parameterized WZW model

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#### $\infty$-Wess-Zumino-Witten theory
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#### Quantum field theory
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#### Physics
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## Idea
 {#Idea}

Where a _[[WZW model]]_ is a [[sigma model]] [[quantum field theory]] whose [[target space]] is a [[group]] $G$, a _parameterization_ of such a model is a sigma-model (subject usually to some constraints) whose target space now is the total space of a $G$-[[principal bundle]] such that the [[action functional]] when restricted to fields that take values only in any one [[fiber]], reduces to the given un-parameterized model.

Parameterized WZW models have been argued to provide a more geometric and more complete description of the [[current algebra]]-sector of [[heterotic string]] backgrounds than the traditional construction in terms of worldsheet fermions  ([Gates-Siegel 88](#GatesSiegel88), [Gates-Ketov-Kozenko-Solovev 91](#GKKS91), [Distler-Sharpe 10, section 7](#DistlerSharpe10)). In particular the [[Green-Schwarz anomaly]] of the heterotic string finds a natural interpretation as the [[obstruction]] to parameterizing the given WZW term over the given principal bundle:

A ([[schreiber:∞-Wess-Zumino-Witten theory|higher]]) [[WZW model]] is an $n$-dimensional [[sigma-model]] [[field theory]] whose [[target space]] is a [[group]] $G$ and whose [[interaction]]-part of the [[action functional]] is the [[higher holonomy]] of a [[circle n-bundle with connection]]

$$
  \mathbf{L}_{WZW}
  \;\colon\;
  G \longrightarrow
  \mathbf{B}^n U(1)_{conn}
$$

(whose [[curvature]] is given by the global [[Maurer-Cartan form]] on $G$).

If one has a $G$-[[principal bundle]]

$$
  \array{
    G &\hookrightarrow& X
    \\
    && \downarrow
    \\
    && B
  }
$$

over some base space $B$, then a [[lift of the structure group]] to the [[Heisenberg n-group]] $Heis_G(\mathbf{L}_{WZW})$ of $\mathbf{L}_{WZW}$ regarded as a [[prequantum n-bundle]], is the structure necessary and sufficient for the [[fiber]]-wise copies of $G$ and $\mathbf{L}_{WZW}$ glue fiberwise to a single [[circle n-bundle with connection]]

$$
  \mathbf{L}_{WZW}^X \;\colon\; X \longrightarrow \mathbf{B}^n U(1)_{conn}
$$

on the total space $X$ of this bundle. This hence yields what one may think of as a coherent collection of [[WZW models]] _parameterized_ over base space $B$.

For the case that $G$ is a [[compact Lie group|compact]] [[semisimple Lie group]] and $\mathbf{L}_{WZW}$ its canonical WZW term (the [[circle n-bundle with connection|2-connection]] on the [[string 2-group]]), then

$$
  Heis_G(\mathbf{L}_{WZW})\simeq String(G)
$$

and hence the [[obstruction]] to the existence of a parameterization is precisely a [[string structure]], recovering the traditional statement of the [[Green-Schwarz anomaly]] (see [cwzw](#cwzw) for details of this claim).

## Examples

### The heterotic string

The [[heterotic string]] [[worldsheet]] [[theory (physics)|theory]] is a [[sigma model]] on [[spacetime]] $X$ combined with a chiral $G$-[[WZW model]] for $G = Spin \times E_8 \times E_8$ or similar and with the [[WZW term]] being the difference between the differentially twisted [[looping]] of the [[universal Chern-Simons circle 3-bundle|universal Chern-Simons 3-connection]] of $Spin$ with that of $E_8 \times E_8$.

By ([Fiorenza-Rogers-Schreiber 13, section 2.6.1](#FiorenzaRogersSchreiber13)) we find in this case that the [[Heisenberg 2-group]] is the [[string 2-group]]

$$
  Heis(\mathbf{L}_{WZW^{het}})
  \simeq
  String(G)
  \,.
$$

It follows thus from the above discussion that a consistent heterotic [[sigma model]] requires that the underlying $G$-[[principal bundle]] on [[spacetime]] admits a [[string structure]]. This is the famous [[Green-Schwarz anomaly]] cancellation condition, re-expressed as a consistency condition for parameterized WZW models.

On the level of [[action functionals]] in [[codimension]] 0 this observation is due to ([Distler-Sharpe 10, section 7](#DistlerSharpe10)).

### The Green-Schwarz super $p$-branes on curved super spacetime

By [Fiorenza-Sati-Schreiber 13](#FiorenzaSatiSchreiber13) the super-[[branes]] of [[string theory]]/[[M-theory]] on [[super Minkowski spacetime]] $\mathbb{R}^{d-1,1;N}$ classified by the [[brane scan]]/[[schreiber:The brane bouquet|the brane bouqet]] are [[schreiber:∞-Wess-Zumino-Witten theory]] [[sigma-model]]

$$
  \mathbf{L}_{WZW^{brane}}
  \;\colon\;
  \mathbb{R}^{d-1,1;N} \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}
  \,,
$$

where $\Gamma$ is the group of [[periods]] of the defining [[super L-∞ algebra]] [[L-∞ algebra cohomology|L-∞ cocycle]]

$$
  \mathbf{B} sIso_N(d-1,1) \longrightarrow \mathbf{B}^{p+2}(\mathbb{R}/\Gamma)
  \,.
$$

A necessary structure globalizing this to a curved super-spacetime $X$ is a [[G-structure]] on $X$ for $G = \mathbf{QuantMorph}(\mathbf{L}_{WZW^{brane}}^{inf})$ the restriction of the WZW term to the [[infinitesimal disk]].

Given this then the [[Green-Schwarz action functional]] refines to a [[local prequantum field theory]] datum $\mathbf{L}_{WZW^{brane}}$ globally defined on $X$.

(For [[branes]] on which other branes may end, such as the [[D-branes]] and the [[M5-brane]], here [[super Minkowski spacetime]] is replaced by a [[extended Minkowski super spacetime]], as discussed in ([Fiorenza-Sati-Schreiber 13](#FiorenzaSatiSchreiber13))).

More details are in ([cwzw](#cwzw)).

## Related concepts

* [[gauged WZW model]]

* [[definite globalization of WZW term]]

* [[higher Cartan geometry]]

## References

Parameterized WZW models as [[sigma models]] for the [[heterotic string]] originate in

* {#GatesSiegel88} [[Jim Gates]], [[Warren Siegel]], _Leftons, Rightons, Nonlinear $\sigma$-Models, and Superstrings_, Phys.Lett. B206 (1988) 631 ([spire](https://inspirehep.net/record/251286/))

* [[Jim Gates]], _Strings, superstrings, and two-dimensional lagrangian field theory_, pp. 140-184 in Z. Haba, J. Sobczyk (eds.) _Functional integration, geometry, and strings_, proceedings of the XXV Winter School of Theoretical Physics, Karpacz, Poland (Feb. 1989), , Birkhäuser, 1989.

* {#GKKS91} [[Jim Gates]], S. Ketov, S. Kozenko, O. Solovev, _Lagrangian chiral coset construction of heterotic string theories in $(1,0)$ superspace_, Nucl.Phys. B362 (1991) 199-231 ([spire](http://inspirehep.net/record/314337/?ln=en))

Discussion relating this to equivariant [[elliptic genera]] is in section 7 of

* {#DistlerSharpe10} [[Jacques Distler]], [[Eric Sharpe]], _Heterotic compactifications with principal bundles for general groups and general levels_, Adv. Theor. Math. Phys. 14:335-398, 2010 ([arXiv:hep-th/0701244](http://arxiv.org/abs/hep-th/0701244))

Review of this includes

* [[Eric Sharpe]], _Recent developments in heterotic compactifications_, in [[Eric Sharpe]], [[Arthur Greenspoon]], _[Advances in String Theory: The First Sowers Workshop in Theoretical Physics](http://www.ams.org/bookstore-getitem/item=AMSIP-44)_, AMS 2008 ([pdf slides (39-49)](http://www.phys.vt.edu/sowers/talks/sharpe-sowers.pdf))

The corresponding [[elliptic genera]] had been considered in

* [[Matthew Ando]], _The sigma orientation for analytic circle-equivariant elliptic cohomology_, Geom. Topol. 7 (2003) 91-153 ([arXiv:math/0201092](http://arxiv.org/abs/math/0201092))

and with more emphasis on [[equivariant elliptic cohomology]] in

* [[Matthew Ando]], _Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe_, [talk 2007](http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php) ([lecture notes pdf](http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf))

The discussion of the relevant [[Heisenberg n-group]] theory is in

* {#FiorenzaRogersSchreiber13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:Higher geometric prequantum theory]]_, 2013 ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

with more details in

* {#cwzw} [[Urs Schreiber]], _[[schreiber:Obstruction theory for parameterized higher WZW terms]]_

* [[Urs Schreiber]], section "definite forms" in _[[schreiber:differential cohomology in a cohesive topos]]_

General [[schreiber:∞-Wess-Zumino-Witten theory]] is set up in section 6 of

* {#FiorenzaSatiSchreiber13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, 2013 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

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parametric operator

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Parametric_operator).

[[!redirects parametric operators]]


parametric right adjoint

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### Context
#### Category theory
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#Contents#
* table of contents
{:toc}

## Definition

Let $T\colon A\to B$ be a [[functor]] such that the [[category]] $A$ has a [[terminal object]] $1$.  Then $T$ can canonically be factored as the composite

$$
  A
   \overset{\;\;T_1\;\;}{\longrightarrow}
  B/T1
   \overset{\;\;\Sigma_{T1}\;\;}{\longrightarrow}
  B
$$

of $T$ applied to the [[slice category]] $A \simeq A/1$, followed by [[dependent sum]] (projection on the source).

We say that $T$ is a **parametric right adjoint**, or **p.r.a.**, if the functor $T_1$ is a [[right adjoint]].  Parametric right adjoints are also called **local right adjoints**, though this terminology conflicts with that of [[local adjunctions]] (see [[locally]]). It is equivalent to the notion of right [[multi-adjoint]], but [[multi-adjoints]] can be formulated without assuming $A$ has a terminal object.

A [[monad]] is called **p.r.a.** if its functor part is p.r.a. and moreover its unit and multiplication are [[cartesian natural transformation|cartesian]].  Thus in particular it is a [[cartesian monad]].  A p.r.a. monad is also called a **strongly cartesian monad**.

## Properties

* Since $\Sigma_{T1}$ [[created limit|creates]] [[connected limits]], if $T$ is p.r.a. then it [[preserved limit|preserves]] connected limits, and in particular preserves [[pullbacks]].  It follows that any p.r.a. monad is a [[cartesian monad]].

* Conversely, a [[functor]] between [[presheaf categories]] is p.r.a. if it preserves [[connected limits]]. The reason is that (in the notation above), if $T$ preserves connected limits, then $T_1$ preserves all small limits, and a limit-preserving functor from a [[cototal category]] such as a presheaf category to a locally small category is a right adjoint.

* Any [[polynomial functor]] $\Sigma_h \Pi_g f^*$ is p.r.a., since then $T_1$ can be identified with $\Pi_g f^*$, which has the left adjoint $\Sigma_f \; g^*$.

* If $E$ is a [[presheaf category]] and $T\colon E \to Set$ is p.r.a., then the [[comma category]] $Set/T$ (also called the [[Artin gluing]] in this context) is again a presheaf category.  Conversely, if $Set/T$ is a presheaf category, then $T$ preserves connected limits, and thus is p.r.a.

* A parametric right adjoint functor (with [[locally small category|locally small]] codomain) has in particular a left [[multi-adjoint]], which sends each object $b\in B$ to the family of all units $\eta_{b,i} : b \to T L(b,i)$, where $i$ ranges over all morphisms $b\to T 1$ and $L : B/T1 \to A$ is the left adjoint of $T_1$.  This is because any morphism $b\to T a$ induces a unique composite $i:b \to T a \to T 1$, and hence a unique factorization through $L(b,i)$.  Conversely, if $A$ has a terminal object and $T:A\to B$ has a left multi-adjoint, then it is a parametric right adjoint.

