle]] $T \Sigma$ along $O(n) \to GL(n)$.

This is still the naive space of fields, not yet generally covariant. So we next pass to the general covariant $\mathbf{B}\mathbf{Aut}(T\Sigma)$-[[context]] and form the correct generally covariant space of fields, being the [[type in context]] $\mathbf{B} \mathbf{Aut}(T \Sigma)$ given by

$$
  \mathbf{Conf}
  \coloneqq
  \vdash
   \prod_{D^n : \mathbf{B}GL(n)}
   \sum_{T\Sigma : \mathbf{B}\mathbf{Aut}(T\Sigma)}
   T \Sigma \to GL(n)/O(n)   : Type
  \,,
$$

which is the integrated [[BRST complex]] of Einstein gravity field configurations modulo diffeomorphism ghosts: the [[smooth infinity-groupoid|smooth groupoid]] whose

* [[objects]] are [[vielbein]] fields $e$ on $X$;

* [[morphisms]] are

  1. orthogonal frame transformations of the fibers of the tangent bundle;

  1. general diffeomorphisms of the base $\Sigma$.

We unwind this a bit more.

A slight subtlety in interpreting the above expression is that in

$$
  D^n : \mathbf{B}GL(n) \vdash \mathbf{B}Aut(T \Sigma) : Type
$$

the [[automorphism ∞-group]] of the tangent bundle it to be formed in the [[context]] of $\mathbf{B}GL(n)$. By the discussion at _[[automorphism ∞-group]]_ the [[delooping]] $\mathbf{B}Aut(T \Sigma)$ is the [[∞-image]] of the name

$$
  (* \stackrel{}{\to} Type) \in \mathbf{H}_{/\mathbf{B}GL(n)}
$$

of $(\Sigma \to \mathbf{B}GL(n))$ in the slice. By the discussion at [slice-(∞,1)-topos -- Object classifier](over-%28infinity%2C1%29-topos#ObjectClassifier) the [[object classifier]] in the slice is the [[projection]] $(Type \times \mathbf{B}GL(n) \to \mathbf{B}GL(n))$.

So the name and its pullback are given by a diagram of the form

$$
  \array{
    \Sigma &&\to&& \widehat{Type} \times \mathbf{B}GL(n)
    \\
    {}^{\mathllap{}}\downarrow &&\swArrow&& \downarrow
    \\
    \mathbf{B}GL(n) &&\stackrel{(\vdash \Sigma, id)}{\to}&& Type \times \mathbf{B}GL(n)
    \\
    & {}_{\mathllap{id}}\searrow &\swArrow& \swarrow
    \\
    && \mathbf{B}GL(n)
  }
$$

in $\mathbf{H}$. Here the [[∞-image]] is directly read off to be the factorization in the third column of

$$
  \array{
    T \Sigma
    &\to&
    \Sigma
     &\to&
      (T \Sigma \sslash \mathbf{Aut}(T\Sigma)) \times \mathbf{B}GL(n)
     &\to&
    \widehat{Type} \times \mathbf{B}GL(n)
    \\
    \downarrow
    &&
    {}^{\mathllap{}}\downarrow && \downarrow
     && \downarrow
    \\
    * &\to&
       \mathbf{B}GL(n)
       &\stackrel{}{\to}&
       \mathbf{B}\mathbf{Aut}(T \Sigma) \times \mathbf{B}GL(n)
       &\to&
       Type \times \mathbf{B}GL(n)
    \\
    && & {}_{\mathllap{id}}\searrow &\swArrow& \swarrow
    \\
    && && \mathbf{B}GL(n)
  }
  \,,
$$

where each square and hence each rectangle is an [[(∞,1)-pullback]] in $\mathbf{H}$. This shows that the automorphism $\infty$-group of $T \Sigma$ in the context of $\mathbf{B}GL(n)$ is just the absolute automorphism $\infty$-group freely [[context extension|context extended]]. The [[categorical semantics]] of the [[dependent type]]

$$
  D^n \colon \mathbf{B}GL(n), T \Sigma \colon \mathbf{B}\mathbf{Aut}(T \Sigma)
  \vdash
  T \Sigma \colon Type
$$

is the third column from the left in the above diagram. This means that the dependent sum in

$$
  \mathbf{Conf}
  \coloneqq
  \vdash
   \prod_{D^n : \mathbf{B}GL(n)}
   \sum_{T\Sigma : \mathbf{B}\mathbf{Aut}(T\Sigma)}
   T \Sigma \to GL(n)/O(n)   : Type
  \,,
$$

forms the internal hom in $\mathbf{H}_{/\mathbf{B}GL(n)}$ between the homotopy fiber of that third column formed in $\mathbf{H}_{/\mathbf{B}GL(n)}$, which is the second column (and therefore now does rememeber the $GL(n)$-action on $T \Sigma$) with $\mathbf{orth}$, rememeberting that the result has an $\mathbf{Aut}(T\Sigma)$-action by precomposition.

## References
 {#References}

### History
 {#ReferencesHistory}

The pre-history of the idea of general covariance is reviewed in

* H. G. Alexander (ed.), _The Leibniz-Clarke Correspondence: Together With Extracts from Newton's Principia and Opticks_, Manchester University Press (1998)
 {#Alexander}

The original articles by Einstein on the idea of general covariance include his _Entwurf_ (sketch)

* [[Albert Einstein]], M. Grossmann, _Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation_ Zeitschrift für Math. Phys. 62, 225–259 (1914)
 {#EinsteinGrossmann}

of the theory of [[gravity]] (where the notion of general covariance is still rejected) and then the full development of [[general relativity]]

* [[Albert Einstein]], _Die Grundlage der allgemeinen Relativitätstheorie. Ann. Phys. (Leipzig) 49, 769–822 (1916)
 {#Einstein1916}

where it is fully embraced.

An attempt at a fairly comprehensive review of the history of the idea of general covariance and its reception up to modern days is in

* J. D. Norton, _General covariance and the foundations of general relativity: eight decades of dispute_, Rep. Prog. Phys **56** (1993), ([original pdf](http://www.pitt.edu/~jdnorton/papers/decades.pdf), [reprint pdf](http://www.pitt.edu/~jdnorton/papers/decades_re-set.pdf))
 {#Norton}

### Formalizations of general covariance
 {#ReferencesFormalizations}

A formalization in the context of [[AQFT]] is proposed and discussed in

* [[Klaus Fredenhagen]], [[Rudolf Haag]], _Generally covariant quantum field theory and scaling limits_,  Comm. Math. Phys. Volume 108, Number 1 (1987), 91-115. ([EUCLID](http://projecteuclid.org/euclid.cmp/1104116359))

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Rainer Verch]], _The generally covariant locality principle -- A new paradigm for local quantum physics_, Commun.Math.Phys.237:31-68,2003 ([math-ph/0112041](http://arxiv.org/abs/math-ph/0112041))

A review is in

* [[Romeo Brunetti]], Martin Porrmann, Giuseppe Ruzzi, _General Covariance in Algebraic Quantum Field Theory_ ([arXiv:math-ph/0512059](http://arxiv.org/abs/math-ph/0512059))
 {#BrunettiPorrmannRuzzi}

For more see the references at _[[AQFT on curved spacetimes]]_.

