unbounded topos

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

## Idea

Most [[geometric morphisms]] between [[toposes]] are [[bounded geometric morphism|bounded]], and so most toposes can be considered as either a [[base topos]] - the universe of discourse - such as [[Set]], or as a topos of [[internal sheaves]] over another topos (for example any [[Grothendieck topos]]).

While there are lots of examples of toposes which aren't Grothendieck toposes, that is, bounded toposes over $Set$, for example the topos [[FinSet]] of finite sets, or [[realisability toposes]], this is due to the fact they _do not admit_ a geometric morphism to $Set$. Toposes of this form usually can be considered as a base topos/universe of discourse for non-standard versions of reasoning, for example [[finitism]] can be seen as taking the topos $FinSet$ as base topos.

An **unbounded topos** is a non-Grothendieck topos which _does_ admit a geometric morphism to $Set$ (or some other specified base topos), but which isn't equivalent to the category of sheaves on some small [[site]].

## Definition

A topos $\mathcal{E}$ equipped with a geometric morphism $p\colon \mathcal{E} \to Set$ is **unbounded** if $p$ is not bounded. Equivalently, $\mathcal{E}$ (equipped with $p$) is not equivalent to the category of [[sheaves]] on some small site (equipped with the usual geometric morphism to $Set$).

This means that for any set $\{A_i\}_{i\in I}$ of objects of $\mathcal{E}$, which would aspire to be a [[separator|separating family]] for $\mathcal{E}$, there is a pair of morphisms $f,g\colon X \to Y$ of $\mathcal{E}$ such that for all $i\in I$, and all $t\colon A_i \to X$, we have $f\circ t = g\circ t$, but $f\neq g$.

More generally, we could work over a given base topos $\mathcal{S}$, and then an $\mathcal{S}$-topos $p\colon \mathcal{E} \to \mathcal{S}$ is unbounded if $p$ is not a bounded geometric morphism or equivalently it is not equivalent to the category of sheaves on an [[internal site]] in $\mathcal{S}$.

## Examples

There are relatively few examples of unbounded toposes.

* Given a non-[[locally small category|locally small]] [[groupoid]] $G$ with only a [[set]] of isomorphism classes of objects, the functor category $Cat(G,Set) =: GSet$ is a topos. It has a geometric morphism to $Set$, namely the global sections functor $\Gamma(X) = GSet(1,X)$, which is unbounded. $GSet$ is moreover [[cocomplete category|cocomplete]], [[Boolean topos|Boolean]] and even locally small.

* If $K$ is a topological group, the category $Unif(K)$ of sets with a [[uniformly continuous]] $K$-[[action]] is a $Set$-topos. In the case that $K$ has no smallest open subgroup, then $Unif(K)$ is still Boolean and locally small, but is not cocomplete (it fails to have infinite coproducts), and so not a Grothendieck topos.

* The [[topos of algebras over a monad|topos of coalgebras]] of a pullback-preserving [[comonad]] $M$ on a Grothendieck topos whose functor part is not [[accessible functor|accessible]] is a cocomplete topos $MCoalg$ over $Set$ which is not [[locally presentable category|locally presentable]] hence not a Grothendieck topos.

  * As a particular example, one can take a [[exact functor|left exact]] [[endofunctor]] $F$ on $Set$ and form the corresponding comonad $(X,Y) \mapsto (X,Y\times F(X))$ on $Set\times Set$ and the topos of coalgebras for this (which is equivalent to the [[Artin gluing]] $Gl(F)$ of $F$). In this case however it is not clear that there exist such endofunctors without assuming the existence of [[large cardinals]] (for instance the existence of a [[proper class]] of [[measurable cardinals]] is sufficient to give such an endofunctor).

    ([[David Roberts|DR]]: I think this result has since been improved, see the paper J. Adamek, V. Koubek and V. Trnkova, "How large are left exact functors?" in [TAC](http://www.tac.mta.ca/tac/volumes/8/n13/8-13abs.html). I will have to sort out whether what they are saying applies)

* Any topos that is not locally small, relative to some fixed topos $set$ of sets, when such a setting is properly defined, is not bounded over $set$.

* More coming soon!

## Related topics

* [[internal sheaves]]
* [[internal site]]
* [[bounded geometric morphism]]

## References

The unbounded toposes $GSet$, $Unif(K)$ and $MCoalg$ are mentioned in B3.1.4 of

* [[Peter Johnstone]], [[Sketches of an Elephant]]

as being (at the time) essentially the only examples of unbounded toposes.

[[!redirects unbounded topos]]
[[!redirects unbounded toposes]]
[[!redirects unbounded topoi]]


uncertainty of fluxes

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###Context###
#### Quantum systems
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#### Differential cohomology
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[[!include differential cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In ([[higher gauge theory|higher]]) [[gauge theory]], the total (generalized) electric/magnetic [[fluxes]] through given [[surfaces]] (or higher [[submanifolds]]) should be [[observables]] and hence have [[Poisson brackets]] in the [[classical field theory]] To the extent that these induce non-trivial operator [[commutators]] in the corresponding [[quantum field theory]], this means that the corresponding kinds of fluxes would be subject to the [[Heisenberg uncertainty principle]].

A subtlety here is that the usual [[phase space]]-methods for [[Lagrangian field theories]] directly provide these brackets only for the [[differential forms]] which are the [[flux densities]]. These however, provide only a [[rational homotopy theory|rational]] image of the fluxes, which in general may furthermore have [[torsion subgroup|torsion]]-[[cohomology group|cohomology]]-components.

In plain [[electromagnetism]], the [[canonical momentum]] of the [[gauge potential]] $A$ is proportional (by constants to be ignored in the following) to the electric flux density $\star F$ (where $F$ is the [[Faraday tensor]] [[flux density]] and $\star$ is the [[Hodge star operator]] on the given [[spacetime]] [[pseudo-Riemannian manifold]]), which however means that the the Poisson bracket of the magnetic flux density $F \equiv \mathrm{d}_{dR} A$ with the electric flux density is a total derivative (cf. eg. [Blaschke & Gieres 2021 (5.5)](Yang-Mills+theory#BlaschkeGieres21)), so that the bracket of the integrated fluxes $\Phi_E$ and $\Phi_B$ through given [[orientable manifold|orientable]] [[closed manifold|closed]] surfaces $S_E$ and $S_B$, respectively, actually vanishes (cf. [FMS07b (3.6)](#FreedMooreSegal07b) and Prop. \ref{PoissonBracketBetweenElectricAndMagneticFluxes} below):

\[
  \label{CommutingAbelianFluxes}
  \big\{
    \Phi_E
    ,\,
    \Phi_B
  \big\}
  \;\equiv\;
  \Big\{
  \textstyle{\int}_{S_B} \star F
  ,\,
  \textstyle{\int}_{S_E} F
  \Big\}
  \;=\;
  0
  \,.
\]

The thrust of [FMS 07a](#FreedMooreSegal07a), [07b](#FreedMooreSegal07b) is the claim that this bracket becomes non-vanishing if torsion-components of the fluxes (through their enhancement to [[ordinary differential cohomology]]) are retained, but it seems that this is ultimately by definition ([FMS07a, Def. 1.29](#FreedMooreSegal07a); [FMS07b (3.28)](#FreedMooreSegal07b)) more than by derivation from first principles.

On the other hand, for [[non-abelian group|non-abelian]] [[gauge group]], a careful analysis of the (somewhat subtle) Poisson brackets reveals ([Cattaneo & Perez 2017](#CattaneoPerez17)) that the electric fluxes -- understood as flux densities integrated against [[Lie algebra]] valued functions $\alpha$ -- have a non-trivial bracket among *themselves* (Prop. \ref{PoissonBracketBetweenElectricFluxes}):

$$
 \big\{
   \Phi^\alpha_E
   ,\,
   \Phi^{\alpha'}_E
 \big\}
 \;\propto\;
 \Phi_E^{[\alpha, \alpha']}
$$

(where $[-,-]$ is the pointwise [[Lie bracket]]). This result has found a lot of attention (only) in the context of [[first-order formulations of gravity]].

## Details
 {#Details}

Consider a [[metric Lie algebra]] $\mathfrak{g}$ -- such as [[line Lie algebra|$\mathfrak{u}(1)$]] or  [[su(n)|$\mathfrak{su}(n)$]], [[so(n)|$\mathfrak{so}(n)$]] via their [[Killing forms]], or any [[direct sum]] of these.

For 3+1 dimensional [[Yang-Mills theory]] with [[gauge group|gauge]] Lie algebra $\mathfrak{g}$, we discuss ([below](#ThePoissonBracketOfIntegratedFluxes)) the [[Poisson bracket]] of integrals (against [[Lie algebra]]-valued functions) of [[flux densities]] ([[curvature 2-forms]]) over [[oriented manifold|oriented]] [[closed manifold|closed]] [[surfaces]], following [Cattaneo & Perez 2017](#CattaneoPerez17) (where this is discussed for the special case $\mathfrak{g} =$ [[su(2)|$\mathfrak{su}(2)$]], cf. Ex. \ref{TheLieAlgebrasu2}).

The Poisson bracket of electric fluxes among themselves is equation (7) in [CP17](#CattaneoPerez17), and to warm-up we spell out the relevant computation in detail (Prop. \ref{PoissonBracketBetweenElectricFluxes} below, which also provides a rigorous proof of the abelian case (eq:CommutingAbelianFluxes)).
Then we similarly compute the bracket between electric and magnetic fluxes (Prop. \ref{PoissonBracketBetweenElectricAndMagneticFluxes} below, which seems not to discussed elsewhere in the literature).

The computations are standard and straightforward, but since, as usual, the results depend crucially on delicate signs, we go into full detail.

\linebreak

### Lie theoretic preliminaries

Our [[ground field]] is the [[real numbers]].

Consider a [[Lie algebra]] $\mathfrak{g}$ with [[Lie bracket]]

\[
  \label{LieBracket}
  [-,-]
    \,\colon\,
  \mathfrak{g} \otimes \mathfrak{g}
  \longrightarrow
  \mathbb{R}
  \,.
\]

We aim to mostly write intrinsic expressions, but it is still useful to have the component expression at hand. For that purpose we choose once and for all any [[linear basis]] of the [[underlying]] [[vector space]] of the Lie algebra

$$
  \mathfrak{g}
  \;\simeq_{{}_\mathbb{R}}\;
  \mathbb{R}
  \big\langle
    t_1, \cdots, t_{{}_{dim(\mathfrak{g})}}
  \big\rangle
$$

in terms of which the Lie bracket (eq:LieBracket) is given by the structure constants $\{f^i{}_{j k}\}$ defined by the following equation (using [[Einstein summation convention]], throughout):

$$
  [t_j, t_k]
  \;=\;
  f^i{}_{j k} \, t_i
  \,.
$$

The following discussion is all about [[Lie algebra-valued differential form|$\mathfrak{g}$-valued differential forms]], being elements of the [[tensor product]] $\Omega^\bullet_{dR} \otimes \mathfrak{g}$ of the [[de Rham algebra]] on a given [[smooth manifold]] with the given [[Lie algebra]].

