paracompact topological space

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

# Contents
* table of contents#
{: toc}

## Definition

Recall:

+-- {: .num_defn #LocallyFiniteCover}
###### Definition
**([[locally finite cover]])**

Let $(X,\tau)$ be a [[topological space]].

An [[open cover]] $\{U_i \subset X\}_{i \in I}$ of $X$ is called  _locally finite_ if for each point $x \in X$, there exists a  [[neighbourhood]] $U_x \supset \{x\}$ such that it [[intersection|intersects]] only finitely many elements of the cover, hence such that  $U_x \cap U_i \neq \emptyset$ for only a [[finite number]] of $i \in I$.

=--

+-- {: .num_defn #RefinementOfOpenCovers}
###### Definition
**([[refinement]] of [[open covers]])

Let $(X,\tau)$ be a [[topological space]], and let $\{U_i \subset X\}_{i \in I}$ be a [[open cover]].

Then a _[[refinement]]_ of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an [[open cover]] in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

=--

Now:

+-- {: .num_defn #ParacompactSpace}
###### Definition
**(paracompact topological space)**

A [[topological space]] $(X,\tau)$ is called **paracompact** if every [[open cover]] of $X$ has a [[refinement]] (def. \ref{RefinementOfOpenCovers}) by a [[locally finite open cover]] (def. \ref{LocallyFiniteCover}).

=--

+-- {: .num_remark #DifferingTerminology}
###### Remark
**(differing terminology)**

As with the concept of [[compact topological spaces]] ([this remark](compact#space#DifferingTerminology)), some authors demand a paracompact space to also be a [[Hausdorff topological space]]. See at *[[paracompact Hausdorff space]]*.

=--

## Examples

+-- {: .num_example}
###### Example

Every [[compact space]] is paracompact.

=--

+-- {: .num_example }
###### Example

Every [[locally connected topological space|locally connected]] [[locally compact topological group|locally compact]] [[topological group]] is paracompact ([this prop.](topological+group#ConnectedLocallyCompactTopologicalGroupsAreSigmaCompact)).

=--

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

[[locally compact space|locally compact]] and [[second-countable space|second-countable]] [[Hausdorff space]] are paracompact.

=--

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

[[locally compact and sigma-compact spaces are paracompact]]

=--

+-- {: .num_example #ParacompactEuclideanSpace}
###### Example
**([[Euclidean space]] is paracompact)**

For $n \in \mathbb{N}$, then the [[Euclidean space]] $\mathbb{R}^n$,
regarded with its [[metric topology]]
is paracompact.

=--

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

Euclidean space is evidently [[locally compact topological space|locally compact]] and [[sigma-compact topological space|sigma-compact]].
Therefore the statement follows since [[locally compact and sigma-compact spaces are paracompact]] (prop. \ref{ParacompactFromLocallyCompactAndSigmacompact}).

=--

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

Paracompactness is preserved by forming [[disjoint union spaces]] ([[coproducts]] in [[Top]]).

=--

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

Consider a disjoint union $X = \coprod X_\lambda$ whose components are paracompact.  As the union is disjoint, the components, that is to say, the $X_\lambda$, are open in $X$.  Thus any open cover, say $\mathcal{U}$, of $X$ has a refinement by open sets, say $\mathcal{V}$, such that each $V \in \mathcal{V}$ is contained in some $X_\lambda$.  Thus we can write $\mathcal{V} = \coprod \mathcal{V}_\lambda$.  As each $X_\lambda$ is paracompact, each $\mathcal{V}_\lambda$ has a locally finite refinement, say $\mathcal{W}_\lambda$.  Then let $\mathcal{W} := \coprod \mathcal{W}_\lambda$.  As each $\mathcal{W}_\lambda$ is a refinement of the corresponding $\mathcal{V}_\lambda$, $\mathcal{W}$ is a refinement of $\mathcal{V}$, and hence of $\mathcal{U}$.  As each point of $X$ has a neighbourhood which meets only elements of _one_ of the $\mathcal{W}_\lambda$, and as that $\mathcal{W}_\lambda$ is locally finite, $\mathcal{W}$ is locally finite.  Thus $\mathcal{U}$ has a locally finite refinement.

=--

* [[manifolds]]

  *  finite-dimensional manifolds are locally compact, so we have the results above, but we also have some converses:

     * a finite-dimensional Hausdorff topological [[manifold]] is paracompact precisely if it is [[metric space|metrizable]]

     * a finite-dimensional Hausdorff topological manifold is paracompact precisely if each component is second-countable

   * [[infinite-dimensional manifolds]] are generally not locally compact, but we still have some results:

     * The [[Frechet manifold|Frechet]] [[smooth loop space]] of a compact finite dimensional manifold is paracompact.

     * More generally, if $E$ is the sequential [[limit]] of separable Hilbert spaces $H_n$, such that the canonical projections
       $$
         p_n : E \to H_n
       $$
       satisfy
       $$
         closure(p_n^{-1}(B)) = p_n^{-1}(closure(B))
       $$

       for any open ball $B$ in $H_n$, then $E$ is paracompact, and furthermore admits _smooth_ partitions of unity.

       ( [Brylinski, section I.4](#Brylinski))

* [[CW-complexes are paracompact Hausdorff spaces]] ([Miyazaki 52](#Miyazaki52)), see for instance [Hatcher, appendix of section 1.2](#Hatcher). For a discussion that highlights which [[axiom of choice|choice principles]] are involved, see ([Fritsch-Piccinini 90, Theorem 1.3.5 (p. 29 and following)](#FP)).

* metric spaces

  * every [[separable space|separable]] [[metric space]] is paracompact;

  * every [[metric space]] whatsoever is paracompact, assuming the [[axiom of choice]]; see at _[[metric spaces are paracompact]]_

  * pseudometric spaces are paracompact under the same conditions, if one does not require Hausdorffness;

* In particular we have the following implications

  * [[second-countable space]] and regular Hausdorff space
    $\Rightarrow$ [[metric space|metrizable space]]
    $\Rightarrow$ paracompact space

    (the first is Urysohn's metrization theorem, the second is due to [Stone 48](#Stone48), see also at _[[second-countable regular spaces are paracompact]]_ and _[[metric spaces are paracompact]]_)

  * paracompact space and locally metric space
    $\Rightarrow$
    [[metric space|metrizable space]]

    (this is due to Smirnov)

* special cases

  * the [[Sorgenfrey line]] is a good example of a paracompact space that doesn\'t fit into other general classes of paracompact spaces (in particular, it is not [[metrisable topological space|metrisable]], locally compact, or a manifold);

* counterexamples

  * the [[long line]] is *not* paracompact, even though it is a [[manifold]] (unless one specifically requires paracompactness of manifolds) but it fails to be [[second-countable space|second-countable]] (even though it is connected) or metrisable.

  * the [[Sorgenfrey plane]] (a product of two [[Sorgenfrey lines]]) is not paracompact. This shows that the product of paracompact spaces need not be paracompact.

## Properties

### General
 {#PropertiesGeneral}

\begin{proposition}\label{ClosedSubspacesOfParacompactsAreParacompact}
A [[closed subspace]] of a [[paracompact topological space]] is itself paracompact.
\end{proposition}
(e.g. [here](https://www.uio.no/studier/emner/matnat/math/MAT4520/v17/beskjeder/para.pdf#page=1));

* __Dieudonne's theorem__: [[paracompact Hausdorff spaces are normal]]

* every paracompact [[finite number|finite]]-[[dimension of a manifold|dimensional]] [[topological manifold]] has a [[partition of unity]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

  Care should be taken as to in which [[category]] one considers [[partitions of unity]] on paracompact spaces: For example, analytic [[partitions of unity]] generally do not exist on ([[finite number|finite]] -[[dimension of a manifold|dimensional]]) [[smooth manifolds]], even when smooth ones do.

* For paracompact Hausdorff spaces, all [[open covers]] are [[numerable open cover|numerable open covers]].

* [[Michael's theorems]]

+-- {: .num_example #CountableCoverOfUnionsofOpenSubsetsInsideGivenCover}
###### Lemma

Let $X$ be a [[paracompact Hausdorff space]], and let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]]. Then there exists a [[countable cover]]

$$
  \{V_n \subset X\}_{n \in \mathbb{N}}
$$

such that each element $V_n$ is a [[union]] of [[open subsets]] of $X$ each of which is contained in at least one of the elements $U_i$ of the original cover.

=--

(e.g. [Hatcher, lemma 1.21](#Hatcher))

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

Let $\{f_i \colon  X \to [0,1]\}_{i \in I}$ be a [[partition of unity]] subordinate to the original cover, which exists since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]].

For $J \subset I$ a [[finite set]], let

$$
  V_J
    \;\coloneqq\;
  \left\{
    x \in X
    \;\vert\;
    \underset{j \in J}{\forall}
    \left(
       \underset{k \in I \setminus J}{\forall}
       \left(
         f_j(x) \gt f_k(x)
       \right)
    \right)
  \right\}
  \,.
$$

By local finiteness there are only a [[finite number]] of $f_k(x)$ greater than zero, hence the condition on the right is a finite number of strict inequalities. Since the $f_i$ are continuous, this implies that $V_J$ is an [[open subset]].

Moreover, $V_J$ is contained in $supp(f_j)$ for $j \in J$ and hence in one of the $U_i$.

Now for $n \in \mathbb{N}$ take

$$
  V_n
    \;\coloneqq\;
  \underset{ {J \subset I} \atop { {\vert J\vert} = n } }{\cup}
  V_J
$$

to be the union of the $V_J$ over all subset $J$ with precisely $n$ elements.

The set $\{V_n \subset X\}_{n \in \mathbb{N}}$ is a cover because for any $x \in X$ we have $x \in V_{J_x}$ for

$$
 J_x \coloneqq \{ i \in I \;\vert\; f_i(x) \gt 0  \}
$$

(which is finite by local finitness of the partition of unity).

=--

### Colimits

See at _[[colimits of paracompact Hausdorff spaces]]_.

### Homotopy and Cohomology
 {#HomotopyAndCohomology}

* On paracompact spaces, abelian [[Čech cohomology]] does compute [[abelian sheaf cohomology]],

  i.e. the canonical morphism $\check{H}(X,A) \to H(X,A)$ for  $A$ any [[chain complex]] of [[sheaf|sheaves]] is an [[isomorphism]] when the [[topological space]] underlying $X$ is paracompact.

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

On a paracompact space $X$, every [[hypercover]] of finite height is refined by the [[Čech nerve]] of an ordinary [[open cover]].

=--

For a hypercover of height $n \in \mathbb{N}$, this follows by intersecting the open covers that are produced by the following lemma for $0 \leq k \leq n$.

+-- {: .num_lemma }
###### Lemma

For $X$ a paracompact topological space, let $\{U_\alpha\}_{\alpha \in A}$ be an [[open cover]], and let each $(k+1)$-fold intersection $U_{\alpha_0, \cdots, \alpha_{k}}$ be equipped itself with an open cover $\{V^{\alpha_0, \cdots, \alpha_k}_{\beta}\}$.

Then there exists a refinement $\{U'_{\alpha'}\}$ of the original cover, such that each $(k+1)$-fold intersection $U'_{\alpha'_0, \cdots, \alpha'_k}$ for all indices distinct is contained in one of the $V_\beta$.

=--

This appears as ([[Higher Topos Theory|HTT, lemma 7.2.3.5]]).