### Formalization in homotopy type theory
 {#FormalizationInHomtopyTypeTheory}

Background and context for the above formalization of [[classical field theory|classical]]/[[quantum field theory]] in [[homotopy type theory]] see

* [[Urs Schreiber]], _[[twisted smooth cohomology in string theory]]_, Lectures at _[ESI program on K-theory and quantum fields](http://maths-old.anu.edu.au/esi/2012/)_ (2012)
 {#SchreiberLectures}

* [[Urs Schreiber]], [[Mike Shulman]], _[[schreiber:Quantum gauge field theory in Cohesive homotopy type theory]]_ (2012)
 {#SchreiberShulman}

See also _[[higher category theory and physics]]_.


General Discussion

see at *[[HomePage]]* or *[[HowTo]]*


General Discussion > history

< [[General Discussion]]


general linear group

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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[field]] $k$, the **general linear group** $GL(n,k)$ (or $GL_n(k)$) is the [[group]] of [[invertible morphism|invertible]] [[linear maps]] from the [[vector space]] $k^n$ to itself.  It may canonically be identified with the group of $n\times n$ [[matrix|matrices]] with entries in $k$ having nonzero [[determinant]].

### As a topological group
 {#AsAtopologicalGroup}

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

Let $k = \mathbb{R}$ or $= \mathbb{C}$ be the [[real numbers]] or the [[complex numbers]] equipped with their [[Euclidean topology]].

#### Definition

+-- {: .num_defn #GLnAsTopologicalGroup}
###### Definition
**(general linear group as a topological group)**

For $n \in \mathbb{N}$, as a [[topological group]] the _general linear group_ $GL(n, k)$ is defined as follows.

The underlying group is the [[group]] of [[real number|real]] or [[complex numbers|complex]] $n \times n$ [[matrices]] whose [[determinant]] is non-vanishing

$$
  GL(n,k)
  \;\coloneqq\;
  \left(
    A \in Mat_{n \times n}(k)
    \; \vert \;
    det(A) \neq 0
  \right)
$$

with group operation given by [[matrix multiplication]].

The [[topological space|topology]] on this set is the [[subspace topology]] as a subset of the [[Euclidean space]] of [[matrices]]

$$
  Mat_{n \times n}(k) \simeq k^{(n^2)}
$$

with its [[metric topology]].

=--

+-- {: .num_lemma #GLnTopologicalWellDefined}
###### Lemma
**(group operations are continuous)**

Definition \ref{GLnAsTopologicalGroup} is indeed well defined in that the group operations on $GL(n,k)$ are indeed [[continuous functions]] with respect to the given topology.

=--

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

Observe that under the identification $Mat_{n \times n}(k) \simeq k^{(n^2)}$ [[matrix multiplication]] is a [[polynomial function]]

$$
  k^{(n^2)}
   \times
  k^{(n^2)}
   \simeq
  k^{ 2 n^2 }
    \longrightarrow
  k^{(n^2)}
    \simeq
  Mat_{n \times n}(k)
  \,.
$$

Similarly [[inverse matrix|matrix inversion]] is a [[rational function]]. Now [[rational functions are continuous]] on their [[domain]] of definition,
and since a real matrix is invertible previsely if its determinant is non-vanishing, the domain of definition for matrix inversion is precisely
$GL(n,k) \subset Mat_{n \times n}(k)$.

=--

+-- {: .num_defn}
###### Definition
**(stable general linear group)**

The evident [[tower]] of embeddings

$$
  k \hookrightarrow k^2 \hookrightarrow k^3 \hookrightarrow \cdots
$$

induces a corresponding [[tower diagram]] of embedding of the general linear groups (def. \ref{GLnAsTopologicalGroup})

$$
  GL(1,k)
    \hookrightarrow
  GL(2,k)
    \hookrightarrow
  GL(3,k)
    \hookrightarrow
  \cdots
  \,.
$$

The [[colimit]] over this [[diagram]] in the [[category]] of [[topological group]] is called the _stable general linear group_ denoted

$$
  GL(k) \;\coloneqq\; \underset{\longrightarrow}{\lim}_n GL(n,k)
  \,.
$$

=--

#### Properties

+-- {: .num_example #AsSubspaceOfTheMappingSpace}
###### Proposition
**(as a [[subspace]] of the [[compact-open topology|mapping space]])

The [[topological space|topology]] induced on the real general linear group when regarded as a [[topological subspace]] of [[Euclidean space]] with its [[metric topology]]

$$
  GL(n,\mathbb{R})
    \subset
  Mat_{n \times n}(\mathbb{R})
    \simeq
  \mathbb{R}^{(n^2)}
$$

(as in def. \ref{GLnAsTopologicalGroup}) coincides with the topology induced by regarding the general linear group as a [[subspace]] of the [[mapping space]] $Maps(k^n, k^n)$,

$$
  GL(n,\mathbb{R})
    \subset
  Maps(k^n, k^n)
$$

i.e. the set of all [[continuous functions]] $k^n \to k^n$ equipped with the [[compact-open topology]].

=--

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

On the one had, the [[universal property]] of the [[mapping space]] ([this prop.](Introduction+to+Topology+--+1#UniversalPropertyOfMappingSpace)) gives that the inclusion

$$
  GL(n, \mathbb{R}) \to Maps(\mathbb{R}^n, \mathbb{R}^n)
$$

is a [[continuous function]] for $GL(n,\mathbb{R})$ equipped with the [[Euclidean space|Euclidean]] [[metric topology]], because this is the [[adjunct]] of the defining continuous [[action]] map

$$
  GL(n, \mathbb{R}) \times \mathbb{R}^n \to \mathbb{R}^n
  \,.
$$

This implies that the [[Euclidean space|Euclidean]] [[metric topology]] on $GL(n,\mathbb{R})$ is equal to or [[finer topology|finer]] than the subspace topology coming from $Map(\mathbb{R}^n, \mathbb{R}^n)$.

We conclude by showing that it is also equal to or [[coarser topology|coarser]], together this then implies the claims.

Since we are speaking about a subspace topology, we may consider the open subsets of the ambient Euclidean space $Mat_{n \times n}(\mathbb{R}) \simeq \mathbb{R}^{(n^2)}$. Observe that a [[neighborhood base]] of a linear map or matrix $A$ consists of sets of the form

$$
  U_A^\epsilon
  \;\coloneqq\;
  \left\{B \in Mat_{n \times n}(\mathbb{R}) \,\vert\, \underset{{1 \leq i \leq n}}{\forall}\; |A e_i - B e_i| \lt \epsilon \right\}
$$

for $\epsilon \in (0,\infty)$.

But this is also a [[base for the topology|base]] element for the [[compact-open topology]], namely

$$
  U_A^\epsilon
   \;=\;
  \bigcap_{i = 1}^n V_i^{K_i}
  \,,
$$

where $K_i \coloneqq \{e_i\}$ is a [[singleton]] and $V_i \coloneqq B^\circ_{A e^i}(\epsilon)$ is the [[open ball]] of [[radius]] $\epsilon$ around $A e^i$.

