partial map classifier

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### Context
#### Universes
+-- {: .hide}
[[!include universe - contents]]
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=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[partial function]] $f \colon A\rightharpoonup B$ between two [[sets]] can be turned into a [[total function]] $\overline{f} \colon A\to B_\bot$ by adjoining a further [[element]] $\bot$ to $B$ and sending all $a\in A$ that are not in the [[domain]] of definition of $f$ to this new value $\bot$ = "undefined" and the rest just to their value under $f$. Conversely, every function with codomain $B_\bot$ corresponds to a unique partial function to $B$ whence $B_\bot$ is a 'classifying object' for these.

More generally, a **partial function classifier** or **partial map classifier** of an [[object]] $B$ in a [[category]] $\mathcal{C}$ is a [[representing object]] for [[partial maps]] with [[codomain]] $B$.

## Definition

Let $\mathcal{C}$ be a category with [[pullbacks]].  Recall that a **partial map** $A \rightharpoonup B$ in $\mathcal{C}$ is a [[span]] $A \leftarrow D \to B$ in which the map $D\to A$ is a [[monomorphism]].  The subobject $D \hookrightarrow A$ is called the [[domain]] of the partial map.  Two partial maps are considered *equal* if they are related by an isomorphism of spans; in this way we obtain a [[set]] of partial maps $Par_{\mathcal{C}}(A,B)$.

We can compose a partial map $A\rightharpoonup B$ with a map $B\to B'$ in an obvious way.  We can also compose it with a map $A'\to A$ by pulling back the mono $D\hookrightarrow A$ to $A'$.  In this way $Par_{\mathcal{C}}(-,-)$ becomes a [[profunctor]] from $\mathcal{C}$ to itself.  (In fact, it is the hom-set of another category with the same objects as $\mathcal{C}$.)

A **partial map classifier** for $B$ is an object $B_\bot$ together with an isomorphism
$$ \mathcal{C}(A,B_\bot) \cong Par_{\mathcal{C}}(A,B) $$
natural in $A$.  By the [[Yoneda lemma]] this means there is a universal partial map $B_\bot \rightharpoonup B$.

This can be unpacked into the following condition:

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

A partial map classifier for $B$ is uniquely determined by a map $B \to B_\bot$ with the property that, for every partial map $A\rightharpoonup B$ depicted below, there is a unique $A \to B_\bot$ making a pullback square

\begin{tikzcd}
D \arrow[d, hook] \arrow[r]  & B \arrow[d]
\\ A \arrow[r, dotted] & B_\bot
\end{tikzcd}

=--

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

For any monomorphism $B \to B_\bot$, there is a natural transformation $\mathcal{C}(-,B_\bot) \to Par_{\mathcal{C}}(-,B)$ given by forming pullbacks along $B \to B_\bot$. It is a natural bijection iff it satisfies the condition of the proposition.

If $B \to B_\bot$ satisfies the condition of the proposition, then it is a monomorphism, by considering the (total) partial map $\id_B$.

=--

The existence of partial map classifiers $B_\bot$ for all objects $B$ in $\mathcal{C}$ amounts to the existence of a right adjoint for the inclusion $\mathcal{C}\hookrightarrow Par(\mathcal{C})$ where the latter is the usual category of partial maps of $\mathcal{C}$.

## Constructions

In a [[Boolean category|Boolean]] [[extensive category]] (such as a [[Boolean topos]]), we can define the partial map classifier as $B_\bot = B + 1$, where $1$ is the [[terminal object]].  This is because in an extensive category, a map $A\to B+1$ is equivalent to a decomposition of $A$ as a [[coproduct]] $D+E$ together with a map $D\to B$ (the map $E\to 1$ being unique), and in a Boolean category every subobject of $A$ is complemented and hence induces such a coproduct decomposition.  The universal partial map $B+1 \rightharpoonup B$ has domain the summand $B$, on which it is the identity.  Note that $B\mapsto B+1$ is also known as the [[maybe monad]].

Partial map classifiers also exist in every [[elementary topos]], but in the non-Boolean case they are harder to construct. Letting $t : 1 \to \Omega$ denote the top element, the dependent product $t_*(B)$ is precisely the map $B_\bot \to \Omega$ classifying $B \subseteq B_\bot$. To wit, with this definition,

$$
  \begin{aligned}
    \mathcal{C}(A, B_\bot)
    &\cong \coprod_{p : A \to \Omega} \mathcal{C}/\Omega(p, t_*(B))
    \\&\cong \coprod_{p : A \to \Omega} \mathcal{C}(t^*(p), B)
    \\&\cong \coprod_{A' \subseteq A} \mathcal{C}(A', B)
    \\&\cong Par_{\mathcal{C}}(A, B)
  \end{aligned}
$$

Alternatively, in the [[internal logic]], two ways of defining $B_\bot$ that more closely follow the classical construction are:

*  $B_\bot$ is a [[subobject]] of the [[power object]] $\mathcal{P}B$ consisting of the [[subsingleton]] subobjects of $B$.  In this case, the universal partial map $B_\bot \rightharpoonup B$ has domain the set of [[singleton]] subobjects of $B$, on which it is an isomorphism.

*  $B_\bot$ is the object of partial maps $1 \rightharpoonup B$.  In this case, the universal partial map $B_\bot \rightharpoonup B$ has domain the set of partial maps which are total, on which it is an isomorphism.

Note that neither of these constructions is [[predicative mathematics|predicative]].  The second makes more sense in a [[higher category]] (or in its internal logic such as [[homotopy type theory]]).

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

In an [[constructive mathematics|intuitionistic context]], $B + 1$ still classifies something: namely partial maps $A \rightharpoonup B$ whose domain of definition is a [[decidable subset|decidable subobject]] of $A$.

In [[type theory]] one uses the partiality [monad](monad+%28in+computer+science%29) to treat general recursion as an algebraic effect. In this we we obtain a classifier for recursively enumerable subsets. Is [this](http://www.cs.ox.ac.uk/ralf.hinze/WG2.8/22/slides/tarmo.pdf) the first appearance?
It is clear that this idea can be generalized to other classes of propositions.
=--

## Relative partial maps

Given objects $p : X \to S$ and $q : Y \to S$ of $\mathcal{C}/S$, a partial map from $p$ to $q$ can be described as a commutative diagram in $\mathcal{C}$.

\begin{tikzcd}
  A \arrow[r] \arrow[d, hook] & Y \arrow[d, "q"]
  \\ X \arrow[r, "p"] & S
\end{tikzcd}

A partial map classifier $Opt_S(q)$ for $q$ in $\mathcal{C}/S$ can be described as a representing object for the presheaf on $\mathcal{C}$ sending $X$ to the set of diagrams of this form (modulo choice of representatives for subobjects), together with its structure map $Opt_S(q) \to S$, which represents the natural transformation sending this diagram to $p \in \mathcal{C}(X, S)$.

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

If $\mathcal{C}$ is a topos and $p : X \to S$ is an object of $\mathcal{C}/S$, there is a pullback in $\mathcal{C}$

\begin{tikzcd}
  \mathrm{Opt}_S(q) \arrow[r] \arrow[d] & X_\bot \arrow[d, "p"]
  \\ S \times \Omega \arrow[r] & S_\bot
\end{tikzcd}

where $Opt_S(q) \to X_\bot$ corresponds to forgetting the base of a relative partial map, and $Opt_S(q) \to S$ is the structure map, and $Opt_S(q) \to \Omega$ corresponds to giving the domain of definition.

=--

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

Let $A$ be an object of $\mathcal{C}$. It suffices to show this square is a pullback after applying $\mathcal{C}(A, -)$ and the isomorphism to the presheaves these objects represent.

An object of the pullback can be described as an arrow $A \to S$, a subobject $B \hookrightarrow A$, and a partial map $A \hookleftarrow B' \to X$ with the property that the induced partial maps $A \hookleftarrow B \to S$ and $A \hookleftarrow B' \to S$ are the same partial map (up to choice of representative for the subobject).

This is bijective with the set of commutative diagrams of the form

\begin{tikzcd}
B \arrow[r] \arrow[d, hook] & X \arrow[d, "q"]
\\ A \arrow[r] & S
\end{tikzcd}

and the projection to $\mathcal{C}(A, S)$ takes the bottom horizontal arrow.

=--

## Properties

* In a [[topos]], the partial map classifier $B_\bot$ of $B$ is [[injective object|injective]]. The canonical embedding $B\rightarrowtail B_\bot$ shows accordingly that a topos _has enough injectives_!

* In a topos, the [[subobject classifier]] $true:1\rightarrow\Omega$ coincides with the monomorphism $\eta_1:1\rightarrowtail 1_\bot$.

* In a topos, the assignment of $B_\bot$ to $B$ is functorial.

## References
 {#References}

* [[Francis Borceux]], sec.5.5 in: *[[Handbook of Categorical Algebra]]* vol. 3, Cambridge UP (1994)

* [[Peter Johnstone]], Section 1.2 of: _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp. (Dover reprint 2014)

* [[Peter Johnstone]], Section A2.4 in: *[[Sketches of an Elephant]]*

For related topics in [[homotopy type theory]]:

* [[Thorsten Altenkirch]], [[Nils Anders Danielsson]], [[Nicolai Kraus]], *Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type* [[abs:1610.09254](https://arxiv.org/abs/1610.09254)]

* [[Martín Escardó]], Cory Knapp, *Partial Elements and Recursion via Dominances in Univalent Type Theory* [[doi:10.4230/LIPIcs.CSL.2017.21](https://doi.org/10.4230/LIPIcs.CSL.2017.21)]

Discussion for [[topological spaces]]:

* {#MaySigurdsson06} [[Peter May]], [[Johann Sigurdsson]], Def. 1.3.6 in: _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006   ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

[[!redirects partial function classifier]]
[[!redirects partial function classifiers]]

[[!redirects partial map classifier]]
[[!redirects partial map classifiers]]


partial model category

## Idea

Partial model categories are one of the many intermediate notions
between [[relative categories]] and [[model categories]].