> {#DiffFormsWithValuesInAdjointBundle} More generally, we should be dealing with differential forms with values in [[sections]] of a $\mathfrak{g}$-[[adjoint bundle]] $\mathfrak{g}_P$ [[associated bundle|associated]] to a [[principal bundle]] $P$ ([[underlying]] a [[principal connection]] modelling a [[background field|background]] Yang-Mills [[gauge potential]]). Since the [[adjoint bundle]] $\mathfrak{g}_P$ is a [[Lie algebra]]-[[fiber bundle]], we may think of $\Omega^\bullet_{dR}(-; \mathfrak{g})$ as $\Omega^\bullet(-;\mathfrak{g}_P)$ in all of the following, without otherwise changing much of the discussion (the definition of the flux density $F_A$ picks up one correction term).
But since, in the end, we consider only [[adjoint action|$\mathrm{ad}$-]][[invariant]] combinations of such $\mathfrak{g}_P$-valued forms (namely the invariant flux densities $\langle \alpha, E\rangle$ and $\langle \beta, F_A\rangle$ from Prop. \ref{TheProperFluxObservables} below) the result of the following discussion is actually independent of the background field $P$, so that we may as well take it to be trivial --- as usual in most of the physics literature on the subject.

The primary example are $\mathfrak{g}$-valued 1-forms

\[
  \label{GaugePotential}
  A
    \;\equiv\;
  A^i \otimes t_i
  \;\in\;
  \Omega^1_{dR} \otimes \mathfrak{g}
  \,,
\]

which are going to represent the (local) [[gauge potential]] of $\mathfrak{g}$-[[Yang-Mills theory]].

In writing (Lie- or Poisson-)brackets of such [[Lie algebra-valued differential form|$\mathfrak{g}$-valued differential forms]], we mean to (1.) take the [[wedge product]] of the bare [[differential forms]] *in the given order* and (2.) apply the bracket to the [[coefficients]],

\[
  \label{LieBracketOnLieAlgebraValuedDifferentialForms}
  \omega, \gamma \in \Omega^\bullet_{dR} \otimes \mathfrak{g}
  \;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;
  [ \omega , \gamma ]
  \,\coloneqq\,
  \omega^i \wedge \gamma^j
  \otimes
  [t_i, t_j]
  \,.
\]

This bracket makes [[Lie algebra-valued differential forms|$\mathfrak{g}$-valued forms]] into a [[super Lie algebra]]:

\[
  \label{GradedSkewSymemtryOfLieBracketOnLieAlgValuedForms}
  \begin{array}{l}
    \omega \,\in\, \Omega_{dR}^p \otimes \mathfrak{g}
    \\
    \omega' \,\in\, \Omega_{dR}^{p'} \otimes \mathfrak{g}
  \end{array}
  \;\;\;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;\;\;
  \big[ \omega , \omega' \big]
  \;=\;
  -(-1)^{p p'}
  \big[ \omega' , \omega \big]
  \,.
\]

For example, the Lie bracket of the [[gauge potential]] 1-form  (eq:GaugePotential) with itself is:

\[
  \label{BracketOfGaugePotentialWithItself}
  [ A , A ]
  \;\equiv\;
  A^j \wedge A^k \otimes [t_i, t_j]
  \;=\;
  f^i{}_{j k}
  \,
  A^j \wedge A^k \otimes t_i
  \,.
\]

The [[Jacobi identity]] of $\mathfrak{g}$ says that $[t_i,-]$ is a [[derivation]] of the Lie bracket:

\[
  \label{JacobiIdentity}
  \big[t_i, [t_j, t_k] \big]
  \;=\;
  \big[ [t_i, t_j], t_k] \big]
  \,+\,
  \big[ t_j, [t_i, t_k] \big]
  \;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;
  f^\bullet{}_{i l} f^{l}{}_{j k}
  \;=\;
  f^\bullet{}_{l k}  f^l{}_{i j}
  \,+\,
  f^\bullet{}_{ j l } f^l{}_{ i k }
  \,.
\]

For any $X \in \Omega^n_{dR} \otimes \mathfrak{g}$ this means for instance, with (eq:BracketOfGaugePotentialWithItself), that

\[
  \label{AAX}
  \big[A, [A, X]\big]
  \;=\;
  \big[ [A,A], X \big]
  -
  \big[A, [A, X]\big]
  \;\;\;\;
  \text{and hence}
  \;\;\;\;
  \big[A, [A, X]\big]
  \;=\;
  \tfrac{1}{2}
  \big[ [A,A], X \big]
  \,.
\]

Given a [[gauge potential]] $A$ as above (eq:GaugePotential), the *[[covariant derivative]]* is the sum of the [[de Rham differential]] with the operation $[A,-]$ (eq:LieBracketOnLieAlgebraValuedDifferentialForms):

\[
  \label{CovariantDerivative}
  \mathrm{d}_{A} X
  \;\equiv\;
  \mathrm{d} X + [A, X]
  \;=\;
  \mathrm{d} X^\bullet + f^\bullet{}_{j k} A^j \wedge X^k
  \,.
\]

Since the [[de Rham differential]] $\mathrm{d}$ is a degree=1 [[graded derivation]] on the [[de Rham algebra]] $\Omega^\bullet_{dR}$, the [[Jacobi identity]] (eq:JacobiIdentity) implies that the [[covariant derivative]] (eq:CovariantDerivative) is a degree=1 [[graded derivation]] on the [[Lie bracket]] (eq:LieBracketOnLieAlgebraValuedDifferentialForms) of [[Lie algebra-valued differential forms]]:

\[
  \label{CovariantDerivativeIsDerivationOfLieBracket}
  \begin{array}{l}
    \omega \,\in\, \Omega^p_{dR} \otimes \mathfrak{g}
    \\
    \gamma \,\in\, \Omega^\bullet_{dR} \otimes \mathfrak{g}
  \end{array}
  \;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;
  \mathrm{d}_A [ \omega, \gamma ]
  \;=\;
  \big[
    \mathrm{d}_A \omega
    ,\,
    \gamma
  \big]
  +
  (-1)^{p}
  \big[
    \omega
    ,\,
    \mathrm{d}_A \gamma
  \big]
  \,.
\]

Another key point is that applying the [[covariant derivative]] (eq:CovariantDerivative) twice is equal to applying the Lie bracket $[F_A, -]$ with the [[curvature 2-form]]

\[
  F_A
  \;\equiv\;
  \mathrm{d}A + \tfrac{1}{2}[A, A]
  \,,
\]

representing the [[magnetic charge|magnetic]] [[flux density]] carried by the [[gauge field]] $A$, because:

\[
  \label{CurvatureFromCovariantDerivativeSquared}
  \begin{array}{l}
    \mathrm{d}_A \mathrm{d}_A X
    \\
     \;=\;
     \mathrm{d}_A \big(
       \mathrm{d}X + [A,X]
     \big)
     \\
     \;=\;
     [A, \mathrm{d}X]
     +
     \mathrm{d}[A,X]
     +
     \big[ A, [A, X] \big]
     \\
     \;=\;
     [A, \mathrm{d}X]
     +
     [\mathrm{d} A,X]
     -
     [A, \mathrm{d} X]
     +
     \tfrac{1}{2}\big[ [A, A],  X \big]
     \\
     \;=\;
     \big[
       \underset{
         \equiv \, F_A
       }{
         \underbrace{
           \mathrm{d} A + \tfrac{1}{2}[A, A]
         }
       }
       ,\,
       X
     \big]
     \,,
  \end{array}
\]

where in the third step we used (eq:AAX).

Similarly, from applying the covariant derivative three times follows the [[Bianchi identity]], $\mathrm{d}_A F_A = 0$:

\[
  \label{BianchiIdentity}
  \begin{array}{l}
    \mathrm{d}_A \mathrm{d}_A \mathrm{d}_A X
    \\
    \;=\;
    \mathrm{d}_A \big[ F_A ,\, X \big]
    \\
    \;=\;
    \big[ \mathrm{d}_A F_A ,\, X \big]
    +
    \big[ F_A ,\, \mathrm{d}_A X \big]
    \\
    \;=\;
    \big[ \mathrm{d}_A F_A ,\, X \big]
    +
    \mathrm{d}_A \mathrm{d}_A \mathrm{d}_A X
   \end{array}
   \;\;\;\;\;\;\;\;\;
   \Rightarrow
   \;\;\;\;\;\;\;\;\;
   \mathrm{d}_A F_A \,=\, 0
   \,.
\]

\linebreak

Our assumption that $\mathfrak{g}$ is a [[metric Lie algebra]] means that it is equipped with an binary [[invariant polynomial]], namely a [[bilinear map]]

$$
  \langle -,-\rangle
  \;\colon\;
  \mathfrak{g} \otimes \mathfrak{g}
  \to
  \mathbb{R}
  \,,
  \;\;\;\;\;\;
  k_{i j}
  \;\equiv\;
  \langle t_i, t_j\rangle
$$

which is symmetric

\[
  \label{SymmetryOfInvariantPolynomial}
  \langle t_i, t_j\rangle
  \;=\;
  \langle t_j, t_i\rangle
\]

and [[adjoint action|ad-]][[invariant]], in that

\[
  \label{AdInvarianceOfInvariantPolynomial}
  \big\langle
    [t, -]
    ,\,
    -
  \big\rangle
  +
  \big\langle
    -
    ,\,
    [t,-]
  \big\rangle
  \;=\;
  0
  \,.
\]

For example:

\[
  \label{AdInvarianceOfPairingOnLieAlgValuedForms}
  X, Y
   \,\in\,
  \Omega^0 \otimes \mathfrak{g}
  \;\;\;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;\;\;
  \langle
    [A, X] ,\, Y
  \rangle
  +
  \langle
    X ,\, [A , Y ]
  \rangle
  \;=\;
  0
  \,.
\]

The induced trilinear pairing

\[
  \label{TrilinearPairing}
  \langle -,-,- \rangle
  \;\coloneqq\;
  \big\langle -,[-,-] \big\rangle
  \;\colon\;
  \mathfrak{g} \otimes \mathfrak{g} \otimes \mathfrak{g}
  \to
  \mathbb{R}
  \,,
  \;\;\;\;\;\;\;
  f_{i j k}
  \;\equiv\;
  \langle t_i, t_j, t_k\rangle
  \;=\;
  k_{i i'} f^{i'}{}_{j k}
\]

is also ad-invariant:

\[
  \label{AdInvarianceOfTrilinearPairing}
  \begin{array}{l}
    \big\langle
      [t,-], -,-
    \big\rangle
    +
    \big\langle
      -, [t,-], -
    \big\rangle
    +
    \big\langle
      -, -, [t,-]
    \big\rangle
    \\
    \;\equiv\;
    \big\langle
      [t,-], [-,-]
    \big\rangle
    +
    \Big\langle
      -, \big[ {[t,-]}, -\big]
    \Big\rangle
    +
    \Big\langle
      -, \big[-, [t,-]\big]
    \Big\rangle
    \\
    \;=\;
    \big\langle
      [t,-], [-,-]
    \big\rangle
    +
    \Big\langle
      -, \big[ t, [-, -] \big]
    \Big\rangle
    \\
    \;=\;
    0
    \,,
  \end{array}
\]

(where we used first the Jacobi identity (eq:JacobiIdentity) and then the invariance (eq:AdInvarianceOfInvariantPolynomial) of the bilinear pairing)

and is invariant under [[cyclic permutation]] of its arguments:

\[
  \label{CyclicInvarianceOfTrilinearPairing}
  \begin{array}{l}
    \langle
      t_i, t_j , t_k
    \rangle
    \\
    \;\equiv\;
    \phantom{-\;}
    \big\langle
      t_i, [t_j , t_k]
    \big\rangle
    \\
    \;=\;
    -
    \big\langle
       [t_j ,  t_i ],  t_k
    \big\rangle
    \\
    \;=\;
    \phantom{-\;}
    \big\langle
       [t_i,  t_j ],  t_k
    \big\rangle
    \\
    \;=\;
    \phantom{-\;}
    \big\langle
      t_k, [t_i,  t_j ]
    \big\rangle
    \\
    \;\equiv\;
    \phantom{-\;}
    \langle
      t_k, t_i, t_j
    \rangle
    \,,
  \end{array}
\]

(where we used ad-invariance (eq:AdInvarianceOfInvariantPolynomial), and symmetry (eq:SymmetryOfInvariantPolynomial) of the pairing and skew-symmetry of the Lie bracket).