## Generic morphisms

Central to the theory of parametric right adjoints is the notion of *$T$-generic* morphisms.  For any [[functor]] $T$, a morphism $f \colon B\to T A$ is (strictly) **$T$-generic** if any [[commutative square]] of the following form:
$$
  \array{
    B
    &
    \overset{\alpha}{\longrightarrow} &
    T X
    \\
    \mathllap{{}^f}
    \big\downarrow
    &&
    \big\downarrow
    \mathrlap{{}^{T\gamma}}
    \\
    T A
    &
    \underset{T \beta}{\longrightarrow}
    &
    T Z
  }
$$
has a unique [[lift|filler]] of the form $T\delta \colon T A \longrightarrow T X$.
A **generic factorization** of a map $f\colon B\to T A$ is a factorization
$$
  B
    \overset{g}{\longrightarrow}
  T D
    \overset{T h}{\longrightarrow}
  T A
$$
such that $g$ is $T$-generic.  Note that by the definition of genericity, generic factorizations are unique whenever they exist.  If $T$ is a monad and any map $B \to T A$ has a generic factorization, then there is an induced [[orthogonal factorization system]] on the [[Kleisli category]] of $T$ in which $T$-generic maps are the left class and the right class are the "free" maps, i.e. those which factor through the unit of $T$.

+-- {: .un_prop}
###### Proposition
A functor $T$ is a parametric right adjoint iff every map $B\to T A$ has a generic factorization.
=--
+-- {: .proof}
###### Proof
This is Proposition 2.6 of [(Weber08)](#Weber08).  In fact, the generic factorizations are precisely the universal maps in the left multi-adjoint of $F$ mentioned above.
=--

P.r.a. functors between presheaf categories have an especially nice form.

+-- {: .un_prop}
###### Proposition
A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff any map $y(j)\to T 1$ has a generic factorization, where $y(j)$ is the representable presheaf on an object $j\in J$.
=--
+-- {: .proof}
###### Proof
This is Proposition 2.10 of [(Weber08)](#Weber08). The "only if" direction is the previous proposition, while for the "if" direction, the given hypothesis allows us to define the functor
$$ E_T \colon y/T1 \to [I^{op},Set] $$
sending an object $(y(j) \to T 1)$ to the object occurring in its generic factorization.  Note that $y/T1$ is equivalently the opposite of the [[category of elements]] of $T1$.  The definition of genericity, along with the Yoneda lemma, then shows that
$$ T(Z)(j) = \coprod_{x\in T1(j)} [I^{op},Set](E_T(x),Z) $$
which preserves connected limits, since it is a coproduct of representables.
=--

In particular, a p.r.a. functor $T\colon [I^{op},Set] \to [J^{op},Set]$ is determined by giving the object $T1\in [J^{op},Set]$ together with the functor $E_T\colon y/T1 = el(T1)^{op} \to [I^{op},Set]$.  We can think of $T1(j)$ as the setof all possible "shapes" which $T$ allows us to "glue together" to obtain an element of shape $j$, and $E_T$ as specifying exactly what each of those shapes looks like.  Then the above formula for $T(Z)(j)$ says that we look at all possible shapes $x\in T1(j)$ we can glue to get something of shape $j$, and for each such $x$ we look at all the "diagrams" in $Z$ of the corresponding shape $E_T(x)$.

We can extract from this a description that is clearly a generalization of a [[polynomial functor]].

+-- {: .num_prop #AsPolynomial}
###### Proposition
A functor $T\colon [I^{op},Set] \to [J^{op},Set]$ between presheaf categories is p.r.a. iff when expressed in terms of [[discrete fibrations]], it is the composite
\[ DFib/I \xrightarrow{d^*} DFib/E \xrightarrow{c_*} DFib/K \xrightarrow{p_!} DFib/J \]
for a polynomial in $Cat$
\[ I \xleftarrow{d} E \xrightarrow{c} K \xrightarrow{p} J \]
where $p$ is a discrete fibration and $(d,c)$ is a [[two-sided discrete fibration]] (with in particular $d$ a fibration and $c$ an opfibration).
=--
+-- {: .proof}
###### Proof
Let $p$ be the [[Grothendieck construction]] of $T1$, so that $K = el(T1)^{op}$, and $(d,c)$ the two-sided Grothendieck construction of $E_T\colon el(T1)^{op} \to [I^{op},Set]$ regarded as a profunctor from $K$ to $I$.  The above formula tells us that $T = Lan_p \circ Hom(E_T,-)$, and when rewritten in terms of discrete fibrations this gives the above formula.  More details are in Remark 2.12 of [(Weber08)](#Weber08).
=--

That is, a p.r.a. functor between presheaf categories is the restriction to discrete fibrations of a certain kind of polynomial functor between slices of $Cat$.  When $I$ and $J$ are discrete categories, then so are $K$ and $E$, so that p.r.a. functors between presheaf categories are a direct generalization of polynomial functors between slices of $Set$.  But on the other hand, we can also say that polynomial functors between slices of *Cat* are a direct generalization of p.r.a. functors between presheaf categories.

## Examples

### Free categories

Consider the [[free category]] monad $T$ on the category $Quiv$ of [[quivers]], such that $T A$ is the quiver with the same objects as $A$ and whose arrows are finite composable strings of arrows in $A$..  Here $T 1$ is the monoid $\mathbb{N}$ regarded as a one-object category, and thus an object of $Quiv/T1$ is a quiver together with a natural number assigned to each edge.  For any quiver $A$, the natural augmentation $T A \to T 1$ assigns to each composable string of arrows its length.

The left adjoint of this functor $T_1\colon Quiv \to Quiv/T1$ takes as input a quiver with natural number "lengths" assigned to each of its arrows, and creates a new quiver by gluing together a copy of the quiver $[n] = (0 \to 1 \to\dots \to n)$ (with no arrows other than those drawn) for each arrow of "length" $n$.  Thus $T$ is a parametric right adjoint.

$Quiv$ is of course a presheaf category $[Q^{op},Set]$, where $Q$ is the category $0 \rightrightarrows 1$.  The category $y/T1$, i.e. the opposite of the category of elements of $T1$, has objects $\mathbb{N} \sqcup \{\bot\}$ and nonidentity arrows $\bot \rightrightarrows n$ for all $n\in\mathbb{N}$.  Finally, the functor $E_T \colon y/T1 \to Quiv$ sends $\bot$ to the quiver with one object and no arrows, and $n$ to the quiver $[n] = (0 \to 1 \to\dots \to n)$ described above.

## References

* [[Aurelio Carboni]] and [[Peter Johnstone]], _Connected limits, familial representability and Artin glueing_, Mathematical Structures in Computer Science **5(4)**, 1995, pp. 441 - 459. ([MR:1377312](http://www.ams.org/mathscinet-getitem?mr=1377312), [doi:10.1017/S0960129500001183](https://doi.org/10.1017/S0960129500001183))

Parametric right adjoints were introduced in:

* [[R. Street]], _The petit topos of globular sets_ , Journal of Pure and Applied Algebra **154**, 2000, pp. 299-315. (doi:10.1016/S0022-4049(99)00183-8)

See also:

* [[Mark Weber]], _Generic morphisms, parametric representations, and weakly cartesian monads_, Theory and Applications of Categories, Vol. 13, 2004, No. 14, pp 191-234. ([tac](http://www.tac.mta.ca/tac/volumes/13/14/13-14abs.html))

* {#Weber08} [[Mark Weber]], _Familial 2-functors and parametric right adjoints_, Theory and Applications of Categories, Vol. 18, 2007, No. 22, pp 665-732. ([tac](http://www.tac.mta.ca/tac/volumes/18/22/18-22abs.html))

* {#BMW} [[Clemens Berger]], [[Paul-André Melliès]], [[Mark Weber]], _Monads with Arities and their Associated Theories_, Journal of Pure and Applied Algebra **216**, 2011. ([arXiv:1101.3064](http://arxiv.org/abs/1101.3064), [doi:10.1016/j.jpaa.2012.02.039](https://doi.org/10.1016/j.jpaa.2012.02.039))

In [[database theory]] p.r.a.s between copresheaf categories, known as _data migration functor_, are treated in

* {#Spivak10} [[David Spivak]], _Functorial Data Migration_, Information and Computation **217**, 2012. ([arXiv:1009.1166](https://arxiv.org/abs/1009.1166), [doi:10.1016/j.ic.2012.05.001](https://doi.org/10.1016/j.ic.2012.05.001))

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[[!redirects p.r.a. functors]]
[[!redirects p.r.a. monad]]
[[!redirects p.r.a. monads]]

[[!redirects local right adjoint]]
[[!redirects local right adjoints]]
[[!redirects strongly cartesian monad]]
[[!redirects strongly cartesian monads]]


Parametrized Higher Category Theory and Higher Algebra

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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

##Idea

This entry relates to a series of papers, beginning with [Barwick-Dotto-Glasman-Nardin-Shah 16](#PHCTIntro) which aim to provide common foundations for several different parts of [[homotopy theory]], among which [[equivariant homotopy theory]], [[parametrized homotopy theory]], [[global homotopy theory]] and [[Goodwillie calculus]].

For $G$ a [[finite group]], various concepts in [[equivariant homotopy theory]] are constructed as [[indexed (∞,1)-category|indexed]] over an [[orbit category]], such as the [[homotopy theory]] of [[topological G-spaces]] or of $G$-[[equivariant spectra]], when regarded via [[Elmendorf's theorem]].

{#AtomicOrbital}
 An important ingredient of this program is the concept of an _atomic orbital $\infty$-category_ which is defined in terms of two important properties of the [[orbit category]] of $G$:

1. **Orbital**: Fiber products of representable presheaves are finite disjoint unions of representable presheaves, a restatement of the fact that the category of finite $G$-sets has pullbacks ([Mackey decomposition](double+coset#MackeyFormula)), so a version of the [[Beck-Chevalley condition]];

1. **Atomic**: The triviality of retracts (that is every retraction is an equivalence).

Examples of such $(\infty, 1)$-categories satisfying these two properties include:

1. [[orbit categories]] of finite groups;

1. more generally, orbit categories of [[profinite groups]] (where the [[stabilizers]] are required to be [[open subset|open]]);

1. locally finite groups (where the stabilizers are required to be finite);

1. any [[∞-groupoid]];

1. the cyclonic orbit 2-category (see at _[[cyclotomic spectrum]]_);

1. the 2-category of connected finite groupoids and [[covering maps]];

1. the category of [[finite sets]] of [[cardinality]] $\leq n$ and [[surjective functions]];

1. the [[topological categories]] of [[finite number|finite]]-[[dimension|dimensional]] [[inner product spaces]] (over $\mathbb{R}$ and $\mathbb{C}$) of dimension $\leq n$ and orthogonal [[projections]].

The program looks to generate results which hold for all atomic orbital $\infty$-categories, for any instance of which, $T$, there are the associated concepts of $T$-$\infty$-category, $T$-space and $T$-spectrum.

For many cases of these atomic orbital $\infty$-categories there is a [[conservative (∞,1)-functor]] to a [[poset]], and so they are [[EI (∞,1)-categories]].