=--

+-- {: .num_prop #ConnectednessOfGeneralLinearGroup}
###### Proposition
**(connectedness properties of the general linear group)**

For all $n \in \mathbb{N}$

1. the complex general linear group $GL(n,\mathbb{C})$ is [[path-connected topological space|path-connected]];

1. the real general linear group $GL(n,\mathbb{R})$ is not [[path-connected topological space|path-connected]].

=--

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

First observe that $GL(1,k) = k \setminus \{0\}$ has this property:

1. $\mathbb{C} \setminus \{0\}$ is path-connected,

1. $\mathbb{R} \setminus \{0\} = (-\infty,0) \sqcup (0,\infty)$ is not path connected.

Now for the general case:

1. For $k = \mathbb{C}$: every invertible complex matrix is diagonalizable by a sequence of elementary matrix operations ([this prop.](inverse+matrix#FundamentalTheoremOfLinearAlgebra)). Each of these is clearly path-connected to the identity. Finally the subspace of invertible [[diagonal matrices]] is the [[product topological space]] $\underset{ \{1, \cdots, n\} }{\prod} (\mathbb{C} \setminus \{0\})$ and hence connected (by [this prop.](ProductSpaceOfConnectedSpacesIsConnected), since each factor space is).

1. For $k = \mathbb{R}$: the [[determinant]] function is a [[continuous function]] $GL(n,k) \to \mathbb{R} \setminus \{0\}$, and since the [[codomain]] is not path connected, the domain cannot be either.

=--

+-- {: .num_prop #TopologicalGeneralLinearGroup}
###### Proposition
**(compactness properties of the general linear group)**

The [[topological group|topological]] general linear group $GL(n,k)$ (def. \ref{GLnAsTopologicalGroup}) is

1. not [[compact topological space|compact]];

1. [[locally compact topological space|locally compact]];

1. [[paracompact Hausdorff topological space|paracompact Hausdorff]].

=--

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

Observe that

$$
  GL_n(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}
$$

is an [[open subset|open]] [[subspace]], since it is the [[pre-image]] under the [[determinant]] function (which is a [[polynomial]] and hence continuous, as in the proof of lemma \ref{GLnTopologicalWellDefined}) of the of the open subspace $k \setminus \{0\} \subset k$.

As an open subspace of Euclidean space, $GL(n,k)$ is not compact, by the [[Heine-Borel theorem]].

As Euclidean space is Hausdorff, and since every [[subspace]] of a Hausdorff space is again Hausdorff, so $Gl(n,k)$ is Hausdorff.

Similarly, as Euclidean space is [[locally compact topological space|locally compact]] and since an open subspace of a locally compact space is again locally compact, it follows that $GL(n,k)$ is locally compact.

From this it follows that $GL(n,k)$ is paracompact, since locally compact topological groups are paracompact ([this prop.](topological+group#ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact)).

=--

### As a Lie group
 {#AsALieGroup}

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

Since the general linear group as a [[topological group]] (def. \ref{GLnAsTopologicalGroup}) is an  [[open subspace]] of [[Euclidean space]] (proof of prop. \ref{TopologicalGeneralLinearGroup}) it inherits the structure of a [[smooth manifold]] (by [this prop.](differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds)). The group operations (being [[rational functions]]) are [[smooth functions]] with respect to this smooth structure. This is the general linear group $GL(n,\mathbb{R})$ as a _[[Lie group]]_.

=--

### As an algebraic group

This group can be considered as a (quasi-affine) sub[[variety]] of the [[affine scheme|affine space]] $M_{n\times n}(k)$ of square matrices of size $n$ defined by the condition that the [[determinant]] of a matrix is nonzero. It can be also presented as an affine subvariety of the affine space $M_{n \times n}(k) \times k$ defined by the equation $\det(M)t = 1$ (where $M$ varies over the factor $M_{n \times n}(k)$ and $t$ over the factor $k$).

This [[variety]] is an algebraic $k$-group, and if $k$ is the field of real or complex numbers it is a [[Lie group]] over $k$.

One may in fact consider the set of invertible matrices over an arbitrary unital [[ring]], not necessarily commutative. Thus $GL_n: R\mapsto GL_n(R)$ becomes a [[presheaf]] of [[group]]s on $Aff=Ring^{op}$ where one can take rings either in commutative or in noncommutative sense. In the commutative case, this functor defines a [[group scheme]]; it is in fact the affine group scheme represented by the commutative ring $R = \mathbb{Z}[x_{11}, \ldots, x_{n n}, t]/(det(X)t - 1)$.

Coordinate rings of general linear groups and of special general linear groups have [[quantum group|quantum deformations]] called [[quantum linear groups]].

## Examples

Over [[finite fields]]:

* [[GL(2,3)]]

## Properties

### Representation theory

See at *[[representation theory of the general linear group]]*.

## Related concepts

* [[translation group]], [[affine group]]

* [[Gauss decomposition]],

* [[special linear group]], [[projective linear group]]

* [[orthogonal group]], [[unitary group]],

* [[Borel subgroup]], [[flag variety]]

* [[group of units]]

* [[jet group]]

* [[general linear supergroup]], [[quantum linear group]]

## References

* O.T. O'Meara, _Lectures on Linear Groups_, Amer. Math. Soc., Providence, RI, 1974.

* B. Parshall, J.Wang, _Quantum linear groups_, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

[[!redirects general linear group]]
[[!redirects general linear groups]]

[[!redirects GL(n)]]


general linear Lie algebra

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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $V$ a [[vector space]], the **general linear Lie algebra** or **[[endomorphism Lie algebra]]** $\mathfrak{gl}(V)$ of $V$ is the [[Lie algebra]] whose elements are [[linear map|linear]] [[endomorphisms]] $V \to V$ and whose Lie bracket is given by the [[commutator]] of endomorphisms.

This is also the [[endomorphism L-∞ algebra]] of $V$

If $V$ is a real vector space that carries an [[inner product]] there are the sub-Lie algebras

$$
  \mathfrak{so}(V) \hookrightarrow \mathfrak{o}(V) \hookrightarrow \mathfrak{gl}(V)
$$

the

* [[orthogonal Lie algebra]]

* and [[special orthogonal Lie algebra]].

If $V$ is a complex vector space with an inner product there is

$$
  \mathfrak{so}(V) \hookrightarrow \mathfrak{o}(V) \hookrightarrow \mathfrak{gl}(V)
$$

the

* [[unitary Lie algebra]]

* and [[special unitary Lie algebra]].

## Properties

### Loday-Quillen-Tsygan theorem

The _[[Loday-Quillen-Tsygan theorem]]_ ([Loday-Quillen 84](#LodayQuillen84), [Tsygan 83](#Tsygan83)) states that for any [[associative algebra]], $A$ in [[characteristic zero]], the [[Lie algebra homology]] $H_\bullet(\mathfrak{gl}(A))$  of the infinite [[general linear Lie algebra]] $\mathfrak{gl}(A)$ with [[coefficients]] in $A$ is, up to a degree shift, the [[exterior algebra]] $\wedge(HC_{\bullet - 1}(A))$ on the [[cyclic homology]] $HC_{\bullet - 1}(A)$ of $A$:

$$
  H_\bullet(\mathfrak{gl}(A))
  \;\simeq\;
  \wedge( HC_{\bullet - 1}(A) )
$$

(see e.g [Loday 07, theorem 1.1](#Loday07)).