## Related concepts

* [[countably paracompact topological space]]

* [[compact topological space]], [[countably compact topological space]], [[locally compact topological space]], [[strongly compact topological space]], [[sequentially compact topological space]]

## References

The notion of paracompact space was introduced in

* [[Jean Dieudonné]], _Une généralisation des espaces compacts_, Journal de Mathématiques Pures et Appliquées, Neuvième Série, 23: 65–76 (1944)

That [[fully normal spaces are equivalently paracompact]] is due to

* {#Stone48} A. H. Stone, _Paracompactness and product spaces_, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. ([Euclid](http://projecteuclid.org/euclid.bams/1183512390))

General accounts include

* R. Engelking, _General topology_, chapter 5 is dedicated to paracompact spaces

* [[Brian Conrad]], _Paracompactness and local compactness_, [pdf](http://math.stanford.edu/~conrad/diffgeomPage/handouts/paracompact.pdf)

* D. K. Burke, _Covering properties_, in: [[K. Kunen]], J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland (1984) Ch. 9, 347–422

* {#Hatcher} [[Alan Hatcher]], section 1.2 of _Vector bundles & K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* {#Junnia07} Heikki Junnila, pp. 73 in: *A second course in general topology* (2007) [[[Junnila-ParacompactSpaces.pdf:file]]]

*  English Wikipedia: [paracompact space](http://en.wikipedia.org/wiki/Paracompact_space)

* Springer eom: [paracompact space](http://eom.springer.de/p/p071300.htm), [paracompactness criteria](http://eom.springer.de/p/p071310.htm)

A basic discussion with an eye towards [[abelian sheaf cohomology]] and abelian [[Čech cohomology]] is around page 32 of

* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes geoemetric quantization_
{#Brylinski}

* {#FP} [[Rudolf Fritsch]], Renzo A. Piccinini, _Cellular structures in topology_, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). ([pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf))

Discussion of paracompactness of [[CW-complexes]] includes

* {#Miyazaki52} Hiroshi Miyazaki, _The paracompactness of CW-complexes_, Tohoku Math. J. (2) Volume 4, Number 3 (1952), 309-313. 1952 [Euclid](https://projecteuclid.org/euclid.tmj/1178245380)

[[!redirects paracompact]]
[[!redirects paracompactness]]
[[!redirects Paracompact space]]
[[!redirects Paracompact spaces]]
[[!redirects paracompact space]]
[[!redirects paracompact spaces]]
[[!redirects paracompact topological space]]
[[!redirects paracompact topological spaces]]
[[!redirects paracompactum]]
[[!redirects paracompactums]]
[[!redirects paracompacta]]

[[!redirects Dieudonne's theorem]]
[[!redirects Dieudonne\'s theorem]]
[[!redirects Dieudonne's theorem]]
[[!redirects Dieudonne theorem]]
[[!redirects Dieudonné's theorem]]
[[!redirects Dieudonné\'s theorem]]
[[!redirects Dieudonné's theorem]]
[[!redirects Dieudonné theorem]]


paraconsistent arithmetic

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **paraconsistent arithmetic** is a [[model]] of [[arithmetic]] in [[paraconsistent mathematics]], mostly using [[first order logic|first-order]] [[paraconsistent logic]]. Also called **inconsistent arithmetic**.

## See also

* [[paraconsistent mathematics]]

* [[finite mathematics]]

## References

* Manuel Bremer, *Inconsistent Mathematics*, ([slides](https://www.mbph.de/Logic/Para/InconsistentMathematics.pdf))

[[!redirects paraconsistent arithmetics]]
[[!redirects inconsistent arithmetic]]
[[!redirects inconsistent arithmetics]]


paraconsistent logic

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

# Paraconsistent logic
* table of contents
{: toc}

## Overview

A **paraconsistent logic** is a [[logic]] in which it can happen that a [[contradiction]] is true, in the sense that both $A$ and $\neg A$ hold for some [[proposition]] $A$, without the logic becoming *trivial* in the sense that *all* propositions are true.  Put differently, a paraconsistent logic is one in which the schema _ex contradictione quodlibet_:
$$A,\, \neg{A} \;\vdash\; B $$
(for all formulas $A$ and $B$) does not hold.  Logics which are not paraconsistent are sometimes said to be **explosive**: a single contradiction explodes the entire system into triviality.

Two related schemata are _ex falso quodlibet_:
$$\bot\; \vdash\; B$$
and the _law of non-contradiction_
$$A,\, \neg{A} \;\vdash\; \bot. $$
Either of these may hold in a paraconsistent logic, but of course (assuming transitivity of entailment) not both.

Another pair of schemata which cannot both hold in a paraconsistent logic are _additivity_
$$ A \;\vdash\; A \vee B $$
and the _disjunctive syllogism_:
$$ \neg A ,\, A \vee B \;\vdash\; B. $$
The fact that the two of these together imply _ex contradictione quodlibet_ is known the _Lewis independent argument_.

## Examples

Here we survey some paraconsistent logics.  In-depth development of particular paraconsistent logics should be done on separate individual pages.

### Aristotelian syllogistic
 {#AristotelianSyllogistic}

Paraconsistent logic may seem bizarre to a modern mathematician, but the universal adoption of _ex contradictione quodlibet_ is actually quite recent, dating roughly to the invention in the 19th century of what is now called (clearly inappropriately) [[classical logic]].

One of the oldest formalizations of logic is Aristotelian syllogistic, which is not as expressive as modern predicate logic, but could be regarded as paraconsistent.  A syllogistic analogue of _ex contradictione quodlibet_ would be a "syllogism" such as

* All men are mortal.
* Some men are not mortal.
* Therefore, some pigs can fly.

But this is not one of the valid forms of syllogism.

### Minimal logic

In [[minimal logic]], one simply omits _ex falso quodlibet_ from any of the usual presentations of intuitionistic logic.  The law of non-contradiction remains valid.  Minimal logic satisfies the law
$$ A,\, \neg A \;\vdash\; \neg B $$
for any $A$ and $B$.  This is a weaker form of _ex contradictione quodlibet_, which is still strong enough to be undesirable if one wants to do serious things with a paraconsistent logic.

### Dual intuitionistic logic

In a certain sense, _ex contradictione quodlibet_ is "dual" to the law of [[excluded middle]].  The former asserts that $A \wedge \neg A$ is the least truth value (it implies everything else), while the latter asserts that $A\vee \neg A$ is the greatest truth value (namely "true").

In particular, just as [[intuitionistic logic]] is classical logic without LEM (but with ECQ), the _formal dual of intuitionistic logic_ satisfies LEM but not ECQ.  Specifically, here is the law of non-contradiction which fails: _ex falso quodlibet_ is still valid because it is the dual of the intuitionistically valid sequent $A \vdash \top$.

Dual intuitionistic logic may be presented as a [[sequent calculus]] with the restriction that the left-hand side of any sequent has at most one formula.  It can be modeled by the [[closed subsets]] of any topological space, in which $\wedge$ and $\vee$ are intersection and union, and $\neg$ is the closure of the complement.  Thus $A \vee \neg A$ holds because $A \cup \overline{X\setminus A} = X$, but the law of non-contradiction fails because $A\cap \overline{X\setminus A}$ need not be empty (it is one definition of the [[boundary]] of $A$).

### Linear logic

[[linear logic|Linear logic]] is also paraconsistent, although the analysis becomes complicated because connectives such as $\vee$ and $\wedge$, and even $\top$ and $\bot$, bifurcate into positive and negative linear versions.  With the usual definition of linear negation, we have the following.

* _Ex falso quodlibet_ holds for the positive false, $0\vdash B$, essentially by definition.  But it fails for the negative false $\bot$.

* The law of non-contradiction holds for the negative false, $A,\, \neg A \;\vdash\; \bot$.  Since $\neg A$ is equivalent to $A\multimap \bot$, we can regard this as an instance of [[modus ponens]] for the linear implication.  However, LNC fails for the positive false.

* The additivity property holds for the positive disjunction, $A \vdash A\oplus B$, but fails for the negative disjunction $\parr$.

* The disjunctive syllogism holds for the negative disjunction, $\neg A,\, A \parr B \;\vdash\; B$.  Since $\neg A \parr B$ is equivalent to $A\multimap B$, this can also be regarded as an instance of modus ponens for the linear implication.  However, DS fails for the positive disjunction $\oplus$.

Thus, both ways to deduce _ex contradictione quodlibet_ founder precisely on the finer analysis of connectives provided by linearity.  From a linear point of view, one could say that _ex contradictione quodlibet_ arises from a failure to appreciate that $\oplus$ is different from $\parr$, and that $0$ is different from $\bot$.

Note that linear logic does include an alternative negation which *is* explosive, namely $A\multimap 0$.  However, this "negation" does not satisfy some other desirable properties of negation.  For instance, in "classical" linear logic the linear negation $\neg A = A\multimap\bot$ is involutive, $\neg \neg A \dashv\vdash A$, and satisfies [[excluded middle]] for the negative disjunction, $\vdash A \parr \neg A$, but neither of these hold for the explosive negation $A\multimap 0$.

Also worth noting is that _ex contradictione quodlibet_ can alternatively be stated in terms of a conjuction as $A \wedge \neg A \vdash B$.  In linear logic, of course, the meaning of this depends on whether the conjunction $\wedge$ is positive or negative.  The comma separating multiple formulas in a linear sequent is equivalent to a positive conjunction $\otimes$, so in that case the same analysis above applies.  But for the negative conjunction $\&$, even the "explosive" negation is no longer explosive: $A\& (A\multimap 0)$ doesn't entail $0$, since we can only extract $A$ or $A\multimap 0$ but not both.

### Relevance logics

One goal of [[relevance logic]] is to ensure that no sequent can be valid unless the hypothesis and conclusion have a variable in common.  Thus, _ex contradictione quodlibet_ is invalid in general, but special cases such as $p, \neg{p} \vdash p$ or $p \wedge q, \neg{p \wedge q} \vdash p \vee r$ are valid.  Many relevance logics turn out to also be paraconsistent (although this is not a proof).

### Other paraconsistent logics

(There are many other paraconsistent logics one could discuss...)

## Applications

### Paraconsistent set theory

One potential application of paraconsistent logic would be to give a [[set theory]] in which the [[unrestricted comprehension rule]] is valid, but which is nontrivial.  For instance, if $R = \{ x \mid x\notin x \}$ is the set from [[Russell's paradox]], then a paraconsistent logic could admit that both $R\in R$ and $R\notin R$ without trivializing.

It should be noted, however, that not all paraconsistent logics remain nontrivial in the presence of full comprehension.  If the [[contraction rule]] is permitted (as it is in many [[relevance logics]], then it seems hard to avoid [[Curry's paradox]] in the following form.  For any statement $P$, consider the Curry-Russell set $C = \{ x \mid (x\in x) \to P \}$.  Suppose $C\in C$; then by definition of $C$ we have $(C\in C)\to P$, and so (using $C\in C$ again, by contraction) we have $P$.  Therefore, $(C\in C) \to P$.  But then by definition of $C$ we have $C\in C$, and hence by [[modus ponens]] we have $P$.  Note that this argument does not require ECQ.

## See also

* [[dialetheism]]

* [[paraconsistent mathematics]]

* [[square of opposition]]

## Links

* [English Wikipedia](http://en.wikipedia.org/wiki/Paraconsistent_logic)
* [Stanford Encyclopedia of Philosophy](http://plato.stanford.edu/entries/logic-paraconsistent/)

* Slater, B. H. (1995). "Paraconsistent Logics?". Journal of Philosophical Logic. 24 (4): 451–454. doi:10.1007/BF01048355. S2CID 12125719.

* Béziau, Jean-Yves (2000). "What is Paraconsistent Logic?". In D. Batens; et al. (eds.). Frontiers of Paraconsistent Logic. Baldock: Research Studies Press. pp. 95–111. ISBN 0-86380-253-2.

Paraconsistent logics (among other topic related to inconsistency) are covered in the book

* Chris Mortensen, _Inconsistent Mathematics_, Kluwer Mathematics and Its Applications Series, Vol 312 (1995)

[[!redirects dual intuitionistic logic]]


paraconsistent mathematics

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

#Contents#
* table of contents
{:toc}

## Idea

**Paraconsistent mathematics** is mathematics done with [[paraconsistent logic]]. Also called **inconsistent mathematics**.