## References

Along with an introduction, nine exposés are planned:

* {#PHCTIntro} [[Clark Barwick]], Emanuele Dotto, [[Saul Glasman]], [[Denis Nardin]], [[Jay Shah]], _Parametrized higher category theory and higher algebra: A general introduction_, ([arXiv:1608.03654](https://arxiv.org/abs/1608.03654))

* {#PHCTI} [[Clark Barwick]], Emanuele Dotto, [[Saul Glasman]], [[Denis Nardin]], [[Jay Shah]], _Parametrized higher category theory and higher algebra: Exposé I – Elements of parametrized higher category theory_, ([arXiv:1608.03657](https://arxiv.org/abs/1608.03657))

* {#Shah18} [[Jay Shah]], _Parametrized higher category theory and higher algebra: Exposé II - Indexed homotopy limits and colimits_, ([arXiv:1809.05892](https://arxiv.org/abs/1809.05892))

* [[Denis Nardin]], _Parametrized higher category theory and higher algebra: Exposé IV - Stability with respect to an orbital ∞-category_, ([arXiv:1608.07704](https://arxiv.org/abs/1608.07704))

A survey talk is

* [[Clark Barwick]], _Parametrized higher category theory and parameterized higher algebra_, [video](http://www.birs.ca/events/2016/5-day-workshops/16w5133/videos/watch/201602161030-Barwick.html)


Parametrized Homotopy Theory

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#### Homotopy theory
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[[!include homotopy - contents]]
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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
=--
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This entry collects links related to the book

* [[Peter May]], [[Johann Sigurdsson]],

  **Parametrized Homotopy Theory**,

  Mathematical Surveys and Monographs **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)

on [[parameterized stable homotopy theory]], hence on [[stable homotopy theory]] in [[slice (infinity,1)-toposes]] [[Top]]$/X$ for given topological base spaces $X$: the homotopy theory of [[ex-spaces]] and [[parametrized spectra]].

A survey is in the slides

* [[Peter May]], _What are parameterized spectra good for?_ ([pdf slides](http://www.math.uchicago.edu/~may/TALKS/MichiganPrint.pdf))

## Subtleties
 {#Subtleties}

Beware that section 4.4 claims a new proof of the [[Strøm model structure]], but relying on

* [[Michael Cole]], Prop. 5.3 in:  *Many homotopy categories are homotopy categories*, Topology and its Applications 153 (2006) 1084–1099 ([doi:10.1016/j.topol.2005.02.006](https://doi.org/10.1016/j.topol.2005.02.006))

which later was noticed to be false, by [[Richard Williamson]]; for details see p. 2 and Rem 5.12 and Sec. 6.1 in:

* {#BarthelRiehl13} [[Tobias Barthel]], [[Emily Riehl]], _On the construction of functorial factorizations for model categories_, Algebr. Geom. Topol. 13 (2013) 1089-1124 ([arXiv:1204.5427](http://arxiv.org/abs/1204.5427), [doi:10.2140/agt.2013.13.1089](http://dx.doi.org/10.2140/agt.2013.13.1089), [euclid:agt/1513715550](https://projecteuclid.org/euclid.agt/1513715550))

## Applications

One application is [[twisted cohomology]]: instead of cocycles given by maps $X \to A$, twisted cocycles are given by [[sections]] $X \to P$ of a bundle $P \to X$ of spectra over $X$.

A discussion of some of these issues using tools from [[(infinity,1)-category theory]] are in

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ , section 7: _Parametrized spectra, units and Thom spectra via $\infty$-categories_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))
 {#ABGHR}

See also

* [[John Lind]], _Bundles of spectra and algebraic K-theory_ ([arXiv:1304.5676](http://arxiv.org/abs/1304.5676))

A general abstract context for parameterized spectra are [[tangent (infinity,1)-toposes]].

category: reference


parametrized spectrum

+-- {: .rightHandSide}
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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Goodwillie calculus
+--{: .hide}
[[!include Goodwillie calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _parameterized spectrum_ is a [[bundle]] of [[spectra]] ([May-Sigurdsson 06](#MaySigurdsson06)), hence a [[stable homotopy type]] in [[parameterized homotopy theory]].

Specifically, for $X$ a [[homotopy type]] thought of as  an [[∞-groupoid]], then a spectrum parameterized over $X$ is equivalently an [[(∞,1)-functor]] $X \longrightarrow Spec$ from $X$ to the [[stable (∞,1)-category of spectra]] ([Ando-Blumberg-Gepner 11](#AndoBlumbergGepner11)): this assigns to each [[object]] of $X$ a [[spectrum]], to each [[morphism]] an [[equivalence in an (infinity,1)-category|equivalence]] of spectra, to each [[2-morphism]] a [[homotopy]] between such equivalences, and so forth.

More generally, given an [[(∞,1)-topos]] $\mathbf{H}$, then its [[tangent (∞,1)-topos]] $T\mathbf{H}$ is the [[(∞,1)-category]] of all [[spectrum objects]] in $\mathbf{H}$ parameterized over any object of $\mathbf{H}$ (see also at *[[Joyal locus]]*).

The intrinsic [[cohomology]] of such a [[tangent (∞,1)-topos]] of parameterized spectra is [[twisted generalized cohomology]] in $\mathbf{H}$, and generally is [[twisted bivariant cohomology]] in $\mathbf{H}$.

For more see also at _[[tangent cohesive (∞,1)-topos]]_.

## Properties

### As (co)module spectra

For $X$ a connected homotopy type, then the $X$-parameterized spectra are equivalently the [[module spectra]] over the [[∞-group ∞-ring]] $\mathbb{S}[\Omega X]$ of the [[∞-group]] corresponding to the [[loop space]].

To see this, use first that $X$-parameterized spectra are equivalently [[∞-functors]] of [[(∞,1)-categories]] of the form

$$
  B \Omega X \longrightarrow Spectra
$$

from the [[delooping]] [[∞-groupoid]] of $\Omega X$ to the [[(∞,1)-category of spectra]], then use that these are equivalenty Spectrum [[enriched functors]] out of the one-object spectrum enriched $\infty$-category with hom-spectrum $\mathbb{S}[\Omega X] \simeq \Sigma^\infty_+ \Omega X$.

Moreover $\mathbb{S}[\Omega X]$-[[module spectra]] are equivalent to [[comodule spectra]] over the coalgebra $\mathbb{S}[X] = \Sigma^\infty_+ X$ induced from the [[coalgebra object]] structure of $X$ in the [[Cartesian monoidal (∞,1)-category]] [[∞Grpd]] via the [[diagonal]] ([here](cartesian+monoidal+infinity%2C1-category#CoalgebraObjects)), and using that $\Sigma^\infty$ is a [[strong monoidal functor]]:

$$
   CoModSpectra_{\mathbb{S}[X]}
  \;\simeq\;
   ModSpectra_{\mathbb{S}[\Omega X]}
$$

([Hess-Shipley 14, theorem 1.2 with prop. 5.18](#HessShipley14))

See also at _[[A-theory]]_.

### Six operations yoga
 {#SixOperationsYoga}

For any map $f\colon X\longrightarrow Y $ between [[∞-groupoids]], the parameterized spectra form a [[Wirthmüller context]] version of the [[yoga of six functors]], in that

$$
  [X,Spectra]
    \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}}
  [Y,Spectra]
$$

in that $f^\ast$ is not only a [[strong monoidal functor]] but also a [[strong closed functor]], hence that [[Frobenius reciprocity]] holds.

Moreover, along ([[cospan|co-]])[[spans]] of morphisms pull-push $(f_!\dashv f^\ast)$ satisfies the [[Beck-Chevalley condition]] ([Hopkins-Lurie 14, prop. 4.3.3](#HopkinsLurie14)).

One way to summarize all this is to say that parameterized spectra over  [[∞Grpd]] constitute a [[linear homotopy type theory]] ([Schreiber 14](#Schreiber14)).

## Applications

In [[twisted cohomology]].

## Related concepts

* [[retractive space]]

* [[model structure for parameterized spectra]]

* [[excisive functor]]

* [[tangent (infinity,1)-category]], [[Joyal locus]]

* [[cartesian doubly closed monoidal (infinity,1)-category]]

* [[sheaf of spectra]]

* [[rational parameterized stable homotopy theory]]

* [[ex-space]]

* the [[K-theory]] of spectra parameterized over a connected $X$ is the _[[A-theory]]_ of $X$.

* [[twisted differential cohomology]]

## References
 {#References}

### Over a fixed base

> See also the further references at _[[(∞,1)-module bundle]]_.

Parameterized spectra over a fixed base (in any suitable [[model category]]) are discussed in

* {#Schwede97} [[Stefan Schwede]], section 3 of _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

A comprehensive textbook account of [[model structures for parameterized spectra]] in [[∞Grpd]] $\simeq$ $L_{whe}$[[Top]]:

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], _[[Parametrized Homotopy Theory]]_, Mathematical Surveys and Monographs **132**, AMS (2006) [[arXiv:math/0411656](https://arxiv.org/abs/math/0411656)]

* {#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)]

Alternative formulation 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 of the [[Beck-Chevalley condition]]:

* {#HopkinsLurie14} [[Michael Hopkins]], [[Jacob Lurie]], prop. 4.3.3 in: _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

Discussion of the [[Koszul duality]] between $\mathbb{S}[\Omega X]$-module spectra and $\mathbb{S}[X]$-comodule spectra is in

* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], _Waldhausen K-theory of spaces via comodules_, Advances in Mathematics 290 (2016): 1079-1137 ([arXiv:1402.4719](https://arxiv.org/abs/1402.4719))

### Over varying bases

Discussion of parameterized spectra over varying bases, hence of [[tangent (infinity,1)-category|tangent $\infty$-categories]]:

Discussion of suitable [[model structures for parameterized spectra]]:

* {#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]], 2018)

* {#HebestreitSagaveSchlichtkrull20} [[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)]

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 spectra]]

[[!redirects parameterized spectrum]]
[[!redirects parameterized spectra]]

[[!redirects parametrised spectrum]]
[[!redirects parametrised spectra]]

[[!redirects spectrum bundle]]
[[!redirects spectrum bundles]]

[[!redirects bundle of spectra]]
[[!redirects bundles of spectra]]


parastatistics

# Parastatistics
* table of contents
{: toc}

## Related concepts

* [[braid group statistics]]

## References

* H.S. Green, _A generalized method of field quantization_,  Phys. Rev. __90__, 270–273 (1953)

* wikipedia: [parastatistics](http://en.wikipedia.org/wiki/Parastatistics), [Klein transformation](http://en.wikipedia.org/wiki/Klein_transformation)

* Y. Ohnuki, S. Kamefuchi, _Quantum field theory and parastatistics paperback_, Springer 2011, 508 pp.

* [[Jean-Louis Loday]], Todor Popov, _Parastatistics algebra and super semistandard Young tableaux_, Lie Theory and Its Applications in Physics VII, ed. V.K. Dobrev et al, Heron Press, Sofia 2008, <a href="http://www-irma.u-strasbg.fr/~loday/PAPERS/2008LodayPopov(Parastat).pdf">

* Weimin Yang, Sicong Jing, _Graded Lie algebra generating of parastatistical algebraic atructure [math-ph/0212009](arxiv.org/pdf/math-ph/0212009)

* Tolstoy, V. N., _Algebraic structure of mixed parastatistics_, Proc. XV Int. Conf. on Symmetry Methods in Physics, []()

* Lee Brekke, Sterrett J. Collins, Tom D. Imbo, _Nonabelian vortices on surfaces and their statistics_, [hep-th/9701056](arxiv.org/pdf/hep-th/9701056)

> If we could "turn off" all interactions
in the universe which couple to spin, then different spin states would be in principle
indistinguishable (although still not identical), and the particles would obey parastatistics
at the fundamental level. We can play the same game with color, isospin and numerous
other internal degrees freedom. (Indeed, the parastatistical treatment of quarks [3] predates
the introduction of color [4].)

* Fred Cooper, Avinash Khare, Uday Sukhatme, _Parasupersymmetry in quantum mechanics_, J. Math. Phys. 1993

category: physics


paraunital algebra

\tableofcontents

## Idea

Given a commutative ring $R$, a **paraunital $R$-algebra** is a [[nonunital algebra|nonunital $R$-algebra]] $A$ where there is an element $\iota \in A$ and an [[involution]] $x \mapsto \overline{x}$ such that $x \cdot \iota = \iota \cdot x = \overline{x}$ for all $x \in A$. A paraunital $\mathbb{Z}$-algebra is also called a **paraunital ring**.