## References

The [[Loday-Quillen-Tsygan theorem]] is originally due, independently, to

* {#LodayQuillen84} [[Jean-Louis Loday]], [[Daniel Quillen]], _Cyclic homology and the Lie algebra homology of matrices_ Comment. Math. Helv., 59(4):569–591, 1984.

and

* {#Tsygan83} [[Boris Tsygan]], _Homology of matrix algebras over rings and the Hochschild homology_, Uspeki Math. Nauk., 38:217–218, 1983.

Lecture notes include

* {#Loday07} [[Jean-Louis Loday]], _Cyclic Homology Theory, Part II_, notes taken by Pawe l Witkowsk (2007) ([pdf](https://www.impan.pl/swiat-matematyki/notatki-z-wyklado~/loday_cht_2.pdf))

[[!redirects general linear Lie algebras]]


general linear supergroup

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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The general linear [[supergroup]] $GL(p|q)$ is the [[supergroup]] of invertible linear transformation on $\mathbb{R}^{p|q}$.

## Related concepts

* [[orthosymplectic supergroup]]

* [[Euclidean supergroup]], [[super Poincaré group]]

## References

* [[Manfred Scheunert]], chapter II, 4.2 of _The theory of Lie superalgebras. An introduction_, Lect. Notes Math. 716 (1979)


general relativity

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###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The theory called _the theory of general relativity_ is a [[classical field theory]] that in [[physics]] describes the field of [[gravity]].

In general relativity, physical [[spacetime]] is modeled in terms of [[differential geometry]] as a [[Lorentzian manifold]] whose [[pseudo-Riemannian metric]] -- or rather the [[Levi-Civita connection]] that corresponds to it -- encodes the field of [[gravity]].

The [[action functional]] describing the dynamics of this field is the [[Einstein-Hilbert action]], in which the field of gravity enters in terms of the integral of the [[scalar curvature]] of the [[Levi-Civita connection]] over spacetime.

As is usual in [[classical field theory]], the physically realized configurations -- here: Levi-Civita connections -- are those that extremize this functional. The [[Euler-Lagrange equation]]s characterizing these extrema are the [[Einstein equations]].

Historically the theory of general relativity was developed by [[Albert Einstein]] based on the theory known as [[special relativity]]. Given the conceptual simplicity of the [[Einstein-Hilbert action]], there are several variations of his original version of the theory that are immediately obtained by adding certain terms to the action functional. One of these generalization is [[supergravity]], which extends the theory from ordinary [[differential geometry]] to [[supergeometry]].

## Related concepts

* [[general covariance]]

* [[equivalence principle (physics)]]

* [[relativistic particle]]

* [[relativistic field theory]]

* [[spacetime]]

* [[black hole]]

* [[ADM mass]]

* [[Penrose-Hawking theorem]]

* [[cosmic censorship]]

## References
 {#References}

### General

Textbook accounts:

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]], _[[Gravitation]]_ (1973)

* [[Robert Wald]], *General Relativity*, University of Chicago Press (1984) [[doi:10.7208/chicago/9780226870373.001.0001](https://doi.org/10.7208/chicago/9780226870373.001.0001), [pdf](http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Wald_GR.pdf)]

Lecture notes:

* {#Blau} [[Matthias Blau]], _Lecture notes on general relativity_ ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* Emil T. Akhmedov, _Lectures on General Theory of Relativity_ ([arXiv:1601.04996](https://arxiv.org/abs/1601.04996))

* Pietro Menotti, _Lectures on gravitation_ ([arXiv:1703.05155](https://arxiv.org/abs/1703.05155))

* [[Daniel Baumann]], *[General Relativity](http://cosmology.amsterdam/general-relativity/)*, 2021 ([pdf](http://cosmology.amsterdam/wp-content/uploads/2021/10/GR-Oct19.pdf))

and with focus on [[causal structure]], the [[Penrose singularity theorem]] and related aspects:

* [[Edward Witten]], *Light Rays, Singularities, and All That*, Rev. Mod. Phys. **92** 45004 (2020) [[arXiv:1901.03928](https://arxiv.org/abs/1901.03928), [doi:10.1103/RevModPhys.92.045004](https://doi.org/10.1103/RevModPhys.92.045004), lecture: [video 1](https://www.youtube.com/watch?v=-vS_WIa0vYo), [2](https://www.youtube.com/watch?v=4kEiNotDhck), [3](https://www.youtube.com/watch?v=Qje1ak6XF20), [4](https://www.youtube.com/watch?v=tzt25nkGg3Q), [5](https://www.youtube.com/watch?v=yC5BHqPw38Q), [6](https://www.youtube.com/watch?v=GHPL2_aomo8), [7](https://www.youtube.com/watch?v=tDeQ9ght3aI), [8](https://www.youtube.com/watch?v=fsmcGpKFbFU), [9](https://www.youtube.com/watch?v=Z_crNwSKpW0)]

Emphasis on [[mathematical physics]]

* [[Alan Coley]], _Mathematical General Relativity_ ([arXiv:1807.08628](https://arxiv.org/abs/1807.08628))

* José Natário, _Mathematical relativity_ ([arXiv:2003.02855](https://arxiv.org/abs/2003.02855))

Introductory exposition:

* [[John Baez]], [The General Relativity Tutorial](http://math.ucr.edu/home/baez/gr/gr.html)

See also

* [[Edward Witten]], _Light Rays, Singularities, and All That_ ([arXiv:1901.03928](https://arxiv.org/abs/1901.03928))

See also the references at _[[gravity]]_.

### In relation to thermodynamics

In relation to [[thermodynamics]]

* {#IglesiasSouriau84} [[Patrick Iglesias-Zemmour]], [[Jean-Marie Souriau]] _Heat, cold and Geometry_, in: M. Cahen et al (eds.) Differential geometry and mathematical physics, 37-68, D. Reidel 1983 ([web](http://www.jmsouriau.com/Heat_Cold_And_Geometry_1983.htm), [pdf](http://www.jmsouriau.com/Publications/JMSouriau-PIglesias-HeatColdAndGeometry1983.pdf))

  (on [[thermodynamics]] and [[general relativity]])

### History

* {#Coles01} Peter Coles, _Einstein, Eddington, and the 1919 Eclipse_ ([arxIv:astro-ph/0102462](https://arxiv.org/abs/astro-ph/0102462))

  (on the [[experiment|experimental]] confirmation of [[general relativity]])

[[!redirects General Relativity]]

[[!redirects theory of general relativity]]


General Theory of Natural Equivalences

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This page is about:

* [[Samuel Eilenberg]] and [[Saunders Mac Lane]]:

  \linebreak

  **General theory of natural equivalences**

  \linebreak

  Transactions of the American Mathematical Society

  **58** (1945) 231–294

  [doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6)

  [jstor:1990284](http://www.jstor.org/stable/1990284)

introducing basic notions of [[category theory]] ([[category]], [[functor]], [[natural transformations]]...).

The first page says

> Presented to the Society, September 8, 1942; received by the editors May 15, 1945.

This paper is regarded as the foundational document of [[category theory]]. Notably, in it the term [[natural transformation]] was defined, while the concept of a [[category of categories]] was not mentioned (all the constructions necessary to define it being introduced in the paper).