## See also

* [[paraconsistent logic]]

* [[paraconsistent arithmetic]]

* [[finite mathematics]]

## References

* Manuel Bremer, *Inconsistent Mathematics*, ([slides](https://www.mbph.de/Logic/Para/InconsistentMathematics.pdf))

* McKubre-Jordens, M. and Weber, Z. (2012). "Real analysis in paraconsistent logic". *Journal of Philosophical Logic* 41 (5):901–922. doi:10.1017/S1755020309990281

* Mortensen, C. (1995). *Inconsistent Mathematics.* Dordrecht: Kluwer. ISBN:0-7923-3186-9

* Weber, Z. (2010). "Transfinite numbers in paraconsistent set theory". *Review of Symbolic Logic* 3 (1):71–92. doi:10.1017/S1755020309990281

See also:

* Wikipedia, [Paraconsistent mathematics](https://en.wikipedia.org/wiki/Paraconsistent_mathematics)

* Stanford Encyclopedia of Philosophy, [Inconsistent Mathematics](https://plato.stanford.edu/entries/mathematics-inconsistent/)

[[!redirects inconsistent mathematics]]


paracyclic object

## Definition

A __paracyclic__ (synonym: __$\mathbf{Z}$-cyclic__)
object in a category $C$ is a [[simplicial object]] $F_\bullet$ together with a sequence of isomorphisms $t_n : F_n \rightarrow F_n$, $n\geq 1$, such that
$$\array{
\partial_i t_n = t_{n-1} \partial_{i-1},\,\, i \gt 0, &
\sigma_i t_n = t_{n+1} \sigma_{i-1},\,\, i \gt0, \\
\partial_0 t_n = \partial_n, & \sigma_0 t_n = t_{n+1}^2 \sigma_n,
}$$
where $\partial_i$ are boundaries, $\sigma_i$ are degeneracies. If $t_n^{n+1} = \mathrm{id}:F_n\to F_n$ then the paracyclic object is [[cyclic object|cyclic]].

For example, a paracyclic object in [[Set]] is a [[paracyclic set]].

## Related concepts

* [[cyclic object]]

[[!redirects paracyclic objects]]


paracyclic set

## Idea

A __paracyclic set__ is a [[paracyclic object]] in the [[Set]].

## Related concepts

* [[cyclic set]]

[[!redirects paracyclic sets]]


paradox

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

# Paradoxes
* table of contents
{: toc}

## Idea

A _paradox_ is an example of reasoning that seems perfectly valid on first sight, but which results in a [[contradiction]] or [[inconsistency]]. For example:

> The following statement is false.

> The previous statement is true.

Either sentence alone seems perfectly reasonable, and could meaningfully be said in another context; but together they form a paradox.  It is not just that the sentences contradict each other (in which case one or both might simply be false); but the *existence* of the sentences contradicts the assumption that they are meaningful at all (at least using [[classical logic]]).

Mathematical paradoxes of this form (such as [[Russell's paradox]]) are resolved by requiring stronger well-formedness rules to admissible sentences, for instance by requiring strict [[type theory|typing]] and/or [[guarded recursion]] which prevents infinite self-reference.

In contrast, paradoxes in [[physics]] may not necessarily be logical inconsistencies and tend to be resolved by rejecting unjustified assumptions and adopting more accurate laws of nature. For instance *[[Zeno's paradox of motion]]* is resolved by recognizing the notion of [[limit of a sequence]]; and the *[[hole paradox]]* is (or was) effectively a misunderstanding of the rules of [[differential geometry]] applied to [[spacetime]].

## Examples

### From logic and foundations

* [[Russell's paradox]]

* [[Curry's paradox]]

* [[Burali-Forti's paradox]]

* [[Cantor's paradox]]

* [[Banach-Tarski paradox]]

### From natural science

In [[physics]]:

* [[Zeno's paradox of motion]]

* [[hole paradox]]

* [[Einstein-Podolsky-Rosen paradox]]

* [[ultraviolet catastrophe]]

* [[black hole information paradox]]

## Related concepts

[[!include mathematical statements --- contents]]

[[!redirects paradox]]
[[!redirects paradoxes]]


parafermion

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

> *The introduction of the $Z_k$ parafermions $[$[1](#ZamolodchikovFateev85)$]$ in the context of statistical models and conformal field theory [2] is perhaps one of the most significant conceptual advances in modern theoretical physics* $[$[DGZ02](#DGZ02)$]$

(...)

> *We caution the reader that the level-$k$ Read-Rezayi wavefunction $[$[10](#ReadRezayi99)$]$ is constructed from a conformal field theory (CFT) known as the $Z_k$ parafermions of Zamolodchikov-Fateev $[$[32](#ZamolodchikovFateev85)$]$ type which is closely related to the $SU(2)_k$ Chern-Simons theory, and which is a different object from the parafermions of the Fradkin-Kadanoff-Fendley type $[$[46](#CFRS00InFact)$]$. To avoid confusion we emphasise at this point that within this paper, all mention of parafermions will refer to Fradkin-Kadanoff-Fendley type.* $[$[CFRS00, p. 1](#CFRS00)$]$

> {#CFRS00InFact} *In fact the Zamolodchikov-Fateev parafermions represent a critical point at the transition into a phase described by Fradkin-Kadanoff-Fendley parafermions.* $[$[CFRS00, p. 9](#CFRS00)$]$

(...)

Related to the [[fractional superstring]].

## References

### General

#### ZF type

Introducing parafermion [[2d CFT]]:

* {#ZamolodchikovFateev85} [[Alexander B. Zamolodchikov]], [[Vladimir A. Fateev]] *Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems*, Sov. Phys. JETP **62** 2 (1985) 215-225 $[$[pdf](http://www.jetp.ras.ru/cgi-bin/dn/e_062_02_0215.pdf), [osti:5929972](https://www.osti.gov/biblio/5929972)$]$

Further discussion:

* {#DGZ02} Xiang-Mao Ding, Mark. D. Gould, Yao-Zhong Zhang, *Twisted Parafermions*, Phys.Lett. B **530** (2002) 197-201 $[$[arXiv:hep-th/0110165](https://arxiv.org/abs/hep-th/0110165)$]$

#### FKF type

* [[Eduardo Fradkin]], [[Leo P. Kadanoff]], *Disorder variables and para-fermions in two-dimensional statistical mechanics*, Nuclear Physics B **170** 1 (1980) 1-15 $[$doi:10.1016/0550-3213(80)90472-1$]$

* [[Paul Fendley]], *Free parafermions*, J. Phys. A **47** (2014) 075001 $[$[arXiv:1310.6049](https://arxiv.org/abs/1310.6049), [doi:10.1088/1751-8113/47/7/075001](https://doi.org/10.1088/1751-8113/47/7/075001)$]$

* [[Paul Fendley]], *Parafermionic edge zero modes in $\mathbb{Z}_n$-invariant spin chains*,  J. Stat. Mech. (2012) P11020 $[$[doi:10.1088/1742-5468/2012/11/P11020](https://iopscience.iop.org/article/10.1088/1742-5468/2012/11/P11020)$]$

### Relation to $SU(2)$-WZW model

#### For ZF parafermions

Relating parafermions  to the [[affine Lie algebra|affine]] [[su(2)]]-[[current algebra]]/[[WZW model]] ([[su(2)-anyons]]):

* T. Jayaraman, [[Kumar S. Narain]], M. H. Sarmadi, *$SU(2)_k$ WZW and parafermion models on the torus*, Nuclear Physics B **343** 2 (1990) 418-449 $[$doi:10.1016/0550-3213(90)90477-U$]$

* [[Daniel Cabra]], *Spinons and parafermions in fermion cosets*, talk at International Seminar dedicated to the memory of D. V. Volkov, Kharkov (January 5-7, 1997) $[$[arXiv:hep-th/9704015](https://arxiv.org/abs/hep-th/9704015)$]$

* {#CFRS00} [[Daniel C. Cabra]], [[Eduardo Fradkin]], G. L. Rossini, F. A. Schaposnik, Section 4 of: *Non-Abelian fractional quantum Hall states and chiral coset conformal field theories*, International Journal of Modern Physics A **15** 30 (2000) 4857-4870 $[$[doi:10.1142/S0217751X00002354](https://doi.org/10.1142/S0217751X00002354), [arXiv:cond-mat/9905192](https://arxiv.org/abs/cond-mat/9905192)$]$

On [[boundary conditions]] ([[BCFT]]/[[D-branes]]) for the [[gauged WZW model]] via parafermions:

* [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], *Geometrical interpretation of D-branes in gauged WZW models*, JHEP 0107:046 (2001) $[$[arXiv:hep-th/0105038](https://arxiv.org/abs/hep-th/0105038), [doi:10.1088/1126-6708/2001/07/046](https://doi.org/10.1088/1126-6708/2001/07/046)$]$

See also:

* Gor Sarkissian, pp. 75 in: *Two-dimensional conformal field theories with defects
and boundaries* [RTN](http://training.hepi.tsu.ge/rtn/activities/theses.shtml) (2016) $[$[pdf](http://training.hepi.tsu.ge/rtn/activities/sources/disser-gor.pdf)$]$

#### For FKF parafermions

* Richard Fern, Johannes Kombe, Steven H. Simon, *How $SU(2)_4$ Anyons are $\mathbb{Z}_3$ Parafermions*, SciPost Phys. **3** 037 (2017) $[$[arXiv:1706.06098](https://arxiv.org/abs/1706.06098), [doi:10.21468/SciPostPhys.3.6.037](https://doi.org/10.21468/SciPostPhys.3.6.037)$]$

### Relation to anyonic topological order

As models for [[su(2)-anyon]] [[wavefunctions]]:

* {#ReadRezayi99} [[Nicholas Read]], [[Edward Rezayi]], *Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level*, Phys. Rev. B **59** (1999) 8084 $[$[doi:10.1103/PhysRevB.59.8084](https://doi.org/10.1103/PhysRevB.59.8084)$]$

* David J. Clarke, [[Jason Alicea]], Kirill Shtengel . *Exotic non-Abelian anyons from conventional fractional quantum Hall states*, Nature Communications **4** 1348 (2013) $[$[doi: 10.1038/ncomms2340 (2013)](https://www.nature.com/articles/ncomms2340)$]$

An [[integrable model]] for $\mathbb{Z}_N$-parafermions:

* {#Tsvelik14a} [[Alexei M. Tsvelik]], *An integrable model with parafermion zero energy modes*, Phys Rev. Lett. **113** 066401 (2014) $[$[arXiv:1404.2840](https://arxiv.org/abs/1404.2840), [doi:10.1103/PhysRevLett.113.066401](https://doi.org/10.1103/PhysRevLett.113.066401)$]$

* {#Tsvelik14b} [[Alexei M. Tsvelik]], *$\mathbf{Z}_N$ parafermion zero modes without Fractional Quantum Hall effect* $[$[arXiv:1407.4002](https://arxiv.org/abs/1407.4002)$]$

* {#Borcherding18} Daniel Borcherding, *Non-Abelian quasi-particles in electronic systems*, Hannover 2018 $[$[doi:10.15488/4280](https://doi.org/10.15488/4280)$]$

Relation to [[topological phases of matter]] with [[topological order]]:

* [[Jason Alicea]], [[Paul Fendley]], *Topological phases with parafermions: theory and blueprints*, Annual Review of Condensed Matter Physics **7** (2016) 119-139 $[$[doi:10.1146/annurev-conmatphys-031115-011336](https://doi.org/10.1146/annurev-conmatphys-031115-011336)$]$

* Fernando Iemini, Christophe Mora, Leonardo Mazza, *Topological phases of parafermions: a model with exactly-solvable ground states*, Phys. Rev. Lett. **118** 170402 (2017) $[$[arXiv;1611.00832](https://arxiv.org/abs/1611.00832), [doi:10.1103/PhysRevLett.118.170402](https://doi.org/10.1103/PhysRevLett.118.170402)$]$