A [[unital algebra]] is a paraunital algebra in multiple different ways:

* where the element is given by the unit $1$ and the involution is given by the [[identity function]] $x \mapsto x$.

* where the element is given by the negation of the unit $-1$ and the involution is given by negation $x \mapsto -x$

These were first defined in the specific context of [[composition algebras]] in the [[generalized Hurwitz theorem]] in [Elduque 2021](#Elduque21) but could be generalized from composition algebras to any $R$-algebra.

## Generalizations

This concept could be generalized from the category of $R$-modules to any [[monoidal category]]:

A **paraunital algebra object** in a [[monoidal category]] $(C, I, \otimes)$ is an object $A \in C$ with morphisms $\iota:I \to A$, $\pi:A \otimes A \to A$, and $j:A \to A$ such that $j \circ j = \mathrm{id}_A$ and the following [[commuting diagram|diagrams commute]]:

   $$
     \array{
       I \otimes A
         &\overset{\iota \otimes \mathrm{id}_A}{\longrightarrow}&
       A \otimes A
       \\
       \downarrow^{\lambda_A} & & \downarrow^{\pi}
       \\
       A &\overset{j}{\longrightarrow}& A
     }
     \,,
   $$

   $$
     \array{
       A \otimes I
         &\overset{\mathrm{id}_A \otimes \iota}{\longrightarrow}&
       A \otimes A
       \\
       \downarrow^{\rho_A} & & \downarrow^{\pi}
       \\
       A &\overset{j}{\longrightarrow}& A
     }
     \,,
   $$

where $\lambda_A:I \otimes A \to A$ and $\rho_A:A \otimes I \to A$ are the left and right [[unitors]] of the [[monoidal category]].

In the [[category of sets]], paraunital algebra objects are called **paraunital magmas**, and in the [[category of abelian groups]], paraunital algebra objects are called **paraunital rings**.

## Related concepts

* [[magma]], [[unital magma]]

* [[nonunital algebra]], [[nonunital ring]]

* [[composition algebra]]

* [[generalized Hurwitz theorem]]

## References

* {#Elduque21} [[Alberto Elduque]], *Composition algebras*, Chapter 2 in:  Abdenacer Makhlouf (ed.), *Algebra and Applications I: Non-associative Algebras and Categories*, Sciences-Mathematics, ISTEWiley (2021) 27-57 [[arXiv:1810.09979](https://arxiv.org/abs/1810.09979), [ISBN:978-1-789-45017-0](https://www.wiley.com/en-ae/Algebra+and+Applications+1:+Non+associative+Algebras+and+Categories-p-9781789450170)]

[[!redirects paraunital algebra]]
[[!redirects paraunital algebras]]
[[!redirects para-unital algebra]]
[[!redirects para-unital algebras]]

[[!redirects paraunital ring]]
[[!redirects paraunital rings]]
[[!redirects para-unital ring]]
[[!redirects para-unital rings]]

[[!redirects paraunital algebra object]]
[[!redirects paraunital algebra objects]]
[[!redirects para-unital algebra object]]
[[!redirects para-unital algebra objects]]

[[!redirects paraunital magma]]
[[!redirects paraunital magmas]]
[[!redirects para-unital magma]]
[[!redirects para-unital magmas]]


parent action functional

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#### Physics
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[[!include physicscontents]]
=--
#### Duality
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[[!include duality - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] one speakes of a _parent action functional_ ([Deser-Jackiw 84](#DeserJackiw84)) when referring to an [[action functional]] $S_{par}$ which, by solving the [[equations of motion]] of two different sets of its "[[auxiliary fields|auxiliary]]" [[field (physics)|fields]], is classically equivalent to two different-looking action functionals $S_A$ and $S_B$, thereby exhibiting $S_A$ and $S_B$ as classically equivalent to each other. Hence a parent action functional constitutes a [[correspondence]] between two action functionals

$$
  \array{
    && S_{par}
    \\
    & {}^{\mathllap{\simeq_{clas}}}\swarrow &&  \searrow^{\mathrlap{\simeq_{clas}}}
    \\
    S_A && && S_B
  }
  \,.
$$

The method is used notably to exhibit "[[duality in physics|duality]]" between $S_A$ and $S_B$ of one form or another, in particular [[electric-magnetic duality]]. (See e.g. [Miao-Ohta 03](#MiaOhta03)).

Beware that some authors also speak of _master actions_ instead of _parent actions_ in this context, but there is not relation to "master actions" in the sense of [[BV-BRST formalism]] (except, of course, that both are [[action functionals]] of some sort).

## References

* {#DeserJackiw84} [[Stanley Deser]], [[Roman Jackiw]], Phys. Lett. B, 139 (1984), p. 2366

* {#MiaOhta03} Yan-Gang Miao, Nobuyoshi Ohta, _Parent Actions, Dualities and New Weyl-invariant Actions of Bosonic p-branes_ ([arXiv:hep-th/0301233](http://arxiv.org/abs/hep-th/0301233))

[[!redirects parent action functionals]]

[[!redirects parent action]]
[[!redirects parent actions]]


parenthesized braid operad

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=--
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#Contents#
* table of contents
{:toc}

## Idea

The _parenthesized braid operad_ is an [[operad]] in [[Grpd]] modelled on the [[braid group]].

## Definition

Let $PaB_n$ denote the category defined as follows:

* its set of [[objects]] is the [[free construction|free]] [[magma]] on one generator, or equivalently the set of [[rooted binary tree]]s.

* the set of morphisms between two objects $s,t$ is given by the [[braid group]] $B_n$ whenever $s$ and $t$ are words of the same legnth $n$, and is [[empty set|empty]] otherwise.

Then the collection $PaB$ of the $PaB_n$'s is a [[braided operad]].

The [[composition]]

$$\circ_i:PaB_n \times PaB_m \rightarrow PaB_{m+n-1}$$

is given by replacing the $i$th strand of the first braid, by the second braid made very thin.

$PaB$ also carries an obvious structure of a [[braided monoidal category]]. In fact:

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

$PaB$ is the [[free object|free]] [[braided monoidal category]] on one object. As a consequence, it is an [[initial object]] in the [[category]] of
[[braided monoidal categories]].

=--

## Colored/ordered version

let $CPaB_n$ be the [[groupoid]] defined as follows:

* it set  [[objects]] are parenthesized [[permutations]] of $\{1,\dots,n\}$, that is non-associative, non-commutative monomials on this set in which every letter appears exactly once.

* [[morphisms]] between two objects $s,t$ are braids connecting each letter in $s$ to the same letter in $t$. In other words, let $p:B_n\rightarrow S_n$ be the canonical projection from the braid group to the symmetric group whose kernel is the pure braid group. Then, forgetting the parenthesization and viewing $s,t$ as permutations:

$$Hom(s,t)=p^{-1}(\{s^{-1}t\})$$

Then $CPaB$ is an (ordinary) [[operad]], the operadic structure being the same as for the non-colored version.

A topological interpretation of $CPaB$ is as follows:

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

$CPaB$ may be identified with a full sub-operad of the [[fundamental groupoid]] of the [[little 2-disk operad]].

=--

## References

$PaB$ was originally defined in

* [[Dror Bar-Natan]], _On Associators and the Grothendieck-Teichmuller Group I_ [[arXiv:q-alg/9606021](http://arxiv.org/abs/q-alg/9606021)]

The operad structure was pointed out in

* [[Dmitry Tamarkin]], _Formality of Chain Operad of Little Discs_, [Letters in Mathematical Physics 66:65–72, 2003.](https://sites.math.northwestern.edu/~tamarkin/Papers1/Formality.pdf)


pariah group

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+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _pariah groups_ ([Griess 82](#Griess82)) constitute the [[finite set]] of those 6 [[sporadic finite simple groups]], which are _not_ [[subquotients]] of the [[monster group]], in contrast to the remaining 20 sporadic groups that constitute the [[Happy Family]].

The pariah groups are: [[J1]], [[J3]], [[J4]] (three of the four Janko groups), [[Ru]] (the Rudvalis group), [[ON]] (the O'Nan, or O'Nan–Sims, group), [[Ly]] (the Lyons group). Coincidentally, $J_1$ and $J_4$ were the first and last sporadic groups discovered (after the five [[Mathieu groups]] in the 19th century).

## References

* {#Griess82} [[Robert Griess]], Jr. _The Friendly Giant_, Invent. Math. 69 (1982) 1-102 ([doi:10.1007/BF01389186](https://doi.org/10.1007/BF01389186))

See also

* Wikipedia, _[Pariah group](https://en.wikipedia.org/wiki/Pariah_group)_

[[!redirects pariah groups]]


parity

The term _parity_ means different things in different contexts:

## In superalgebra

In discussion of [[homological algebra|homological]] [[superalgebra]], _parity_ refers to the total $\mathbb{Z}/2$-[[graded object|grading]] which in [Bernstein's convention](signs+in+supergeometry#SuperOddConvention) controls the [[signs in supergeometry|sign rule]]. See at _[[chain complex of super vector spaces]]_ the remark [on parity](chain+complex+in+super+vector+spaces#Parity).

## In particle physics

(...)

* [[CP violation]]

* [[PCT theorem]]

(...)

[[!redirects parities]]


parity complex

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=--
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# Contents
* table of contents
{: toc}

## Idea

The notion of _parity complex_, introduced by [[Ross Street]], is a notion of [[pasting diagram]] shape. It is based on some combinatorial axioms on subshapes of codimension at most 2 which permit the construction of a (strict) $\omega$-[[strict omega-category|category]] [[free construction|freely generated]] from the shape.

## Definition

+-- {: .num_defn}
###### Definition
A **parity structure** is a [[graded set]] $\{C_n\}_{n \geq 0}$ together with, for each $n \geq 0$, functions
$$\partial^+_n \colon C_{n+1} \to P(C_n), \qquad \partial^-_n \colon C_{n+1} \to P(C_n);$$
we assume throughout this article that $\partial^+_n(c)$, $\partial^-_n(c)$ are finite, nonempty, and disjoint.
=--

Following Street, we abbreviate $\partial^+_n(c)$ to $c^+$, and $\partial^-_n(c)$ to $c^-$. The Greek letters $\varepsilon$, $\eta$ refer to values in the set $\{+, -\}$.

+-- {: .num_defn}
###### Definition
A parity structure is a **parity complex** if it satisfies the following axioms:

1. $c^{--} \cup c^{++} = c^{-+} \cup c^{+-}$

1. If $c \in C_1$, then $c^-$ and $c^+$ are both singletons.

1. If $x, y \in c^\eta$ are distinct $n$-cells, then $x^+ \cap y^+ = \emptyset$ and $x^- \cap y^- = \emptyset$.

1. Define a [[relation]] $\lt$ by $x \lt y$ whenever $x^+ \cap y^- \neq \emptyset$, and let $\prec$ be the reflexive transitive closure of $\lt$. Then $\prec$ is antisymmetric, and if $x \prec y$ for $x \in c^\varepsilon$ and $y \in c^\eta$, then $\varepsilon = \eta$.
=--

## Examples

* [[simplex|Simplexes]]

* [[globe|Globes]]

* [[cube|Cubes]]

* Products of parity complexes

* [[polyhedron|Polyhedra]]

## Basic results

(...)