The authors also make careful linguistic, logical and [[foundations|foundational]] comments, some of them remarkable prescient, such as on page 247 where one reads

> Any given system of foundations will then legitimize those subcategories which are allowable classes in the system in question. (...) One might choose to adopt the (unramified) theory of types as a foundation for the theory of classes.

Note that here, "theory of types" is *not* the same as [[type theory]] in the contemporary sense.

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

The numbering in this paper is somewhat unusual in that the arabic numerals to not start over again when a new roman numeral has been introduced.

## Introduction

### I Categories and functors

#### I.1 Definition of categories

#### I.2 Examples of categories

#### I.3 Functors in two arguments

#### I.4 Examples of functors

#### I.5 Slicing of functors

#### I.6 Foundations

### II Natural equivalence of functors

#### II.7 Transformations of functors

#### II.8 Categories of functors

and several sections more.

The paper ends with an appendix making a sweeping representability theorem about (in the author's words) "any category" that to compare with the concept of [[concrete category|concreteness]] can be instructive.

category: reference


general topology

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#Contents#
* table of contents
{:toc}

## Idea

What is called _general topology_ or _point-set topology_ is the study of the basic concepts of [[topology]] based on the concept of _[[topological spaces]]_ considered as [[sets]] (of points) equipped with a topology. The term is to contrast with other areas of topology, such as _[[algebraic topology]]_ or [[differential topology]], and specifically to contrast with [[homotopy theory]], where only the (weak) [[homotopy type]] of a topological space matters, not the [[homeomorphism]] type of its underlying topologized point-set, and where other models/presentations for these [[homotopy types]] may be used (other than "topological point-sets"), notably [[simplicial sets]].

The study of generalizations of topological spaces in the guise of [[sets]] with [[extra structure]], such as to  [[nearness spaces]], [[uniform space|uniformities]], [[bitopological spaces]] and so on, may still be regarded as the subject of "point-set topology".

There is also the point-less formulation of generalized topological spaces in terms of [[locales]]. This might still be regarded as part of _general topology_, but it is manifestly not to be counted as "point-set topology",
and is known insteasd as [[pointfree topology]] (or [[pointless topology]]).

Other non-point-set approaches include [[formal topology]]
and [[abstract Stone duality]].

There is also the term _set-theoretic topology_, but that tends to allude to additional [[foundations|foundational axioms]] being considered or other logical (say [[constructive mathematics|intuitionistic]]) or set-theoretical considerations ([[large cardinals]] for example).

## Related entries

* [[Introduction to Topology -- 1|Basic introduction to general topology]]
* [[geometry]], [[higher geometry]]

## References

* [[Nicolas Bourbaki]], *General topology*, Elements of Mathematics, Springer (1971, 1990, 1995) [[doi:10.1007/978-3-642-61701-0](https://doi.org/10.1007/978-3-642-61701-0)]

For more see the references at _[[topology]]_.

For introduction to general or point-set topology see at _[[Introduction to Topology -- 1]]_.

[[!redirects set-theoretical topology]]
[[!redirects set-theoretic topology]]
[[!redirects point-set topology]]


generalisation as an adjunction

# Generalisation as an adjunction
* the following line creates the automatic table of contents
{: toc}

## Introduction ##

The hypothesis I want to explore is that generalisation can be represented as
an [[adjunction]]. More precisely, that generalisation and instantiation can be represented as an [[adjoint pair]], generalisation being the [[left adjoint]].

By "generalisation", I mean learning concepts from
examples, e.g. in machine learning. I believe that this applies to many
different topics in machine learning, including statistical curve-fitting
(and its implementation in various kinds of neural net), symbolic vector
architectures, instance-based learning, and logical induction. If I'm
right, this could be an important unification of these apparently
unrelated topics.

Because I don't have funding to do research, I've had no time to develop these ideas. So the best I can do
so far is to justify the hypothesis by appealing to various intuitions, and to related
mathematical and computational phenomena. I've put this material into nLab because I think it's interesting, and may well be important to machine learning: the categorists and machine-learning experts I've asked agree.  So perhaps other readers can supply technical substantiation or refutation. If I can ever get funding then of course I'll do so as well. By the way, most of them have also said that even if this result is known to categorists, it isn't known within machine learning, and that publicising it would therefore be useful.

## Potential uses in machine learning ##

These are similar to those that [[Joseph Goguen]]
proposed for category theory in computing science
in [[A Categorical Manifesto]].
If much concept learning can be represented by the same phenomenon, namely an
adjunction, this should make it easier to compare learning algorithms and
to translate algorithms from one branch of machine learning to another.
Moreover, algorithms that can be represented as adjunctions may be better
— if only because they're easier to reason about — than those that can't.

## Potential use in category theory ##

This is very speculative. But suppose that many kinds of generalisation can be formulated as adjunctions. Then what properties distinguish those adjunctions from others? If none, does this mean that all adjunctions can be regarded as generalisations? If so, does that tell us anything useful about adjunctions? Perhaps an information-theoretic way of thinking about them?

## Idea ##

Let $C$ be a category of concepts, and $E$ be a category of sets of examples.

Let $G \colon E \to C$ be a "generalisation" functor that maps each set of
examples to its generalisation as a concept.

Conversely, let
$F \colon C\to E$ be a forgetful functor that maps each concept to some
canonical set of examples.

Then my hypothesis is that $F$ and $G$ form an adjunction,
$G$ being left adjoint to $F$.

### Interpretation for non–category-theorists ###

Suppose we have a set
$E$ of sets
of examples, and a set $C$ of concepts. Then we can define a function $G$ that
maps each set of examples $e$ in $E$ to a concept $c$ in $C$, by generalisation.
Conversely, we can define a function $F$ that maps each concept
$c$ in $C$ to a
set of examples $e$ in $E$.

I hope that we can then regard
$G$ as finding what might be called the smallest
generalisation, or most efficient representation, of its argument.
Conversely,
$F$ ought to
"forget" whatever extra structure $e$ gained by becoming a
concept, turning it back into a "canonical" set of examples. $F$ and $G$ may
not be inverses, but they are, in some sense, the nearest one can get to
inverses. I say what's in this paragraph because (a) these are properties
that I believe generalisation and forgetting should have; (b)
adjoint functors often (always?) seem to work like this.

### What is a concept? ###

I haven't said anything about what a concept is. One possibility, just to
illustrate the idea, is the statistical regression line given by a least-squares
fit.  $E$'s objects would then be sets of 2-d points to be fitted, and C's
would be regression lines, probably equipped with goodness-of-fit
information. This is a simple example; instances of other, more complicated, statistical models could also be concepts.

Some other possibilities for concepts are: high-dimensional vectors formed
by superimposing vectors of property-value pairs in so-called "symbolic
vector architectures" or "holographic reduced representations";  regions
of classification space; logical propositions learnt by induction.

### What is generalisation? ###

Here are some examples.

* The examples are two-dimensional points (members of
$R^2$). Generalisation is least-squares fitting, as above. The
concept is a line giving the best least-squares fit to the points.

* Generalisation is fitting of some other statistical model. The concept is an instance of that model.