They axiomatize those properties of [[model categories]]
that only involve weak equivalences.

## Definition

A __partial model category__ is a [[relative category]] $(C, W)$
that satisfies the [[2-out-of-6 property]]
(if $s r$ and $t s$ are weak equivalences, then so are $r$, $s$, $t$, $t s r$)
and admits a [[3-arrow calculus]],
i.e., there are [[subcategories]]
$U, V \subseteq W$ of the weak equivalences (which can be thought of as analogues of acyclic cofibrations
and acyclic fibrations)
such that $U$ is closed under [[cobase changes]] along arbitrary morphisms of $C$ (which are required to exist),
$V$ is closed under [[base changes]] along arbitrary morphisms of $C$,
and any weak equivalence can be functorially factored as the composition $v u$
for some $u\in U$ and $v\in V$.

## Properties

If $(C,W)$ is a partial model category,
then any Reedy fibrant replacement
of the [[Rezk nerve]] $N(C,W)$ is a [[complete Segal space]].

## Related concepts

* [[model category]]

* [[relative category]]

* [[semimodel category]]

* [[weak model category]]

* [[premodel category]]

* [[homotopical category]]

## References

* [[Clark Barwick]], [[Daniel M. Kan]], _Partial model categories and their simplicial nerves_, [arXiv](https://arxiv.org/abs/1102.2512v2).


partial order

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### Context
#### $(0,1)$-Category theory
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[[!include (0,1)-category theory - contents]]
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#### Category theory
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[[!include category theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

A _partial order_ on a set is a way of ordering its elements to say that some elements precede others, but allowing for the possibility that two elements may be incomparable without being the same.  This is the fundamental notion in [[order theory]].

By regarding the ordering $\leq$ as the existence of a unique [[morphism]], preorders may be regarded as certain [[categories]] (namely, [[thin categories]]).  This category is sometimes called the _order category_ associated to a partial order; see [below](#AsACategoryWithExtraProperties) for details.

## Definitions

### As a set with extra structure

A poset may be understood as a [[set]] with [[extra structure]].

Given a [[set]] $S$, a __partial order__ on $S$ is a (binary) [[relation]] $\leq$ with the following properties:

* [[reflexive relation|reflexivity]]: $x \leq x$ always;

* [[transitive relation|transitivity]]: if $x \leq y \leq z$, then $x \leq z$;

* [[antisymmetric relation|antisymmetry]]: if $x \leq y \leq x$, then $x = y$.

A __poset__ is a set equipped with a partial order.

### As a preorder with antisymmetry

A poset is precisely a [[proset]] satisfying the extra condition that
$x \leq y \leq x$ implies that $x = y$.

#### In dependent type theory

In [[dependent type theory]], the identity type $x = y$ representing equality is not necessarily valued in [[h-propositions]]. As a result, a naive translation of the set theoretic definition of a partial order from a preorder above doesn't work. Instead, one has to postulate an [[equivalence of types]] between the identity type $x = y$ and the condition $x \leq y \leq x$:

A **partial order** is a preorder $X$ with an [[equivalence of types]] $\mathrm{antisym}(x, y):(x =_X y) \simeq (x \lt y) \times (y \lt x)$ for all $x:X$ and $y:X$.

Equivalently, one could [[inductively define]] a function

$$x:X, y:X \vdash \mathrm{idToOrd}(x, y):(x =_X y) \to (x \lt y) \times (y \lt x)$$

on reflexivity of the [[identity type]] by reflexivity of the preorder

$$\mathrm{idToSymCon}(x, x)(\mathrm{id}_X(x)) \coloneqq \mathrm{refl}_{\leq}(x)$$

Then, a partial order is a preorder such that the defined function $\mathrm{idToOrd}(x, y)$ is an [[equivalence of types]] for all $x:X$ and $y:X$.

$$\mathrm{antisym}:\prod_{x:X} \prod_{y:X} \mathrm{isEquiv}(\mathrm{idToOrd}(x, y))$$

In either case, since the preorder is valued in propositions, the antisymmetry axiom ensures that the partial order is an [[h-set]].

### As a category with extra properties
 {#AsACategoryWithExtraProperties}

A poset [[relation between preorders and (0,1)-categories|may be understood]] as a [[category]] with [[extra property]], sometimes called its _order category_.

A __poset__ [[relation between preorders and (0,1)-categories|is]] a [[category]] such that:

* for any [[pair]] of [[objects]] $x, y$, there is at most one morphism from $x$ to $y$

* if there is a morphism from $x$ to $y$ and a morphism from $y$ to $x$ (which by the above implies that $x$ and $y$ are [[isomorphic]]), then $x = y$.

[[relation between preorders and (0,1)-categories|Equivalently]], this says that a poset is a [[skeletal category|skeletal]] [[thin category|thin]] category, or equivalently a skeletal [[category enriched]] over the [[cartesian monoidal category]] of [[truth values]] or equivalently a skeletal [[(0,1)-category]]. (See also at *[[enriched poset]]*).

When we do this, we are soon [[relation between preorders and (0,1)-categories|led to contemplate]] a slight generalization of partial orders: namely [[preorder|preorders]]. The reason is that the antisymmetry law, saying that $x \le y$ and $y \le x$ imply $x = y$, violates the [[principle of equivalence]] in a certain sense.  (On the other hand, it does not violate it if taken as a *definition* of [[equality]].)

### Monotone functions

The morphisms of partially ordered sets are [[monotone functions]]; a __monotone function__ $f$ from a poset $S$ to a poset $T$ is a [[function]] from $S$ to $T$ (seen as [[structured sets]]) that preserves $\leq$:
$$ x \leq y \;\Rightarrow\; f(x) \leq f(y) .$$
Equivalently, it is a [[functor]] from $S$ to $T$ (seen as certain categories).

In this way, posets form a [[category]] [[Pos]].

### Intervals

A (closed bounded) **[[interval]]** in a poset $C$ is a set of the form
$$[x,y] = \{r\in C|x\le r\le y\}.$$

A poset is **[[locally finite poset|locally finite]]** if every closed bounded interval is finite.

### Kinds of posets

A poset with a [[top element]] and [[bottom element]] is called __bounded__.  (But note that a *[[subset]]* of a poset may be bounded without being a bounded as a poset in its own right.)  More generally, it is __bounded above__ if it is has a top element and __bounded below__ if it has a bottom element.

A poset with all finite [[meets]] and [[joins]] is called a __[[lattice]]__; if it has only one or the other, it is still a __[[semilattice]]__.

A poset in which every finite set has an upper bound (but perhaps not a *least* upper bound, that is a join) is a __[[directed set]]__.

As remarked above, a poset in which each interval $[x,y]$ is a [[finite set]] is called __locally finite__ or a __[[causet]]__.

A poset with a bounding [[countable set|countable subset]] is called __$\sigma$-bounded__.  That is, the poset is $\sigma$-bounded above if there exists a sequence $(x_n)_{n=1}^{N}$ (where $N$ is a natural number or infinity) such that for every $y$ in the poset there is an $x_m$ with $y \leq x_m$.  (The poset is $\sigma$-bounded below if we have $x_m \leq y$ instead.)  Note that every bounded poset is $\sigma$-bounded, but not conversely.  Note that some authorities require $N = \infty$; this makes a difference only for the empty poset (we say it is $\sigma$-bounded, they say it is not).

### In higher category theory

A poset can be understood as a [[(0,1)-category]]. This suggests an obvious [[vertical categorification]] of the notion of poset to that of [[n-poset]].

## Properties

### Extension to linear order
 {#ExtensionToLinearOrder}

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

On a [[finite set]], every [[partial order]] may be
[[linear extension of a partial order|extended]] to a [[linear order]].

For non-finite sets this still holds with the [[axiom of choice]].

=--

See at _[[linear extension of a partial order]]_ [this Prop.](linear+extension+of+a+partial+order#LinearExtensionOfAPartialOrder).

### Locales from posets -- Alexandroff topology
{#LocalesFromPosets}

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

For $P$ a poset, write $Up(P)$ for the [[topological space]] whose underlying [[set]] is the underlying set of $P$ and whose [[open subset]]s are the _upward closed subsets_ of $P$: those subsets $U \subset P$ with the property that

$$
  ((x \in U) and (x \leq y)) \Rightarrow (y \in U)
  \,.
$$

This is called the **[[Alexandroff topology]]** on $P$.

=--

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

This construction naturally extends to a
[[full and faithful functor]].

$Up : $ [[Poset]] $\to$ [[Top]] $\to$ [[Locale]].

=--

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

For $P$ a poset, there is a [[natural equivalence]]

$$
  Sh(Up(P)) \simeq [P,Set]
$$

between the [[category of sheaves]] on the [[locale]] $Up(P)$ and the
category of [[copresheaves]] on $P$.

=--

For more see [[Alexandroff topology]].

### Cauchy completion

Every poset is a [[Cauchy complete category]]. Posets are the Cauchy completions of [[prosets]]. ([Rosolini](#Rosolini))

### In univalent universes

In [[homotopy type theory]], every [[type]] with a partial order in a [[univalent universe]] is a poset. ([Tosun](#Tosun), [Ahrens, North, Shulman, Tsementzis](#ANST))

## Examples

* The [[function algebra]] of [[rational number|rational]]-valued [[functions]] on any [[set]] $X$ is a partial order.

* The [[function algebra]] of [[Cauchy real number|Cauchy real]]-valued functions on $X$ is a partial order. In fact, it is the [[Cauchy completion]] of the function algebra of [[rational number|rational]]-valued [[functions]] on $X$.