## Related notions

* [[directed complex]]

* [[direct category]]

* [[test category]]

* [[oriental]]

* [[pasting scheme]]

## References

* [[Ross Street]], _Parity complexes_, Cahiers Top. Géom Diff. Catégoriques 32 (1991), 315-343. ([link](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_4/CTGDC_1991__32_4_315_0/CTGDC_1991__32_4_315_0.pdf)) Corrigenda, Cahiers Top. Géom Diff. Catégoriques 35 (1994), 359-361. ([link](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1994__35_4/CTGDC_1994__35_4_359_0/CTGDC_1994__35_4_359_0.pdf))

[[!redirects parity complexes]]


Parmenides dialogue

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=--
=--
=--

This page collects material related to

* [[Plato]]

  _Parmenides_

  370 BC

  English translation by Benjamin Jowett, 1871

  ([plain text](http://www.sacred-texts.com/cla/plato/parmeni.htm))

Named after the Eleatic philosopher, the **Parmenides** is one of the later and most influential dialogues of [[Plato]]. The first part concerns the [[doctrine of ideas]], the second part concerns a [[dialectic]] between "the one" and "the many", and is commonly perceived as being mysterious (for a dialetheic interpretation see [Priest 2012](#Priest12)). It concludes as follows:

> $[$...$]$ whether one is or is not, one and the others in
relation to themselves and one another, all of them, in every way, are
and are not, and appear to be and appear not to be.

> Most true.

The Parmenides dialogue is also the main source on _[[Zeno]]_.

#Contents#
* table of contents
{:toc}

## Hegel's interpretation

The Parmenides dialogue has influenced, and its topics (and mysterious style) have been picked up, by [Hegel WdL](#HegelWdL), eg. in [§357](Science%20of%20Logic#357).

> Hegel's interpretation of Plato's Parmenides during the early Jena period focuses largely on its methodological value as a radical exercise in negative skepticism, and, as such, as introduction to proper philosophizing. In his Relation of Skepticism to Philosophy, for example, Hegel characterizes Plato's dialogue as exhibiting "the negative side of the knowledge of the absolute" (die negative Seite der Erkenntnis des Absoluten). According to this interpretation, the dialogue's role in the history of philosophy is twofold. On the one hand, the negative dialectic of ideas that constitutes its backbone would exhibit the inadequacy of the understanding to provide true cognition. By showing that concepts (here understood as "determinations of the understanding [Verstandes-bestimmungen]") like similar and dissimilar, older and younger, continuous and discrete, or, more crucially, same and other, are intimately connected with their respective contradictory, Plato would demonstrate that to deny or attribute these opposites simultaneously to "finite" or "badly infinite," that is, non-self-reflexive, objects of thinking leads to utter unintelligibility. On the other hand, Hegel believes also that the dialogue works as indirect proof of the validity of a different cognitive mode - namely, reason - that Plato intends to display and account for in a separate trilogy: the Sophistes, the Politicos, and, in definitive form, the Philosophos. The Parmenides would then tacitly imply that "truly infinite," self-reflexive objects of thinking may actually be made intelligible precisely by the dialectic contradictions of which the dialogue shows only the negative results. ... ([de Laurentiis 04](#deLaurentiis04))

## Related entries

* [[Timaeus dialogue]]

* [[Republic dialogue]]

## References

* Wikipedia _[Parmenides (dialogue)](http://en.wikipedia.org/wiki/Parmenides_%28dialogue%29)_

* Stanford Encyclopedia of Philosophy, _[Plato's Parmenides](http://plato.stanford.edu/entries/plato-parmenides/)_

* {#HegelWdL} [[Georg Hegel]], _[[Science of Logic]]_

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]]_

  -- _[Eleatic school -- Parmenides](#Lectures+on+the+History+of+Philosophy#Parmenides)_

  -- _[Plato -- Dialectic -- Parmenides dialogue](Lectures+on+the+History+of+Philosophy#ParmenidesDialogue)_

* {#deLaurentiis04} A. de Laurentiis, _On Hegel's reading of Plato's Parmenides_, 2004

* {#Priest12} G. Priest, _The Parmenides: a Dialetheic Interpretation_,  Plato J.Int. Plato Soc.**12** (2012)([link](http://gramata.univ-paris1.fr/Plato/article120.html))

[[!redirects Parmenides]]

[[!redirects Parmenides (dialogue)]]

category:reference


Parsa Bonderson

* [homepage](https://www.microsoft.com/en-us/research/people/parsab/)

## Selected writings

On potential [[quantum measurement]] of non-abelian [[anyons]]:

* {#BondersonShtengelSlingerland07} [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Decoherence of Anyonic Charge in Interferometry Measurements*, Phys. Rev. Lett. **98** (2007) 070401 [[doi:10.1103/PhysRevLett.98.070401](https://doi.org/10.1103/PhysRevLett.98.070401), [arXiv:quant-ph/0608119](https://arxiv.org/abs/quant-ph/0608119)]

* [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Interferometry of non-Abelian Anyons*, Annals Phys. **323** (2008) 2709-2755 [[doi:10.1016/j.aop.2008.01.012](https://doi.org/10.1016/j.aop.2008.01.012), [arXiv:0707.4206](https://arxiv.org/abs/0707.4206)]

On [[symmetry protected topological phases]] and their [[topological orders]]:

* {#BBCW19} [[Maissam Barkeshli]], [[Parsa Bonderson]], [[Meng Cheng]], [[Zhenghan Wang]], *Symmetry Fractionalization, Defects, and Gauging of Topological Phases*, Phys. Rev. B **100** 115147 (2019) $[$[arXiv:1410.4540](https://arxiv.org/abs/1410.4540), [doi:10.1103/PhysRevB.100.115147](https://doi.org/10.1103/PhysRevB.100.115147)$]$

On [[anyon|anyonic]] [[topological order]] expressed via [[unitary fusion categories]]/[[modular tensor categories]]:

* [[Parsa Bonderson]], *Measuring Topological Order*, Phys. Rev. Research **3**, 033110 (2021) $[$[arXiv:2102.05677](https://arxiv.org/abs/2102.05677), [doi:10.1103/PhysRevResearch.3.033110](https://doi.org/10.1103/PhysRevResearch.3.033110)$]$

category: people


Parseval's theorem

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### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--

#Contents##
* table of contents
{:toc}

## Idea

In [[harmonic analysis]], _Parseval' theorem_ states that [[Fourier transform]] is a [[unitary operator]].

For Fourier transform over [[Cartesian spaces]] see e.g. [Hörmander 90, theorem 7.1.6](#Hoermander90), [this prop.](BasicPropertiesOfFourierTransformOverCartesianSpaces).

## Related theorems

* [[Fourier inversion theorem]]

* [[Paley-Wiener-Schwartz theorem]]

## References

For Fourier transform over [[Cartesian spaces]]:

* {#Hoermander90} [[Lars Hörmander]], theorem 7.1.6 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

See also

* Wikipedia, _[Parseval's theorem](https://en.wikipedia.org/wiki/Parseval%27s_theorem)_

[[!redirects Parseval theorem]]

[[!redirects Parseval's formula]]

[[!redirects Parseval formula]]


partial combinatory algebra

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### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A partial combinatory algebra is a generalization of a model of the [[untyped lambda calculus]] allowing for application to be only partially defined.  It is often expressed as an algebraic abstraction of [[combinatory logic]].

## Definition

The following definitions are taken from [Hofstra](#Hofstra).

A _partial applicative structure_ is a set equipped with a partial binary operation

$$m \colon A \times A \to A$$

An [[function application|application]] of $m$ is indicated by juxtaposition: $a b$ denotes $m(a, b)$ if (and only if) $m(a, b)$ is defined, i.e., if $(a, b)$ belongs to the [[domain]] of $m$. _Homomorphisms_ between partial applicative structures $A, B$ are defined to be total functions $f \colon A \to B$ such that $f(a b) = f(a)f(b)$, meaning the right side is defined whenever the left side is. In other words, in the locally posetal [[bicategory]] of sets and [[relation|relations]], we have a 2-cell

$$\array{
A \times A & \stackrel{f \times f}{\to} & B \times B \\
m \downarrow & \leq & \downarrow m \\
A & \underset{f}{\to} & B
}$$

By convention, unbracketed juxtapositions are associated to the left, so that for example $x y z$ means $(x y)z$.

If $A$ is a partial applicative structure and $\mathbf{Magma}(A)$ is the free [[magma]] on the underlying set of $A$, then we have an evident partial function

$$\alpha \colon \mathbf{Magma}(A) \to A$$

which evaluates binary words in $A$ as elements in $A$, whenever possible, using the partial applicative structure and induction on the structure of the word.

+-- {: .num_defn}
###### Definition
A partial applicative structure $A$ is a _partial combinatory algebra_ (a **PCA**) if there are elements $k, s \in A$ such that

* For all $a, b \in A$, $k a$ and $k a b$ are defined and $k a b = a$.
* For all $a, b \in A$, $s a$ and $s a b$ are defined and for all $c$, $(a c)(b c)$ is defined if and only if $s a b c$ is defined, and $s a b c = (a c)(b c)$.

A _homomorphism_ of PCA's is a homomorphism of the underlying partial applicative structures.

A _total_ combinatory algebra is a PCA $A$ whose application $m \colon A \times A \to A$ is a total function.
=--

Note that appropriate elements $k$, $s$ are not considered part of the structure (to be preserved under homomorphism) -- here one is interested only in the property that such elements exist. Indeed, they need not be uniquely determined within the PCA.

## Functional completeness

The definition of PCA given above is traditional but somewhat opaque at first glance. The real point of the definition is that a PCA is the same thing as a partial applicative structure $A$ that enjoys a _functional completeness_ property, in the following sense.

Let $X$ be a countably infinite set of "variables", and let $A[X]$ denote the [[magma]] freely generated from the disjoint sum $A + X$. Each term $t$, i.e., each element $t \in A[X]$, has finitely many $x_i \in X$ occurring inside it; these are called the _free variables_ of $t$, and we write $FV(t)$ for the set of free variables. If $FV(t) \subseteq \{x_0, x_1, \ldots, x_n\}$, then we can think of $t$ as giving a partially defined function in $n+1$ variables. That is, we may consider $t$ as a term of $\mathbf{Magma}(A + \{x_0, \ldots, x_n\})$, and (partially) define the [[substitution]] $t[a_0/x_0, \ldots, a_n/x_n]$ where $a_0, \ldots, a_n$ are elements of $A$, by evaluating at $t$ of the composite

$$\mathbf{Magma}(A + \{x_0, \ldots, x_n\}) \stackrel{\mathbf{Magma}(\phi)}{\to} \mathbf{Magma}(A) \stackrel{\alpha}{\to} A$$

where $\phi(x_i) = a_i$ and is elsewhere the identity.

Then, we say $A$ is _functionally complete_ if every term $t \in A[X]$, viewed as a partial operation on $A$, is represented by some element in $A$. To be precise: for each term $t$ with $FV(t) \subseteq \{x_0, \ldots, x_n\}$, there exists an element $a \in A$ such that

* For all $a_0, \ldots, a_{n-1} \in A$, $a a_0 \ldots a_{n-1}$ is defined;

* For all $a_0, \ldots, a_n \in A$, $t[a_0/x_0, \ldots, a_n/x_n]$ is defined precisely when $a a_0 \ldots a_n$ is defined, and these two elements are equal.

For example, if $k$, $s$ are elements which realize $A$ as a PCA, then $k$ represents the term $t = x_0$ viewed as belonging to $\mathbf{Magma}(A + \{x_0, x_1\})$, and $s$ represents the term $(x_0 x_2)(x_1 x_2)$ viewed as belonging to $\mathbf{Magma}(A + \{x_0, x_1, x_2)\}$.

+-- {: .num_theorem}
###### Theorem
A partial combinatory algebra is functionally complete.
=--

+-- {: .proof}
###### Proof
The idea is to simulate $\lambda$-abstraction $\lambda x. t$, where $x \in X$ is a variable and $t$ is a term, by induction on $t$. Indeed, given elements $k, s$ that realize $A$ as a PCA, put

$$\array{
\lambda x. x & \coloneqq & s k k & \\
\lambda x. t & \coloneqq & k t & if  x \notin FV(t) \\
\lambda x. t t' & \coloneqq & s(\lambda x. t)(\lambda x. t')
}$$

One then proves equality of partial functions $(\lambda x. t)(a) = t[a/x]$, by induction on $t$.
=--

[Hofstra](#Hofstra) _defines_ a PCA to be a functionally complete partial applicative structure. This is an unbiased definition, in the sense that it does not privilege certain elements $k$, $s$ (or, for that matter, any other system of combinators that provide functional completeness).