On twisted parafermions:

* Xiang-Mao Ding, Mark Gould, and Yao-Zhong Zhang. *Twisted Parafermions*. (2001). [arXiv:hep-th/0110165](https://arxiv.org/abs/hep-th/0110165)

[[!redirects parafermions]]

[[!redirects para-fermion]]
[[!redirects para-fermions]]


parallel computing

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[computing]]

* [[distributed computing]]

* [[concurrency theory]]

* [[quantum parallelism]], [[quantum computing]]

* [[nondeterministic computing]]

* [[reversible computing]]

## References

See also:

* Wikipedia, *[Parallel computing](https://en.wikipedia.org/wiki/Parallel_computing)*

[[!redirects parallel computation]]
[[!redirects parallel computations]]


parallel morphism > history

< [[parallel morphism]]
[[!redirects parallel morphism -- history]]


parallel morphisms

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

#Contents#
* table of contents
{:toc}

## Idea

Two [[morphisms]] in a category $C$ (or just [[edges]] in a [[directed graph]]) are __parallel__ if they have the same [[source]] and [[target]]. Equivalently a pair of __parallel morphisms__ in $C$ consists of an [[object]] $x$, and object $y$, and two morphisms $f, g: x \to y$.

$$
  \array{
    x
      &
      \underoverset
        {\underset{g}{\longrightarrow}}
        {\overset{f}{\longrightarrow}}
        {}
      &
    y
  }
$$

This may be extended to a family of any number of morphisms, but the morphisms are always compared pairwise to see if they are parallel. Degenerate cases: a family of one parallel morphism is simply a morphism; a family of zero parallel morphisms is simply a pair of objects.

## The walking parallel pair

The above considerations can be formalized in the following definition.

\begin{definition}
The __walking parallel pair__ category $P$
has two objects, 0 and 1,
and two nonidentity arrows, $f,g\colon 0\to 1$.
\end{definition}

Now functors $P\to C$ are pecisely pairs of parallel morphisms.

## Limits and colimits

The [[limit]] of a pair (or family) or morphisms is called their __[[equalizer]]__; the [[colimit]] is their __[[coequalizer]]__. (Of course, these do not always exist.)

## Related concepts

* [[fork]]

[[!include free diagrams -- table]]

[[!redirects parallel morphism]]
[[!redirects parallel morphisms]]
[[!redirects parallel pair of morphisms]]
[[!redirects parallel pairs of morphisms]]

[[!redirects parallel arrow]]
[[!redirects parallel arrows]]
[[!redirects parallel pair of arrows]]
[[!redirects parallel pairs of arrows]]

[[!redirects parallel pair]]
[[!redirects parallel pairs]]

[[!redirects walking parallel pair]]
[[!redirects walking parallel pairs]]

[[!redirects pair of parallel morphisms]]
[[!redirects pairs of parallel morphisms]]

[[!redirects parallel edge]]
[[!redirects parallel edges]]

[[!redirects walking equalizer category]]
[[!redirects walking coequalizer category]]


parallel postulate

## Related concepts

| [[synthetic geometry]]  |
|-------------------------|
| [[Euclidean geometry]]  |
| [[hyperbolic geometry]] |
| [[elliptic geometry]]   |

## References

* Wikipedia, _[Parallel postulate](http://en.wikipedia.org/wiki/Parallel_postulate)_


parallel transport

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A [[connection on a bundle]] $\nabla$ for $\pi : P \to X$ a $G$-[[principal bundle]] encodes data that assigns to each path $\gamma : [0,1] \to X$ a [[homomorphism]]

$$
  tra_\nabla(\gamma) : P_{\gamma(0)} \to P_{\gamma(1)}
$$

between the [[fiber|fibers]] of the bundle, such that this assignment depends well (e.g. smoothly) on the choice of path and is compatible with composition of paths.

This assignment is called the _parallel transport_ of the connection.

### The idea of parallelism

The term "parallel" comes from one of the many equivalent definitions of the notion of [[connection on a bundle]]: the original formulation of [[Ehresmann connection|Ehresmann connections]].

In that formulation, the connection is encoded at each point $p \in P$ in the total space by a decomposition of the [[tangent space]] $T_p P$ as a [[direct sum]] $T_p P \simeq V_p \oplus H_p$ of [[vector space|vector spaces]], such that

* $V_p = \ker \pi_*|_p$ is the [[kernel]] of the projection map that sends vectors in the total space to vectors in base space (this part is fixed by the choice of $p : P \to X$);

* $H_p \subset T_p P$ is a _choice_ of complement, such that this choice varies smoothly over $P$ in an evident sense and is compatible with the $G$-[[action]] on $P$.

The vectors in $V_p$ are called _vertical_ , the vectors in $H_p$ are called _horizontal_ . One may think of this as defining locally in which way the base space sits horizontally in the total space, equivalently as identifying locally a "smoothly varying local trivialization" of $P$.

More precisely, given such a choice of horizontal subspaces, there is for every path $\gamma : [0,1] \to X$ and every choice of lift $\hat \gamma(0) \in P$ of the start point $\gamma(0)$ to the total space of the bundle, a _unique lift_ $\hat \gamma : [0,1] \to P$ of the entire path to the total space:

$$
  \array{
     \hat \gamma(0) &\stackrel{\hat \gamma}{\to}& \hat \gamma(1)
     && \in & P
     \\
     && &&& \downarrow^{\mathrlap{\pi}}
     \\
     \gamma(0) &\stackrel{\gamma}{\to}& \gamma(1)
     && \in & X
  }
  \,
$$

such that $\hat \gamma$ is everywhere _parallel_ (to $X$) in that all its tangent vectors sit in the horizontal subspaces chosen:

$$
  (\partial_\sigma \hat \gamma)(\sigma) \in H_{\gamma(\sigma)} \subset T_{\gamma(\sigma)} P
  \,.
$$

In other words, this means that given a path $\gamma$ down in $X$, we may _transport_ any point $p \in P_{\gamma(0)}$ above its start point _parallely_ (with respect to the notion of parallelism determined by $\nabla$) along $\gamma$, to find a uniquely determined point $tra_\nabla(\gamma)(p) \in P_{\gamma(1)}$ over the endpoint.

### The category-theoretic perspective

The parallel transport-assignment of fiber-homomorphisms to paths

$$
  (x \stackrel{\gamma}{\to} y) \mapsto ( P_x \stackrel{tra_\nabla(\gamma)}{\to} y )
$$

enjoys the following properties:

* it is invariant under [[thin homotopy]] of paths;

* it is compatible with composition of paths and sends constant paths to identity homomorphisms;

* it sends smooth families of paths to compatible smooth families of homomorphisms.

This may be equivalently but more succinctly be formulated as follows:

We say _diffeological groupoid_ for an [[internal groupoid]] in the category of [[diffeological space|diffeological spaces]].

The smooth paths in a smooth manifold $X$ naturally form the diffeological groupoid called the [[path groupoid]] $P_1(X)$.  Objects are points in $X$, morphisms are [[thin homotopy]]-classes of smooth paths which are constant in a neighbourhood of their boundary, composition is concatenation of paths.

For $P \to X$ any $G$-bundle, there is also naturally the diffeological groupoid $At(P)$ -- the [[Atiyah Lie groupoid]]  of $P$. Objects are points in $X$, morphisms are homomorphisms of $G$-[[torsor|torsors]] between the [[fiber|fibers]] over these points.

Then the above properties of parallel transport are equivalent to saying that we have an [[internal functor]]

$$
  tra : P_1(X) \to At(P)
$$

that is the identity on objects. Moreover, this functor _uniquely_ characterizes the [[connection on a bundle|connection]] on $P$ that it comes from. This means that we may identify connections on $P$ with their parallel transport functors.

But even the bundle $P$ itself is encoded in such functors. If instead of looking at the category of internal groupoids and internal functors, we look at the larger [[2-topos]] of _diffeological stacks_  -- [[stack|stacks]] over [[CartSp]].

Then we can take simply the diffeological [[delooping]] groupoid $\mathbf{B}G$, which has a single object and $G$ as its [[hom-set]] and consider morphisms

$$
  tra : P_1(X) \to \mathbf{B}G
$$

in the 2-topos. These are now given by [[anafunctors]] of internal groupoids, and one finds that they encode a [[Cech cohomology|Cech cocycle]] for a $G$-principal bundle $P$ together with the parallel transport of a connection over it.

This is discussed in more detail at

* [[∞-Chern-Weil theory -- preparatory concepts]].

There is also the diffeological groupoid incarnation of the [[fundamental groupoid]] $\Pi_1(X)$ of $X$. Its morphisms are full [[homotopy]]-classes of paths. There is a canonical projection $P_1(X) \to \Pi_1(X)$ that sends a thin-homotopy class of paths to the corresponding full-homotopy class.

A parallel transport functor $tra : P_1(X) \to G$ factors through $\Pi_1(X)$ precisely if the corresponding connection is _flat_ in that its [[curvature form]] vanishes.

### In physics

In [[physics]], a [[connection on a bundle]] over $X$ models a [[gauge field]] such as the [[electromagnetic field]] or more generally a [[Yang-Mills field]] or the field of [[gravity]] on a [[spacetime]] $X$.

The [[force|forces]] exerted by such gauge fields on  charged particles propagating on $X$ (i.e. [[electron|electrons]], [[quark|quarks]] and generally massive particles, respectively) are encoded precisely in the parallel transport assignment of the gauge field connection to their trajectories.

More precisely, the exponentiated [[action functional]] for the electron propagating on $X$ in the presence of an electromagnetic field $\nabla$ is the functional on the space of paths in $X$ given by

$$
  \gamma \mapsto \exp(i S_{kin}(\gamma)) \cdot tra_\nabla(\gamma)
  \,,
$$

where the first term is the standard [[kinetic action]]. If $\nabla$ is a (nontrivial) connection on a trivial bundle, then, as described [below](#Of1Form) it is encoded by a [[differential form]] $A \in \Omega^1(X)$ -- called the _vector potential_  in physics -- and we have

$$
  tra_\nabla(\gamma) = \exp\big(i \int_{[0,1]} \gamma^* A\big)
  \,.
$$

The [[Euler-Lagrange equations]] induced by this functional express precisely the [[Lorentz force]] encoded by $A$ acting on the particle.

If instead of looking at the [[quantum mechanics]] of the quantum particle charged under a fixed background gauge field look at the [[quantum field theory]] of that gauge field itself, we can use the action functional of particles to _probe_ these background fields and obtain quantum observables for them.

This converse assignment where we _fix_ a path $\gamma$ and regard the parallel transport then as a functional over the space of all connections over $X$

$$
  tra_{(-)}(\gamma) : connections \to fiber-homomorphisms
$$

is called the **[[Wilson line]]-observable** of the theory. Or rather its expectation value in the [[path integral]] weighted by the action functional of the gauge theory is called such, schematically:

$$
  W_\gamma := \langle tra_{(\nabla)}(\gamma)\rangle
  =
  \int D \nabla \; \exp(i S_{gauge\;theory}(\nabla))
    tra_\nabla(\gamma)
  \,.
$$

## Special cases

### Trivial bundle: parallel transport of a 1-form {#Of1Form}

Of $P \to X$ is a _trivial bundle_ in that $P = X \times G$, then a connection on this is equivalently encoded in a [[Lie-algebra valued 1-form]]

$$
  A \in \Omega^1(X, \mathcal{g})
$$

on $X$.

In terms of this, parallel transport is a solution to a [[differential equation]].