\linebreak

\begin{example}
\label{TheLieAlgebrasu2}
In the case
$\mathfrak{g} = $ [[su(2)|$\mathfrak{su}(2)$]] equipped with its [[Killing form]] and with linear basis taken to be the [[Pauli matrices]], we have:

* $f_{i j k} = \epsilon_{i j k}$ the [[Levi-Civita symbol]],

* $k_{i j} = \delta_{i j}$ the [[Kronecker delta]].

This is the case considered in [Cattaneo & Perez 2017](#CattaneoPerez17).
\end{example}

\linebreak

### The phase space of Yang-Mills theory
 {#ThePoissonBracketOfCanonicalVariables}

We consider now such [[Lie algebra-valued differential forms]] on a given [[smooth manifold|smooth 3-manifold]] $X$, thought of as a chosen [[Cauchy surface]] in a 3+1-dimensional [[Minkowski spacetime]].

On the space of $\mathfrak{g}$-valued 1-forms $A$ \eqref{GaugePotential} and 2-forms $E$, which on any [[coordinate chart]] $\mathbb{R}^3 \hookrightarrow X$ we may expand as

$$
  A
  \,=\,
  A_a^i \, \mathrm{d}x^a \otimes t_u
  \,,
  \;\;\;\;
  E \,=\,
  E_{a b}^i \,
  \mathrm{d} x^a \wedge \mathrm{d} x^b \otimes t_u
  \,,
$$

we declare the [[distribution|distributional]] [[Poisson bracket]]:

\[
  \label{ThePoissonBracketOnCanonicalCoordinates}
  \begin{array}{l}
    \big\{ E^i_{a b}(x), E^j_{c d}(x') \big\}
    \;=\; 0
    \\
    \big\{ A^j_a(x), A^j_b(x') \big\}
    \;=\; 0
    \\
    \big\{ E^i_{a b}(x), A^j_c(x') \big\}
    \;\equiv\;
    k^{ i j } \epsilon_{a b c} \delta(x,x')
  \end{array}
\]

together with the [[first class constraint]] that the [[covariant derivative]] (eq:CovariantDerivative) of $E$ vanishes:

\[
  \label{GaussLawConstraint}
  \mathrm{d}_A E \;\approx\; 0
  \,.
\]

This is the [[phase space]] of [[Yang-Mills theory]] [eg. [Friedman & Papastamatiou 1983, §3](Yang-Mills+theory#FriedmanPapastamatiou83); [Bassetto, Lazzizzera & Soldati 1984, §2](Yang-Mills+theory#BassettoLazzizzeraSoldati84)] with:

* the [[canonical coordinate]] being the [[gauge potential]] $A$ in [[temporal gauge]],

* its [[canonical momentum]] being the [[electric field]] [[flux density]] $E$,

* the [[constrained mechanics|constraint]] (eq:GaussLawConstraint) expressing the [[Gauss law]].

\linebreak

On this phase space, we are concerned with [[observables]] expressing electromagnetic flux through [[closed manifold|closed]] [[oriented manifold|oriented]] [[surface]] [[submanifolds]] (not necessarily [[connected topological space|connected]])

\[
  \label{TheEmbeddedSurfaces}
  S \xhookrightarrow{\phantom{---}} X
  \,.
\]

Taken at face value, the linear such observables are the surface [[integration of differential forms|integrals]] of the [[flux densities]] over $S$ against $\mathfrak{g}$-valued "smearing"-functions:

\[
  \label{NaiveFluxObservables}
  \begin{array}{c}
  \Phi_E^\alpha
  \;\overset{?}{\equiv}\;
  \int_S \langle \alpha,  E \rangle
  \,,
  \\
  \Phi_B^\beta
  \;\overset{?}{\equiv}\;
  \int_S \langle \alpha, F_A \rangle
  \,,
  \end{array}
  \;\;\;\;\;\;\;\;
  \text{for}
  \;\;
  \alpha, \beta
    \;\in\;
  \Omega^0(S) \otimes \mathfrak{g}
\]

and more general flux observables ought to be taken to be the [[polynomials]] in these linear observables.

However, these expressions (eq:NaiveFluxObservables) need to be corrected ("regularized") in order to become actual observables, since as given they do not have associated smooth [[Hamiltonian vector fields]]. This is the point explained in [Cattaneo & Perez 2017](#CattaneoPerez17): Instead, one needs to consider 3-dimensional "smearing" of the canonical observables in Prop. \ref{TheProperFluxObservables} below.

Using the orientation of $S$ we consider any one-sided [[tubular neighbourhood]] $\hat S$ of $S$ inside $X$, extending to the "exterior" of $S$ (a non-compact [[submanifold]] [[manifold with boundary|with boundary]] $S$),

> ([CP17](#CattaneoPerez17) speak of extending to the "interior" and seem to imagine that $S$ is actually the boundary of a compact sub-manifold -- which however is generally not what one wants to assume in discussion of fluxes: Typical choices of $S$ coincide with actual ([[asymptotic boundary|asymptotic]]) boundaries of the ambient spacetime, such as around the singular locus of a [[magnetic monopole]] that has been removed from spacetime. Our choice of extending to a neighbourhood of $S$ towards the exterior, instead of the interior, works just as well and allows $S$ to actually sit on a boundary of spacetime.)

and extending the coefficient functions to this neighbourhood with [[compact support]]  (i.e. such that they vanish some finite distance from $S$):

\begin{proposition}
\label{TheProperFluxObservables}
**([CP17 (6)](#CattaneoPerez17))**
Well-defined linear flux-observables equivalent to
the naïve observables (eq:NaiveFluxObservables)
are of this form:
\[
  \label{TheFluxObservables}
  \begin{array}{l}
  \Phi_E^\alpha
  \;\equiv\;
  \int_{\widehat S}
  \big\langle
    \mathrm{d}_A \alpha
    ,\,
    E
  \big\rangle
  \,,
  \\
  \Phi_B^\beta
  \;\equiv\;
  \int_{\widehat S}
  \big\langle
    \mathrm{d}_A \alpha
    ,\,
    F_A
  \big\rangle
  \,,
  \end{array}
  \;\;\;\;
  \text{for}
  \;\;
  \alpha, \beta
    \;\in\;
  \Omega^0\big(\widehat S \big)_{cpt}
    \otimes \mathfrak{g}
  \,.
\]
\end{proposition}
\begin{proof}
  To check that these are indeed equivalent to the naïve observables, in that the difference is proportional to the left hand side of the [[Gauss law]] (eq:GaussLawConstraint), so that they coincide on the locus where the Gauss law holds:
$$
  \begin{array}{l}
  \int_{\widehat{S}}
    \big\langle
      \mathrm{d}_A \alpha
      ,\,
      E
    \big\rangle
    \\
    \;\approx\;
    \int_{\widehat{S}}
    \Big(
    \big\langle
      \mathrm{d}_A \alpha
      ,\,
      E
    \big\rangle
    +
    \underset{
      \approx 0
    }{
    \underbrace{
    \big\langle
      \alpha
      ,\,
      \mathrm{d}_A E
    \big\rangle
    }
    }
    \Big)
    \\
    \;=\;
    \int_{\widehat{S}}
    \Big(
    \big\langle
      \mathrm{d} \alpha
      ,\,
      E
    \big\rangle
    \,+\,
    \big\langle
      \alpha
      ,\,
      \mathrm{d} E
    \big\rangle
    \Big)
    \\
    \;=\;
    \int_{\widehat{S}}
    \mathrm{d}
    \big\langle
      \alpha
      ,\,
      E
    \big\rangle
    \\
    \;=\;
    \int_{S}
    \big\langle
      \alpha
      ,\,
      E
    \big\rangle
    \mathrlap{\,,}
  \end{array}
$$
where we used:

1. the Gauss law

1. (eq:AdInvarianceOfPairingOnLieAlgValuedForms)

1. derivation property of $\mathrm{d}$

1. [[Stokes theorem|Stokes]]

\end{proof}

\linebreak

In the computation [below](#ThePoissonBracketOfIntegratedFluxes), of the Poisson brackets of these flux observables, we repeatedly need the following identities, which also serve as good examples for how to compute with the Poisson brackets (eq:ThePoissonBracketOnCanonicalCoordinates):

\begin{example}
A key consequence of the corrected flux observables (eq:TheFluxObservables), is a non-trivial Poisson bracket between electric field observables and plain smearing functions:
\[
  \label{PoissonBracketBetweenElectricFieldAndSmearingFunction}
  \begin{array}{l}
    \Big\{
      \textstyle{\int}
      \big(\mathrm{d}_A \alpha_i\big)
      E^i
      ,\,
      \mathrm{d}_A \beta
    \Big\}
    \\
    \;\equiv\;
    \Big\{
      \textstyle{\int}
      \big(\mathrm{d}_A \alpha_i\big)
      E^i
      ,\,
      \mathrm{d}\beta
      + [A, \beta]
    \Big\}
    \\
    \;=\;
    \textstyle{\int}
    \big(\mathrm{d}_A \alpha_i\big)
    \Big[
      \big\{
        E^i
        ,\,
        A
      \big\}
      ,\,
      \beta
    \Big]
    \\
    \;=\;
    \big(\mathrm{d}_A \alpha_i\big)
    \big[
      t^i
      ,\,
      \beta
    \big]
    \\
    \;=\;
    \Big[
      \big(\mathrm{d}_A \alpha\big)
      ,\,
      \beta
    \Big]
    \,.
  \end{array}
\]
\end{example}
This is the relation needed for the computation of the bracket among electric fluxes in Prop. \ref{PoissonBracketBetweenElectricFluxes} below.

\begin{example}
\label{BracketOfElectricFluxWithMagneticFluxDensity}
The Poisson bracket of an electric flux observable with a magnetic flux density results in the *covariant derivative* of the electric smearing function:
\[
  \label{TheBracketOfElectricFluxWithMagneticFluxDensity}
  \begin{array}{l}
    \Big\{
      \textstyle{\int}
      \big(\mathrm{d}_A \alpha_i\big) \wedge E^i
      ,\,
      F_A(x')
    \Big\}
    \;=\;
    d_A \big(\mathrm{d}_A \alpha\big)(x')
  \end{array}
\]
\end{example}
This is the main relation needed in the computation of the bracket between electric and magnetic fluxes, in Prop. \ref{PoissonBracketBetweenElectricAndMagneticFluxes} below.
\begin{proof}
For the first summand in $F_A \,\equiv\, \mathrm{d}A + \tfrac{1}{2}[A, A]$ we have simply:
$$
  \begin{array}{l}
    \Big\{
      \textstyle{\int}
      \big(\mathrm{d}_A \alpha_i \big) \wedge E^i
      ,\,
      \mathrm{d} A(x')
    \Big\}
    \\
    \;=\;
    \mathrm{d}
      \Big\{
      \textstyle{\int}
      \big(\mathrm{d}_A \alpha_i \big) \wedge E^i
      ,\,
       A(x')
    \Big\}
    \\
    \;=\;
    \mathrm{d}
    \big(\mathrm{d}_A \alpha(x') \big)
    \,.
  \end{array}
$$
For the second summand, considering any $\omega \in \Omega^1_{dR} \otimes \mathfrak{g}$, we have:

$$
  \begin{array}{l}
    \Big\{
      \textstyle{\int}
       \omega_i E^i
       ,\,
       \tfrac{1}{2} [A, A]
    \Big\}
    \\
    \;=\;
    \tfrac{1}{2}
    \Big[
      \big\{
        \textstyle{\int}
        \omega_i E^i
        ,\,
        A
      \big\}
      ,\,
      A
    \Big]
    +
    \tfrac{1}{2}
    \Big[
      A
      ,\,
      \big\{
        \textstyle{\int}
        \omega_i E^i
        ,\,
        A
      \big\}
    \Big]
    \\
    \;=\;
    \big[
      A
      ,\,
      \omega
    \big]
    \,.
  \end{array}
$$

Together this yields the claim.