* The examples are logical sentences, classified as positive or negative.
Generalisation is logical induction. The concept is a new sentence from which we
can infer as many of the positive examples as possible (hopefully all) and as
few of the negative examples (hopefully none) as possible.

* The examples are pairs $\langle{v,w}\rangle$ associating an element of the
vector space $V$ with an element of the vector space
$W$. Generalisation is training a linear-associator neural net. The concept is a linear transformation which, if possible, maps all the $v$
to the $w$, and if that's not possible, does the best it can by finding
an optimum transform which minimises interference between different
$\langle{v,w}\rangle$ associations.

* The examples are pairs $\langle{p,l}\rangle$ where $p$ is a point in
$R^n$ and $l$ is a member of some set
$L$ of symbolic labels. For instance, the $p$ might be
elements of some space whose dimensions measure a voter's preference for
various policies; the $l$ could then be the possible parties,
$\{green,labour,libdem,tory\}$. Generalisation is nearest-neighbour
prediction. The concept is a function which extends the set of pairs
$\langle{p,l}\rangle$, mapping every point in $R^n$ to a
label. It can therefore be used to predict the preferred party of other
voters. I've taken this example from _Truth
from Trash_ by Chris Thornton.

* The examples are the instances of generalisation given in this write-up.
Generalisation is fitting each to a notion in category theory. The concept is a categorical construct...!

### What other structure might the category of examples have? ###

Although I introduced $E$ as a category of sets of
examples, I'm sure there are other possibilities. An obvious one, though
still set-related, is that the objects are weighted sets, where each
weight tells the generaliser how important the corresponding example is.
This probably ties up with fuzzy logic, fuzzy set theory, and stochastic logic programming. See also the following section.

## Some simple examples of generalisation in more detail##

These are taken from my short write-up [_Generalisation is an adjunction_](http://www.j-paine.org/generalisation.html), recast into the naming convention I'm using here.

### 1. Logical induction by conjunction ###

Let $C$ be the
category whose objects are the atomic propositions
$P$, $Q$, $R$, ... and their conjunctions. Let there be an arrow from
$P$
to $P'$ if $P$ implies $P'$. This makes $C$ into
a partial ordering defined by implication.

Let $E$ be the category
whose objects are non-empty sets of the above atomic propositions. It has the obvious partial ordering by inclusion.

Let $G$ map
each set of propositions in $E$ to their conjunction in $C$; let $F$ be its inverse. $G$ and $F$ reverse arrows, as in the next example.

The above is trivial, but I find it suggestive. Because it seems reasonable to say that forming the conjunction of a set of
propositions as $G$ does is one (crude) way of generalising from them. Informally speaking, the conjunction contains just enough information to imply
them all, but none of the others in the category (unless they were implied by
the originals anyway). Now, we also know that in $C$, their
conjunction is their limit. (More correctly, it's a limit of the diagram
containing the propositions and the implication arrows between them.) But this
formalises the notion expressed in the "just enough information" sentence,
because of the universal mapping property of the limit. (That is, any other
proposition which implies the originals also implies their conjunction.)

### 2. Logical induction by quantification ###

Let $E$'s objects be the non-empty sets of sentences $e(I)$ where $I$ is an integer. So one
object would be ${ e(1), e(2) }$. Interpret $e(I)$ as meaning "the
integer $I$ is an example of the concept". Interpret the arrows in
$E$ as set inclusion.

Let $C$ be the category whose objects are sentences:
either the atomic sentences $e(I)$ or the universally-quantified sentence
$\forall X \colon e(X)$. (Unlike the category of sentences in the
earlier
example, this category does not contain conjunctions.) Interpret the arrows
as implication.

Now define $G$
as follows. $G$ maps each singleton $e(I)$ to the sentence $e(I)$. It maps sets with more than one element
to the universally-quantified sentence. It also reverses arrows, mapping set
inclusion to reverse implication.

We could say that $G$ implements a simple form of logical induction,
rather more interesting than the earlier
one. And there are two languages, that of $C$
restricted compared to that of $E$, because
$C$ cannot express conjunctions, and so has to
approximate them by quantifying over all possible cases. The functor $G$ is
"doing the best it can" in these circumstances.

### 3. Least-squares fitting ###

Let $E$'s objects be the non-empty sets
of colinear elements of $R^2$. Once again, let the arrows
be set inclusion.

Let $C$ be the category whose objects are either the
singletons $\{ x \in R^2 \}$ or infinite lines in
$R^2$. Let the arrows be set inclusion.

Then let $G$ map each singleton to itself, and map each set
$S$ with more than one element to the line going through all $S$'s
points. $G$ maps inclusions to inclusions. As with the previous instance,
$G$ flattens most of $E$ into
$C$.

### Where are the adjunctions? ###

All the instances above can be formalised as adjunctions. Here's a summary of the proof, via [[Galois connection|Galois connections]]:
1. $E$ and $C$ are posets.
2. $G$ and $F$ are order-preserving, which is a necessary condition in the definition of Galois connection.
3. $G$ and $F$ satisfy the Galois connection condition.
4. A Galois connection is a special case of an adjunction.

The first point follows from the orderings I imposed on $E$ and $C$. The second holds for $F$, because it's either an identity, as in the least-squares example, or equivalent to one, as in the conjunction and quantification examples. It holds also for $G$, because it can't "cross over". If $e \lt e'$, then $G e$ may equal $G e'$, but it can't be greater. The third point follows by simple calculation with these orders. The fourth is a standard result.

$G$ can be regarded as a functor which maps a set of examples to an object which is in some
sense the "completion" of that set: the good old "free completion". It acquires a right adjoint $F$ which
maps this object back to the set of all possible examples derivable from this
completion object.

I think of this
in terms of the units and counits: an adjunction $E$
to $C$ is determined by the two functors $G$ and
$F$ and by two natural transformations
$i \colon I_E \Rightarrow G;E$
and
$e \colon E;G \Rightarrow I_G$. Given any object $c$ in
$C$, there is a morphism taking every $G F c$ to
$c$. Since $F$ maps $g$ to the set of all possible examples, and
$G$ should map that back to the original generalisation, this is the
identity. Hence we get one natural transformation.

In the other direction, given any object $e$ in
$E$, the functor $F G$ will map it to the set of all
possible examples of which $E$ is a part. There is an inclusion from
$e$ to $F G e$, and since this respects the other inclusions in
$E$, once again we get a natural transformation,
$i \colon I_E \Rightarrow G;E$.

(I need to relate this to the notion of limiting amount of information,
showing how that arises from the adjunction.)

### The problem of noise, and of different languages ###

In the least-squares example, I stipulated that the sets of points in $E$
must be colinear. This isn't very realistic: in real life, most sets of
examples will not exactly fit a line.

In general, given any useful generalisation method, some sets of examples
will not have an exact generalisation. You can say that this is due to
experimental error, or to an inadequate statistical model, or to an
insufficiently powerful description language, or to restricted storage capacity,
or whatever. See also sections on "Change of language" below.

I thought of fixing this by extending
$G$ so that it maps each set of non-colinear points
to the line with the least least-squares distance from them.
(But what if, in this
or other examples, there is no unique best generalisation?)