For $\gamma : [0,1] \to X$ we have the pull-back 1-form $\gamma^* A \in \Omega^1([0,1])$. For $f \in C^\infty([0,1], G)$ a smooth function with values in the [[Lie group]] $G$, consider the differential equation

$$
  d f + \rho(f)_*(\gamma^*A) = 0
  \,,
$$

where $d f : T [0,1] \to T G$ is the differential of $f$ and where $\rho : G \times G \to G$ is the left [[action]] of $G$ on itself (i.e. just the multiplication on $G$) and $r(f)_* : T G \to T G$ its differential and using the defining identification $\mathfrak{g} \simeq T_e G$ we take $r(f)_*(A)$ to be the composite $T [0,1] \stackrel{\gamma^* A}{\to} \mathfrak{g} \hookrightarrow T G \stackrel{r(f)_*}{\to} T G$.

If $G$ is a [[matrix Lie group]] such as the [[orthogonal group]] $O(n)$ or the [[unitary group]] $U(n)$, then also its Lie algebra identifies with matrices, and we may write this simply as

$$
  d f + \gamma^*(A) \cdot f  = 0
  \,,
$$

where the dot is matrix multiplication.

By general results on [[differential equations]], this type of equation has a unique solution for each choice of value of $f(0)$.

+-- {: .un_defn}
###### Definition
The parallel transport of $A \in \Omega^1(X,\mathfrak{g})$ along a path $\gamma : [0,1] \to X$ which we write

$$
  tra_A(\gamma) := P \exp(\int_{[0,1]} \gamma^* A) \in G
$$

is the value $f(1) \in G$ for the unique solution of the equation $d f  + \rho(f)_*(A) = 0$ with initial value $f(0) = e$ (the neutral element in $G$).
=--

The notation here is motivated from the special case where $G = \mathbb{R}$ is the group of [[real number|real numbers]]. In that case the [[Lie algebra]] $\mathfrak{g} \simeq \mathbb{R}$ is abelian, the differential equation above is simply

$$
  d f = \gamma^*(A) \wedge f
$$

for a real valued function $f \in C^\infty([0,1])$, and the unique solution to that with $f(0) = e = 0$ is literally the exponential of the integral of $A$:

$$
  tra_A(\gamma) = \exp(\int_{[0,1]} \gamma^* A)
  \,.
$$

In the case of general nonabelian $\mathfrak{g}$ this simple exponential formula gives the wrong result. One can see that a slightly better approximation to the correct result is given by

$$
  \exp(\int_{[0,1/2]} \gamma^* A) \cdot \exp(\int_{[1/2,1]} \gamma^* A)
$$

and an even a bit more better approximation by

$$
  \exp(\int_{[0,1/3]} \gamma^* A) \cdot \exp(\int_{[1/3,2/3]} \gamma^* A)
   \cdot \exp(\int_{[2/3,1]} \gamma^* A)
$$

and so on, with the correct result being the limit of this sequence -- if one defines it carefully -- as we integrate piecewise over ever smaller pieces of the path.

This is called a **path-ordered integral**. The "P" in the above formula is short for "path ordering". Possibly this notation originates in [[physics]] where the above is known as the [[Dyson formula]].

## Higher parallel transport

The notion of [[connection on a bundle]] generalizes to that of [[connection on a 2-bundle]]. [[connection on a 3-bundle]] and generally to that of [[connection on an ∞-bundle]]. The come with a notion of _higher parallel transport_ over manifolds of dimension greater than 1.

See [[higher parallel transport]] for details.

## Related concepts

* [[connection on a bundle]], [[connection on a 2-bundle]], [[connection on an infinity-bundle]],

* **parallel transport**, [[higher parallel transport]], [[super parallel transport]]

  * [[nonabelian Stokes theorem]]

* [[holonomy]]

  * [[holonomy group]]

  * [[special holonomy]]

* [[identity transport]] in [[homotopy type theory]]

  (for the relation to parallel transport see [there](transport#RelationToParallelTransport))

## References

### General

(...)

### In terms of parallel transport
 {#ReferencesInTermsOfParallelTransport}

There are many equivalent statements of the ordinary definition of a [[connection on a bundle]]. The following lists references related to the statement that the connection is equivalently encoded in terms of its [[parallel transport]].

Apparently one of the oldest occurrences of the idea that a  [[principal bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$ may be reconstructed from its [[holonomy|holonomies]] around all smooth loops for any fixed base point in the connected base space $X$ appears in

* S. Kobayashi, Comptes Rendus, 238, 443-444. (1954). ([web](http://gallica.bnf.fr/ark:/12148/bpt6k31909/f443.item))

A more detailed and more general discussion has then been given in

* [[John Milnor]], *Construction of Universal Bundles, I*,  Annals of Mathematics **63** (1956) 272-284 [[doi:10.2307/1969609](https://doi.org/10.2307/1969609)]

A detailed discussion of the differentiable case appears is

* Costake Teleman  _Généralisation du groupe fondamental_, Annales Scientifiques de l'école Normale Supérieure **3** 77  (1960) 195-234 [[numdam:ASENS_1960_3_77_3_195_0](http://www.numdam.org/item?id=ASENS_1960_3_77_3_195_0)]

* Costake Teleman, Annali di Matematica, Pura ed Applicata, LXII, 379-412. (1963).

In the special case of [[flat connections]] (see also *[[local systems]]*) the statement is classical, a canonical early account being:

* [[Pierre Deligne]], §I.1 of: *Equations différentielles à points singuliers réguliers*, Lecture Notes  Math. **163**, Springer (1970) $[$[publications.ias:355](https://publications.ias.edu/node/355)$]$

This history is recollected in the introduction of:

* [[John Barrett]], _Holonomy and path structures in general relativity and Yang-Mills theory_, Int. J. of Theor. Phys., Iiol. 30, No. 9, 1991 ([doi:10.1007/BF00671007](https://doi.org/10.1007/BF00671007))

who himself gives a proof. Barret implicitly uses the  [[diffeological space]] structure on the space of loops.

Followups:

* [[Piotr Hajac]], _Axiomatic holonomy maps and generalized Yang-Mills moduli space_, Letters in Mathematical Physics volume 27, pages301–309 (1993) ([doi:10.1007/BF00777377](https://doi.org/10.1007/BF00777377))

A note on how a 1-form is encoded in the parallel transport that it induces along paths is also in

* [[Daniel Freed]], [appendix B](http://arxiv.org/PS_cache/hep-th/pdf/9206/9206021v1.pdf#page=56) of: _Classical Chern-Simons theory, Part I_ ([arXiv:hep-th/9206021](http://arxiv.org/abs/hep-th/9206021))

In

* Jerzy Lewandowski _Group of loops, holonomy maps, path bundle and pathconnection_, Class. Quantum Grav. 10 (1993) 879-904 ([pdf](http://ej.iop.org/links/r8XqKR99r/8rbzxFDZ2xGeFw29av5vpA/cq930508.pdf))

the statement of the equivalence is attributed to

* Jeeva Anandan, _Holonomy groups in gavity and gauge fields_, Pmc. Conf Differential Geometric Methods in Plystcs (fieste 1981) ed. G Denardo and H. D. Doebner (Singapore, World Scientific) ([spire:169409](https://inspirehep.net/literature/169409))

Therein it is shown that smoothness of the [[parallel transport]] is a necessary condition for it to come from a smooth bundle with connection. Barrett also shows that this is sufficient.

Lewandowski adds to this a formulation of an equivalence of bundles with connections and the subset of loops around which the corresponding parallel transport is trivial.

Around the same time appeared

* A. Caetano, [[Roger Picken]], _An axiomatic definition of holonomy_, Int. Journ. Math. 5 (1994) 835 ([scan](http://ccdb4fs.kek.jp/cgi-bin/img_index?9405501), [doi:10.1142/S0129167X94000425](https://doi.org/10.1142/S0129167X94000425))

that generalizes these ideas from loops to general paths. These authors introduced the idea of sitting instants of paths and noticed that the most elegant way to (re)state the maximal equivalence relation on paths which is respected by parallel transport is in terms of [[thin homotopy]].

Barrett originally had something very similar but slightly different. With Caetano and Picken's relation, the space of thin homotopy classes of paths in $X$ becomes an groupoid $\mathbf{P}_1(X)$ -- the [[path groupoid]] -- internal to [[diffeological space]].

Discussion in terms of local functorial data that lends itself to generalization to [[higher parallel transport]] is given in

* [[Urs Schreiber]], [[Konrad Waldorf]], _Parallel Transport and Functors_, J. Homotopy Relat. Struct. 4, 187-244 (2009) ([arXiv:0705.0452](https://arxiv.org/abs/0705.0452))

A quick proof that bundles with connections are encoded in their parallel transport along paths was noted in

* [[Florin Dumitrescu]], _Connections and Parallel Transport_, Journal of Homotopy and Related Structures **5** 1 (2010) 171–175 [[arXiv:0903.0121](http://arxiv.org/abs/0903.0121)]

For more see the references at *[[connection on a bundle]]*.

A discussion of parallel transport in the [[tangent bundle]] in terms of [[synthetic differential geometry]] (motivated by a discussion of [[gravity]]) is in

* [[Gonzalo Reyes]], _General Relativity: Affine connections, parallel transport and sprays_ ([pdf](https://marieetgonzalo.files.wordpress.com/2009/01/affineconnections.pdf))


parallelogram identity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
#### Functional analysis
### Contents {: .clickToReveal}
### Contents {: .clickToHide tabindex="0"}
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

# The Parallelogram Identity
* table of contents
{: toc}

## Idea

The **parallelogram identity** is an identity which characterises those [[norms]] which are the norms associated with [[inner products]].  An inner product can be considered as being the structure required to define the angle between two vectors and a norm can be considered as being the structure required to define the length of a vector.  From standard Euclidean geometry, lengths and angles (almost) determine each other so knowing one, we should be able to define the other.

If length is known, angle can be defined by the *cosine law*:

$$
c^2 = a^2 + b^2 - 2 a b \cos C
$$

Although this formula could be used to define "angle" for any length (that is, norm), not every length admits a *sensible* notion of an angle.  There are several ways to describe what the _fundamental properties_ of angles should be.  One is to say that angles should add:

$$
\theta(u,v) + \theta(v,w) = \theta(u,w)
$$

This needs careful interpretation since the angle between two vectors is slightly ambiguous: one can choose the internal angle or the external angle and it is not possible to make a consistent choice.  However, modulo that uncertainty, the above is a reasonable property to insist on.

A special case of this is when $w = -u$.  This leads to the following diagram.

+-- {: style="text-align: center"}
<svg width="356" height="155" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="59338">
 <g>
  <title>Layer 1</title>
  <path id="svg_59338_1" d="m41,145l195,-122l81,122l-276,0z" stroke-width="2" stroke="#000000" fill="none"/>
  <line id="svg_59338_2" y2="145" x2="178" y1="24" x1="236" stroke-width="2" stroke="#000000" fill="none"/>
  <path id="svg_59338_3" d="m197,108c14,10 19,19 21,36" stroke-width="2" stroke="#000000" fill="none"/>
  <path id="svg_59338_4" d="m191,116c-19,-14 -51,7 -51,29" stroke-width="2" stroke="#000000" fill="none"/>
  <foreignObject height="20" width="48" font-size="16" id="svg_59338_5" y="133" x="0">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mo rspace="0em" lspace="verythinmathspace">−</mo>
      <mi>u</mi>
     </mrow>
     <annotation encoding="application/x-tex">-u</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject height="20" width="48" font-size="16" id="svg_59338_6" y="135" x="308">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>u</mi>
     </mrow>
     <annotation encoding="application/x-tex">u</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject height="20" width="48" font-size="16" id="svg_59338_7" y="0" x="211">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>v</mi>
     </mrow>
     <annotation encoding="application/x-tex">v</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject height="20" width="48" font-size="16" id="svg_59338_8" y="107" x="201">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>ϕ</mi>
     </mrow>
     <annotation encoding="application/x-tex">\phi</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject height="20" width="48" font-size="16" id="svg_59338_9" y="96" x="131">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>ψ</mi>
     </mrow>
     <annotation encoding="application/x-tex">\psi</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>
</svg>
=--

The cosine law for this special case leads to two formulae:

$$
\begin{aligned}
{\|u + v\|^2} &= {\|u\|^2} + {\|v\|^2} - 2{\|u\|}{\|v\|} \cos(\psi) \\
{\|u - v\|^2} &= {\|u\|^2} + {\|v\|^2} - 2{\|u\|}{\|v\|} \cos(\phi)
\end{aligned}
$$

By imposing the assumption that angles should add correctly, we deduce that $\psi = \pi - \phi$ and thus $\cos (\psi) = - \cos(\phi)$.  Summing the two lines above leads to the parallelogram identity:

$$
{\|u + v\|^2} + {\|u - v\|^2} = 2 {\|u\|^2} + 2 {\|v\|^2}
$$

If a norm satisfies this identity, then the definition of angle (using the cosine identity) satisfies all the basic properties of angles that one can consider.