It may be instructive to state the last computation with more components made explicit:

> (combinatorial prefactors currently not shown properly, but they don't affect the sign, of course)

$$
  \begin{array}{l}
    \Big\{
      \textstyle{\int}
      \omega^i
      E_i
      ,\,
      \tfrac{1}{2}[A, A]
    \Big\}
    \\
    \;=\;
    \textstyle{\int}
    \omega_{i a}
    \epsilon^{a b c}
    \tfrac{1}{2}
    \big\{
      E^i_{b c}
      ,\,
      A^j_{b'}
      A^k_{c'}
    \big\}
    [t_j, t_k]
    \mathrm{d} x^{b'}
      \wedge
    \mathrm{d} x^{c'}
    \\
    \;=\;
    \textstyle{\int}
    \tfrac{1}{2}
    \omega^i_a
    \epsilon^{a b c}
    \big(
      k^{i j}
      \epsilon_{b c b'}
      A^k_{c'}
      +
      A^j_{b'}
      k^{i k} \epsilon_{b c c'}
    \big)
    [t_j, t_k]
    \mathrm{d} x^{b'}
      \wedge
    \mathrm{d} x^{c'}
    \\
    \;=\;
    \tfrac{1}{2}
    \big(
      A^k_{c'}
      \omega^j_{b'}
      +
      A^j_{b'}
      \omega^k_{c'}
    \big)
    [t_j, t_k]
    \mathrm{d} x^{b'}
      \wedge
    \mathrm{d} x^{c'}
    \\
    \;=\;
    A^j_{b'}
    \omega^k_{c'}
    [t_j, t_k]
    \mathrm{d} x^{b'}
      \wedge
    \mathrm{d} x^{c'}
    \\
    \;=\;
    [A, \omega]
    \,.
  \end{array}
$$
\end{proof}

\linebreak

### The sub-phase space of integrated fluxes
 {#ThePoissonBracketOfIntegratedFluxes}

\begin{theorem}
\label{ReducedPhaseSpaceOfIntegratedFluxes}
The [[reduced phase space|reduced]] sub-[[phase space]] of integral fluxes through a given surface $S \hookrightarrow X$ (eq:TheEmbeddedSurfaces) in $\mathfrak{g}$-[[Yang-Mills theory]] is [[isomorphism|isomorphic]] to the [[Fréchet manifold|Fréchet]] [[Lie-Poisson manifold]]

\[
  \label{SemidirectProductLieAlgebraOfMaps}
  C^\infty\bigg(
    S
    ,\,
    \Big(
      \underset{el}{
        \underbrace{
          \big(
            \mathfrak{g}
            ,\,
            [-,-]
          \big)
        }
      }
      \ltimes_{ad}
      \underset{mag}{
        \underbrace{
          \mathfrak{g}
        }
      }
    \Big)^\ast
  \bigg)
\]

given by the Lie algebra of [[smooth maps]] (with pointwise Lie bracket) from $S$ into the [[linear dual space|linear dual]] of the [[semidirect product Lie algebra]] of $\mathfrak{g}$ with its [[underlying]] [[abelian Lie algebra]] via the [[adjoint action]].
\end{theorem}
\begin{proof}
  That the [[underlying]] [[Fréchet manifold]] is as claimed is just a restatement of the form of the linear flux observables in Prop. \ref{TheProperFluxObservables}. We need to check that the Poisson brackets (eq:ThePoissonBracketOnCanonicalCoordinates) of these linear observables is equivalently the Lie bracket of their smearing functions $\alpha_{el}$, $\alpha_{mag}$, regarded as elements of (eq:SemidirectProductLieAlgebraOfMaps).
This is the content of Prop. \ref{PoissonBracketBetweenElectricFluxes} and Prop. \ref{PoissonBracketBetweenElectricAndMagneticFluxes} below.
\end{proof}

\begin{proposition}
\label{PoissonBracketBetweenElectricFluxes}
**([CP17 (7)](#CattaneoPerez17))**
\linebreak
 The Poisson bracket (eq:ThePoissonBracketOnCanonicalCoordinates) of electric flux observables (eq:TheFluxObservables) among each other is the Lie bracket on their smearing functions:
$$
  \Big\{
    \Phi_E^{\alpha}
    ,\,
    \Phi_E^{\beta}
  \Big\}
  \;=\;
  \Phi_E^{ [\alpha,\beta] }
  \,.
$$
\end{proposition}
\begin{proof}
We compute as follows:
\[
  \label{PoissonBracketElectricElectricFluxes}
  \begin{array}{l}
  \Big\{
    \textstyle{\int}
    \big( \mathrm{d}_A \alpha_i \big) E^i
    ,\,
    \textstyle{\int}
    \big( \mathrm{d}_A \beta_j \big) E^j
  \Big\}
  \\
  \;=\;
  \textstyle{\int}
  \Big[
    \big(\mathrm{d}_A \alpha\big)
    ,\,
    \beta
  \Big]_i
  E^i
  -
  \textstyle{\int}
  \Big[
    \big(\mathrm{d}_A \beta\big)
    ,\,
    \alpha
  \Big]_i
  E^i
  \\
  \;=\;
  \textstyle{\int}
  \Big[
    \big(\mathrm{d}_A \alpha\big)
    ,\,
    \beta
  \Big]_i
  E^i
  +
  \textstyle{\int}
  \Big[
    \alpha
    ,\,
    \big(\mathrm{d}_A \beta\big)
  \Big]_i
  E^i
  \\
  \;=\;
  \textstyle{\int}
  \big(
    \mathrm{d}_A
    [\alpha,\beta]_i
  \big)
  E^i
  \mathrlap{\,,}
  \end{array}
\]
where we used, in order of appearance:

1. (eq:PoissonBracketBetweenElectricFieldAndSmearingFunction)

1. (eq:GradedSkewSymemtryOfLieBracketOnLieAlgValuedForms)

1. (eq:CovariantDerivativeIsDerivationOfLieBracket).

\end{proof}

\begin{proposition}
\label{PoissonBracketBetweenElectricAndMagneticFluxes}
 The Poisson bracket (eq:ThePoissonBracketOnCanonicalCoordinates) of an electric flux observable with a magnetic flux observable (eq:TheFluxObservables) is the magnetic flux observable smeared by the Lie bracket of the given smearing functions:
$$
  \Big\{
    \Phi_E^\alpha
    ,\,
    \Phi_B^\beta
  \Big\}
  \;=\;
  \Phi_B^{[\alpha, \beta]}
  \,.
$$
\end{proposition}
\begin{proof}
We compute as follows:
\[
  \label{PoissonBracketElectricMagneticFluxes}
  \begin{array}{l}
  \Big\{
    \textstyle{\int}
    \big( \mathrm{d}_A \alpha_i \big)
    \wedge
    E^i
    ,\,
    \textstyle{\int}
    \big( \mathrm{d}_A \beta_j \big)
    \wedge
    F_A^j
  \Big\}
  \\
  \;=\;
  \textstyle{\int}
  \Big\langle
    \big[
      \mathrm{d}_A \alpha
      ,\,
      \beta_j
    \big]
    ,\,
    F_A^j
  \Big\rangle
  +
  \textstyle{\int}
  \Big\langle
    \big(
      \mathrm{d}_A \beta
    \big)
    ,\,
    \mathrm{d}_A
    \big(
      \mathrm{d}_A \alpha
    \big)
  \Big\rangle
  \\
  \;=\;
  \textstyle{\int}
  \big\langle
    \mathrm{d}_A \alpha
    ,\,
    \beta
    ,\,
    F_A
  \big\rangle
  +
  \textstyle{\int}
  \big\langle
    \mathrm{d}_A \beta
    ,\,
    F_A
    ,\,
    \alpha
  \big\rangle
  \\
  \;=\;
  \textstyle{\int}
  \big\langle
    \mathrm{d}_A \alpha
    ,\,
    \beta
    ,\,
    F_A
  \big\rangle
  +
  \textstyle{\int}
  \big\langle
    \alpha
    ,\,
    \mathrm{d}_A  \beta
    ,\,
    F_A
  \big\rangle
  \\
  \;=\;
  \textstyle{\int}
  \big\langle
    \mathrm{d}_A [\alpha, \beta]
    ,\,
    F_A
  \big\rangle
  \,.
  \end{array}
\]
Here we used, in order of appearance:

1. (eq:PoissonBracketBetweenElectricFieldAndSmearingFunction) and (eq:TheBracketOfElectricFluxWithMagneticFluxDensity);

1. (eq:CyclicInvarianceOfTrilinearPairing) and (eq:CurvatureFromCovariantDerivativeSquared)

1. (eq:CyclicInvarianceOfTrilinearPairing)

1. (eq:CovariantDerivativeIsDerivationOfLieBracket).

\end{proof}

### The algebra of quantum observables on fluxes
 {#AlgebraOfQuantumObservablesOnFluxes}

By Thm. \ref{ReducedPhaseSpaceOfIntegratedFluxes}, the [[quantum observables]] on fluxes in Yang-Mills theory should constitute a [[quantization]] of the [[Lie-Poisson structure|Lie-Poisson]] [[phase space]] (eq:SemidirectProductLieAlgebraOfMaps).

The [[strict deformation quantization|strict]] ([[non-perturbative field theory|non-perturbative]]) [[deformation quantization|deformation]] [[quantization]] of [[phase spaces]] which are [[Lie-Poisson manifolds]] of a [[Lie algebra]] is well-known [[Rieffel 1990](C-star+algebraic+deformation+quantization#Rieffel90); [Landsman 1999, Ex. 2](C-star+algebraic+deformation+quantization#Landsman99); [Landsman & Ramazan 2001, Ex. 11.1](C-star+algebraic+deformation+quantization#LandsmanRamazan01) ]: The [[C-star algebra|$C^\ast$-algebra]] of [[algebra of quantum observables|quantum observables]] is a [[group algebra]] of a corresponding [[Lie group]] (i.e. with the [[convolution]] product), with due care about [[analysis|analytic]] issue.

Hence consider $G$ a [[Lie group]] (not necessarily [[connected topological space|connected]]) with [[Lie algebra]] $\big(\mathfrak{g}, [-,-]\big)$ which serves as the actual [[gauge group]] of our [[Yang-Mills theory]].

Given a [[linear representation]] of $G$ on $\mathfrak{g}$ which on the [[connected component]] $G_{\mathrm{e}} \subset G$ of the [[neutral element]] restricts to the [[adjoint action]] and which preserves a [[lattice (discrete subgroup)|lattice]] $\Lambda \subset \mathfrak{g}$ (possibly empty, generally not maximal-dimensional), we may take the [[Lie integration]] of the [[underlying]] [[abelian Lie algebra]] of $\mathfrak{g}$ to be the [[torus]] $\mathfrak{g}/\Lambda$ equipped with a [[group action]] by $G$.