The problem with
this is that $G$ then no longer respects the arrows between the sets in $E$. Originally, if $S$ was a
subset of $S'$, then $G(S)$ was a subset of $G(S')$. But this is
no longer necessarily true: if the points in $S$ are colinear, and we make
$S'$ from $S$ by adding a point that isn't on the line, then the
best-fit line to $S'$ will be different from that to $S$, so the
inclusion won't hold.

Maybe the way to fix this is not to decide _a priori_ that the
morphisms in $E$ should be inclusions, but to let them
be determined by the morphisms in $C$. Philosophically
speaking, perhaps this is reasonable — we don't perceive the raw data directly,
but always fit it to a preconceived model. But I feel I'm missing something else.

What's the essence? A least-squares fit maps a set of points $e_i$ as follows. Some points $e_good$ fall exactly onto the regression line. Others, $e_bad$, don't. In effect, it's splitting the set into two parts, one of which needs added error information to describe it. Equivalently, it's adding error information to every point, even though that's 0 for some. Consider in the light of the FCA example $\{apple,orange\},
\{apple,pear\}$ vs. $\{round\},\{green\}$, and extension vs. intension. What _property_ is common to all points on a regression line?

... I need fonts to distinguish between sets of examples and their elements...

That's about as far as I've got; the examples I've so far constructed have the same structure as in 18, and seem to be missing something. I need to think about the structure of $C$ and $E$, and probably get away from $E$ being only a subcategory of Set. One possibility which feels promising (thanks to Victor Winschel for encouraging me on this) is to see whether stochastic logic programming can be formulated in terms of adjoints.

### Another example: learning linear transformations ###

(I need to complete this and the following examples.)
In this, I'm leading up to generalisation in simple neural networks. Think of learning a linear transformation. $E$ is the category of sets of pairs $\langle v,w \rangle$. $C$ is the category of linear transformations between vector spaces $V$ and $W$. $G$ puts pairs together to make a linear transformation. (What does $F$ do, and is it unique?) As with the least-squares example, this only works exactly if I restrict the sets in $E$.

### Another example: linear associators ###

The idea is as in the previous example, but where sets of examples aren't restricted to being expressible as linear transformations. In this case, generalisation has to "do the best it can", modulo interference between learnt associations. Question: what should $C$ be? Ought it to be some kind of completion of $C$ above? See http://www.syndar.org/PAdams/The%20Linear%20Associator.pdf for how the associator works. $C$ will depend very tightly on how the associator is trained. I'll start by considering how a single learning phase, in the terminology of the above reference, maps pairs to a matrix: the matrix $W$ in the notation of the reference's Equations 1 onwards. (Also to be completed.)

### Another example: averages ###

Let $E$ be the category of real pairs $\langle h, t \rangle$. Let $C$ be the category of real pairs whose elements sum to 1. Let $G$ map $\langle h, t \rangle$ to $\langle h/(h+t), t/(h+t) \rangle$. Let $F$ be the identity.
(Are $E$ and $C$ discrete?) (I'm working towards the way that induction in stochastic logic programming induces weights for the learnt clauses.)

### Another example: generalisation against a background ###

I don't know what to call this, but it feels relevant. It's a picture that came to me. Imagine that $E$ is the category of subsets of $R^2$. Now imagine that we have a real plane hatched by irregular intersecting lines, as if someone has been slashing a pencil across it again and again, or Jackson Pollock has been painting it. (I'm being intentionally vague.) Now let $E'$ and $C$ also be the category of subsets of $R^2$. Let $G_1 \colon E \to E'$ distort each set of examples by moving each point in it to the nearest intersection of these lines. (Assume there is a unique intersection.) Then let $G_2 \colon E' \to C$ find the smallest enclosing grid square as in my section "Galois connections and the idea of smallest enclosing concept" below. Let $G = G_1 ; G_2$.

A little more generally, let $C$ be a category whose objects are either also subsets of $R^2$, or something that's easy to picture as derived, using elementary plane geometry, from these subsets. For example, their boundaries, or their centroids. Let $G$ translate each .....

Informally, the idea is that there's a "background" which is not related to the examples, .....

## Informal justification ##

### 1. Colimits ###

Imagine various views of a physical object such as a leather
armchair. A view of the top of the left arm; a view of the seat; and so
on. Then we can merge these to make a composite view. If the views are
sets, the merging can be described as a [[colimit]] in the category of sets.

By the way, I mention the leather armchair because Goguen used it
as an example of
merging views in a paper on [[sheaf semantics of concurrent interacting objects]]: [_Sheaf
Semantics for Concurrent Interacting Objects_](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.4296).

### 2. When different views overlap ###

Such a colimit automatically handles the situation where different
views overlap and agree on their overlaps. In other words, it's a way to
reconstruct as much of the armchair as possible from partial views of it.

(The chart and atlas representation of manifolds is an example of this too
E.g. page 38 of [_Category theory Lecture 6: Colimits_, Jonathan Kirby](http://maths.dept.shef.ac.uk/magic/course_files/143/ctlecture6.pdf). )

#### Interpretation for non–category-theorists ####

I should explain that colimit is a categorical
construction. Set union is a special case of it. Goguen introduced the
notion that colimits can model "putting together" components into a
system. Many computer-science researchers use them to put together
specifications or program modules into bigger specifications or modules.

### 3. When different overlaps disagree ###

But in real life, different views might disagree on their overlaps. In
this case, we must find a way to combine these views with as little
disagreement as possible. That also feels like a colimit.

### 4. Adjoints as the solution to optimisation problems ###

This ties up with the idea that left adjoints can be regarded as
finding the optimum, or most efficient, solution to a problem.

(This is an
interpretation I came across in the Wikipedia entry for [_Adjoint functors_]
(http://en.wikipedia.org/wiki/Adjoint_functors): possibly authored by the
Wikipedian called "Functor salad". We can regard the solution as being
an initial object in a category of solutions. Its initiality models the
fact that it contains the least disagreement about overlaps, or what I
think of as the least "strain", of any possible solution.)

### 5. But concepts are not physical objects ###

In the above, I've been talking about reconstructing a single physical
object from different, possibly incompatible, views of it: six blind men
feel an elephant. I don't know whether one should call such reconstruction
"generalisation". But I strongly suspect that there's an analogy with the
construction of a concept from examples of it, if we think of the examples
as different views of the concept.

### 6. Which may be related to... ###

This may be related to the use of colimits for merging ontologies,
and for sensor fusion.

### 7. Categorical perception ###

I'll note that this also feels relevant to the topic of "categorical
perception" introduced by cognitive scientist Stevan Harnad, and to
similar methods of learning classifications. [_Neural network modeling
of categorical perception_, by R.I. Damper and S.R. Harnad](http://eprints.ecs.soton.ac.uk/397/) ;
[_Categorical Perception_, by S.R.
Harnad ] (http://cogprints.org/3017/1/catperc.html). This
is something I need to think about further: for the moment, I'll
just quote from Harnad's abstract to the second paper:

      Differences can be perceived as gradual and quantitative, as with
      different shades of gray, or they can be perceived as more abrupt and
      qualitative, as with different colors. The first is called continuous
      perception and the second categorical perception. Categorical
      perception (CP) can be inborn or can be induced by learning. Formerly
      thought to be peculiar to speech and color perception, CP turns
      out to be far more general, and may be related to how the neural
      networks in our brains detect the features that allow us to sort the
      things in the world into their proper categories, "warping" perceived
      similarities and differences so as to compress some things into the
      same
      perception and the second categorical perception. Categorical
      perception (CP) can be inborn or can be induced by learning. Formerly
      thought to be peculiar to speech and color perception, CP turns
      out to be far more general, and may be related to how the neural
      networks in our brains detect the features that allow us to sort the
      things in the world into their proper categories, "warping" perceived
      similarities and differences so as to compress some things into the
      same
      category and separate others into different categories.