## Inner Products

A [[norm]] which satisfies the parallelogram identity is the norm associated with an  [[inner product]].
Using the parallelogram identity, there are three commonly stated equivalent forumlae for the inner product; these are called the [[polarization identities]].  (The notation can vary a little, but it is usually some form of round or angled brackets.)  These formulae hold for vector spaces over $\mathbb{R}$; there are similar formulae for $\mathbb{C}$, but they have more terms.

$$
\begin{aligned}
\langle u, v \rangle &= \frac{1}{2} \left({\|u + v\|^2} - {\|u\|^2} - {\|v\|^2} \right) \\
&= \frac{1}{2} \left({\|u\|^2} + {\|v\|^2} - {\|u - v\|^2}\right) \\
&= \frac{1}{4} \left({\|u + v\|^2} - {\|u - v\|^2}\right)
\end{aligned}
$$

It is possible to show directly that this is an inner product.  Certain properties are easy to deduce directly from the formulae.

+-- {: .num_lemma #lemipprop}
###### Lemma
The function $(u,v) \to \langle u, v\rangle$ satisfies the following properties:

1. $\langle u, u \rangle  \ge 0$
2. $\langle u, u \rangle = 0$ if and only if $u = 0$
3. $\langle u, 0 \rangle = 0$
4. $\langle u, v \rangle = \langle v, u \rangle$
5. $\langle \lambda u, \lambda v \rangle = \lambda^2 \langle u, v \rangle$
6. $\langle u, v + w \rangle = \langle u, v \rangle + \langle u, w\rangle$

=--

+-- {: .proof}
###### Proof
All but the last of these is a simple deduction from the formulae. The last is as well, but takes a little work to get it in the correct form.  We multiply by $4$ to simplify the notation.

$$
\begin{aligned}
4\langle u, v + w \rangle &= {\|u + v + w\|^2} - {\|u - v - w\|^2} \\
&={\left\|\left(\frac{1}{2}u + v\right) + \left(\frac{1}{2}u + w\right)\right\|^2} - {\left\|\left(\frac{1}{2}u - v\right) + \left(\frac{1}{2}u - w \right)\right\|^2} \\
&={\left\|\left(\frac{1}{2}u + v\right) + \left(\frac{1}{2}u + w\right)\right\|^2} + {\left\|\left(\frac{1}{2}u + v\right) - \left(\frac{1}{2}u + w\right)\right\|^2} \\
&\qquad -  {\left\|\left(\frac{1}{2}u - v\right) - \left(\frac{1}{2}u - w\right)\right\|^2} - {\left\|\left(\frac{1}{2}u - v\right) + \left(\frac{1}{2}u - w \right)\right\|^2} \\
&=2{\left\|\frac{1}{2}u + v\right\|^2} + 2{\left\|\frac{1}{2}u + w\right\|^2} -  2{\left\|\frac{1}{2}u - v\right\|^2} - 2{\left\|\frac{1}{2}u - w\right\|^2} \\
&= 2{\left\|\frac{1}{2}u + v\right\|^2}  -  2{\left\|\frac{1}{2}u - v\right\|^2} \\
&\quad+ 2{\left\|\frac{1}{2}u + w\right\|^2} - 2{\left\|\frac{1}{2}u - w\right\|^2}\\
&= 8\langle \frac{1}{2}u,v \rangle + 8\langle \frac{1}{2}u,w\rangle
\end{aligned}
$$

To get it as stated, we then apply this in the special case of $w = 0$ to deduce that $\langle u, v \rangle = 2\langle \frac{1}{2} u, v\rangle$.  Substituting this back in to the above, we obtain the form in the statement.
=--

The properties above were all reasonably straightforward deductions from the definition.  There is one more property that is needed which is a little more complicated.  First, we note a useful result about the continuity of the supposed inner product.

+-- {: .num_lemma #ipcts}
###### Lemma
The map $(u,v) \mapsto \frac{1}{2}({\|u + v\|^2} - {\|u\|^2} - {\|v\|^2})$ is continuous.
=--

+-- {: .proof}
###### Proof
This is a simple consequence of composition of continuous maps.
=--

Using that, we can prove the following property of the proposal for an inner product.

+-- {: .num_proposition #iplinear}
###### Proposition
The map $\langle u,v\rangle$ satisfies the identity

$$
\langle u,\lambda v \rangle = \lambda \langle u, v \rangle
$$

for all vectors $u$, $v$, and real numbers $\lambda$.
=--

+-- {: .proof}
###### Proof
We first use the formula $\langle u, v + w \rangle = \langle u, v \rangle + \langle u, w\rangle$ to prove that for any $n \in \mathbb{N}$

$$
\langle u, n v \rangle = n \langle u, v\rangle.
$$

We prove this by induction.  It is clearly true for the case $n = 1$.  Assume that it holds for $n$, then

$$
\langle u, (n+1)v \rangle = \langle u, n v + v \rangle = \langle u, n v\rangle + \langle u, v \rangle = n \langle u, v \rangle + \langle u, v \rangle = (n + 1)\langle u, v \rangle
$$

whence, by induction, it holds for all $n \in \mathbb{N}$.

Next we observe that it holds for all $n \in \mathbb{Z}$.  It clearly holds for $n = 0$, and for $n \in \mathbb{N}$ then

$$
0 = \langle u, 0 \rangle = \langle u, v - v \rangle = \langle u, v \rangle + \langle u, - v \rangle
$$

whence

$$
\langle u, -v \rangle = - \langle u, v \rangle.
$$

Next, we prove it for $\frac{p}{q} \in \mathbb{Q}$.  In fact, it is sufficient to prove it for $\frac{1}{n}$ with $n \in \mathbb{N}$.  For that we observe that

$$
\langle u, \frac{1}{n} v \rangle = \frac{n}{n} \langle u, \frac{1}{n} v \rangle = \frac{1}{n} \langle u, \frac{n}{n} v \rangle = \frac{1}{n} \langle u, v \rangle.
$$

To get to $\lambda \in \mathbb{R}$ we need to appeal to continuity.  We find a sequence $(q_n) \in \mathbb{Q}$ such that $(q_n) \to \lambda$ and see that by continuity

$$
\langle u, \lambda v \rangle = \langle u, (\lim q_n) v \rangle = \lim \langle u, q_n v \rangle = \lim q_n \langle u, v \rangle = \lambda \langle u, v \rangle.
$$
=--

+-- {: .num_corollary #scangip}
If ${\|{-}\|}$ is a norm on a vector space satisfying the parallelogram law then the  function

$$
(u,v) \mapsto \frac{1}{2}\left({\|u + v\|^2} - {\|u\|^2} - {\|v\|^2}\right)
$$

is an [[inner product]].
=--

## Isometries on inner product spaces

Because lengths determine inner products, and because inner products respect linear structure, isometries must also respect linear (or affine) structure:

+-- {: .un_thm}
######Theorem
Let $V$ be a vector space with a norm satisfying the parallelogram law. Then an isometry (distance-preserving function) $f \colon V \to V$ is an injective linear affine map.
=--

+-- {: .proof}
######Proof
Put $v = f(0)$. Then the map $x \mapsto f(x) - v$ is also an isometry, so that without loss of generality we may assume $f(0) = 0$, and prove that $f$ is a linear isomorphism.

Letting $d(x, y)$ denote the distance between $x$ and $y$, we have

$${\|x-y\|} = d(x, y) = d(f(x), f(y)) = {\|f(x)-f(y)\|}$$

and then, putting $y=0$, we have ${\|x\|} = {\|f(x)\|}$. Hence

$$\array{
\langle x, y\rangle & = & \frac1{2}\left({\|x\|^2} + {\|y\|^2} - {\|x-y\|^2}\right) \\
 & = & \frac1{2}\left({\|f(x)\|^2} + {\|f(y)\|^2} - {\|f(x)-f(y)\|^2}\right) \\
 & = & \langle f(x), f(y) \rangle
}$$

i.e., $f$ preserves the inner product.

Let $W$ be the linear subspace spanned by the image $f(V)$. Then $W$ inherits an inner product, and if $\{e_i\}$ is an orthonormal basis of $V$ (without restriction on cardinality), then $\{u_i = f(e_i)\}$ is also an orthonormal basis of $W$, since $f$ preserves the inner product. Next, for any linear combination $a x + b y$,

$$\array{
\langle f(a x + b y), u_i \rangle_W & = & \langle a x + b y, e_i \rangle_V \\
 & = & a \langle x, e_i\rangle_V + b\langle y, e_i\rangle_V \\
 & = & a \langle f(x), u_i \rangle_W + b \langle f(y), u_i \rangle_W \\
 & = & \langle a f(x) + b f(y), u_i \rangle_W
}$$

for each $u_i$, whence $f(a x + b y) = a f(x) + b f(y)$. Thus $f$ is a linear map, and a linear isomorphism onto the subspace $W$ because it carries a basis $e_i$ of $V$ to a basis $u_i$ of $W$.

Summarizing, any isometry $f \colon V \to V$ is of the form

$$f(x) = M x + b$$

where $M$ is a linear injection and $b$ is a vector.
=--

For information on related results, see [[isometry]] and [[Mazur-Ulam theorem]].

[[!redirects parallelogram identity]]
[[!redirects parallelogram law]]


parameterized homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Parameterized ([[stable homotopy theory|stable]]) [[homotopy theory]] is ([[stable homotopy theory|stable]]) [[homotopy theory]] of [[bundles]] of [[homotopy types]]/[[stable homotopy types]] over a given base space.

For formalizations see also at

* [[ex-space]]

* [[parameterized spectrum]], [[excisive functor]]

* [[(infinity,1)-module bundle]]

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

* [[twisted cohomology]]

## Parameterized point-set topology

> The [[point-set topology]] of parametrized spaces is surprisingly subtle. [[May & Sigurdsson 2006, p. 15](#MaySigurdsson06)]

Write:

* $kTop$ for the [[convenient category of topological spaces|convenient category]] of [[compactly generated topological spaces]] (k-spaces)

* $kwHaus \subset Top_k$ for that of [[compactly generated weak Hausdorff spaces]].

For $B \,\in\, \mathcal{C}$ an [[object]] in any [[category]], write $\mathcal{C}_{/B}$ for the [[slice category]] over it.