With these choices, the [[semidirect product group]]

$$
  G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda)
$$

exists and has as [[Lie algebra]] the [[semidirect product Lie algebra]] defining the [[Lie-Poisson structure]] (eq:SemidirectProductLieAlgebraOfMaps).

Therefore a [[non-perturbative quantum field theory|non-perturvative]] [[algebra of quantum observables]] is given by the [[C-star algebra|$C^\ast$-]][[group algebra]], suitably defined, of the group of maps $C^\infty\big(S, G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda)\big)$.

To isolate the topological sector of fluxes, consider the [[connected components]] of this group given by the [[homotopy classes]] of maps

$$
  \pi_0
  \Big(
    C^\infty\big(
      S
      ,\,
      G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda)
    \big)
  \Big)
  \;\simeq\;
  \pi_0
  \bigg(
    Map\Big(
      \esh S
      ,\,
      \esh
      \big(
        G \,\ltimes_{ad}\, (\mathfrak{g}/\Lambda)
      \big)
    \Big)
  \bigg)
$$

This is now a [[discrete group]]. Its [[group algebra]] serves as an algebra of observables on the topological sectors of fluxes.

(...)

\linebreak

## Related concepts

* **[[flux]]**

  * [[flux tube]]

  * [[flux quantization]]

  * [[flux compactification]]

## References

In [[electromagnetism]], with focus on torsion components that are argued to not generally commute:

* {#FreedMooreSegal07a} [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], *The Uncertainty of Fluxes*, Commun. Math. Phys. **271**  (2007) 247-274 [[arXiv:hep-th/0605198](http://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3)]

* {#FreedMooreSegal07b} [[Daniel Freed]], [[Greg Moore]], [[Graeme Segal]], *Heisenberg Groups and Noncommutative Fluxes*,  Annals Phys. **322** (2007) 236-285 [[arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200), [doi:10.1016/j.aop.2006.07.014](https://doi.org/10.1016/j.aop.2006.07.014)]

In [[SU(2)]]-[[gauge theory]] (highlighted for the case of the [[first-order formulation of gravity]] but applying verbatim also to [[Yang-Mills theory]], cf. [there](Yang-Mills+theory#ReferencesPhaseSpaceAndCanonicalQuantization)):

* {#CattaneoPerez17} [[Alberto S. Cattaneo]], Alejandro Perez, *A note on the Poisson bracket of 2d smeared fluxes*, Class. Quant. Grav. **34** (2017) 107001 [[arXiv:1611.08394](https://arxiv.org/abs/1611.08394), [doi:10.1088/1361-6382/aa69b4](https://doi.org/10.1088/1361-6382/aa69b4)]

Proposal of [[experiments]] potentially measuring [[Heisenberg uncertainty relations]] of fluxes:

* [[Alexei Kitaev]], [[Gregory W. Moore]], [[Kevin Walker]], *Noncommuting Flux Sectors in a Tabletop Experiment* [[arXiv:0706.3410](https://arxiv.org/abs/0706.3410)]

See also:

* Mikhail A. Savrov, *Commutator of Electric Charge and Magnetic Flux* [[arXiv:2003.02225](https://arxiv.org/abs/2003.02225)]


uncertainty principle

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[quantum observables]]

* [[commutator]]

* [[uncertainty of fluxes]]

## References

* Wikipedia, _[Uncertainty principle](http://en.wikipedia.org/wiki/Uncertainty_principle)_

[[!redirects uncertainty principles]]

[[!redirects Heisenberg uncertainty principle]]
[[!redirects Heisenberg uncertainty principles]]

[[!redirects Heisenberg uncertainty relation]]
[[!redirects Heisenberg uncertainty relations]]


uncountable ordinal

[[!redirects uncountable ordinals]]

#Contents#
* table of contents
{:toc}

## Idea

An _uncountable ordinal_ is an [[ordinal]] which is not a [[countable ordinal]].

## Examples

* [[first uncountable ordinal]]


under category

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

* table of contents
{: toc}

## Definition

Given a [[category]] $C$ and an [[object]] $c \in C$, the __under category__ (also called __coslice category__) $c \downarrow C$ (alternative notations include $C^{c/}$ and $c/C$ and sometimes, confusingly, $c\backslash C$) is the category whose

* objects are morphisms in $C$ starting at $c$; $c \to d$

* morphisms are commuting triangles
$
 \array{
    && c
    \\
    & \swarrow && \searrow
    \\
    d_1 &&\to && d_2
  }
  \,.
$

The under category $c\downarrow C$ is a kind of [[comma category]]; it is the strict pullback

$$
 \array{
  c \big\downarrow C
  &\to&
  pt
  \\
  \big\downarrow
  &&
  \big\downarrow^{\mathrlap{pt \mapsto c}}
  \\
  [I,C]
  &\stackrel{d_0}{\to}&
  C
 }
$$
in [[Cat]], where

* $I$ is the [[interval category]] $\{0 \to 1\}$;

* $[I,C]$ is the [[closed monoidal category|internal hom]] in [[Cat]], which here is the [[arrow category]] $Arr(C)$;

* the functor $d_0$ is evaluation at the left end of the interval;

* $pt$, the point, is the terminal category, the 0th [[oriental]], the 0-globe;

* the right vertical morphism maps the single object of the point to the object $c$.

The left vertical morphism $c \downarrow C \to C$ is the forgetful morphism which forgets the tip of the triangles mentioned above.

The [[duality|dual]] notion is an [[over category]].

## Examples

\begin{example}
If $0$ is an [[initial object]] in $\mathbf{C}$, then $0\downarrow\mathbf{C}$ is isomorphic to $\mathbf{C}$.
\end{example}

\begin{example}\label{PointedObjects}
**([[pointed objects]])** \linebreak

If $C$ has a [[terminal object]] $\ast \,\in\, C$, then the coslice $C^{\ast/}$ is known as the category of *[[pointed objects]]* in $C$. For instance:

* the category of [[pointed sets]], is the coslice of [[Set]] under the [[singleton]] set,

* the category of [[pointed topological spaces]] is the coslice of [[Top]] under the [[point space]],

* the category of [[pointed simplicial sets]] is the coslice of [[sSet]] under the [[0-simplex]].

If $C$ is a [[monoidal category]] with [[tensor unit]] $I \,\in\, C$, then the coslice $I \downarrow C$ is also known as the category of *[[pointed objects in a monoidal category]]*. For instance:

* the category of [[pointed abelian groups]] is the coslice of [[Ab]] under the additive group of [[integers]],

* the category of [[pointed modules]] is the coslice of [[Mod]] under the additive module of the [[ground ring]]

* the category of [[bi-pointed object|bi-pointed sets]] is the coslice of $Set_*$, the [[category of pointed sets]], under the [[boolean domain]].

* the category of [[pointed endofunctors]] is a coslice of and [[endofunctor|endo]]-[[functor category]] under the [[identity functor]].

Generally, for any $c \in C$ one may think of the $c$-coslice category as the category of "$c$-pointed objects".
\end{example}

\begin{example}
The category of [[commutative algebras]] over a [[field]] $F$ is the coslice under $F$ of the category [[CRing]] of [[commutative rings]].
\end{example}

## Properties

### Limits
 {#LimitsAndColimits}

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

If $C$ is a [[category]] with all [[limits]], then a  [[limit]] in any of its [[under categories]] $t/C$ is computed as a limit in the underlying category $C$.

In detail:

Let $F \colon D \to t/C$ be any [[functor]].

Then, the [[limit]] over $p \circ F$ in $C$ is the image under the evident projection $p \colon t/C \to C$ of the limit over $F$ itself:

$$
  p(\lim F)  \simeq \lim (p F)
$$

and $\lim F$ is uniquely characterized by $\lim (p F)$.

=--

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

Over a morphism $\gamma : d \to d'$ in $D$ the limiting cone over $p F$  (which exists by assumption) looks like

$$
  \array{
    && \lim p F
    \\
    & \swarrow && \searrow
    \\
    p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d')
  }
$$

By the universal property of the limit this has a unique
lift to a cone in the [[under category]] $t/C$ over $F$:

$$
  \array{
    && t
    \\
    & \swarrow &\downarrow & \searrow
    \\
    && \lim p F
    \\
    \downarrow & \swarrow && \searrow & \downarrow
    \\
    p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d')
  }
$$

It therefore remains to show that this is indeed a limiting cone over $F$. Again, this is immediate from the universal property of the limit in $C$. For let $t \to Q$ be another cone over $F$ in $t/C$, then $Q$ is another cone over $p F$ in $C$ and we get in $C$ a universal morphism $Q \to \lim p F$

$$
  \array{
    && t
    \\
    & \swarrow & \downarrow & \searrow
    \\
    && Q
    \\
    \downarrow & \swarrow &\downarrow & \searrow & \downarrow
    \\
    && \lim p F
    \\
    \downarrow & \swarrow && \searrow & \downarrow
    \\
    p F(d) &&\stackrel{p F(\gamma)}{\to}&& p F(d')
  }
$$

A glance at the diagram above shows that
the composite $t \to Q \to \lim p F$ constitutes a morphism
of cones in $C$ into the limiting cone over $p F$. Hence it must equal our morphism $t \to \lim p F$, by the universal property of $\lim p F$, and hence the above diagram does commute as indicated.

This shows that the morphism $Q \to \lim p F$
which was the unique one giving a cone morphism on $C$ does lift to a
cone morphism in $t/C$, which is then necessarily unique, too.  This demonstrates the required universal property of $t \to \lim p F$ and
thus identifies it with $\lim F$.

=--

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

One often says "$p$ [[reflected limit|reflects limits]]" to express the conclusion of this proposition. A conceptual way to consider this result is by appeal to a more general one: if $U: A \to C$ is [[monadic functor|monadic]] (i.e., has a left adjoint $F$ such that the canonical comparison functor $A \to (U F)-Alg$ is an equivalence), then $U$ both reflects and preserves limits. In the present case, the projection $p: A = t/C \to C$ is monadic, is essentially the category of algebras for the monad $T(-) = t + (-)$, at least if $C$ admits binary coproducts. (Added later: the proof is even simpler: if $U: A \to C$ is the underlying functor for the category of algebras of an _endofunctor_ on $C$ (as opposed to algebras of a monad), then $U$ reflects and preserves limits; then apply this to the endofunctor $T$ above.)

=--

## Related concepts

* [[over-category]]

  * [[slice category]]

  * **under category**

  * [[over topos]]

* [[over (∞,1)-category]],

  * [[model structure on an over category]]

  * [[over-(∞,1)-topos]]

## References

Discussion in the generality of [[(infinity,1)-categories|$(\infty,1)$-categories]]:

* [[Jacob Lurie]], Rem. 1.2.5.9 in: *[[Higher Topos Theory]]*, Annals of Mathematics Studies **170**, Princeton University Press (2009) [[pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf)]

[[!redirects under category]]
[[!redirects under-category]]
[[!redirects undercategory]]
[[!redirects coslice category]]
[[!redirects co-slice category]]
[[!redirects under categories]]
[[!redirects under-categories]]
[[!redirects undercategories]]
[[!redirects coslice categories]]
[[!redirects co-slice categories]]
[[!redirects co-slice]]


underlying ordinary category

# Underlying ordinary categories

* table of contents
{: toc}

## Definition

Let $(V,\otimes,I)$ be a [[monoidal category]] and $C$ a $V$-[[enriched category]].  The **underlying ordinary category** of $C$ is the ordinary [[category]] $C_0$ with the same objects and with hom-sets

$$C_0(x,y) = V(I,C(x,y)).$$

This is an instance of [[change of enriching category]].