* More generally, if $A$ is a [[vector space]] on the [[rational numbers]] with a [[quadratic form]], then $A$ is a partial order, and its Cauchy completion is a vector space of same dimension on the [[Cauchy real numbers]] with a quadratic form. In particular, the Cauchy [[complex numbers]] and the [[Gaussian rationals]] are partial orders.

## Related concepts

* [[relation between preorders and (0,1)-categories]]

* [[preorder]]

* [[continuous poset]]

* [[Noetherian poset]]

* [[enriched poset]]

* [[specialization topology]]

* [[prefix order]]

* [[partially ordered object]]

* [[Thomason model structure]]

## References

* Richard P. Stanley, Enumerative [[combinatorics]], vol I [pdf](http://www-math.mit.edu/~rstan/ec/ec1.pdf)

[[Cauchy completion]] of prosets and posets is discussed in

* G. Rosolini, _A note on Cauchy completeness for preorders_ ([pdf](http://rivista.math.unipr.it/fulltext/1999-2s/06.pdf))
 {#Rosolini}

Here are some references on [[directed homotopy theory]]:

* [[Marco Grandis]], _Directed homotopy theory, I. The fundamental category_ ([arXiv](http://arxiv.org/abs/math.AT/0111048))

* [[Tim Porter]], _Enriched categories and models for spaces of evolving states_, Theoretical Computer Science, 405, (2008), pp. 88--100.

* [[Tim Porter]], _Enriched categories and models for spaces of dipaths. A discussion document and overview of some techniques_ ([pdf](http://drops.dagstuhl.de/opus/volltexte/2007/898/pdf/06341.PorterTimothy.Paper.898.pdf))

Posets in [[univalent universes]] in [[homotopy type theory]] are discussed in

* {#Tosun} Ayberk Tosun, Formal Topology in Univalent Foundations [pdf](https://odr.chalmers.se/handle/20.500.12380/301098)

* {#ANST} Benedikt Ahrens, Paige Randall North, Michael Shulman, Dimitris Tsementzis, The Univalence Principle ([abs:2102.06275](https://arxiv.org/abs/2102.06275))

On posets that are [[cofibrant objects]] in the Thomason model structure:

* [[Roman Bruckner]], [[Christoph Pegel]], _Cofibrant objects in the Thomason Model Structure_, [[arXiv:0808.4108](http://arxiv.org/abs/1603.05448)]

* [[Roman Bruckner]], *Abstract Homotopy Theory and the Thomason Model Structure*, PhD thesis, Bremen 2016 [[gbv:46-00105527-15](http://nbn-resolving.de/urn:nbn:de:gbv:46-00105527-15), [pdf](https://media.suub.uni-bremen.de/bitstream/elib/1120/1/00105527-1.pdf)]

[[!redirects poset]]
[[!redirects posets]]
[[!redirects partial orders]]
[[!redirects partial ordering]]
[[!redirects partial orderings]]
[[!redirects partially ordered]]
[[!redirects partially ordered set]]
[[!redirects partially ordered sets]]
[[!redirects partially-ordered]]
[[!redirects partially-ordered set]]
[[!redirects partially-ordered sets]]

[[!redirects bounded poset]]
[[!redirects bounded posets]]

[[!redirects order category]]
[[!redirects order categories]]
[[!redirects poset category]]
[[!redirects poset categories]]


partial recursive function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Deduction and induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

A **partial recursive function** (often "[[computable function]]", but see there for disambiguation) is a [[partial function]] of [[natural numbers]] which can be [[definition|defined]] by an [[algorithm]] or computer [[program]] (e.g., a [[Turing machine]]), taking finitely many [[natural numbers]] as inputs, and which on input may run forever, but otherwise eventually halts and returns a natural number as output.

This idea as described is vague until it is circumscribed by a specific notion of computer program (Turing machines, register machines, abaci, etc.). There is a standard article of faith called the "[[Church-Turing thesis]]", identifying functions on natural numbers that are algorithmically computable with those that are computable using a Turing machine (or some variant class of machines that is Turing-complete).

A purely mathematical definition of the intended class of functions is given below.

## Definition

+-- {: .num_defn #recursive}
###### Definition
A _partial recursive function_ is a [[partial function]] from $\mathbb{N}^k$ to $\mathbb{N}$ (where $\mathbb{N}$ is the [[set]] of [[natural numbers]] and $k \geq 0$ is finite) that belongs to the smallest [[class]] $\mathcal{C}$ of partial functions that

* includes all [[constant functions]] $1 \to \mathbb{N}$;

* includes all [[projection]] maps $\pi_i: \mathbb{N}^k \to \mathbb{N}$, $i = 1, \ldots, k$;

* includes the successor function $s: \mathbb{N} \to \mathbb{N}$;

* is closed under [[composition]]: if $f_1: \mathbb{N}^{k} \to \mathbb{N}, \ldots, f_n: \mathbb{N}^{k} \to \mathbb{N}$ and $g: \mathbb{N}^n \to \mathbb{N}$ belong to $\mathcal{C}$, then so does $g \circ (f_1, \ldots, f_n): \mathbb{N}^{k} \to \mathbb{N}$;

* is closed under primitive [[recursion]]: if $g: \mathbb{N}^k \to \mathbb{N}$ and $h: \mathbb{N}^{k+2} \to \mathbb{N}$ belong to $\mathcal{C}$, then the function $f: \mathbb{N}^{k+1} \to \mathbb{N}$ defined recursively by the equations (for $y \in \mathbb{N}$ and $\mathbf{x} \in \mathbb{N}^k$)
$$f(0, \mathbf{x}) = g(\mathbf{x})$$
$$\,$$
$$f(y+1, \mathbf{x}) = h(y, f(y, \mathbf{x}), \mathbf{x})$$
also belongs to $\mathcal{C}$;

* is closed under minimization: given any _[[total function|total]]_ function $f: \mathbb{N}^{k+1} \to \mathbb{N}$ belonging to $\mathcal{C}$, the partial function $g: \mathbb{N}^k \to \mathbb{N}$, defined by $g(\mathbf{x}) = c$ iff $f(c, \mathbf{x}) = 0$ and $f(d, \mathbf{x}) \gt 0$ whenever $0 \leq d \leq c-1$, also belongs to $\mathcal{C}$.
=--

+-- {: .num_defn}
###### Definition
A _primitive recursive function_ is a function that belongs to the smallest class of functions of the form $\mathbb{N}^k \to \mathbb{N}$ that contains constants, projection maps, the successor map, is closed under composition, and is closed under primitive recursion.
=--

Clearly the primitive recursive functions are a subclass of partial recursive functions. Notice that primitive recursive functions are not merely partial functions, but actual (*total*) functions.

+-- {: .num_remark #Lawvere}
###### Remark
Both the class of partial recursive functions and the class of primitive recursive functions yield [[Lawvere theories]] $Th(Comp)$ and $Th(PrimRec)$, where a morphism $k \to 1$ of the Lawvere theory is a function $\mathbb{N}^k \to \mathbb{N}^1$ belonging to the class. It is straightforward to check that in either case, the generating object $1$ of the Lawvere theory (or $\mathbb{N}$ if you prefer) is a parametrized [[natural numbers object|NNO]] in that category: this is the essence of primitive recursion.
=--

+-- {: .num_defn}
###### Definition
A _recursive relation_ (or what is more usual nowadays, a *computable relation*) is a subset of $\mathbb{N}^k$ whose characteristic function $\chi: \mathbb{N}^k \to \{0, 1\}$ is recursive. Similarly, a primitive recursive relation is a relation whose characteristic function is primitive recursive.
=--

## Properties

We may build up a stock of functions and relations that are primitive recursive as follows. (All closure properties mentioned will clearly also apply to partial recursive functions and relations.)

1. Addition, multiplication, and exponentiation on $\mathbb{N}$. The factorial function $n \mapsto n!$ is primitive recursive. Binomial coefficients.

1. The predecessor function $Pred: \mathbb{N} \to \mathbb{N}$, defined by the recursion $Pred(0) = 0$, $Pred(n+1) = n$.

1. *Truncated subtraction*, where $x \stackrel{\cdot}{-} y = x - y$ if $x \geq y$, else $0$, is primitive recursive. It is defined by the recursion $x \stackrel{\cdot}{-} 0 = x$, $x \stackrel{\cdot}{-} (y+1) = Pred(x \stackrel{\cdot}{-} y)$.

1. The distance function ${|x-y|} = (x \stackrel{\cdot}{-} y) + (y \stackrel{\cdot}{-} x)$. The function $\alpha(x) = 1 \stackrel{\cdot}{-} x$; this is the characteristic function of the unary relation or predicate "equals zero". So "equals zero" is a computable relation. So is "doesn't equal zero", since this has characteristic function $\alpha(\alpha(x))$.

1. Therefore the equality relation ${|x-y|} = 0$ is a primitive recursive relation. So is $x \lt y$, being the relation "$y \stackrel{\cdot}{-} x$ doesn't equal zero". If $f$ is a (primitive) recursive function, then its graph defined by the relation $y = f(x)$ is a (primitive) recursive relation.

1. (Boolean combinations) Similarly, if $R$ is a primitive recursive relation (so that its characteristic function $\chi_R$ is primitive recursive), then so is its negation $\neg R$, since $\chi_{\neg R} = \alpha(\chi_R)$. If $P$, $Q$ are primitive recursive relations, so is $P \wedge Q$, since $\chi_{P \wedge Q} = \chi_P \cdot \chi_Q$. Hence Boolean combinations of primitive recursive relations are primitive recursive.