(The "categorical" here is not connected with category theory but
with categorisation.)

### 8. Concepts as sheaves ###

Goguen's leather-armchair example actually applies not just to views of
the shape itself, but to views of attributes defined on it: e.g. colour,
temperature. This is how he introduces his [[sheaf semantics of concurrent interacting objects]], in which he represents objects as
[[sheaf|sheaves]] of local observations, these being
mappings from a base space (e.g. the surface of the chair) to attributes.
Perhaps there is a way to represent concepts as sheaves of properties of
examples, or something related.

### 9. The least general generalisation ###

Another idea that I want to capture is that of least
general generalisation. This phrase is used in inductive logic
programming, and was introduced by
[[Gordon Plotkin]] to describe anti-unification
(_A note on inductive generalization_. In B. Meltzer
and D. Michie, editors, "Machine Intelligence", volume 5, 1970). But I
want to think of it more generally. For instance, if we have a [[poset]]
of concepts, the least general generalisation of two concepts $c1$ and
$c2$ is
a new concept $c$ which implies as much as possible about $c1$ and
$c2$, but
as little as possible about anything else. It contains the information we
need to reconstruct the examples, but as little else as possible.

### 10. Implying as little as possible ###

Similarly, in a category of logical propositions, the product of two
propositions (their conjunction) implies each one, but nothing else. It
contains the information we need to reconstruct the examples, but nothing
else. This property too is something that I believe a generalisation
should have.

### 11. Generalisation as limit-formation ###

This leads me to the intuition that generalisation can be represented
as a limit.

### 12. Change of language. ###

But, contrary to the previous three paragraphs, the
language we normally use to describe concepts is different from that used
to describe examples. For example, in statistical least-squares
regression, the concept is a line plus error information, but the examples
are points.

### 13. Change of language because we want to learn new stuff ###

Moreover, we usually want the concept to be compressed or otherwise
different from just being a conjunction or something analogous. I suppose
there are various reasons for this:
* Saving memory.
* In some generalisers, including most neural nets, stored examples
will interfere just because of the architecture. This may not be a
good thing, but we need to be able to model it.
* Automated discovery. We often want the discovered concept
to say something new about the examples and their similarity. A
mere conjunction can't do that.
* A special case of (c) is when, in inductive logic programming,
we introduce new vocabulary: so-called background knowledge.

### 14. Two different languages, therefore two different categories ###

This difference between the language of examples and the language of
concepts is why I have two categories, $E$ and $C$. And it's also why I want
to model generalisation as an adjunction, rather than as only a limit or
colimit.

### 15. Universal quantification as generalisation? ###

This may be related to the fact that in categorical logic, universal
quantification can be represented as an adjunction. (See the section on
"Lawvere quantifiers: Quantification as adjunction" in [[quantifier]]).
Can one regard that as a kind of generalisation?

### 16. The impossibility of generalising exactly ###

Now that I've introduced the change of language, a very important
point is that the generalisation functor $G$ from examples to concepts, and
the forgetful functor $F$ from concepts to examples, may
not be able to translate exactly. In particular, the
language of concepts may not be able to summarise every set of examples
exactly. For example, a linear regression line can't represent
non-colinear points exactly. A categorisation of animals into a class of
animals that resemble sparrows and blackbirds, and another class of
animals that resemble dogs, cats, and horses, will struggle to place
duck-billed platypuses.

### 17. Generalisation as a left adjoint which "does the best it can" ###

In such cases, I think of the adjunction as "doing the
best it can" to create concepts, or to reconstruct examples from concepts.
I think intuitively of mapping from examples to concepts as creating a
kind of "strain". Imagine the sets of examples as points on a plane, and
generalisation as trying to enclose each set with a springy piece of metal
like an artist's flexicurve. The flexicurve can't bend sharply, so the
"flexicurve concept language" makes the set boundaries smoother and wider
than they should be. (Is this intuition worth formalising?)

### 18. Galois connections and the idea of smallest enclosing concept ###

This is related to an explanation of adjunction in terms of [[Galois
connection|Galois connections]], from a chapter in "The Logical foundations of cognition"
edited by John Macnamara and Gonzalo E. Reyes (once accessible on Google
Books, but they've blocked access). The authors describe a category $E$
whose objects are sets in $R^2$, and a subcategory $C$ of
$E$ constructed as
follows. Draw a grid on $R^2$ of cells of side 1. Then $C$'s objects are shapes
made by combining these grid cells: in other words, unions of squares of
side 1. Let $G:E\to C$ map each set to the smallest union of squares enclosing
it, and $F:C\to E$ map each union of squares to itself. Then $F$ and
$G$ form a
[[Galois connection]]. That's well known, but it's a nice explanation, and
perhaps a good "intuition pump" for concepts and examples. It makes me
think of generalisation as finding the "smallest enclosing concept" for a
set of examples.

### 19. Galois Connections and intension versus extension ###

Jeremy Gibbons reminded me that
[[Galois connection|Galois connections]] are
used in [[Formal Concept Analysis]] to map
between the intension (set of
characterising attributes) and the extension (set of examples) of a
concept. See e.g.
[http://www.alpheccar.org/fr/posts/show/32](http://www.alpheccar.org/fr/posts/show/32) (URL not currently accessible).
Jeremy felt
that this is morally the same as my machine learning scenario — inferring
a concept from a set of examples and/or a set of properties.

### 20. Syntax versus semantics ###

My idea probably also related to Lawvere's comment about an adjunction between syntax
and semantics, explained in Peter Smith's article [_The Galois connection between syntax and semantics_](http://www.logicmatters.net/resources/pdfs/Galois.pdf). As [_Patterns in Functional Programming_](http://patternsinfp.wordpress.com)'s posting [_Universal properties and Galois connections_](http://patternsinfp.wordpress.com/2011/02/15/universal-properties-and-galois-connections/) puts it:

    This construction can be used to translate a problem about
    the extension  of a concept (that is, an enumeration of
    its instances) into one about the   intension (that is, the
    characteristic properties of its instances). It is related to
    the  observation that "syntax and semantics are adjoint"—under the
    analogy that "objects" are sets of mathematical structures,
    "properties"   are axioms, and the relation is "satisfaction",
    the models of an axiomatic theory $T$ are included in a set of structures
    $S$
    if and only if the theory $T$ logically entails the minimal
    axiomatization of $S$.

Thanks to Jeremy and to Viktor Winschel for pointing this out.

### 21. The adjunction between minimal realization and behaviour ###

Hendrik Hilberdink asked whether my generalisation adjunction is related to the adjunction between minimal realization and behaviour of an automaton. This is worth looking into. Perhaps the structure of those categories will give some clues about that of my categories of concepts and examples.