[[!redirects underlying ordinary categories]]


underlying set

Given a [[concrete category]], that is a [[category]] $C$ equipped with a [[functor]] $U$ from $C$ to the [[category of sets]] (satisfying certain conditions), we call $U$ a [[forgetful functor]] and call $U(x)$, for $x$ an [[object]] of $C$, the __underlying set__ of $x$.

In the case where $C$ is explicitly a category of [[structured sets]], then every object $x$ of $C$ is a [[set]] ${|x|}$ equipped with some [[extra structure]].  In that case, the underlying set of $x$ is precisely this set ${|x|}$.

More generally, if $C$ is concrete over $D$, so we have $U\colon C \to D$ faithful, then given any object $x\colon C$, its __underlying object__ of $D$ is $U(x)$.

[[!redirects underlying set]]
[[!redirects underlying sets]]

[[!redirects underlying object]]
[[!redirects underlying objects]]

[[!redirects underlying]]


underlying topological space

In [[topology]], the notion of [[topological space]] is generally taken as basic, and other related structures (such as a [[metric]], a [[uniform structure]], etc) are typically seen as [[extra structure]] on that space.  Then it\'s natural to speak of the [[underlying]] topological space of a given metric space etc.

One can take other perspectives.  For example, topological spaces, uniform spaces, and metric spaces can all be viewed as [[quasigauge spaces]] with [[extra property]], and then there is almost no overlap between these.  However, [[Top]] is a [[reflective subcategory]] of [[QGau]], and the underlying topological space of a metric or uniform space is the reflection in $Top$ of its underlying quasigauge space.

In any case, given any notion of [[space]] $X$ with an [[underlying set]] ${|X|}$ of [[points]] and a [[frame]] of [[open subsets]] (i.e., a [[subframe]] $\mathcal{O}X$ of the [[power set]] $\mathcal{P}{|X|}$), we do obtain a topological space $({|X|},\mathcal{O}X)$ that may be said to underlie $X$.

[[!redirects underlying topological space]]
[[!redirects underlying topological spaces]]

[[!redirects underlying topology]]
[[!redirects underlying topologies]]


underlying type of free infinity-group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[dependent type theory]] with [[uniqueness of identity proofs]], the [[free group]] on a type $A$ is the ([[higher inductive type|higher]]) [[inductive type]] generated by an element $\mathrm{unit}:\mathrm{FreeGroup}(A)$ and a function $\mathrm{gen}:A \to (\mathrm{FreeGroup}(A) \simeq \mathrm{FreeGroup}(A))$. However, in dependent type theory without [[uniqueness of identity proofs]], the resulting type is no longer a [[free group]], because the resulting type is no longer [[0-truncated]]. Instead, it is the underlying type of the free [[infinity-group|$\infty$-group]] on a [[type]] $A$, and denoted as  $\mathrm{UTFIG}(A)$. Nonetheless, the underlying type of the free infinity group on a [[set]] $A$ is a group.

This is the invertible version of the type of [[lists]] $\mathrm{List}(A)$ on a type $A$, which in the context of [[uniqueness of identity proofs]] is the [[free monoid]] on a type $A$, but without [[uniqueness of identity proofs]] is the underlying type of the free [[A-infinity space|$A_\infty$-space]] on a type $A$; i.e. a free [[monoid in a monoidal (infinity,1)-category]], and $\infty$-groups are the invertible versions of $A_\infty$-spaces.

## Definitions

Assuming that [[identification types]], [[equivalence types]] and [[dependent product types]] exist in the type theory, the underlying type of the free infinity-group on a type $A$ of [[generators]] is the ([[higher inductive type|higher]]) [[inductive type]] $\mathrm{UTFIG}(A)$ generated by an element $\mathrm{unit}:\mathrm{UTFIG}(A)$ and a function $\mathrm{gen}:A \to (\mathrm{UTFIG}(A) \simeq \mathrm{UTFIG}(A))$:

Formation rules for the underlying type of the free infinity-group:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{UTFIG} \; \mathrm{type}}$$

Introduction rules for the underlying type of the free infinity-group:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{unit}:\mathrm{UTFIG}(A)} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{gen}:A \to (\mathrm{UTFIG}(A) \simeq \mathrm{UTFIG}(A))}$$

Elimination rules for the underlying type of the free infinity-group:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{unit}:C(\mathrm{unit}) \quad \Gamma \vdash c_\mathrm{gen}:\prod_{a:A} \prod_{x:\mathrm{UTFIG}(A)} C(x) \simeq C(\mathrm{gen}(a)(x)) \\
      \Gamma \vdash g:\mathrm{UTFIG}(A))
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, g):C(g)}$$

\linebreak

Computation rules for the underlying type of the free infinity-group:

* Judgmental computation rules

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{unit}:C(\mathrm{unit}) \quad \Gamma \vdash c_\mathrm{gen}:\prod_{a:A} \prod_{x:\mathrm{UTFIG}(A))} C(x) \simeq C(\mathrm{gen}(a)(x))
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, 0) \equiv c_\mathrm{unit}:C(\mathrm{unit})}$$

\linebreak

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{unit}:C(\mathrm{unit}) \quad \Gamma \vdash c_\mathrm{gen}:\prod_{a:A} \prod_{x:\mathrm{UTFIG}(A))} C(x) \simeq C(\mathrm{gen}(a)(x)) \\
      \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{UTFIG}(A))
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, \mathrm{gen}(b)(g)) \equiv c_\mathrm{gen}(b)(g)(\mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, g)):C(\mathrm{gen}(b)(g))}$$

\linebreak

* Typal computation rules

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{unit}:C(\mathrm{unit}) \quad \Gamma \vdash c_\mathrm{gen}:\prod_{a:A} \prod_{x:\mathrm{UTFIG}(A))} C(x) \simeq C(\mathrm{gen}(a)(x))
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{UTFIG}(A)}^\mathrm{unit}(c_\mathrm{unit}, c_\mathrm{gen}):\mathrm{Id}_{C(\mathrm{unit})}(\mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, 0), c_\mathrm{unit})}$$

\linebreak

$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \\
      \Gamma \vdash c_\mathrm{unit}:C(\mathrm{unit}) \quad \Gamma \vdash c_\mathrm{gen}:\prod_{a:A} \prod_{x:\mathrm{UTFIG}(A))} C(x) \simeq C(\mathrm{gen}(a)(x)) \\
      \Gamma \vdash b:A \quad \Gamma \vdash g:\mathrm{UTFIG}(A))
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{UTFIG}(A)}^\mathrm{gen}(c_\mathrm{unit}, c_\mathrm{gen}, b, g):\mathrm{Id}_{C(\mathrm{gen}(b)(g))}(\mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, \mathrm{gen}(b)(g)), c_\mathrm{gen}(b)(g)(\mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c_\mathrm{unit}, c_\mathrm{gen}, g)))}$$

\linebreak

Uniqueness rules for the underlying type of the free infinity-group:

* Judgmental uniqueness rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{UTFIG}(A)} C(x) \quad \Gamma \vdash g:\mathrm{UTFIG}(A)}{\Gamma \vdash \mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c(\mathrm{unit}), \lambda a:A.\lambda x:\mathrm{UTFIG}(A).c(\mathrm{gen}(a)(x)), g) \equiv c(g):C(g)}$$

* Typal uniqueness rules

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c:\prod_{x:\mathrm{UTFIG}(A)} C(x) \quad \Gamma \vdash g:\mathrm{UTFIG}(A)}{\Gamma \vdash \eta_{\mathrm{UTFIG}(A)}(c, n):\mathrm{Id}_{C(g)}(\mathrm{ind}_{\mathrm{UTFIG}(A)}^C(c(\mathrm{unit}), \lambda a:A.\lambda x:\mathrm{UTFIG}(A).c(\mathrm{gen}(a)(x)), g), c(g))}$$

The elimination, typal computation, and typal uniqueness rules for the underlying type of the free infinity-group state that the underlying type of the free infinity-group satisfy the **dependent universal property of the underlying type of the free infinity-group**. If the dependent type theory also has [[dependent sum types]] and [[product types]], allowing one to define the [[uniqueness quantifier]], the dependent universal property of the underlying type of the free infinity-group could be simplified to the following rule:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:\mathrm{UTFIG}(A) \vdash C(x) \; \mathrm{type} \quad \Gamma \vdash c_\mathrm{unit}:C(\mathrm{unit}) \quad \Gamma \vdash c_\mathrm{gen}:\prod_{a:A} \prod_{x:\mathrm{UTFIG}(A))} C(x) \simeq C(\mathrm{gen}(a)(x))}{\Gamma \vdash \mathrm{up}_\mathbb{Z}^C(c_\mathrm{unit}, c_\mathrm{gen}):\exists!c:\prod_{x:\mathrm{UTFIG}(A))} C(x).\mathrm{Id}_{C(\mathrm{unit})}(c(\mathrm{unit}), c_\mathrm{unit}) \times \prod_{a:A} \prod_{x:\mathrm{UTFIG}(A)} \mathrm{Id}_{C(\mathrm{gen}(a)(x))}(c(\mathrm{gen}(a)(x)), c_\mathrm{gen}(a)(c(x)))}$$

## Properties

### H-space structure on the underlying type of the free infinity-group

\begin{definition}
The binary operation $\mu$ on the underlying type of the free infinity-group is defined by induction on the underlying type of the free infinity-group

Introduction rules for $\mu$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:\mathrm{UTFIG}(A) \vdash \mu(x):\mathrm{UTFIG}(A) \simeq \mathrm{UTFIG}(A)}$$

By [[uncurrying]] the [[equivalence of types]] one gets the binary operation

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, x:\mathrm{UTFIG}(A), y:\mathrm{UTFIG}(A) \vdash \mu(x, y):\mathrm{UTFIG}(A)}$$

Computation rules for $\mu$:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, g:\mathrm{UTFIG}(A) \vdash \beta_{\mathrm{UTFIG}(A)}^{\mu, \mathrm{unit}}(g):\mu(\mathrm{unit})(g) =_{\mathrm{UTFIG}(A)} g}$$

$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, a:A, x:\mathrm{UTFIG}(A), g:\mathrm{UTFIG}(A) \vdash \beta_\mathbb{Z}^{\mu, \mathrm{gen}}(a, x, g):\mu(\mathrm{gen}(a)(x))(g) =_\mathbb{Z} \mathrm{gen}(a)(\mu(x, g))}$$
\end{definition}

\begin{theorem}
The underlying type of the free infinity-group of $A$ is a non-coherent [[H-space]] with respect to the unit $\mathrm{unit}:\mathrm{UTFIG}(A)$ found in the [[introduction rule]] of the underlying type of the free-infinity group and the binary operation $x:\mathrm{UTFIG}(A), y:\mathrm{UTFIG}(A) \vdash \mu(x)(y):\mathrm{UTFIG}(A)$ defined above.
\end{theorem}

\begin{proof}
The left [[homotopy]] comes from the computation laws for the operation $\mu$ expressed above.
$$\lambda x:\mathrm{UTFIG}(A).\beta_{\mathrm{UTFIG}(A)}^{\mu, \mathrm{unit}}(x):\prod_{x:\mathrm{UTFIG}(A)} \mu(\mathrm{unit})(x) =_{\mathrm{UTFIG}(x)} g$$

It remains to construct the right homotopy
\end{proof}

## See also

* [[free group]]

* [[infinity-group]]

* [[list]] (non-invertible version)

* [[A-infinity space]]

* [[integers type]] (underlying type of the free infinity-group on the [[unit type]]

[[!redirects underlying type of the free infinity-group]]
[[!redirects underlying type of the free infinity-group]]


Understanding colimits in Set

#Contents#

* automatic table of contents goes here
{:toc}

## Idea

On this page, we work work through several of the key examples of **[[colimits]]** in the category [[Set]]. This is part of a bigger project: [[Understanding Constructions in Categories]].