1. If $f(x, y, \mathbf{z})$ and $h(y)$ are primitive recursive, so is
$$g(y, \mathbf{z}) \coloneqq \sum_{x=0}^{h(y)} f(x, y, \mathbf{z})$$
and similarly with the sum replaced by a product. It follows that primitive recursive predicates are closed under _bounded quantification_; e.g., if $R(x, y, \mathbf{z})$ is a primitive recursive predicate and $h$ is a primitive recursive function, then the predicate $Q$ defined by
$$Q(y, \mathbf{z}) = \forall_{0 \leq x \lt y} R(x, y, \mathbf{z}) \coloneqq \bigwedge_{x=0}^{y-1} R(x, y, \mathbf{z})$$
is primitive recursive since $\chi_Q(y, \mathbf{z}) = \prod_{x=0}^{y-1} \chi_R(x, y, \mathbf{z})$.

1. (Bounded least choice operator) If $R(x, y, \mathbf{z})$ is a primitive recursive relation and $h$ is a primitive recursive function, then the function $f(y, \mathbf{z})$ defined to be "the least $x \leq h(y)$ such that $R(x, y, \mathbf{z})$ if such $x$ exists, else $y$" is primitive recursive. Indeed,
$$f(y, \mathbf{z}) = [\sum_{x=0}^{h(y)} (x \cdot \chi_R(x, y, \mathbf{z}) \cdot (\prod_{w=0}^{x-1} \chi_{\neg R}(w, y, \mathbf{z}))] + y \cdot \prod_{x=0}^{h(y)} \chi_{\neg R}(x, y, \mathbf{z}).$$

1. (Pairing and unpairing) There is an isomorphism $\mathbb{N}^2 \to \mathbb{N}$ in the Lawvere theory $Th(PrimRec)$, i.e., there is a primitive recursive function $f: \mathbb{N}^2 \to \mathbb{N}$ with a primitive recursive inverse $g: \mathbb{N} \to \mathbb{N}^2$. For example, take $f$ to be the function
$$f: (m, n) \mapsto \binom{m+n+1}{2} + n$$
Its inverse $g$ takes $m$ to the pair
$$(a \stackrel{\cdot}{-} (m - \binom{a \stackrel{\cdot}{-} 1}{2}+2), m - \binom{a \stackrel{\cdot}{-} 1}{2})$$
where $a$ is the least element less than $m+2$ such that $\binom{a}{2} \gt m$. By the aforementioned properties, this $g$ is manifestly primitive recursive.

As a consequence of this last property, there exist primitive recursive [[isomorphisms]] between $\mathbb{N}$ and $\mathbb{N}^k$ for any $k \gt 0$. Since partial/primitive recursive functions are closed under composition, it is sufficient (and sometimes convenient) to consider only partial/primitive recursive functions on $\mathbb{N}$ itself.  (The exception is the case $k = 0$, but this is trivial, since every partial function $1 \to \mathbb{N}$ is recursive.)

+-- {: .num_remark}
###### Remark
To show that $\mathbb{N}^2 \cong \mathbb{N}$ in the category of primitive recursive functions, it is not enough just to exhibit a primitive recursive bijection $f: \mathbb{N}^2 \to \mathbb{N}$, because it is not true that every primitive recursive bijection possesses a primitive recursive inverse. In other words, it is not true that the forgetful functor $Th(PrimRec) \to Set$ (see Remark \ref{Lawvere}) reflects isomorphisms. See Example \ref{inverse} below.
=--

+-- {: .num_remark}
###### Remark
Similarly, it is not always true that if a graph of a function is a primitive recursive relation, then the function itself is primitive recursive. (For example, the graph of the Ackermann function, discussed below, is a primitive recursive relation.) However, we do have a sample theorem as follows.
=--

+-- {: .num_theorem #bound}
###### Theorem
If a graph of a function $f$ is a primitive recursive relation, and if $f \leq g$ for some primitive recursive $g$, then $f$ is itself primitive recursive.
=--

The proof can be roughly expressed as follows: if $R(x, y)$ is the functional computable relation, then let $f(x)$ be the least $y \leq g(x)$ such that $R(x, y)$. The bounded least choice property shows that $f(x)$ is primitive recursive.

The point hovering in the background is that there are some computable functions which grow faster (in fact, much much much faster) than any primitive recursive function. This underscores the important role of the minimization axiom for partial recursive functions, which allows unboundedly large searches to take place. Indeed, we have the following crucial fact:

+-- {: .num_theorem #graph}
###### Theorem
If $R \subseteq{N}^2$ is a graph of a function $f$ and is a recursive relation, then $f$ is a (total) recursive function. (The converse holds by one of the properties listed above.)
=--

+-- {: .proof}
###### Proof
According to the minimization axiom in Definition \ref{recursive}, given that $\chi_R: \mathbb{N}^2 \to \{0, 1\}$ is a recursive function, the function that takes $x$ to the least $y \in \mathbb{N}$ such that $1 - \chi_R(x, y) = 0$ is also recursive. But this function is simply $f$. This completes the proof.
=--

+-- {: .num_corollary}
###### Corollary
The inverse of a recursive bijection $f$ is also recursive.
=--

+-- {: .proof}
###### Proof
The graph of $f$ is a recursive relation, and so is the opposite graph, which is the graph of the function $f^{-1}$. Apply the preceding theorem.
=--

Before passing on to general recursive functions, it is good to have some idea of the scope of primitive recursive functions. Some examples:

* The function $f(n) = p_n$, the $n^{th}$ prime, is primitive recursive.

## The Ackermann function

+-- {: .num_defn}
###### Definition
For each $n$, we define a function $A_n: \mathbb{N} \to \mathbb{N}$ by

* $A_0(m) = 2 m$;

* $A_{n+1}(m) = (A_n)^m(1)$

where $f^m$ denotes the composition of $m$ copies of $f$. The **Ackermann function** $A: \mathbb{N} \to \mathbb{N}$ is defined by $A(m) = A_m(m)$.
=--

(The function is named after [[Wilhelm Ackermann]]. There are several variations of this function around, and one common version actually puts $A_0(m)=m+1$ and $A_{n+1}(m)=(A_n)^m(f(1))$.)

We show that while each $A_n$ is primitive recursive, the function $A$ grows faster than any primitive recursive function on $\mathbb{N}$, hence is not itself primitive recursive. It does however belong to the class of partial recursive functions.

By property 1 above, $A_0$ is primitive recursive. Supposing that $A_n$ is primitive recursive, $\phi = A_{n+1}$ is also primitive recursive because it satisfies the recursion $\phi(0) = 1$, $\phi(m+1) = A_n (\phi(m))$.

By induction one may easily show $A_n(1) = 2$ for all $n$ and $A_n(2) = 4$ for all $n$. We have $A_{n+1}(3) = A_n(A_n(A_n(1))) = A_n(A_n(2)) = A_n(4)$ for all $n$. The function $A_1$ gives $n^{th}$ powers of $2$, the function $A_2$ gives tetrations (iterated exponentials stacked $n$ high) of $2$, the function $A_3$ gives iterated tetrations, and so on.

Some routine inductions establish the following facts:

* For all $n$, the function $A_n$ is strictly increasing, and with the sole exception of $A_0(0) = 0$, we have $m \lt A_n(m)$ for all $m$, i.e. $A_n$ is strictly inflationary in arguments $m \gt 0$. We also have $n \lt A_n(3)$ for all $n$.

+-- {: .num_lemma}
###### Lemma
If $f: \mathbb{N}^k \to \mathbb{N}$ is primitive recursive, then there exists $n$ such that $f(x_1, \ldots, x_k) \leq A_n(\max \{3, x_1, \ldots, x_k\})$ for all $x_1, \ldots, x_k$. (We say $f$ is **dominated** by $A_n$, for short.)
=--

+-- {: .proof}
###### Proof
In the case where $f$ is constant with value $m$, take $n=m$. For $f$ the successor, use $n = 0$. For each projection $\pi_i: \mathbb{N}^k \to \mathbb{N}$, again use $n = 0$:

$$x_i = \pi_i(x_1, \ldots, x_k) \leq A_0(\max \{3, x_1, \ldots, x_k\}).$$

Now proceed by induction on the construction of primitive recursive functions. Given that $f: \mathbb{N}^k$ is dominated by $A_n$ and $g_1, \ldots, g_k: \mathbb{N}^m \to \mathbb{N}$ are dominated by $A_{n+1}$, we calculate that $h = f \circ (g_1, \ldots, g_k): \mathbb{N}^m \to \mathbb{N}$ is dominated by $A_{n+2}$:

$$\array{
h(x_1, \ldots, x_m) & = & f(g_1(x_1, \ldots, x_m), \ldots, g_k(x_1, \ldots, x_m)) \\
 & \leq & A_n(\max \{3, g_1(x_1, \dots, x_m), \ldots, g_k(x_1, \ldots, x_m)\}) \\
 & \leq & A_n(A_{n+1}(\max\{3, x_1, \ldots, x_m\})) \\
 & = & A_{n+1}(\max\{3, x_1, \ldots, x_m\} + 1) \\
 & \leq & A_{n+2}(\max\{3, x_1, \ldots, x_m\}).
}$$