In the following all bases spaces are assumed (as in [MaSi06, p. 19](#MaySigurdsson06)) to be [[compactly generated weak Hausdorff spaces]] regarded among k-spaces:

\[
  \label{kwHausBaseSpace}
  B \;\in\; kwHaus \xhookrightarrow{\;} kTop
  \,.
\]

Notice that for $(X,p_X), (Y,p_Y) \,\in\, kTop_{/B}$, their [[Cartesian product]] in the [[slice category]] is given by the [[fiber product]] in [[compactly generated topological space|$kTop$]]:

$$
  (X,p_X) \times (Y, p_Y)
  \;\simeq\;
  \big(
    X \times_B Y
    ,\,
    p_X \circ pr_X
    =
    p_Y \circ pr_Y
  \big)
  \;\;\;
  \in
  \;
  kTop_{/B}
$$

For $f \,\colon\, B_1 \xrightarrow{\;\;} B_2$ a [[continuous map]] between such base spaces,
notice the usal [[base change]] [[adjoint triple]]:

\[\label{BaseChangeAdjointTriple}\]
\begin{tikzcd}
  \mathrm{kTop}_{/B_1}
    \ar[
      rr,
      shift left=20pt,
      "{f_!}"{description}
    ]
    \ar[
      from=rr,
      "{f^\ast}"{description}
    ]
    \ar[
      rr,
      shift right=20pt,
      "{f_\ast}"{description}
    ]
    \ar[rr, phantom, shift left=10pt, "{\scalebox{.6}{$\bot$}}"]
    \ar[rr, phantom, shift right=10pt, "{\scalebox{.6}{$\bot$}}"]
  &&
  \mathrm{kTop}_{/B_2}
  \,,
\end{tikzcd}

where

1. $f^\ast$ is [[pullback]] in $kTop$ along $f$;

1. $f_!$ is post-[[composition]] with $f$;

Notice the "[[Frobenius reciprocity]] law" (in its [[cartesian monoidal category|cartesian]] version [here](Frobenius+reciprocity#InCategoryTheory)) which follows immediately by the [[pasting law]] in $kTop$, namely the following [[natural isomorphism]]:

\[
  \label{CartesianFrobeniusReciprocity}
  f_!
  \big(
    (U,p_U)
    \times
    f^\ast (X,p_X)
  \big)
  \;\simeq\;
  \big(
    f_!(U,p_U)
  \big)
  \times
  (X,p_X)
  \,.
\]

In the special case where $f$ is the [[terminal object|terminal]] map to the [[point space]], which we denote

$$
  p_B \,\colon\, B \xrightarrow{\;} \ast
  \,.
$$

we have $kTop_{/\ast} \,\simeq\, kTop$ and the above [[base change]] [[adjoint triple]] becomes

\[\label{TerminalBaseChange}\;\]
\begin{tikzcd}
  \mathrm{kTop}_{/B}
    \ar[
      rr,
      shift left=20pt,
      "{(p_B)_!}"{description}
    ]
    \ar[
      from=rr,
      "{(p_B)^\ast}"{description}
    ]
    \ar[
      rr,
      shift right=20pt,
      "{(p_B)_\ast}"{description}
    ]
    \ar[rr, phantom, shift left=10pt, "{\scalebox{.6}{$\bot$}}"]
    \ar[rr, phantom, shift right=10pt, "{\scalebox{.6}{$\bot$}}"]
  &&
  \mathrm{kTop}
  \,,
\end{tikzcd}

In this case

1. the functor $(p_B)^\ast$ is the [[Cartesian product]] with $B$, regarded as the trivial fibration:

   \[
     \label{TrivialFibration}
     (p_B^\ast) X_0
     \;=\;
     B \times X_0 \xrightarrow{\; pr_B \;} B
     \,,
   \]

1. $(p_B)_!$ gives the total space of a fibration:

   $$
     (p_B)_! \big( X \to B\big) \;=\; X
   $$

1. $(p_B)_*$ gives the [[space of sections]] of a fibration.

Eventually we consider [[pointed objects]]

$$
  (X, p_X, \sigma_X) \;\in\; \big( kTop_{/B}\big)^{B/} \;=\; kTop_{/B}^{B/}
$$

in the [[slice category]] of such a base space -- hence topological "[[bundles]]" $X \xrightarrow{p_X} B$ (in the most general sense, without any condition on the bundle projection, except [[continuous function|continuity]]) equipped with a fixed [[section]] $\sigma_X$ (sometimes called "ex-spaces", see [[May & Sigurdsson 2006, p. 19, footnote 1](#MaySigurdsson06)]).

### Fiberwise mapping spaces
 {#FiberwiseMappingSpaces}

> Parametrized mapping spaces are especially delicate [[May & Sigurdsson 2006, p. 15](#MaySigurdsson06), see Rem. \ref{FibMapSpaceDoesNotPreserveWeakHausdorffness} below]

\begin{definition}\label{PartialMapClassifierSpace}
**(partial map classifier space)**
\linebreak
  For $X \,\in\, kTop$, write $\widetilde X \,\in\, kTop$ for its continuous [[partial map classifier]]: The result of forming the [[disjoint union]] of the [[underlying]] [[set]] of $X$ with a [[singleton set]] $\{\bot\}$  and declaring the [[closed subsets]] on the result to be those of $X$ under the defining [[injection]]
$$
  X \xhookrightarrow{\;\; \iota_X \;\;} \widetilde{X}
$$
together with $\widetilde{X}$ itself.

\end{definition}

([MaSi06, Def. 1.3.6](#MaySigurdsson06))

\begin{definition}\label{FiberwiseMappingSpace}
**(fiberwise mapping space)**
\linebreak
For
$$
  (X, p_X),
  \,
  (A,p_A)
  \;\in\;
  kTop_{B}
$$
a [[pair]] of [[k-spaces]] over $B$ (eq:kwHausBaseSpace)
their *fiberwise mapping space* is the [[pullback]] (in $kTop$):
\[
  \label{FiberwiseMappingSpaceAsPullback}
  \array{
    \phantom{---}
    \mathclap{
    Map
    \big(
       (X,p_X)
       ,\,
       (A,p_A)
    \big)
    }
    \phantom{---}
    &\xrightarrow{\phantom{-------}}&
    Map
    \big(
      X,
      \widetilde{A}
    \big)
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{\mathrlap{
      Map
      \big(
        X, \widetilde{p_A}
      \big)
    }}
    \\
    B
    &
    \underset{
     b \mapsto
     \left(
       x
       \mapsto
       \left\{
         \begin{array}{ll}
           b & if\, x \in X_b
           \\
           \bot & otherwise
         \end{array}
       \right.
     \right)
    }{\longrightarrow}
    &
    Map
    \big(
      X,
      \widetilde{B}
    \big)
    \,,
  }
\]
regarded as an object of $kTop_{/B}$.

Here $Map(X,\widetilde{A})$ denotes the ordinary [[compact-open topology|mapping space]] into the continuous partial map classifier from Def. \ref{PartialMapClassifierSpace}.
\end{definition}

(This is  [May & Sigurdsson 2006, Def. 1.3.7](#MaySigurdsson06), following [Booth & Brown 1978a](#BoothBrown78a)).

\begin{remark}
**(on notation)**
\linebreak
Contrary to most references, Def. \ref{FiberwiseMappingSpace} is intentionally not using a subsript "${}_B$" in the notation for the fiberwise mapping space: This is because "$Map(X,A)_B$" is also standard notation for $Map(X,A) \underset{Map(X,B)}{\times} \{p_B\}$ (see e.g. at *[[space of sections]]*), which is crucially different. Instead, with the above notation, $Map(-,-)$ is always of the same [[type]] as its arguments, as befits an [[internal hom]].
\end{remark}

\begin{proposition}\label{FiberwiseMappingSpaceSatisfiesExponentialLaw}
**(fiberwise mapping space satisfies the exponential law)**
\linebreak
  With $B$ as above (eq:kwHausBaseSpace),
  the fiberwise mapping space (Def. \ref{FiberwiseMappingSpace}) is an [[exponential object]] (satisfies the [[exponential law for spaces|exponential law]]) in that there is a [[natural isomorphism]] of [[hom-sets]]
\[
  \label{ExponentialLawForFiberwiseMappingSpace}
  kTop_{/B}
  \Big(
    (X,p_X)
    ,\,
    Map
    \big(
      (Y,p_Y)
      ,\,
      (A,p_A)
    \big)
  \Big)
  \;\;
  \simeq
  \;\;
  kTop_{/B}
  \big(
    (X,p_X) \times (Y,p_Y)
    ,\,
    (A,p_A)
  \big)
\]
(where on the right we have the [[Cartesian product]] in the [[slice category|slice]], given by the [[fiber product]] $X \times_B Y$ in $kTop$).
\end{proposition}
([Booth & Brown 1978a, Thm. 3.5](#BoothBrown78a), see [May & Sigurdsson 2006, (1.3.9)](#MaySigurdsson06))

\begin{example}\label{FiberwiseMappingSpaceBetweenTrivialFibrations}
**(fiberwise mapping space between trivial fibrations)**
\linebreak
  The fiberwise mapping space (Def. \ref{FiberwiseMappingSpace})
between trivial fibrations (eq:TrivialFibration) is the trivial fibration with fiber the ordinary [[compact-open topology|mapping space]] between the fibers:
$$
  Map
  \big(
     p_B^\ast X_0
     ,\,
     p_B^\ast A_0
  \big)
  \;\simeq\;
  p_B^\ast
  Map\big(X_0,\, A_0\big)
  \,.
$$
\end{example}
\begin{proof}
  This may be gleaned concretely from [[point-set topology|point-set]]-analysis of the defining pullback diagram (eq:FiberwiseMappingSpaceAsPullback), but it also follows abstractly by [[adjunction|adjointness]] from the [[exponential law]] (Prop. \ref{FiberwiseMappingSpaceSatisfiesExponentialLaw}):

For any $(U, p_U) \,\in\, kTop_{/B}$ we have the following sequence of [[natural isomorphisms]]:
$$
  \begin{aligned}
    &
    kTop_{/B}
    \Big(
      (U, p_U)
      ,\,
      \Map
      \big(
        p_B^\ast X_0
        ,\,
        p_B^\ast A_0
      \big)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B}
    \big(
      (U, p_U) \times (B \times X_0, pr_{B})
      ,\,
      p_B^\ast A_0
    \big)
    \\
    &
    \;\simeq\;
    kTop_{/B}
    \big(
      (U \times X_0, p_U \circ pr_U)
      ,\,
      p_B^\ast A_0
    \big)
    \\
    &
    \;\simeq\;
    kTop
    \Big(
      U \times X_0
      ,\,
      A_0
    \big)
    \\
    &
    \;\simeq\;
    kTop
    \Big(
      U
      ,\,
      Map
      \big(
        X_0
        ,\,
        A_0
      \big)
    \Big)
    \\
    &
    \;\simeq\;
    kTop
    \Big(
      (p_B)_!
      (U,p_U)
      ,\,
      Map
      \big(
        X_0
        ,\,
        A_0
      \big)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B}
    \Big(
      (U,p_U)
      ,\,
      p_B^\ast
      Map
      \big(
        X_0
        ,\,
        A_0
      \big)
    \Big)
  \end{aligned}
$$
Here most steps are [Hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism) of the various [[adjoint functors]]: (eq:TerminalBaseChange) and (eq:ExponentialLawForFiberwiseMappingSpace).
Since this holds naturally for all $(U, p_U)$, the claim follows by the [[Yoneda lemma]] (over the [[large category]] $\big(kTop_{/B}\big)^{op}$).
\end{proof}

Similarly:

\begin{proposition}\label{PullbackOfFiberwiseMappingSpace}
**(pullback of fiberwise mapping space)**
\linebreak
For $f \,\colon\, B' \longrightarrow B$ a [[map]] of base spaces (eq:kwHausBaseSpace), the [[pullback]] (eq:BaseChangeAdjointTriple) along $f$ of the fiberwise mapping space (Def. \ref{FiberwiseMappingSpace}) is the fiberwise mapping space of the pullback of the arguments:
$$
  f^\ast
  Map
  \big(
    (X,p_X)
    ,\,
    (A, p_A)
  \big)
  \;\;
  \simeq
  \;\;
  Map
  \big(
    f^\ast
    (X,p_X)
    ,\,
    f^\ast
    (A, p_A)
  \big)
  \,.
$$
In other words: Pullback $f^\ast$ is a [[closed functor]] with respect to fiberwise mapping spaces.
\end{proposition}
\begin{proof}
For any $(U, p_U) \,\in\, kTop_{/B'}$ we have the following sequence of [[natural isomorphisms]]:
$$
  \begin{aligned}
    &
    kTop_{/B'}
    \Big(
      (U,p_U)
      ,\,
      f^\ast
      Map
      \big(
        (X,p_X)
        ,\,
        (A,p_A)
      \big)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B}
    \Big(
      f_! (U,p_U)
      ,\,
      Map
      \big(
        (X,p_X)
        ,\,
        (A,p_A)
      \big)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B}
    \Big(
      \big(
        f_!(U, p_U)
      \big)
      \times
      (X, p_X)
      ,\,
      (A,p_A)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B}
    \Big(
      f_!
      \big(
        (U, p_U)
        \times
        f^\ast (X, p_X)
      \big)
      ,\,
      (A,p_A)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B'}
    \Big(
        (U, p_U)
        \times
        f^\ast (X, p_X)
      ,\,
      f^\ast
      (A,p_A)
    \Big)
    \\
    &
    \;\simeq\;
    kTop_{/B'}
    \Big(
      (U, p_U)
      ,\,
      Map
      \big(
        f^\ast (X, p_X)
        ,\,
        f^\ast (A,p_A)
      \big)
    \Big)
  \end{aligned}
$$
Here the crucial step, besides various [Hom-isomorphisms](adjoint+functor#InTermsOfHomIsomorphism), is the use of Cartesian "[[Frobenius reciprocity]]" (eq:CartesianFrobeniusReciprocity).