Recall that a colimit, or universal cocone, over a [[diagram]] $F:J\to Set$ is a [[cocone]] $T$ over $J$ such that, given any cocone $T'$, there is a unique [[cone function]] from $T$ to $T'$.

## Initial Object

An [[initial object]] is a universal cocone over the empty diagram. In this section, we demonstrate how this leads us to the statement:

+-- {: .standout}
The empty set $\varnothing$ is the [[initial object]] in $Set$.
=--

To demonstrate, first note that a cocone over an empty diagram is just a set and a corresponding cocone function is just a function. Therefore, we are looking for a "universal set" $\bullet$ such that for an other set $C$, there is a unique function

$$f:\bullet\to C.$$

The empty set fills the bill because we have the [[empty function]] from $\varnothing$ to $C$ for all $C$.

Therefore the empty set is an initial object in $Set$.

## Binary Coproducts

Binary [[coproducts]] correspond to disjoint unions in $Set$.

**Under Construction**

## Arbitrary Small Coproducts

arbitrary (but small) [[coproducts]]

## Coequalizers

[[coequalizers]]

## Pushouts

[[pushouts]]

## Cofibred Products

[[cofibered coproduct|cofibred coproducts]]

[[!redirects Understanding Colimits in Set]]


Understanding constructions in categories

This is an experiment. As with any experiment, it may flop.

The idea is to create a series of "Understandings", i.e. tutorial references that work through category-theoretic constructions for various categories. Hopefully, each Understanding will be fairly homogeneous in structure, e.g. we define the category briefly with a reference to its formal page and then proceed to work through specific constructions in gory detail with specific examples.

These pages are to be friendly in nature with a similar spirit as [[John Baez]]' [This Week's Finds in Mathematical Physics](http://math.ucr.edu/home/baez/TWF.html).

Here is a list of pages we hope to get started on:

* [[Understanding limits in Set]]
* [[Understanding colimits in Set]]
* [[Understanding M-Set]]

Here is an initial list of constructions to include (originally designed for [[Set]]):

Under [[limits]]:

* [[terminal object]]
* binary [[products]]
* arbitrary (but small) [[products]]
* [[equalisers]]
* [[pullbacks]]
* [[fibred products]]

Under [[colimits]]:

* [[coterminal object]]
* binary [[coproducts]]
* arbitrary (but small) [[coproducts]]
* [[coequalisers]]
* [[pushouts]]
* [[cofibered coproduct|cofibred coproducts]]

Other constructions:

* [[exponential objects]]
* [[dependent products]]
* [[power objects]]
* probably more

category: reference

[[!redirects Understanding Constructions in Categories]]
[[!redirects Understanding Constructions on Categories]]
[[!redirects Understanding Categories]]


Understanding Constructions in Set

**Warning: This page is under construction. Feedback welcome.**

See [[Understanding Constructions in Categories]]

* [[Understanding Limits in Set]]
* [[Understanding Colimits in Set]]

***

## Other Constructions ##

#### Exponential Objects

[[exponential objects]]

#### Dependent Products

[[dependent products]]

#### Power Objects

[[power objects]]

## Discussion ##

+--{.query}
[[Eric]]: How is this (note: terminal object) the universal cone over the empty diagram?

_Toby_:  It seems to me that this is really a question about terminal objects in general than about terminal objects in $Set$.  A cone over the empty diagram is simply an object, and a morphism of cones over the empty diagram is simply a morphism.  A universal cone over a diagram $J$ is a cone $T$ over $J$ such that, given any cone $C$, there is a unique cone morphism from $C$ to $T$.  So a univeral cone over the empty diagram is an object $T$ such that, given any object $C$, there is a unique morphism from $C$ to $T$.  In other words, a universal cone over the empty diagram is a terminal object.

I don\'t see the point of the last paragraph before this query box.  Already at the end of the previous paragraph, we\'ve proved that $\bullet$ is a terminal object, since there is a unique function (morphism) to $\bullet$ from any set (object) $C$.  It almost looks like you wrote that paragraph by modifying the paragraph that *I* had written in that place, but that paragraph did something different: it proved that ${!}$ was unique.  Apparently, you thought that this was obvious, since you simply added the word 'unique' to the previous paragraph.

Alternatively, if you *want* to keep a paragraph that proves unicity, then you can remove 'unique' and rewrite my original unicity proof in terminology more like yours, as follows:
> Now let ${!}'\colon C \to \bullet$ be any function. Then
> $$ {!}'(z) = * = {!}(z)$$
> for any element $z$ of $C$, so ${!}' = {!}$.

[[Eric]]: Thanks Toby. I think what I'm looking for is a way to understand that a singleton set is the universal cone over the empty diagram. All these items should be seen as special cases of [[limit]]. Unfortunately, I don't understand limit well enough to explain it. In fact, that is one of the reasons to create this page, i.e. so that I can understand limits :)

The preceding paragraph was my attempt to make it look like a limit, but I obviously failed :)

Ideally, this section would show how terminal object is a special case of limit somehow.

=--

[[!redirects Understanding Set]]
[[!redirects Understanding Constructions on Set]]
[[!redirects understanding constructions in Set]]
[[!redirects understanding Set]]
[[!redirects understanding constructions on Set]]


Understanding limits in Set

[[!redirects Understanding Limits in Set]]
#Contents#

* automatic table of contents goes here
{:toc}

## Idea ##

On this page, we work work through several of the key examples of **[[limits]]** in the category [[Set]]. This is part of a bigger project: [[Understanding Constructions in Categories]].

Recall that a limit, or universal cone, over a [[diagram]] $F:J\to Set$ is a [[cone]] $T$ over $J$ such that, given any cone $T'$, there is a unique [[cone function]] from $T'$ to $T$.

## Terminal Object ##

A [[terminal object]] is a universal cone over the empty diagram. In this section, we demonstrate how this leads us to the statement:

+-- {: .standout}
Any [[singleton]] (a one-element set) is a [[terminal object]] in $Set$.
=--

To demonstrate, first note that a cone over an empty diagram is just a set and a corresponding cone function is just a function. Therefore, we are looking for a "universal set" $\bullet$ such that for an other set $C$, there is a unique function

$$f:C\to\bullet.$$

Singletons fill the bill because for any element $c\in C$ we have the unique function defined by

$$f(c) = *.$$

Therefore any singleton is a terminal object in $Set$.

## Product ##

A [[product]] is a universal cone over a discrete diagram. In this section, we demonstrate how this leads us to the statement:

+--{: .standout}
Any [[cartesian product]] is a [[product]] in $Set$.
=--

To demonstrate, first note that a discrete diagram $F:J\to Set$ produces a [[family of sets]] $(A_i)$ with no functions between them. A cone over the discrete diagram consists of a set $T$ and a single component

$$\f_i:T\to A_i$$

for each set $A_i$. Therefore, we are looking for a universal cone $\prod_i A_i$ such that for any other cone $T$, there is a unique cone function

$$f:T\to\prod_i A_i.$$

Cartesian product fills the bill because for any $t\in T$, we have the unique function defined by

$$f(t) = \prod_i f_i(t).$$

This function is unique because with any other function

$$g:T\to\prod_i A_i$$

with $\pi_i\circ g = f_i$, then for any element $t\in T$

$$g(t) = \prod_i f_i(t) = \left(\prod_i f_i\right)(t) = f(t).$$

Therefore $f = g$.

## Equalizer ##

An [[equalizer]] is the universal cone over a parallel diagram
$$\bullet\rightrightarrows\bullet.$$
In this section, we demonstrate how this leads us to the statement:

+--{: .standout}
The [[equalizer]] of two functions is the subset on which both functions coincide.
=--

To demonstrate, first note that a parallel diagram $F:J\to Set$ produces two sets $X,Y$ with two [[parallel morphism|parallel functions]] $f,g:X\to Y$. A cone over the parallel diagram consists of a set $T$ and two components

$$T_X:T\to X\quad\text{and}\quad T_Y:T\to Y.$$

This is the first example we encounter where the diagram contains morphisms so recall that with a cone we also want the the component diagrams to commute, i.e. we want

$$f\circ T_X = T_Y\quad\text{and}\quad g\circ T_X = T_Y.$$

Therefore, we are looking for a universal cone $Eq$ such that for any other cone $T$, there is a unique function

$$f:T\to Eq.$$

Let's first define the set
$$Eq = \left\{x\in X|f(x)=g(x)\right\}$$
with two functions $Eq_X:Eq\to X$ and $Eq_Y:Eq\to Y$ defined by
$$Eq_X(x) = x\quad\text{and}\quad Eq_Y(x) = f(x)$$
and verify that it is indeed a cone. The only thing we need to check is that
$$g\circ Eq_X = Eq_Y$$
which amounts to showing that $f(x) = g(x)$ for all $x\in Eq$, but that is the definition of $Eq$, so we do have a cone.

Now let $\phi,\psi: T\to Eq$ be two cone functions. For any element $t\in T$, we have

$$T_X(t) = Eq_X\circ\phi(t) = Eq_X\circ\psi(t)\implies \phi(t)=\psi(t)$$

so that $\phi = \psi$ and the cone function is unique.

Since every cone function $\phi:T\to Eq$ is unique, it follows that $Eq$ together with $Eq_X:Eq\to X$ and $Eq_Y:Eq\to Y$ is a universal cone, i.e. $Eq$ is an equalizer.

## Pullbacks ##

A [[pullback]] is a universal cone over a [[cospan]]
$$
  \array{
     && \bullet &&&& \bullet
      \\
      &
      && \searrow
       &
       & \swarrow
      &&
     \\

     &&&&
     \bullet
     &&&&
  }.
$$
In this section, we demonstrate how this leads us to the statement:

+--{.standout}
The [[pullback]] of two functions of sets is the set of pairs $(x,y)$ such that $f(x)=g(y)$.
=--

To demonstrate, first note that a cospan $F:J\to Set$ produces three sets $X,Y,Z$ with two functions $f:X\to Z$ and $g:Y\to Z$. A cone over the cospan consists of a set $T$ and three components

$$T_X:T\to X,\quad T_Y:T\to Y,\quad\text{and}\quad T_Z:T\to Z$$

satisfying

$$f\circ T_X = T_Z\quad\text{and}\quad g\circ T_Y = T_Z,$$

which implies, of course,

$$f\circ T_X = g\circ T_Y.$$

Therefore, we are looking for a universal cone $Pb$ such that for any other cone $T$, there is a unique function

$$f:T\to Pb.$$

Let's first define the set
$$Pb = \left\{(x,y)|f(x)=g(y)\right\}$$
with three functions $\pi_X:Pb\to X$, $\pi_Y:Pb\to Y$, and $T_X:Pb\to Z$ defined by
$$\pi_X(x,y) = x,\quad\text{and}\quad \pi_Y(x,y) = y,\quad\text{and}\quad T_Z(x,y) = f(x)$$
and verify that it is indeed a cone. The only thing we need to check is that
$$f\circ\pi_X = g\circ \pi_Y = T_Z$$
which amounts to showing that $f(x) = g(y)$ for all $(x,y)\in Pb$, but that is the definition of $Pb$, so we do have a cone.