### Examples of combinators
 {#ExamplesOfCombinators}

1. Let us check that $s k k$ indeed represents the identity function $I$. This is trivial: we have, for any $a \in A$, equalities between defined terms
$$s k k a = (k a)(k a) = a.$$

1. Consider the second projection function, corresponding to $x_1 \in \mathbf{Magma}(A + \{x_0, x_1\})$. Thinking in terms of $\lambda$-terms, this is represented by $\lambda x. \lambda y. y = \lambda x. s k k = k(s k k)$, or $k I$. In other words, we calculate
$$k I a b = ((k I) a) b = I b = b.$$

1. Following the proof of functional completeness, we have
$$\lambda x. x x = s(\lambda x. x)(\lambda x. x) = s I I$$

1. {#FinallyClassicalConstruction} Finally, consider the classical construction of the [[fixed-point combinator]], $Y = \lambda y. (\lambda x. y(x x))(\lambda x. y(x x))$. We have firstly
$$\lambda x. y(x x) = s(\lambda x. y)(\lambda x. x x) = s(k y)(s I I)$$
which means
$$\array{
Y & = & \lambda y. (s I I)(s(k y)(s I I)) \\
 & = & s(\lambda y. s I I)\big(\lambda y. s(k y)(s I I)\big) \\
 & = & s(k (s I I)) \big(s(\lambda y. s(k y))(\lambda y. s I I)\big) \\
 & = & s(k (s I I)) \big(s(s(k s)k)(k (s I I))\big)
}$$

## Examples of PCAs

1. (Kleene's first algebra.) Suppose given a coding of all [[partial recursive functions]] of the form  $\mathbb{N}^k \to \mathbb{N}$
(ranging over $k = 0, 1, 2, \ldots$) by elements of $\mathbb{N}$, and a coding of elements of $\mathbb{N}^k$ over all $k$ by elements of $\mathbb{N}$. Define a partial applicative structure
$$\mathbb{N} \times \mathbb{N} \to \mathbb{N}$$
where $p q = \{p\}([q])$ whenever the right side is defined, where $\{p\}$ is the $p^{th}$ partial recursive function and $[q]$ is the $q^{th}$ tuple. It may be checked that $k$ and $s$, defined extensionally in the obvious way, are partial recursive functions, so we get in this way a PCA.

1. Suppose we have a [[cartesian closed category]] generated by an object $X$ such that $X \times X$ and $X^X$ are retracts of $X$. Then elements $1 \to X$ form a PCA, in fact a total combinatory algebra, where the applicative structure is
$$X \times X \stackrel{r \times 1_X}{\to} X^X \times X \stackrel{eval}{\to} X$$
for some chosen retraction $r$. In other words, models of the untyped [[lambda calculus]] give PCA's.

1. [[Kleene's second algebra]].

## Realizability toposes

From any PCA, a corresponding "[[realizability]] [[tripos]]" can be constructed, from which, in turn, a corresponding "realizability topos" can be constructed, as outlined in the article on [[tripos|triposes]].

A preliminary technical task is to encode pairing and unpairing functions by elements of $A$. By this we mean functions $p \colon A \times A \to A$, $l \colon A \to A$, $r \colon A \to A$ such that for all $(a, a') \in A \times A$, we have $(a, a') = (l(p(a, a')), r(p(a, a')))$. One way of doing this is to put

* $p = \lambda x. \lambda y. \lambda z. z x y$

* $l = \lambda p. p(\lambda x. \lambda y. x)$

* $r = \lambda p. p(\lambda x. \lambda y. y)$

whereupon we may calculate

$$\array{
p a a' & = & \lambda z. z a a' \\
l(p a a') & = & (\lambda z. z a a')(\lambda x. \lambda y. x) \\
 & = & (\lambda x. \lambda y. x) a a' \\
 & = & (\lambda y. a) a' = a
}$$

$$\array{
r(p a a') & = & (\lambda z. z a a')(\lambda x. \lambda y. y) \\
 & = & (\lambda x. \lambda y. y) a a' \\
 & = & (\lambda y. y) a' = a'
}$$

That out of the way, let $P(A)$ be the power set of $A$ and let $X$ be a set.
Put a preorder structure on $P(A)^X$ as follows: given $f, g \in P(A)^X$, let $Hom(f, g)$ be the set of $a$ in $A$ such that for all $x$ in $X$ and $a'$ in $f(x)$, $a a'$ is defined (that is, $a$ is applicable to $a'$), and $a a'$ is an element of $g(x)$. We can turn this into a preorder by taking $f \leq g$ just in case $Hom(f, g)$ is inhabited.

+-- {: .num_theorem}
###### Theorem
The preorder $P(A)^X$ has finite products, finite coproducts, and is cartesian closed. In other words, the preorder $P(A)^X$ is a [[Heyting prealgebra]].
=--

+-- {: .proof}
###### Proof
An initial object is given by the constant function taking each $x$ to the empty subset $\emptyset \subseteq A$, and a terminal object is given by the constant function taking each $x$ to the full subset $A \subseteq A$.

Take $f, g \colon X \to P(A)$. For binary products, define

$$(f \wedge g)(x) = \{p a a' | a \in f(x) \wedge a' \in g(x)\}$$

Notice that $l$ realizes $f \wedge g \leq f$ (since $l (p a a') = a$), and similarly $r$ realizes $f \wedge g \leq g$. Furthermore, suppose given $h \colon X \to P(A)$, and that $t \in A$ realizes $h \leq f$ and $u \in A$ realizes $h \leq g$. Then

$$v = \lambda b. p (t b)(u b)$$

realizes $h \leq f \wedge g$. Thus $f \wedge g$ is a product in the preorder.

For binary coproducts, define

$$(f \vee g)(x) = \{p l a | a \in f(x)\} \cup \{p r a' | a' \in g(x)\}$$

Then $p l$ realizes $f \leq f \vee g$ and $p r$ realizes $g \leq f \vee g$. Furthermore, suppose $t$ realizes $f \leq h$ and $u$ realizes $g \leq h$. Then

$$v = \lambda b. l(b)(p(t(r b))(u(r b)))$$

realizes $f \vee g \leq h$. Indeed, we have

$$\array{
v(p l a) & = & l(p l a)(p(t(r(p l a)))(u(r(p l a)))) \\
 & = & l(p(t a)(u a)) \\
 & = & t a
}$$

and similarly $v(p r a') = u a'$. In either case, we see $v(b)$ lives in $h(x)$ for any $b \in (f \vee g)(x)$.

For cartesian closure, define

$$(f \Rightarrow g)(x) = \{a' | \forall a \in f(x) a' a \downarrow \wedge a' a \in g(x)\}$$

where $a' a \downarrow$ is standard shorthand for "$a' a$ is defined". Then for any $f$ and $g$, the combinator $\lambda b. l(b)r(b)$ realizes $(f \Rightarrow g) \wedge f \leq g$, and the combinator $\lambda b. \lambda a. p a b$ realizes $g \leq f \Rightarrow (f \wedge g)$.
=--

For any function $f \colon X \to Y$, it is immediate that

$$P(A)^f \colon P(A)^Y \to P(A)^X$$

is a functor preorder map that preserves Heyting prealgebra structure.

Furthermore, in the case of a projection map $f \colon Z \times Y \to Y$, there will be left and right adjoints to $P(A)^f \colon P(A)^Y \to P(A)^{Z \times Y} (\cong (P(A)^Z)^Y)$, as induced by the union and intersection maps from $P(A)^Z$ to $P(A)$.

## Related concepts

* [[combinatory logic]]
* [[realizability]]
* [[Turing category]]

[[!include computable mathematics -- table]]

## References

* {#Hofstra} Pieter J.W. Hofstra, _Partial Combinatory Algebras and Realizability Toposes_,
([web](http://mysite.science.uottawa.ca/phofstra/)) ([pdf](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CF8QFjAB&url=http%3A%2F%2Fmysite.science.uottawa.ca%2Fphofstra%2FKananaskis.pdf&ei=georUITcEebV0QGctIGwDg&usg=AFQjCNFywYRYAfzrgHFq5AsaBZ5li_qfyQ&sig2=BCT05ts3GBLntD2VZazC4w))

Lecture notes include

* {#Bauer05} [[Andrej Bauer]], section 2 of _Realizability as connection between constructive and computable mathematics_, in T. Grubba, P. Hertling, H. Tsuiki, and [[Klaus Weihrauch]], (eds.) _CCA 2005 - Second International Conference on Computability and Complexity in Analysis_, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. ([pdf](http://math.andrej.com/data/c2c.pdf))

[[!redirects combinatory algebra]]

[[!redirects partial combinatory algebra]]
[[!redirects partial combinatory algebras]]

[[!redirects Kleene's first algebra]]
[[!redirects Kleene's first partial combinatory algebra]]

[[!redirects realizability tripos]]

[[!redirects PCA]]
[[!redirects PCAs]]


partial differentiation

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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=--
=--

# Partial differentiation
* table of contents
{: toc}

## Idea

When a [[multifunction]] is differentiated with respect to any one of its [[variable|arguments]] alone, holding the others fixed, then we are engaged in _partial_ [[differentiation]].

## Definition

Very generally, let $(X_i)_i$ be a [[family]] of [[differentiable spaces]] (in some sense), let $Y$ be another such space, and let $f$ be a [[differentiable map]] to $Y$ from a [[subspace]] $U$ of the [[cartesian product]] $\prod_i X_i$.  Let $d$ be a relevant [[differential]] or [[derivative]] operator, and let $x_i$ be the composite
$$ U \hookrightarrow \prod_i X_i \twoheadrightarrow X_i $$
of the [[inclusion map]] of $U$ and the $i$th [[product projection]] (the $i$th [[coordinate]]).  Then under good conditions, we have
$$ d{f} = \sum_i \partial_i{f} \,d{x_i} $$
for a unique family $(\partial_i{f})_i$ of [[linear operators]], the __partial derivatives__ of $f$ with respect to this decomposition of $U$.  The term $\partial_i{f} \,d{x_i}$, which may be denoted $d_i{f}$, is similarly a __partial differential__ of $f$.

More precisely, we choose a [[category]] of [[differentiable spaces]] and differentiable maps between them, on which there is an [[endofunctor]] that takes each space $U$ to a notion of [[tangent bundle]] $T{U}$, which is assumed to be a [[vector bundle]] over $U$, and takes a map $f\colon U \to Y$ to $d{f}\colon T{U} \to T{Y}$.  (Note that this isn't the case for [[generalised smooth spaces]], but we could take [[microlinear spaces]], as well as more familiar examples such as [[differentiable manifolds]].)  Then $d{x_i}\colon T{U} \to T{X_i}$, $\partial_i{f}_p\colon T_{x_i(p)}{X_i} \to T_{f(p)}{Y}$ is a linear operator between [[stalks]] (for $p$ a [[point]] in $U$), and the sum takes place in the [[vector space]] $T_{f(p)}{Y}$.