And given that $g: \mathbb{N}^k \to \mathbb{N}$ is dominated by $A_{n+1}$ and $h: \mathbb{N}^{k+2} \to \mathbb{N}$ is dominated by $A_n$, if we define $f: \mathbb{N}^{k+1} \to \mathbb{N}$ by recursion by $f(0, x_1, \ldots, x_k) = g(x_1, \ldots, x_k)$ and $f(y+1, x_1, \ldots, x_k) = h(y, f(y, x_1, \ldots, x_k), x_1, \ldots, x_k)$, we calculate that $f$ is dominated by $A_{n+3}$, in two steps. First we claim

$$f(y, x_1, \ldots, x_k) \leq A_{n+1}(y + \max\{3, x_1, \ldots, x_k\}).$$

Indeed, this is true by assumption for $y=0$. And then

$$\array{
f(y+1, x_1, \ldots, x_k) & = & h(y, f(y, x_1, \ldots, x_k), x_1, \ldots, x_k) \\
 & \leq & A_n(\max\{3, y, f(y, x_1, \ldots, x_k), x_1, \ldots, x_k)\}) \\
 & \leq & A_n(\max\{3, y, A_{n+1}(y + \max\{3, x_1, \ldots, x_k\}), x_1, \ldots, x_k\}) \\
 & \leq & A_n(A_{n+1}(y + \max\{3, x_1, \ldots, x_k\})) \\
 & = & A_{n+1}(y+1+\max\{3, x_1, \ldots, x_k\})
}$$

which justifies the claim. Finally, we have

$$\array{
f(y, x_1, \ldots, x_k) & \leq & A_{n+1}(y + \max\{3, x_1, \ldots, x_k\}) \\
 & \leq & A_{n+1}(2\max\{3, y, x_1, \ldots, x_k\}) \\
 & \leq & A_{n+1}(A_{n+2}(\max\{3, y, x_1, \ldots, x_k\})) \\
 & = & A_{n+2}(\max\{3, y, x_1, \ldots, x_k\} + 1) \\
 & \leq & A_{n+3}(\max\{y, x_1, \ldots, x_k\})
}$$

so that $f$ is dominated by $A_{n+3}$. This completes the proof.
=--

+-- {: .num_corollary}
###### Corollary
The Ackermann function $A$ is not primitive recursive.
=--

+-- {: .proof}
###### Proof
If $A: x \mapsto A_x(x)$ were recursive, then so would be the function $\phi: x \mapsto A_x(x) + 1$. In that case, $\phi$ is dominated by $A_n$ for some $n \geq 3$. We then arrive at the contradiction

$$A_n(n) + 1 = \phi(n) \leq A_n(\max\{3, n\}) = A_n(n).$$
=--

+-- {: .num_prop}
###### Proposition
The _graph_ of the Ackermann function _is_ a primitive recursive relation.
=--

+-- {: .proof}
###### Proof
The rough idea is to let $z$ bound the search for solutions $(x, y)$ to $A_x(y) = z$. For $x, y \gt 0$ we have $A_x(y) = A_{x-1}(A_x(y-1)) = A_x(y')$ where $y' \coloneqq A_x(y-1)$. We have $y' \lt A_{x-1}(y') = z$. Starting with $y_0 = y \lt z$, iterate this procedure so that

$$z = A_x(y_0) = A_{x-1}(y_1) = A_{x-2}(y_2) = \ldots = A_0(y_x) = 2 y_x$$

with each $y_i$ less than $z$. (Explicitly, the iteration is $y_{i+1} \coloneqq A_{x-i}(y_i - 1)$.)

Thus, after disposing of some trivial low number cases, WLOG we may take $z \gt 4$, where the ternary predicate

$$(z \gt 4) \wedge (A_x(y) = z)$$

is equivalent to

$$(x \gt 0) \wedge (y \gt 2) \wedge (x \lt z) \wedge \exists_{y_0, y_1, \ldots y_x \lt z} (y = y_0) \wedge (\forall_{i \lt x} y_{i+1} = A_{x-i}(y_i - 1)) \wedge (2 y_x = z)$$

where the quantifications are manifestly bounded by $z$. This shows that the ternary predicate $A_x(y) = z$ is primitive recursive.

From this it follows that the binary relation $A_x(x) = z$ is also primitive recursive, as claimed.
=--

Combining this result with Theorem \ref{graph}, we have

+-- {: .num_corollary}
###### Corollary
The Ackermann function $A: x \mapsto A_x(x)$ is recursive.
=--

+-- {: .num_example #inverse}
###### Example
Here we exhibit a primitive recursive bijection whose inverse is not primitive recursive. The graph $y = A_x(x)$ of the Ackermann function is primitive recursive binary predicate, and the image $I = \{y: \exists_x y = A_x(x)\}$ is a primitive recursive unary predicate, because the existential quantifier is a bounded quantifier applied to a primitive recursive relation:

$$I = \bigvee_{x=0}^{y-1} [A_x(x) = y].$$

Observe that both $I$ and its complement $\neg I$ in $\mathbb{N}$ are infinite.

For any infinite set $J \subseteq \mathbb{N}$, let $p_J: \mathbb{N} \to \mathbb{N}$ be the function taking $n$ to the $n^{th}$ element of $J$ (with respect to the usual order $\lt$). For example, $p_I(x) = A_x(x)$. Now define
a function $f: \mathbb{N} \to \mathbb{N}$ by

$$\array{
f(m) & = & 2p_I^{-1}(m) & \text{if} \; m \in I \\
     & = & 2p_{\neg I}^{-1}(m) + 1 & \text{if} \; m \in \neg I
}$$

Then $f(m) \leq 2m + 1$ and the graph of $f$ is primitive recursive, so by Theorem \ref{bound}, $f$ is a primitive recursive function. It is a bijection by construction. But $f^{-1}$ is not primitive recursive, because $f^{-1}(2x) = p_I(x) = A(x)$, and $A$ is not primitive recursive.
=--

## Busy Beaver function

## Related concepts

* [[recursive subset]]

[[!include computable mathematics -- table]]

## References

A standard textbook in recursive functions is

* Hartley Rogers, Jr., _Theory of recursive functions and effective computability_, McGraw-Hill, 1967.

Quite a bit of the arrangement and proofs above were taken from clear and useful lecture notes of Stephen Simpson,

* Stephen G. Simpson, _Foundations of Mathematics_ (Lecture Notes), October 1, 2009. ([pdf](http://www.math.psu.edu/simpson/notes/fom.pdf))

Of course, there is always good old Wikipedia:

* Wikipedia, _[Computable function](http://en.wikipedia.org/wiki/Computable_function)_

[[!redirects partial recursive function]]
[[!redirects partial recursive functions]]
[[!redirects recursive partial function]]
[[!redirects recursive partial functions]]

[[!redirects Ackermann function]]
[[!redirects Ackermann functions]]

[[!redirects primitive recursive function]]
[[!redirects primitive recursive functions]]


partial section

# Partial sections
* table of contents
{: toc}

## Definition

Given a [[morphism]] $f: A \to B$ in any [[category]], a __partial section__ of $f$ is a [[partial map]] $s: B \nrightarrow A$ such that $f \circ s$ equals the identity on the [[domain]] of $s$. Explictly, this is a [[subobject]] $i: D \to B$ of $B$ together with a morphism $s: D \to A$ such that $f \circ s = i$.

## Examples

* A [[local section]] of a continuous map $f: A \to B$ is a partial section of $f$ whose domain is an [[open subset]] of $B$.

* In [[obstruction theory]], one may be interested in how far one can construct a section of a [[bundle]] (say a [[principal bundle]] with [[group]] $G$), $p: E \to X$, over a [[CW-complex]] $X$. Given a section of the restriction of $E$ over the $(k-1)$-skeleton $X_{k-1}$ (i.e., a partial section of $X$), obstructions to extending to a partial section over $X_k$ are measured by a [[cohomology class|class]] in $H^k(X_k, X_{k-1}; \pi_k(G))$.

[[!redirects partial section]]
[[!redirects partial sections]]


partial sum

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

#Contents#
* table of contents
{:toc}

## Definition

In [[analysis]], given a [[series]]

$$
  \underoverset
    {k = 0}
    {\color{blue}\infty}
    {\sum}
  a_k
$$

its $n$th *partial sum* (for $n \in \mathbb{N}$ any [[natural number]]) is the actual [[sum]]

$$
  \underoverset
    {k = 0}
    {\color{blue}n}
    {\sum}
    a_k
$$
of the first $n+1$ summands.

## Related concepts

* [[series]]

* [[Taylor series]]

## References

See also:

* Wikipedia, *Partial sum*

[[!redirects partial sums]]


partial trace

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A *partial trace* is a [[trace]] applied to (only) one factor in a [[tensor product]].

The partial trace is generalized by [[traced monoidal categories]]. That is, a traced monoidal category is a [[monoidal category]] paired with an operation which behaves like the partial trace.

## Definition

Let $\text{Vec}_k$ denote the category of [[finite-dimensional vector spaces]] over a field $k$. The partial trace gives a compatible collection of linear maps

$$\tr_W \colon \text{End}(V\otimes W)\to \text{End}(V)$$

given by

$$
  \tr_W(f\otimes g) \,=\, \tr(g)\cdot f
  \,,
$$

where "$tr$" on the right denotes the usual [[trace]].

Explicitly, the partial trace can also be defined as follows:

Let $f \colon V\otimes W\to V\otimes W$ be an endomorphism. Let $v_{1}, \ldots, v_{m}$ and $w_{1}, \ldots, w_{n}$ be [[linear bases]] for $V$ and $W$ respectively. Then $f$ has a [[matrix]] representation $\{a_{kl,ij}\}$ where $1 \le k,i \le m$ and $1 \le l,j \le n$ relative to the basis of the space $V \otimes W$ given by $e_{k} \otimes f_{l}$. Consider the sum

$$
  b_{k,i}
  \,=\,
  \sum_{j=1}^{n}a_{k j, i j}
$$

for $k,i$ over $1, \ldots, m$.  This gives the matrix $b_{k,i}$.  The associated linear operator on $V$ is independent of the choice of bases and corresponds to the partial trace.