Now let $\phi,\psi: T\to Eq$ be two cone functions with
$$\phi(t) = (\phi_X(t),\phi_Y(t)\quad\text{and}\quad\psi(t) = (\psi_X(t),\psi_Y(t)$$
for functions
$$\phi_X,\psi_X:T\to X\quad\text{and}\quad\phi_Y,\psi_Y:T\to Y.$$
For any element $t\in T$, we have
$$T_X(t) = \pi_X\circ\phi(t) = \pi_X\circ\psi(t)\implies \phi_X(t)=\psi_X(t).$$
Similarly
$$T_Y(t) = \pi_Y\circ\phi(t) = \pi_Y\circ\psi(t)\implies \phi_Y(t)=\psi_Y(t)$$
so that $\phi = \psi$ and the cone function is unique.

Since every cone function $\phi:T\to Pb$ is unique, it follows that $Pb$ together with $\pi_X:Pb\to X$, $\pi_Y:Pb\to Y$, and $T_Z:Pb\to Z$ is a universal cone, i.e. $Pb$ is a pullback.

## Fibred Products ##

**Under Construction**
[[fibred products]]


Understanding M-Set

This is a page similar to other "Journal Club" pages where we attempt to understand some concept. In this case it is the category [[M-Set]]. Other Journal Club pages include:

* [[An Exercise in Kantization]]
* [[geometric infinity-function theory|Geometric ∞-Function Theory]]

One paper, in particular, we would like to look at is

>[**The category of M-sets**](http://cat.inist.fr/?aModele=afficheN&cpsidt=13517583)

>EBRAHIMI M. Mehdi and MAHMOUDI M.

>**Abstract**

>A topos is a category which looks and behaves very much like the category of sets, and so it may be thought of as a universe for mathematical discourses. One of the very useful topoi in many branches of mathematics as well as in computer sciences is the topos MSet, of sets with an action of a monoid M on them. It is well known that MSet, being isomorphic to the functor category SetM, is a topos. Here, we explicitly give the ingredients of a topos in MSet and investigate their properties for the working scientists and computer scientists. Among other things, we give some equivalent conditions, such as the left Ore condition, to Ω, the subobject classifier of MSet, being a Stone algera. Also the free and the cofree objects, as well as, limits and colimits are discussed in MSet.

Here are some initial questions we hope to better understand:

* What exactly is $M$-$Set$?
* How is it isomorphic to $Set^M$? (Is the '$M$' here the same as the '$M$' in $M$-$Set$? If not, how are the two senses of $M$ related?)
* How does one calculate finite products in $M$-$Set$/$Set^M$?

As [[Todd Trimble]] said

>We can go on from there to discuss general limits and colimits in $M‑Set$, cartesian closure, the subobject classifier, and so on. This could be fun and a very good learning experience, and there are people right here who know this material rather well.

So let's do it!

[[Todd Trimble]]: Okay!

Off the bat, I'm not seeing a clear connection between $M$-sets and multisets. It sounds like you were hoping that (locally finite) multisets were in some way connected with $\mathbb{N}$-sets, so let's talk a little about the latter. As stated in the abstract, an $\mathbb{N}$-set is a set equipped with an action by the monoid $\mathbb{N}$ (natural numbers $0, 1, 2, \ldots$ with the monoid multiplication taken to be ordinary addition).

A first thing to do is get a very clear picture of what $\mathbb{N}$-sets are like; there are other suggestive words that can be used to describe them. Any ideas?

[[Eric]]: So the [[monoid]] $\mathbb{N}$ has one object $\bullet$, an identity morphism labeled "+0" and a morphism labeled "+1" that generates all the other morphisms, e.g. $+2 = (+1)\circ(+1)$? So we need to understand the [[functor category]] $Set^\mathbb{N}$. Hmm...

So $F(\bullet)$ is one set $X$ and we need to understand what is the function $F(+1)$ on that set. Hmm...

I guess it doesn't really matter what $F(+1)$ is. It can be any function $f:X\to X$. We simply have $F(+n) = f^n$ with $f^0 = Id$. Ok!

_Todd_: Right! Very good. So an $\mathbb{N}$-set is tantamount to a set $X$ equipped with an endofunction $f: X \to X$. You could think of $f$ as giving a discrete-time dynamics on $X$.

>Toby: And also equipped with an element $z\colon 1 \to X$.

And you've sort of answered as well the question about the two senses of $M$. One sense is that $M$ as monoid is a set equipped with a monoid multiplication; the other is to think of $M$ as a one-object category. Here in the nLab we often use $B M$ for the latter, to keep the two senses distinguished. So, to be pedantic, we really ought to say:

$$M‑Set \simeq Set^{B M}$$

although I think I'll just elide over the $B$ and let the context dictate the appropriate sense.

If you'd like to stay focused on multisets and shelve the project of doing some topos theory here, that would be fine. I'll leave it up to you. Of course, now that you've started it, other people might wander by and want to continue this discussion...

If you did want to continue, then I think we could use $\mathbb{N}$-sets as a running example, since a lot (not all, but a lot) of the general phenomena associated with $M$-sets are already illustrated in this case, and we have clear mental pictures for this case. We could go on for instance to understand finite products (pretty easy), and then more general limits and colimits, and move on from there.

* [[Understanding Categories]]

category: reference


undirected object

#Idea#

In a category with [[interval object]] one has for every object $X$ a notion of paths in $X$. (Indeed, the very definition of category with interval object is meant to guarantee that for every object there is its [[fundamental category]] in the [[Trimble n-category|Trimblean sense]]).

The idea is that if for all these paths in $X$ there is a _reverse_ path, then the object $X$ is _undirected_ or _groupoidal_. Otherwise it is _strictly directed_ (not to be confused with a [[directed object]]).

#Definition (tentative)#

Let $V$ be a [[category]] with an [[interval object]] $I$. Recall that there is for every object $X$ of $V$ a canonical morphism $X_0 \to [I,X]$ which embeds the points of $X$, $X_0 := [pt,X]$, as the constant paths into the "object of directed paths" $[I,X]$.

An object $X$ in $V$ is called **undirected** or an **undirected object** with respect to $I$, – or **$I$-undirected** – if this morphism is a weak equivalence:
$$
  (X is undirected) \Leftrightarrow
  ([\pt,X] \stackrel{\simeq}{\to} [I,X])
  \,;
$$
otherwise it is called **strictly directed** or a **strictly directed object** with respect to $I$, or **strictly $I$-directed**.

#Examples#

* $V = $  [[Top]] with its standard [[model category]] structure, and $I = [0,1]$ the standard interval. Then all objects are undirected since the interval is contractible.

* $V = \omega Cat$, [[strict omega-category|strict omega-categories]], equipped with the [[folk model structure]] and $I = \{a \to b\}$ the 1-[[globe]], the first [[oriental]]. Then
  * strict $\omega$-[[omega-groupoid]]s are undirected;
  * in particular the interval $I$ itself is strictly directed, as there is no weak equivalence from $I_0 = I$ to $[I,I] = \{a \to b \to c\}$;
  * in general, the undirected objects should be precisely those for which every $j$-morphism is an $\omega$-[[equivalence]].  These should consist of those $\infty$-[[infinity-groupoid]]s that are strict as $\infty$-categories, or those objects that are weakly equivalent to a strict $\omega$-groupoid.  For this purpose it is essential that the internal-hom for the [[Crans-Gray tensor product]] on strict $\omega$-categories contains _lax_ transformations, modifications, and so on; if it only contained _pseudo_ natural transformations, then the undirected objects would just be those for which every 1-morphism is an equivalence.

* Once we have a closed monoidal homotopical structure on the category $dTop$ of [[directed space|directed topological spaces]], it should be true that in $V = dTop$ with $I = I_d$ the standard directed interval, an object $(X, d X)$ is undirected precisely if it contains no nontrivial directed path.

#Remarks#

Whether or not an object is undirected depends on the choice of interval object.

* For instance the terminal object $pt$ itself is always a an interval object, $I = pt$, but in a trivial way. Every object is $pt$-undirected.

* A bit more generally, the interval object may be not equal to the terminal object, but still be _weakly equivalent_ to it.  For instance the standard (undirected) topological  interval $[0,1]$ is not equal to but is weakly equivalent to the point in the standard  [[model structure on topological spaces]]. This implies in particular that with  respect to the standard undirected topological interval even a [[directed space|directed topological space]] should undirected. (Well, we still need to specify the closed structure on $dTop$ to make this true...) It's only the standard _directed_ topological interval which can detect the directedness of a strictly directed topological space.

* Analogous statements are true in the example of [[strict  omega-category|strict omega-categories]], where the analogue of the standard directed topological interval is the 1-[[globe]] $I_{dir} = \{a \to b\}$, while the example of the standard undirected topological interval is the [[groupoid]] version of this, $I_{inv} = \{a \stackrel{\simeq}{\to} b\}$, where the morphism from $a$ to $b$ is an [[isomorphism]]. As opposed to $I_{dir}$ this $I_{inv}$ is weakly equivalent to the terminal category $pt = \{\bullet\}$ (the 0-[[globe]]) (with respect to the [[folk model structure]]). It should be true that every strict $\omega$-category is $I_{inv}$-undirected, even if it is strictly $I_{dir}$-directed.


uniform convergence

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

#Contents#
* table of contents
{:toc}

## Definition

In [[analysis]], _uniform convergence_ refers to a type of [[convergence]] of [[sequences]] $(f_n)_{n \in \mathbb{N}}$ of [[functions]] $f_n \colon X \to \mathbb{R}$  into the [[real numbers]]. In terms of [[epsilontic analysis]] such a sequence converges uniformly to some function $f \colon X \to \mathbb{R}$ if for each [[positive real number]] $\epsilon \gt 0$ there exists a [[natural number]] $N_\epsilon \in \mathbb{N}$ such that if $n \geq N(\epsilon)$ then for _all_ points $x \in X$ the difference (in [[absolute value]]) between the value of $f_n$ at that point and that of $f$ at that point is smaller than $\epsilon$:

$$
  \underset{\epsilon \gt 0}{\forall}
   \;
  \underset{N(\epsilon) \in \mathbb{N}}{\exists}
   \;
  \underset{n \geq N(\epsilon)}{\forall}
   \;
  \underset{x \in X}{\forall}
   \;
  \,
  \vert f_n(x) - f(x) \vert \lt \epsilon
  \,.
$$

What is _uniform_ about this convergence is that the bound $N(\epsilon)$ is required to work for all $x \in X$ (hence uniformly over $x$). This is in contrast to [[pointwise convergence]] where one allows a different bound $N$ to exist for each $\epsilon$ and each point $x \in X$ separately. Since for non-[[finite set|finite]] $X$ the [[maximum]] of all such local choices of $N$ in general does not exist, uniform convergence is a stronger condition than [[pointwise convergence]].

## Examples

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

Let

1. $X$ be a [[set]];

1. $Y$ a [[complete space|complete]] [[metric space]].

Consider the set $F(X,Y)$ of [[functions]] $X \to Y$ as a [[metric space]] via the [[supremum norm]]. Then this is again [[complete space|complete]]: every [[Cauchy sequence]] of functions converges uniformly.

If $X$ is equipped with the structure of a [[topological space]] and if the Cauchy sequence of functions consist of [[continuous functions]], then also the [[limit of a sequence|limit]] function is continuous.

=--

(e.g. [Gamelin-Greene 83, theorem I 2.5 and II 3.5](#GamelnGreene83))

## Related concepts

* [[pointwise convergence]]

* [[uniformly continuous map]]

* [[uniform property]]

* [[topology of uniform convergence]]

## References

* Wikipedia, _[Uniform convergence](https://en.wikipedia.org/wiki/Uniform_convergence)_

## References

* {#GamelinGreene83} Theodore Gamelin, Robert Greene, _Introduction to Topology_, Dover (1983, 199)