We can extend this if we work in a [[cartesian closed category]] of [[generalised smooth spaces]].  As in the above, let $(X_i)_{i \in I}$ be a [[family]] of [[smooth spaces]] and $Y$ another smooth space.  For simplicity, let $f \colon \prod_i X_i \to Y$ be a smooth map (aka [[morphism]] in the category) defined on the whole product (so we take $U = \prod_i X_i$ in the above).  For $i_0 \in I$ we can use the cartesian closed structure to define a morphism

$$
C^\infty(\prod_i X_i, Y) \xrightarrow{\cong} C^\infty(\prod_{i \ne i_0} X_i, C^\infty(X_{i_0},Y)).
$$

Thus given a morphism $f \colon \prod_i X_i \to Y$ we get a parametrised family of morphisms $X_{i_0} \to Y$ which we could write (using parameters) as $f(x_{\widehat{i_0}})(x_{i_0})$.  As taking the derivative is a smooth functor, we can *partially* differentiate the morphisms by applying differentiation to the morphisms $X_{i_0} \to Y$, thus yielding $d f_{i_0}(x_{\widehat{i_0}})(x_{i_0},v)$ as a morphism $\prod_{i \ne i_0} X_i \to C^\infty(T X_{i_0}, T Y)$.  In full, $d f_{i_0}$ is the image of $f$ under the chain of morphisms:

$$
C^\infty(\prod_i X_i, Y) \xrightarrow{\cong} C^\infty(\prod_{i \ne i_0} X_i, C^\infty(X_{i_0},Y)) \xrightarrow{C^\infty(\prod_{i \ne i_0} X_i, -)} C^\infty(\prod_{i \ne i_0} X_i, C^\infty(T X_{i_0}, T Y)).
$$

This is the **partial derivative** of $f$ along $X_{i_0}$.

## Notation

When the coordinates $x_i$ are given individual names $u, v, w, \ldots$, one usually writes $\partial{f}/\partial{u}$ for $\partial_i{f}$ (where $u$ replaces $x_i$); but $(\partial{f}/\partial{u})_{v,w,\ldots}$ is less ambiguous.  Similarly, one can write $(d{f})_{v,w,\ldots}$ for the partial differential $(\partial{f}/\partial{u})_{v,w,\ldots} \,d{u}$, which is $d_i{f}$ when $u$ replaces $x_i$.  (If $d{f}$ is thought of as an [[infinitesimal]] change in $f$, then $(d{f})_{v,w,\ldots}$ is an infinitesimal change subject to the condition that $v,w,\ldots$ are _fixed_.)  Then
$$ \left(\frac{\partial{f}}{\partial{u}}\right)_{v,w,\ldots} = \frac{(d{f})_{v,w,\ldots}}{d{u}} = \frac{(d{f})_{v,w,\ldots}}{(d{u})_{v,w,\ldots}} ,$$
which explains the notation and why '$\partial$' looks like '$d$'.  (The reason for the latter equality is that $\partial_i{x_j}$ is the [[Kronecker delta]] $\delta_{i,j}$.)

## Related concepts

* [[partial differential equation]]

* [[jet bundle]]

* [[differentiable function]], [[smooth function]]

* [[function with rapidly decreasing partial derivatives]]

## References

The [[Kock-Lawvere axiom]] for the [[axiom|axiomatization]] of differentiation in [[synthetic differential geometry]] was introduced in

* {#Kock77} [[Anders Kock]], _A simple axiomatics for differentiation_,  Mathematica Scandinavica Vol. 40, No. 2 (October 24, 1977), pp. 183-193 ([JSTOR](http://www.jstor.org/stable/24491223))

[[!redirects partial differentiation]]

[[!redirects partial derivative]]
[[!redirects partial derivatives]]

[[!redirects partial differential]]
[[!redirects partial differentials]]


partial equivalence relation

# Partial equivalence relations
* table of contents
{: toc}

## Definitions

A __partial equivalence relation__ (sometimes abbreviated __PER__) is a [[binary relation]] satisfying the [[symmetric relation|symmetry]] and [[transitive relation|transitivity]] conditions of an [[equivalence relation]], but not necessarily the [[reflexive relation|reflexivity]] condition. That is, a partial equivalence relation on $X$ is a binary relation $R(x, y) \subseteq X^2$ such that for all $x$ and $y$ in $X$, $R(x, y)$ implies $R(y, x)$, and for all $x$, $y$, and $z$ in $X$, $R(x, y)$ and $R(y, z)$ together imply $R(x, z)$.

Just as [[unary relations]] on a set $X$ correspond to [[subsets]] of $X$ and equivalence relations on $X$ correspond to [[quotient set|quotients]] of $X$, so partial equivalence relations on $X$ correspond to [[subquotients]] of $X$.  That is, the elements satisfying $R(x, x)$ comprise a subset of $X$, on which the relation restricts to a total equivalence relation specifying a further quotient.

## Examples

Consider the set $X$ of all [[infinite sequences]] of [[rational numbers]].  Let such sequences $x$ and $y$ be related if
$$ \lim_{i,j\to \infty} {|x_i - y_j|} = 0 .$$
Then this defines a partial equivalence relation $R$ on $X$; the corresponding subquotient of $X$ is the set of [[Cauchy real numbers]].  Normally, this definition of real number is split into two parts: those sequences satsifying the reflexivity condition of $R$ are the [[Cauchy sequences]] of rational numbers (under the absolute-value metric), and then we impose a total equivalence relation on the Cauchy sequences.  But a single partial equivalence relation does all of the work.  (This example generalises in the usual ways.)

## Category of PERs

If $A$ is a [[partial combinatory algebra]], then the partial equivalence relations on $A$ are the objects of the [[category of PERs]] over $A$, a [[locally cartesian closed category|locally cartesian closed]] [[Heyting category]] that is a [[full subcategory]] of the [[quasitopos]] of [[assemblies]], which is in turn a full subcategory of the [[realizability topos]] over $A$.

[[!redirects partial equivalence relation]]
[[!redirects partial equivalence relations]]
[[!redirects PER]]
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partial evaluation

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## Idea

A partial evaluation is an instruction to evaluate a [[formal expression]] piecewise, without necessarily obtaining the final result.

For a simple example, the sum "$3+4+5$" can be evaluated to "$12$", but also _partially_ evaluated to "$7+5$".

## Definition

Let $(T,\mu,\eta)$ be a [[monad]] on [[Set]], and let $(A,a)$ be an [[algebra over a monad|algebra]] over $T$. Given two elements $p,q\in T A$, a **partial evaluation** from $p$ to $q$ is an element $r\in T T A$ such that $\mu(r)=p$, and $T a(r)=q$.

Equivalently, partial evaluations are the 1-simplices of the [[bar construction]] (considered as a [[simplicial set]]) of the algebra $(A,a)$.

## Examples

* For many [[probability monads]], partial evaluations correspond to [[conditional expectation]] of [[algebra over a monad|algebra]]-valued [[random variables]];
* For the [[free cocompletion]] monad, partial evaluations correspond to pointwise left [[Kan extensions]].

## See also

* [[bar construction]], [[monad]], [[algebra over a monad]], [[operad]], [[algebra over an operad]]
* [[simplicial set]], [[Segal space]], [[Segal conditions]], [[Kan complex]]

## References

* [[Tobias Fritz]] and [[Paolo Perrone]], _Monads, partial evaluations, and rewriting_. ([arXiv](https://arxiv.org/abs/1810.06037))

* Carmen Constantin, [[Tobias Fritz]], [[Paolo Perrone]] and Brandon Shapiro, _Partial evaluations and the compositional structure of the bar construction_, Theory and Applications of Categories 39, 2023. ([arXiv:2009.07302](https://arxiv.org/abs/2009.07302))

* Carmen Constantin, [[Tobias Fritz]], [[Paolo Perrone]] and Brandon Shapiro, _Weak cartesian properties of simplicial sets_, Journal of Homotopy and Related Structures, 2023. ([arXiv:2105.04775](https://arxiv.org/abs/2105.04775))

* [[Paolo Perrone]], [[Walter Tholen]], *Kan extensions are partial colimits*, Applied Categorical Structures, 2022. ([arXiv:2101.04531](https://arxiv.org/abs/2101.04531))

[[!redirects partial evaluations]]
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partial fraction decomposition

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## See also

* [[polynomial]]

* [[rational fraction]]

## References

* Wikipedia, [Partial fraction decomposition](https://en.wikipedia.org/wiki/Partial_fraction_decomposition)


partial function

# Partial functions
* table of contents
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## Idea

A _partial function_ $f: A \to B$ is like a [[function]] from $A$ to $B$ except that $f(x)$ may not be defined for every element $x$ of $A$.  (Compare a [[multi-valued function]], where $f(x)$ may have several possible values.)

In some fields (including secondary-school mathematics even today), functions are often considered to be partial by default, requiring one to specify a _total_ function otherwise.  As category-theoretic and type-theoretic formalisation spreads, this is difficult to treat as the basic concept, and the most modern idea is that a function must be total.  If you want partial functions, then you can get them in terms of total functions as below.

## Definitions

### In type theory

#### As functions

In [[dependent type theory]], given [[h-sets]] $A$ and $B$, a **partial function** $f$ from $A$ to $B$ is an [[h-proposition]]-valued type family $P_f$ on $A$ which states whether $f$ is defined for element $a:A$, and a section

$$x:A, p:P_f \vdash f:B$$

If the dependent type theory has [[function types]] and [[dependent sum types]], then one could define the partial function directly as:

$$f:\left(\sum_{a:A} P_f(a)\right) \to B$$

#### As partial anafunctions

In [[dependent type theory]], given types $A$ and $B$, a **partial anafunction** is a [[type family]] $x\colon A, y \colon B \vdash R(x, y) \; \mathrm{type}$ which comes with a family of witnesses
$$x \colon A \vdash p(x) \colon \mathrm{isProp}\left(\sum_{y:B} R(x, y) \right)$$
that for each [[term|element]] $x\colon A$, the [[dependent sum type]] $\sum_{y \colon B} R(x, y)$ is a [[mere proposition]].

Historically, partial anafunctions are called *[[functional relations]]*, but nowadays in type theory one tends to use the term *[[relation]]* only for those [[type families]] which are valued in [[mere propositions]].

For any partial anafunction, there is a partial function
$$x:A, p:\mathrm{isContr}\left(\sum_{y:B} R(x, y) \right) \vdash f(x, p):B$$

which could be written as the function
$$f:\sum_{x:A} \mathrm{isContr}\left(\sum_{y:B} R(x, y) \right) \to B$$

since $\sum_{y:B} R(x, y)$ is already a proposition, one could simply use
$$x:A, p:\sum_{y:B} R(x, y) \vdash f(x, p):B$$
instead, and by the rules of dependent sum types, it becomes
$$x:A, y:B, p:R(x, y) \vdash f(x, y, p):B$$

on the other side, the function can be written as
$$f:\left(\sum_{x:A} \sum_{y:B} R(x, y) \right) \to B$$

#### Examples

There are multiple examples of partial functions in mathematics.

The [[reciprocal function]] on the [[real numbers]] is only defined on the real numbers apart from zero, and so is given by the following rules:

$$\frac{\Gamma \vdash \mathbb{R} \; \mathrm{type}}{\Gamma, x:\mathbb{R}, p:\vert x \vert \gt 0 \vdash {(-)}^{-1}:\mathbb{R}}$$

$$\frac{\Gamma \vdash \mathbb{R} \; \mathrm{type}}{\Gamma, x:\mathbb{R}, p:\vert x \vert \gt 0 \vdash \mathrm{recipax}:x \cdot {(x, p)}^{-1} =_\mathbb{R} 1}$$

The reciprocal function is given by the partial anafunction $x \cdot y =_\mathbb{R} 1$.

Another example is the [[metric square root]] on the [[real numbers]], which is only defined on the non-negative real numbers, and is given by the following rules:

$$\frac{\Gamma \vdash \mathbb{R} \; \mathrm{type}}{\Gamma, x:\mathbb{R}, p:x \geq 0 \vdash \sqrt{-}:\mathbb{R}}$$

$$\frac{\Gamma \vdash \mathbb{R} \; \mathrm{type}}{\Gamma, x:\mathbb{R}, p:x \geq 0 \vdash \mathrm{sqrtax}:\sqrt{x, p}^2 =_\mathbb{R} x}$$

In both cases, the witness $p$ is usually left implicit in informal notation, and the partial function is written as ${x}^{-1}$ or $\sqrt{x}$.