## String diagram representations

Given vector spaces $A,B$, or more generally, given objects $A,B$ in a traced monoidal category, we will often write the partial trace using [[string diagrams]] as follows:

\begin{imagefromfile}
"file_name": "partial-trace.png",
"width": 200,
"unit": "px"
\end{imagefromfile}

Considering $\text{Vec}_k$ as a [[pivotal category]], this diagram is completely rigorous. To make sure this notation is consistent, we verify

\begin{imagefromfile}
"file_name": "partial-trace-computation.png",
"width": 450,
"unit": "px"
\end{imagefromfile}

Additionally the closed loop can _only_ be trace. This is because the axioms of a pivotal category we can move around the loop, so

\begin{imagefromfile}
"file_name": "composition-commutes.png",
"width": 500,
"unit": "px"
\end{imagefromfile}

It is a standard fact from [[linear algebra]] that the trace is uniquely characterized by the fact that it is linear and acts the same on $f\circ g$ and $g\circ f$ for all $f,g:A\to A$.

## Related concepts

* [[trace]], [[traced monoidal category]]

* [[partial trace quantum channel]]

[[!redirects partial traces]]


partial trace quantum channel

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

#Contents#
* table of contents
{:toc}

## Idea

For $\mathscr{B}$ a [[finite-dimensional Hilbert space]], the operation of [[partial trace]] over $\mathscr{B}$ is a quantum channel

$$
  \array{
    \mathscr{H}
    \otimes
    \mathscr{B}
    \otimes
    \mathscr{B}^\ast
    \otimes
    \mathscr{H}^\ast
    &
    \xrightarrow{
      trace^{\mathscr{B}}
    }
    &
    \mathscr{H} \otimes \mathscr{H}^\ast
    \\
    \left\vert \psi, \beta \right\rangle
    \left\langle \beta', \psi' \right\vert
    &\mapsto&
    \left\vert \psi \right\rangle
    \left\langle \beta' \vert \beta \right\rangle
    \left\langle \psi' \right\vert
    \mathrlap{\,.}
  }
$$

which represents the "operation" of averaging the [[mixed state|mixed]] [[quantum states]] of the [[composite system|composite]] [[quantum system]] $\mathscr{H} \otimes \mathscr{B}$ over the degrees of freedom of $\mathscr{B}$.

## Properties

[[!include quantum channels and decoherence -- section]]

[[!include environment rep of quantum measurement channel - section]]

## Related concepts

* [[decoherence]]

* [[quantum channel]]

  * [[quantum measurement channel]]

  * [[unitary quantum channel]]

[[!redirects partial trace quantum channels]]

[[!redirects averaging quantum channel]]
[[!redirects averaging quantum channels]]


partially ordered dagger-category

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

## Definition

A **locally partially ordered category** is a [[category]] together with [[partial order]] $\subseteq$ on each of its [[hom-set|hom-sets]] such that
$$ f_1 \subseteq f_2 \;\wedge\; g_1 \subseteq g_2 \;\Rightarrow\; g_1 \circ f_1 \subseteq g_2 \circ f_2 $$
for any morphisms $f_1$, $g_1$, $f_2$, $g_2$ such that the composites above are defined. In other words, a category enriched in [[Pos]], the category of [[posets]].

Compare this to the notion of category [[internal category|internal]] to [[Pos]].

Similarly, such a partially ordered category can be considered as a special kind of [[double category]], while a [[locally posetal 2-category|locally partially ordered category]] can be considered as a special kind of $2$-[[2-category|category]].

## Partially ordered $\dagger$-categories ##

For a partially ordered $\dagger$-[[dagger-category|category]] we can (optionally?) require that
\[
  f^{\dagger} \subseteq g^{\dagger} \Leftrightarrow f \subseteq g
\]
for any morphisms $f$ and $g$.

>The following nomenclature points were raised by one of the previous contruibutors. What do people think? Are they justified or in conflict with existing terminology? Are there examples where the conditions come up naturally?

 * For a partially ordered $\dagger$-category I will call _monovalued_ morphism such a morphism $f$ that $f \circ f^{\dagger} \subseteq 1_{\mathrm{Dst} f}$.

 * For a partially ordered category with inverses I will call _entirely defined morphism_ such a morphism $f$ that $f^{\dagger} \circ f \supseteq
1_{\mathrm{Src} f}$.

## Partially ordered groupoids

The particular case of an ordered groupoid (in which each $f^{-1}$ is the inverse of $f$) is called an ordered groupoid.  This has been studied extensively by Mark Lawson, for instance see

* [[ Mark V. Lawson]],  _Constructing ordered groupoids_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46 no. 2 (2005), p. 123-138,
([numdam](http://www.numdam.org/item?id=CTGDC_2005__46_2_123_0))

or his book:

* [[M.V.Lawson]], _Inverse semigroups: the theory of partial symmetries_, World Scientific, 1998.

He makes the point that they were initially studied by Ehresmann and are very closely related to inverse semi-groups.

## See also ##
* [[generalized continuity]]

[[!redirects partially ordered category]]
[[!redirects partially ordered dagger-category]]
[[!redirects partially ordered dagger category]]
[[!redirects partially ordered †-category]]
[[!redirects ordered groupoid]]


partially ordered object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The notion of a *partially ordered object* is the generalization of that of *[[partially ordered sets]]* as one passes from the ambient [[category of sets]] [[internalization|into]] more general ambient  [[categories]] with suitable properties.

## Definitions

In a [[finitely complete category]] $C$, a **partially ordered object** is a [[preordered object]] $(X, R, s, t, \rho, \tau_p)$ such that the internal preorder $R\stackrel{(s,t)}\hookrightarrow X \times X$ is an [[internal antisymmetric relation]].

## See also

* [[preordered object]]

* [[partially ordered set]]

* [[internal antisymmetric relation]]

* [[meet-semilattice object]]

* [[join-semilattice object]]

[[!redirects partially ordered objects]]


partially ordered ring

# Contents
* table of contents
{: toc}

## Definition

A __partially ordered ring__ is a [[ring]] $R$ with a [[partial order]] such that for all elements $a,b,c$ in $R$, $a \leq b$ implies $a + c \leq b + c$, and $0 \leq a$ and $0 \leq b$ implies $0 \leq a \cdot b$.

Due to the [[reflexivity]] of the partial order, partially ordered rings may have [[zero divisors]]. Also, the [[trivial ring]] is an partially ordered ring.

If the relation $\leq$ is only a [[preorder]], then the ring $R$ is said to be a **preordered ring**.

## Related concepts

* [[pseudolattice ordered ring]]

* [[totally ordered ring]]

* [[pseudo-ordered rings]]

* [[ordered field]]

* [[ordered integral domain]]

* [[ordered group]]

* [[protoring]]

## External links

* Wikipedia, _Partially ordered ring_

[[!redirects partially ordered ring]]
[[!redirects partially ordered rings]]

[[!redirects preordered ring]]
[[!redirects preordered rings]]

[[!redirects partially ordered weak local ring]]
[[!redirects partially ordered weak local rings]]

[[!redirects partially ordered weak Heyting field]]
[[!redirects partially ordered weak Heyting fields]]


particle

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## Idea

The [[field (physics)|field]] of [[quantum field theory]] started out as a description of the [[fundamental particles]] that are observed in [[experiment]], such as [[electrons]] and [[photons]].

However, even so, abstractly the formalization of the concept of _particle_ within [[QFT|QFTs]] is somewhat subtle.

If the quantum field theory is on [[Minkowski space]] and comes with a [[Hilbert space|Hilbert]] [[space of states]] on which thus the  [[Poincare group]] of translations, rotations and boosts in Minkowski space acts, the massive _particle_ excitations of the theory can be found in the discrete spectrum of the time translation operator as the [[irreducible representation|irreducible]] [[unitary representations of the Poincare group]].
For QFTs on [[curved spacetime|curved]] [[spacetimes]] the situation is more subtle.

Often, however, QFTs are considered as [[quantizations]] of given [[Lagrangians]]. In these cases one often identifies their particle content with that explicitly encoded by the Lagrangian. Notably when that arises from [[second quantization]] of some 1-dimensional [[sigma-model]], the particles of the theory are those described by these sigma-models.

## What is a particle?
 {#WhatIsAParticle}

The fundamental concept of modern [[physics]] is that of [[quantum field theory]] (QFT); the concept of particle is derived from that, and need not make sense in every case. ("That's why it's called 'field theory'.")

 In the perspective of the [[Schrödinger picture]], a $(d+1)$-dimensional [[QFT]] is given by a [[functor]] $Z$ on a [[category of cobordisms]] (possibly with [[geometry|geometric]] structure, such as [[pseudo-Riemannian metric]] structure) between $d$-dimensional [[manifolds]] ("[[FQFT]]"). It is crucial to notice that one such QFT always has **two different interpretations**:

1. a [first quantized worldvolume perspective](#FirstQuantizedPerspective);

1. a [second quantized spacetime perspective](#SecondQuantizedSpacetimePerspective).

### First quantized worldvolume perspective
 {#FirstQuantizedPerspective}

If we think of a $d$-dimensional manifold as the shape of a some quantum object -- (as such commonly called a $d$-[[brane]]) -- then a [[cobordism]] between two such is thought of as a piece of [[worldvolume]], a way for parts of such an object to interact with other parts. From this perspective the functor $Z$ assigns to a manifold the [[space of quantum states]] that the [[brane]] of this shape may have, and to a [[cobordism]] the linear map which is the time evolution along the cobordism.

 Here if $d = 0$ then the [[brane]] is a "0-brane" and this is a "particle" (or [[D0-brane]]), the worldvolume is the "[[worldline]]" and the QFT encodes the [[worldline theory]] of the particle, its [[quantum mechanics]].

 If instead $d = 1$ then the brane is a 1-brane, for instance a [[string]] of [[D1-brane]], if $d = 2$ then the brane is a 2-brane also called a [[membrane]], and so on.

Given this, one may try to see if this data describes a brane propagating _in_ some [[spacetime]] (the "[[target space]]" of the brane). It is the topic of [[spectral geometry]] (in the sense of [[Alain Connes]]) to try to reconstruct from this data the would-be target spacetime that the brane is propagating in. For instance for $d = 0$ the data of a QFT in this sense here is a [[spectral triple]] and [[noncommutative geometry]] provides a general way to make sense of the target space of the particle. If $d =1$ the QFT data here is that of a [[2-spectral triple]], and so on.