Since these isomorphism are [[natural isomorphism|natural]] in $(U,p_U)$, the claim follows by the [[Yoneda embedding]] (for the [[large category]] $\big( kTop_{/B'}\big)^{op}$).
\end{proof}
,

\begin{example}\label{FiberOfFiberwiseMappingSpace}
**(fiber of fiberwise mapping space is mapping space of fibers)**
\linebreak
  For $b \in B$, the [[fiber]] of the fiberwise mapping space fibration (Def. \ref{FiberwiseMappingSpace}) is [[homeomorphism|homemorphic]] to the ordinary [[compact-open topology|mapping space]] betwee the [[fibers]]:
$$
  Map
  \big(
    (X,p_x)
    ,\,
    (A,p_A)
  \big)_b
  \;\;
  \simeq
  \;\;
  Map
  \big(
    X_b
    ,\,
    A_b
  \big)
  \,.
$$
\end{example}
(e.g. [May & Sigurdsson 2006, p. 21](#MaySigurdsson06))
\begin{proof}
  This is immediate from concrete analysis of the defining [[pullback]]-diagram (eq:FiberwiseMappingSpaceAsPullback) in Def. \ref{FiberwiseMappingSpace}, but it is also the special case of Prop. \ref{PullbackOfFiberwiseMappingSpace} for $B = \{b\}$.
\end{proof}

### Homotopy theory of the fiberwise mapping space

\begin{proposition}\label{FiberwiseMappingSpacePreservesHFibrations}
**(fiberwise mapping space preserves h-fibrations)**
\linebreak
  If $p_X \colon X \to B$
  and $p_A \colon A \to B$
  are [[Hurewicz fibrations]], then so is the map (eq:FiberwiseMappingSpaceAsPullback) out of their fiberwise mapping space (Def. \ref{FiberwiseMappingSpace}):
$$
  p_X, p_A \,\in\, hFib
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  p_{ Map\big( (X,p_X) , (A,p_A)\big) }
  \;\in\;
  hFib
  \,.
$$
\end{proposition}
(This is due to [Booth 1970, §6.1](#Booth70), see [MaSi06, Prop. 1.3.11](#MaySigurdsson06).)

\begin{remark}\label{FibMapSpaceDoesNotPreserveWeakHausdorffness}
**(fiberwise mapping space does not preserve weak Hausdorffness)**
\linebreak
  Even if $X$ and $A$ are [[weak Hausdorff spaces]] over the weak Hausdorff space $B$ (eq:kwHausBaseSpace), their
  fiberwise mapping space (Def. \ref{FiberwiseMappingSpace}) need  not be weak Hausdorff ([Booth & Brown 1974a](#BoothBrown74a)). Sufficient conditions for this to be the case are given in [Lewis 1985, Prop. 1.5](#Lewis85)

On the other hand, the suitable [[cofibrant resolution]] of the fiberwise mapping space will again be weak Hausdorff (see [MaSi06, p. 19](#MaySigurdsson06)).
\end{remark}

## Related concepts

* [[retractive space]]

* [[parameterized spectrum]]

* [[model structure for parameterized spectra]]

* [[rational parametrized homotopy theory]]

  * [[rational fibration lemma]]

* [[rational parameterized stable homotopy theory]]

## References

[[!include exponential law for parameterized topological spaces -- references]]

### Parameterized ("fiberwise") homotopy theory

On the [[homotopy theory]] of such parameterized topological spaces:

* [[Ioan Mackenzie James]], §IV of: *Fibrewise topology*, Cambridge Tracts in Mathematics, Cambridge University Press (1989) [ISBN:9780521360906]

* [[Ioan Mackenzie James]], _[[Introduction to fibrewise homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_ (1995)

* [[Michael C. Crabb]], [[Ioan Mackenzie James]]: *Fiberwise homotopy theory*, Springer Monographs in Mathematics, Springer (1998) [[doi:10.1007/978-1-4471-1265-5](https://doi.org/10.1007/978-1-4471-1265-5), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabbjames.pdf) ,[pdf](https://www.gbv.de/dms/ilmenau/toc/243540361.PDF)]

On [[model structures for parameterized spectra]]:

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006   ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

* {#BraunackMayer19} [[Vincent Braunack-Mayer]], *Combinatorial parametrised spectra*, Algebr. Geom. Topol. **21** (2021) 801-891 [[arXiv:1907.08496](https://arxiv.org/abs/1907.08496), [doi:10.2140/agt.2021.21.801](https://doi.org/10.2140/agt.2021.21.801)]

  > (based on the [[schreiber:thesis Braunack-Mayer|PhD thesis]])

* [[Fabian Hebestreit]], [[Steffen Sagave]], [[Christian Schlichtkrull]], *Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces*, Forum of Mathematics, Sigma **8** (2020) e16 [[arXiv:1904.01824](https://arxiv.org/abs/1904.01824), [doi:10.1017/fms.2020.11](https://doi.org/10.1017/fms.2020.11)]

* {#Malkiewich19} [[Cary Malkiewich]], *Parametrized spectra, a low-tech approach* [[arXiv:1906.04773](https://arxiv.org/abs/1906.04773), user guide: [pdf](https://people.math.binghamton.edu/malkiewich/users_guide_parametrized.pdf), [[Malkiewich-ParametrizedSpectraUserGuide.pdf:file]]]

* {#Malkiewich23} [[Cary Malkiewich]], *A convenient category of parametrized spectra* [[arXiv:2305.15327](https://arxiv.org/abs/2305.15327)]

Discussion in [[(∞,1)-category theory]]:

* {#AndoBlumbergGepner11} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map_, Geom. Topol. 22 (2018) 3761-3825 ([arXiv:1112.2203](http://arxiv.org/abs/1112.2203))

Discussion as a [[linear homotopy type theory]]:

* {#Schreiber14} [[Urs Schreiber]], _[[schreiber:Quantization via Linear homotopy types]]_ [[arXiv:1402.7041](http://arxiv.org/abs/1402.7041)]

  > (intended [[categorical semantics]])

* [[Mitchell Riley]], [[Eric Finster]], [[Daniel R. Licata]], *Synthetic Spectra via a Monadic and Comonadic Modality* [[arXiv:2102.04099](https://arxiv.org/abs/2102.04099)]

  > ([[inference rules]] for the [[classical modality]] $\natural$)

* {#Riley22Thesis} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269),  [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

  > (including [[inference rules]] for the [[multiplicative conjunction]] by bringing in the required [[bunched logic]])

[[!redirects parametrized homotopy theory]]

[[!redirects fiberwise homotopy theory]]
[[!redirects fibrewise homotopy theory]]

[[!redirects parameterized stable homotopy theory]]
[[!redirects parametrised stable homotopy theory]]
[[!redirects parametrized stable homotopy theory]]

[[!redirects parameterized topology]]
[[!redirects fiberwise topology]]
[[!redirects fibrewise topology]]

[[!redirects fiberwise mapping space]]
[[!redirects fiberwise mapping spaces]]


parameterized quantum system

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

#Contents#
* table of contents
{:toc}

## Idea

A *parameterized quantum system* is a [[quantum system]] depending on "external" or "classical" parameters.

In typical examples arising in [[quantum mechanics]], this simply means that the [[Hamiltonian]] $H$ is not just one fixed [[linear operator]] on a given [[Hilbert space]] [[space of quantum states|of quantum states]] $\mathcal{H}$, but a [[continuous function|continuous]] (or suitably [[smooth function|smooth]]) [[map]]

$$
  H(-)
  \;\colon\;
  P
  \longrightarrow
  End(\mathcal{H})
$$

of such, from some [[topological space|topological]] "parameter space" $P$.

More abstractly, where a single [[quantum system]] is described by [[linear type theory]] (see also at *[[quantum logic]]*), a parameterized quantum system is described by *[[dependent linear type theory]]*.

## Properties

### Adiabatic Berry phases and adiabatic quantum computation

The *[[quantum adiabatic theorem]]* says that/how the evolution of a parameterized quantum system under sufficiently slow movement of the external parameters remains in an [[eigenstate]] for a given system of [[commutator|commuting]] [[quantum observables]]. The remaining transformations on these [[eigenspaces]] are known as (non-abelian) *[[Berry phases]]*. These are used as [[quantum gates]] in [[adiabatic quantum computation]].

## Examples

### Time-dependent quantum mechanics

For example, *time-dependent* Hamiltonians/quantum systems are described this way for $P \subset \mathbb{R}$ interpreted as the space of [[time]]-parameters.

See also at *[[Dyson formula]]*.

### Defect anyon braiding

In some models of [[braid group statistics]] (see [there](braid+group+statistics#AsBraidingOfDefects) for more) the [[anyons]] are [[defects]]/[[solitons]] whose positions serve as external parameters of the system, so that the (non-abelian) [[Berry phases]] under their [[quantum adiabatic theorem|adibatic moverment]] constitutes a [[braid group representation]] on the [[quantum system]]'s [[ground states]].

## Related concepts

* [[Dyson formula]]

* [[quantum adiabatic theorem]]

* [[controlled quantum gate]]

* [parameterized quantum computation](quantum+computation#ClassicalControlQuantumData)

* [[parameterized homotopy theory]]

* [[parameterized stable homotopy theory]]

* [[dependent linear type theory]]

* [[subsystem]]

* [[EPR phenomena]]

## References

Discussion of classically-[[parameterized quantum system|parameterized]] quantum circuits:

(...)

Discussion of quantumly-[[parameterized quantum systems|parameterized]] quantum circuits:

* Evan Peters, Prasanth Shyamsundar, *Qarameterized circuits: Quantum parameters for QML* (2021) [[web](https://peterse.github.io/groveropt/)]

following

* Guillaume Verdon, Jason Pye, Michael Broughton, *A Universal Training Algorithm for Quantum Deep Learning* [[arXiv:1806.09729](https://arxiv.org/abs/1806.09729)]

* Prasanth Shyamsundar, *Non-Boolean Quantum Amplitude Amplification and Quantum Mean Estimation* [[arXiv:2102.04975](https://arxiv.org/abs/2102.04975)]

[[!redirects parameterized quantum systems]]

[[!redirects time-dependent Hamiltonian]]
[[!redirects time-dependent Hamiltonians]]