### In category theory

#### In the category of sets
 {#DefinitionInSets}

Given [[set]]s $A$ and $B$, a __partial function__ $f$ from $A$ to $B$ consists a [[subset]] $D$ of $A$ and a (total) function from $D$ to $B$.  In more detail, this is a [[span]]
$$ \array {
  &                & D \\
  & \swarrow_\iota &   & \searrow^f \\
A &                &   &            & B \\
} $$
of total functions, where $\iota: D \to A$ is an [[injection]].  (This condition can be dropped to define a partial [[multi-valued function]], which is simply a span.)

$A$ and $B$ are called the _[[source]]_ and _[[target]]_ of $f$ as usual; then $D$ is the __domain__ of $f$ and $\iota: D \to A $ is the __inclusion__ of the domain into the source.  By abuse of notation, the partial function $f$ is conflated with the (total) function $f: D \to B$.

Notice that the induced function $D \to A \times B$ is an [[injection]], so a partial function is the same as a [[functional relation]] seen from a different point of view.

We consider two partial functions (with the same given source and target) to be __equal__ if there is a [[bijection]] between their domains that makes the obvious diagrams commute.  You can get this automatically in a traditional set theory by requiring $D$ to be literally a [[subset]] of $A$ (with $\iota$ the inclusion map).

#### General abstract
 {#DefinitionGeneralAbstract}

##### In terms of spans

If $C$ is a [[category]] with [[pullbacks]], then a **partial map** from $a$ to $b$ may be defined to be a [[span]]

$$\array{
 & & D & & \\
 & ^\mathllap{i} \swarrow & & \searrow^\mathrlap{f} & \\
A & & & & B
}$$

where $i$ is [[monomorphism|monic]]. Such spans are closed under span composition, and as a [[locally full sub-2-category]] of $Span(C)$, the bicategory of partial maps in $C$ is locally preordered. In more down-to-earth terms, if $(i, f)$ and $(i', f')$ are partial maps from $a$ to $b$, we have $(i, f) \leq (i', f')$ if there exists (necessarily monic) $j$ such that $i = i' \circ j$ and $f = f' \circ j$.

Abstract bicategories of partial maps, parallel to [[bicategory of relations|bicategories of relations]], were introduced by ([Carboni 87](#Carb87)).

##### In terms of the maybe monad

Notice that a partial function $f$ from $A$ to $B$ as [above](#DefinitionInSets) is (in [[classical mathematics]]) equivalently a genuine function from $A$ to the [[disjoint union]] ([[coproduct]]) of $B$ with the point (the singleton)

$$
  \phi \;\colon\; A \longrightarrow B \coprod \ast
  \,.
$$

The [[subset]] $D \hookrightarrow A$ in the [above](#DefinitionInSets) is the [[preimage]] $\phi^*(B)$; for $x$ in this preimage, $f(x) = \phi(x)$.  Conversely, an element $x \in A$ is sent to $\ast$ by $\phi$ if and only if $f$ is undefined at $x$.

This in turn is equivalently a morphism in the [[Kleisli category]] of the [[maybe monad]]. Phrased this way, the concept of partial function makes sense in any [[category]] with [[coproducts]] and with a [[terminal object]]. It comes out as intended when the category is an [[extensive category]] (partial functions with complemented domain).

## Examples

In secondary-school mathematics, one often makes functions partial by fiat, just to see if students can calculate the domains of composite functions and the like.  This is not (only) busywork, as in applications one often has a function given by a formula that is really valid only on a certain domain.  However, in more sophisticated analyses (such as those that Lawvere and his followers propose for physics and synthetic geometry), these domains and the total functions on them become the primary objects of study, with the partial functions being secondary (as $\iota$ is seen as merely a way to place _coordinates_ on $D$).

In analysis, one often considers partial functions whose domains are required to be intervals in the real line, regions of the complex plane, or dense subsets of a [[Banach space]].

In a [[field]], the multiplicative [[inverse]] is a partial function whose domain is the set of non-zero elements of the field.

## The category of sets and partial functions

The [[category]] $Set_part$ of sets and partial functions between them is important for understanding computation.  However, one often replaces this with an [[equivalence of categories|equivalent]] category of sets and total functions.

Specifically, replace each set $S$ with the set $S_\bot$ of all [[subset]]s of $S$ with at most one element, otherwise known as the [[partial map classifier]] of [[Set]].  In this context, we identify an element $x$ of $S$ with the subset $\{x\}$ and write the [[empty set|empty]] subset as $\bot$.  Then a partial function $S \to T$ becomes a total function $S_\bot \to T_\bot$ such that [[inhabited set|inhabited]] subsets of $T$ are assigned only to inhabited subsets of $S$.  Then $Set_part$ is equivalent to the category $Set_\bot$ of such sets and functions.

In [[classical mathematics]] $S_\bot \cong S \amalg \{\bot\}$, although this is not true [[constructive mathematics|constructively]].  In this case, $S\mapsto S \amalg \{\bot\}$ is the [[maybe monad]] and $Set_\bot$ is its [[Kleisli category]].  Moreover, since every algebra for this monad is [[free algebra|free]] this category is also equivalent to its [[Eilenberg-Moore category]], which is the category $Set_*$ of [[pointed set]]s and total point-preserving functions.  Traditionally, one uses the notation of $Set_\bot$ but (unless one is a constructivist) thinks of this as simply different notation for $Set_*$.  It is still true constructively that $S\mapsto S_\bot$ is a monad (the [[partial map classifier]]) and $Set_\bot$ is its Kleisli category, but it is (probably) no longer true that every algebra is free.

For a more sophisticated analysis of computation, $Set_\bot$ can be replaced with a suitable category of domains, such as [[direction|directed]] [[complete lattice|complete]] [[partial order|partially ordered]] sets (DCPOs).  The requirement that $\bot$ be preserved can then be removed to model lazy computation, but now we are hardly talking about partial functions anymore.

## Algebras of partial functions

The functions of high-school mathematics, consisting of real (or complex)-valued functions of one (or two or three) real (or complex) variables, are by default partial functions.  As they take values in a [[field]], one may consider adding or multiplying them.  The usual rule is that $\dom(f + g) = \dom f \cap \dom g$, etc, but this leads to an unusual algebra: a commutative [[semiring]] in which addition has an identity element (the always-defined constant [[zero function]]) and multiplication has an absorbing element (the never-defined [[empty function]]), but it fails to be a [[rig]] because these two elements are not the same.  It has many other interesting properties, such as simultaneous additive and multiplicative [[idempotent element|idempotents]] (the zero functions with arbitrary domains).

An axiomatic treatment of such semirings may be found at the end of [Richman 2012](#Richman12), as well as in the article [[category of partial endofunctions]].

## Related concepts

* [[total function]]
* [[restriction category]]
* [[category of partial endofunctions]]
* [[partial function type]]
* [[multivalued partial function]]

## References
 {#References}

* {#Carb87} [[Aurelio Carboni]], Bicategories of partial maps, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 28 no. 2 (1987), p. 111-126 ([web](http://www.numdam.org/item?id=CTGDC_1987__28_2_111_0))

* [[Robin Cockett]], [[Steve Lack]], _Restriction categories I: categories of partial maps_ ([pdf](http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS_06/RestrictionsI.pdf))

* [[Robin Cockett]], [[Steve Lack]], _Restriction categories II: partial map classification_ ([web](http://maths.mq.edu.au/~slack/papers/restii.html))

* [[Robin Cockett]], [[Steve Lack]], _Restriction categories III: colimits, partial limits, and extensivity_ ([arXiv:math/0610500](http://arxiv.org/abs/math/0610500))

* {#Richman12} [[Fred Richman]], *Algebraic functions, calculus style*. Communications in Algebra, Volume 40, Issue 7, July 2012, Pages 2671-2683 [[doi:10.1080/00927872.2011.584337](https://doi.org/10.1080/00927872.2011.584337)]

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partial function type

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## Idea

In the same way that one could define [[equivalence types]] as types of [[one-to-one correspondences]] and [[function types]] as types of [[anafunctions]], one could define partial function types as types of [[partial anafunctions]].

Partial anafunctions are [[type families]] $x\colon A, y \colon B \vdash R(x, y) \; \mathrm{type}$ which comes with a family of witnesses
$$x \colon A \vdash p(x) \colon \mathrm{isProp}\left(\sum_{y:B} R(x, y) \right)$$
that for each [[term|element]] $x\colon A$, the [[dependent sum type]] $\sum_{y \colon B} R(x, y)$ is a [[mere proposition]]. From every partial anafunction, one could derive the [[partial function]]
$$x:A, p:\sum_{y:B} R(x, y) \vdash f(x, p):B$$
and for every [[mere proposition]]-valued type family $x:A \vdash P(x)$ and every [[partial function]] $x:A, p:P(x), \vdash f(x, p):B$, one could define the [[partial anafunction]] $x:A, y:B \vdash R(x, y)$ as
$$R(x, y) \coloneqq \sum_{p:P(x)} f(x, p) =_B y$$

Defining partial function types requires both [[identity types]] and [[heterogeneous identity types]] being defined first, which we shall write as $a =_A b$ and $x =_{B}^{p} y$ respectively for $a:A$, $b:A$, $p:a =_A b$, $x:B(a)$, and $y:B(b)$. We use the notation $A \rightharpoonup B$ to represent the type of partial functions between $A$ and $B$.

## Rules for partial function types

The rules for partial function types are as follows:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma \vdash A \rightharpoonup B \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \rightharpoonup B, x:A, y:B \vdash \mathcal{P}_{A, B}(f, x, y) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type} \quad \Gamma, x:A, p:P(x), q:P(x) \vdash \tau_{-1}(x, p, q):p =_{P(x)} q \quad \Gamma, x:A, p:P(x) \vdash f(x, p):B}{\Gamma \vdash (x, p) \mapsto f(x, p):A \rightharpoonup B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type} \quad \Gamma, x:A, p:P(x), q:P(x) \vdash \tau_{-1}(x, p, q):p =_{P(x)} q \quad \Gamma, x:A, p:P(x) \vdash f(x, p):B}{\Gamma, x:A, p:P(x) \vdash \alpha(x, p):\mathcal{P}_{A, B}((x, p) \mapsto f(x, p), x, f(x, p))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type} \quad \Gamma, x:A, p:P(x), q:P(x) \vdash \tau_{-1}(x, p, q):p =_{P(x)} q}{\Gamma, f:A \rightharpoonup B, x:A, p:P(x) \vdash \mathrm{ev}(f, x, p):B}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash P(x) \; \mathrm{type} \quad \Gamma, x:A, p:P(x), q:P(x) \vdash \tau_{-1}(x, p, q):p =_{P(x)} q}{\Gamma, f:A \rightharpoonup B, x:A, p:P(x) \vdash \beta(f, x, p):\mathcal{F}_{A, B}(f, x, \mathrm{ev}(f, x, p))}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \rightharpoonup B, x:A, y:B, u:\mathcal{P}_{A, B}(f, x, y), z:B, v:\mathcal{P}_{A, B}(f, x, z) \vdash \kappa(f, x, y, u, z, v):y =_B z}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \rightharpoonup B, x:A, y:B, u:\mathcal{P}_{A, B}(f, x, y), z:B, v:\mathcal{P}_{A, B}(f, x, z) \vdash \eta(f, x, y, u, z, v):u =_{\mathcal{P}_{A, B}(f, x)}^{\kappa(f, x, y, u, z, v)} v}$$

By the rules for [[dependent sum types]] and [[dependent product types]], one could derive that

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type}}{\Gamma, f:A \rightharpoonup B \vdash \eta(f):\prod_{x:A} \mathrm{isProp}\left(\sum_{y:B} \mathcal{P}_{A, B}(f, x, y) \right)}$$

which is precisely the statement that $\mathcal{P}_{A, B}(f)$ is a [[partial anafunction]] for all [[partial functions]] $f:A \rightharpoonup B$.

## See also

* [[partial function]], [[partial anafunction]]
* [[function type]], [[dependent function type]]
* [[equivalence type]]

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