### Second quantized perspective
 {#SecondQuantizedSpacetimePerspective}

On the other hand, we may think of the $d+1$-dimensional [[cobordisms]] here themselves already as [[spacetimes]]. In this case the QFT describes [[field (physics)|fields]] on spacetime. In favorable circumstances this can arise from the previous case by a process of [[second quantization]], meaning that these fields may be thought of as [[condensates]] of branes/particles in the previous sense. Conversely one says that these particles are the _quanta_ of the fields that we start with.

But generally, given a QFT in this perspective, to extract from it the particle content that it comes from under [[second quantization]] is subtle. One of the common definitions of particle quanta only applies to non [[general covariance|generally-covariant]] [[free field theories]] (e.g. [Haag 92, section VI](#Haag92)). This means that already for quantum field theory on a fixed [[curved spacetime]] there is in general no longer any concept of particle-quanta of the fields. This situation would only become worse were one to think of the given QFT as incorporating also [[quantum gravity]]. The concept of [[field (physics)|field]] here is fundamental, that of particle quanta is not.

### Mixed perspectives

Since the formalism of [[FQFT]] does not "know" whether we want to think of a given QFT as a [first-quantized worldvolume theory](#FirstQuantizedPerspective) or as a [second quantized spacetime theory](#SecondQuantizedSpacetimePerspective) in general both perspectives may be sensible at the same time.

This is indeed so, but of course this mixing only becomes relevant once one really dares to consider higher-dimensional [[branes]] in the first place, hence in [[string theory]].

Indeed, [[perturbative string theory]] is all set up this way: one starts with a 2-dimensional QFT which one thinks of as the first-quantized [[worldsheet]] theory of a [[string]]. But this means that one may start to ask which "particles" propagate "on the worldvolume". Notably the "embedding fields" of the string [[sigma-model]] which describes how its worldsheet sits in [[spacetime]] look, from the perspective of the worldsheet theory, like [[scalar fields]]. Their [[superpartners]] look like [[fermion]] fields. If one considers the string worldsheet before gravitational [[gauge fixing]] then there is also a [[graviton]] on the worldsheet, and so on.

Hence in general one may want to/need to consider an intricate pattern of "branes withing branes". For instance the [[worldsheet]] [[gravity]] of the [[string]] may itself arises from quantizing other strings for which that worldsheet is target spacetime ([[world sheets for world sheets|Green 87]]).

## Examples

### Particles and non-particles in 3d TQFT
 {#ParticlesAndNonParticlesIn3dTQFT}

We consider a [[QFT]] which is a [[3d TQFT]] of [[Chern-Simons theory]] type and discuss some aspects of the notion of _particle_ in that context.

Assume for the sake of argument that we agree to think of the 3d TQFT here as a realization of [[3d quantum gravity]], as indicated there.  Then this means that as an [[FQFT]] the system assigns to every closed [[surface]] a [[space of quantum states]] to be thought of as the space of states of the [[observable universe]] in that 2+1-dimensional world. A state in here encodes the field of [[gravity]]. It would be somewhat subtle to extract from just the [[FQFT|functorial]] [[3d TQFT]] here the intrinsic notion of particle-quanta. In general, given an [[TQFT]] in the form of an [[FQFT]], there is essentially no established way to going about determining what the particle-quanta would be that an observer "in this universe" would see.

In fact, an argument due to ([Witten 92](A-model#Witten92)) says that if [[Chern-Simons theory]] [[3d TQFT]] is the [[second quantization]] of anything, then it is not of particles but of [[topological strings]] ([[A-model]]). See also ([Costello 06](A-model#Costello06)) and see at _[TCFT -- Worldsheet and effective background theories](TCFT#ActionFunctionals)_ for more on this.

Beware, again, that this concerns the quanta for the fields of the QFT regarded as a QFT on a 2+1-dimensional [[spacetime]]. However we may change perspective and instead think of the [[3d TQFT]] here as a first-quantized [[worldvolume]] [[theory (physics)|theory]]. As such it would be a [[membrane]] [[theory (physics)|theory]], often called the _[[topological membrane]]_, naturally.

Now if we allow [[boundaries]] of [[worldvolume]], hence consider an [[extended TQFT]] with its [[boundary field theory]], then the boundary theory is a 1+1-dimensional worldsheet theory, hence describes a [[string]]. This way of how a first quantized [[string]] can arise as the [[boundary field theory]] of a first-quantized [[topological membrane]] is an instance of the "[[holographic principle]]" known as _[[AdS3-CFT2 and CS-WZW correspondence]]_.

Moreover, going further up in [[codimension]], the [[3d TQFT]] may have [[defect field theory|defects]] of codimension 2, hence have inside it a 0+1-dimensional [[defect field theory]]. This hence may be thought of as a first-quantized particle. (Notice that it is a _first quantized_ particle, not a quantum of a field of the 3d theory regarded as a spacetime theory, for these particles-as-quanta do not have worldlines given by cobordisms,only their first quantized avatars do, but they are not the first quantized 1d defect theory considered now.)

Indeed, these first quantized codim-2-defects/0-branes/1d-particles in [[Chern-Simons theory]] are famous as having "[[Wilson line]] [[worldline theory]]". See at _[orbit method -- Nonabelian charged particle trajectories](orbit%20method#GaugeAndGravityWilsonLoops)_ for details on their incarnation as [[prequantum field theory]]. After [[quantization]] these first quantized 0-branes/1d-particles are famously represented in the [[Reshetikhin-Turaev construction]] as ribbon lines labeled by [[objects]] in a [[modular tensor category]].

In conclusion, given a [[3d TQFT]] regarded as [[quantum gravity]] of 2+1-dimensional [[spacetimes]], it is at best subtle to extract from it particles in the sense of "quanta of the fields of the spacetime field theory", while extracting from it first quantized codim-2 [[defect field theory|defect]] 0-[[branes]] is a famous step in [[Chern-Simons theory]].

## Related concepts

* [[sigma-model]]

* [[worldline]], [[worldline theory]]

* [[non-relativistic particle]]

* [[relativistic particle]], [[spinning particle]], [[superparticle]]

* [[virtual particle]], [[antiparticle]]

* [[fundamental particle]], [[standard model of particle physics]]

* [[quasiparticle]], [[anyon]]

* [[matter]], [[force]]

* [[brane]], [[string]], [[membrane]]

* [[mechanics]]

* [[vacuum]]

## References
 {#References}

Chapter VI of

* {#Haag92} [[Rudolf Haag]], _[[Local Quantum Physics]]_ (1992)

discusses how to extract notions of particles from a [[local net of observables]] satisfying the [[Haag-Kastler axioms]].

Further discussion of subtleties of the definition of particles _in_ (non-[[free field theory|free]]) [[quantum field theories]]:

* {#Bain00} [[Jonathan Bain]], _Against particle/field duality: Asymptotic particle states and interpolating fields in interacting QFT (or: Who's afraid of Haag's theorem)_, Erkenntnis 53: 375–406, 2000  ([pdf](http://faculty.poly.edu/~jbain/papers/lsz.pdf))

* Benjamin H. Feintzeig, Jonah Librande, Rory Soiffer, *Localizable Particles in the Classical Limit of Quantum Field Theory* ([arXiv:2104.06442](https://arxiv.org/abs/2104.06442))

[[!redirects particles]]

[[!redirects quantum]]
[[!redirects quanta]]


particle accelerator

## References

* Wikipedia, _[Particle accelerator](http://en.wikipedia.org/wiki/Particle_accelerator)_


Particle Data Group

* [home page](https://pdg.lbl.gov/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Particle_Data_Group)

## Selected writings

On [[particle physics]]:

* [[Particle Data Group]], *[The Review of Particle Physics](https://pdg.lbl.gov/)*,  Prog. Theor. Exp. Phys. 2020, 083C01 (2020) ([doi:10.1093/ptep/ptaa104](https://doi.org/10.1093/ptep/ptaa104))

category: people


particle physics

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## Entries on particle physics

* [[fundamental particle]],

* [[field (physics)]], [[quantum field theory]]

  [[perturbative quantum field theory]]

* [[scattering amplitude]], [[S-matrix]]

* [[model (in particle physics)]]

* [[standard model of particle physics]]

  * [[naturalness]]

  * [[Grand Unified Theory]], [[GUT]]

* [[experiment]],  [[phenomenology]]

[[!include fields and quanta - table]]

## Related entries

* [[quantum gravity]]

* [[string theory]]

## References

* [[Robert E. Marshak]], *Conceptual Foundations of Modern Particle Physics*, World Scientific 1993 ([doi:10.1142/1767](https://doi.org/10.1142/1767))

* [[Ulrich Mosel]], *Fields, Symmetries, and Quarks*, Springer 1999 ([doi:10.1007/978-3-662-03841-3](https://link.springer.com/book/10.1007/978-3-662-03841-3))

* James Dodd, [[Ben Gripaios]], _The Ideas of Particle Physics_, Cambridge University Press 2020 ([ISBN:9781108727402](https://www.cambridge.org/gb/academic/subjects/physics/particle-physics-and-nuclear-physics/ideas-particle-physics-4th-edition?format=PB), [doi:10.1017/9781108616270]( https://doi.org/10.1017/9781108616270))

* Wikipedia, _[Particle physics](https://en.wikipedia.org/wiki/Particle_physics)_

* Mark Thomson, _[Particle physics](http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/welcome.html)_ handouts

* [[Particle Data Group]], _Review of Particle Physics_, Progress of Theoretical and Experimental Physics, Volume 2020, Issue 8, August 2020, 083C01 ([doi:10.1093/ptep/ptaa104](https://doi.org/10.1093/ptep/ptaa104))

Relation to [[string theory phenomenology]]:

* [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012 ([doi:10.1017/CBO9781139018951](https://doi.org/10.1017/CBO9781139018951))


particle statistics

Related entries include [[statistical mechanics]],
[[braid group statistics]], [[spin-statistics theorem]], [[Bekenstein-Hawking entropy]], [[parastatistics]]

category: physics


partition

# Partitions
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{: toc}

## Idea

A _partition_ is any of various ways of dividing a mathematical object into nontrivial _parts_ that (in some sense) cover the object while being mutually disjoint.

## Definitions

Each of the following definitions is a special case of a general concept to follow.

### Partitions of sets

Given a [[set]] $S$, a __partition__ of $S$ is a collection $\pi$ of [[inhabited subsets]] of $S$ (called _blocks_) such that

*  For $A, B \in \pi$, if $A \cap B$ is inhabited, then $A = B$;
*  The [[union]] $\bigcup \pi$ is all of $S$ (the [[improper subset]]).

A partition $\pi$ __refines__ a partition $\rho$ (of the same set) if every block of $\pi$ is contained in some block of $\rho$:
$$ \forall A \in \pi,\; \exists B \in \rho,\; A \subseteq B .$$
Refinement is a [[partial order]] on the class of partitions of $S$.

Partitions of $S$ are in [[bijective correspondence]] with [[equivalence relations]] on $S$; a partition is precisely a collection of [[equivalence classes]].  Refinement corresponds to implication between relations.

Partitions of $S$ are also closely related to [[surjections]] out of $S$.
Any partition $\pi$ of $S$ induces a surjection $S \to \pi$ that assigns each element of $S$ to a unique block, and, conversely, any surjection $f : S \to B$ induces a partition of $S$ by taking the blocks to be the [[fiber]]s $\{f^{-1}(x) \mid x \in B\}$.
In general, many different surjections can map to the same partition under this correspondence. However, there is a natural [[preorder]] on surjections defined by taking $f \le g$ just in case there exists a (necessarily unique) $h$ such that $g = h \circ f$.
Then the operations we have just described give an [[equivalence of categories|equivalence]]
$$ Part(S) \rightleftarrows S \downarrow Surj $$
between the partial order of partitions ordered by refinement and the preorder of surjections ordered by factorization.

### Partitions of natural numbers

Given a [[natural number]] $n$,

1. a __composition__ of $n$ is a [[list]] of [[positive number|positive]] natural numbers whose [[sum]] is $n$,

1. a __partition__ of $n$ is an unordered list, or [[multiset]], of such numbers.

That is, different compositions define the same partition if the compositions differ only by order. A partition may also be defined as a [[monotone function|monotone]] composition.

In practice, notably in the [[representation theory of the symmetric group]]/[[representation theory of the general linear group|general linear group]], partitions of natural numbers $n \in \mathbb{N}$ are often given as anti-monotone sequences, denoted as

$$
  \lambda
  \;=\;
  (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_{rows(\lambda)})
  \,,
  \;\;\;\;
  \underoverset
    {rows(\lambda)}
    {i = 1}
    {\sum}
    \lambda_i
  \;=\;
  n
  \,.
$$

#### Young diagrams

Often, especially in [[representation theory]] ([[representation theory of the symmetric group|of the symmetric group]]/[[representation theory of the general linear group|general linear group]]), partitions of a natural number $n$ appear as *[[Young diagrams]]* with $n$ boxes. While Young diagrams in themselves are just an equivalent way of thinking of or depicting partitions, they are used to indicate ("[[concept with an attitude]]") that decorations of partitions to [[Young tableaux]] play a role in a given discussion.

#### Partition function

The __[[partition function (number theory)|partition function]]__ $p$ gives the number of partitions of $n$ as a [[function]] of $n$; this is [OEIS A000041](https://oeis.org/A000041).  Its (ordinary) [[generating function]] is

$$
  \sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}
  \,.
$$

#### Relation to partitions of sets

Every [[natural number]] $n$ corresponds to a [[finite set]] $[n]$, and every partition of $[n]$ (as defined above) gives a partition of $n$, but different partitions of $[n]$ may give the same partition of $n$.  Conversely, a composition of $n$ defines a partition of $[n]$, but not every partition of $[n]$ arises in this way.

More precisely, we have the following, where $\rightarrowtail$ indicates an [[injection]] and $\twoheadrightarrow$ indicates a [[surjection]]:
$$ Comp(n) \rightarrowtail Part([n]) \twoheadrightarrow Part(n) ;$$
the [[composite]] of this is also a surjection, which is [[split surjection|split]] by the definition of a partition as a monotone composition.

One may also speak of __multiplicative__ compositions and partitions of $n$ for positive $n$ (where the above are _additive_), also called (ordered and unordered) __factorizations__; these are (ordered and unordered) lists of natural numbers greater than $1$ whose product is $n$.

### Of intervals

Given [[real numbers]] $a$ and $b$, a __partition__ $\pi$ of the [[interval]] $[a,b]$ is an inhabited finite [[list]] $c_0, \ldots, c_n$ such that:
* $c_0 = a$,
* $c_i \leq c_{i+1}$ for $i \lt n$,
* $c_n = b$.

A partition $\pi$ __refines__ a partition $\rho$ if every element of the list $\rho$ belongs to the list $\pi$.  The __mesh__ (or __norm__) of $\pi$ is
$$ {\|\pi\|} \coloneqq \max_i (c_{i+1} - c_i) .$$
A __tag__ of $\pi$ is a list $t_0, \ldots, t_{n-1}$ such that $c_i \leq t_i \leq c_{i+1}$ for $i \lt n$.  Tagged partitions are used to define the [[Riemann integral]], the [[Darboux integral]], and the [[gauge integral]].

### Of measure spaces

Given a [[measure space]] $S$ (or more generally a [[measurable space]] equipped with a notion of [[null sets]]), a __partition__ of $S$ is a collection $\pi$ of [[positive subset|positive]] (non-null) [[measurable subsets]] of $S$, such that:
*  For $A, B \in \pi$, if $A \cap B$ is positive, then $A$ and $B$ are equal;
*  The [[union]] $\bigcup \pi$ is [[full subset|full]].

(One might merely allow $A$ and $B$ to be equivalent, meaning that their [[symmetric difference]] is null, when $A \cap B$ is positive, but this does not give a good notion of tagged partition; really, we should think of

so insist that $\pi$ be _saturated_: if $A$ and $B$ are equivalent and $A \in \pi$, then $B \in \pi$.  Alternatively, especially if one views the partition an _[[indexed subset|indexed]]_ collection of subsets, then one would still require $A = B$ when $A \cap B$ is positive.)

Then a partition of a set is simply a partition of the measure space given by [[counting measure]].

Partitions in this sense can be used to define the [[Lebesgue integral]] in analogy to the definition of the Riemann integral using partitions of intervals.

### Of unity on topological spaces

Given a [[topological space]] $S$, a __[[partition of unity]]__ on $S$ is a collection $U$ of [[continuous maps]] from $S$ to the [[unit interval]] such that, for each point $x$ in $S$:
* $f(x) \gt 0$ for only finitely many $f$ in $U$;
* $\sum_f f(x) = 1$ (where this sum is defined by the previous clause).

### General

Let $M$ be an [[abelian monoid]] (written additively) with the property that $a + b = 0$ only if $a, b = 0$.  (In other words, the nonzero elements form an [[ideal]]; there\'s probably a name for this kind of monoid, but I don\'t know it.)  For purposes of [[constructive mathematics]], we should start with an ideal of elements of $M$ declared _nonzero_, then require that $\{0\}$ be its [[complement]].  Then a __finite composition__ of an element $x$ of $M$ is a finite [[list]] of nonzero elements of $M$ whose sum is $x$, and a __finite partition__ of $x$ is the same thing without regard to order (so a [[multiset]] instead of a list, which is why $M$ must be abelian).  If we now equip $M$ with some notion of sum of some infinite [[series]], we can generalise to (not necessarily finite) __partitions__ (and even compositions, for ordered series), which really should be viewed as [[indexed families]] of elements of $M$.

Composititions and partitions of natural numbers are immediate examples of the finite notion, using the monoid $\{0, 1, 2, \ldots\}$ under addition; multiplicative compositions and partitions use instead $\{1, 2, \ldots\}$ under multiplication.

A partition of unity is a partition of the [[constant function]] with value $1$ in the monoid of continuous maps to the nonnegative [[real line]].  Here we define an infinite sum of functions when only finitely many of the functions are nonzero at each point.  (We could in principle define more infinite sums than these, which would give a notion more general than the usual one of partition of unity.)

A partition of a set [or measure space] $S$ is a partition of $S$ itself in the monoid of [measurable] multisubsets of $S$ [modulo equivalence almost everywhere].  This monoid has all infinite sums, so there is no finiteness requirement.  (Every multisubset in a partition is in fact a subset, but working in the monoid of subsets under [[union]], rather than multisubsets under multi-union, would allow the parts to overlap.)

Finally, a partition of an interval is a finite partition in the monoid [[generators and relations|generated by]] the compact intervals subject to the relations
$$ [a,b] + [b,c] = [a,c] .$$

In the last few examples, each part of a partition is an [[inhabited set]].  In such cases, a __tagged partitition__ is a partition in which each part has been equipped with one of its elements.

## The lattice of partitions of a finite set

For any finite set $S$ with $n$ elements, the partially ordered set of partitions of $S$ forms a [[lattice]] (called $\Pi_n$). Viewing partitions as equivalence relations on $S$, this lattice structure may be described as follows [(Birkhoff)](#Birkhoff):

* the [[bottom]] element is the [[diagonal relation]] on $S$
* the [[top]] element is the total relation on $S$
* the [[meet]] of two equivalence relations $\pi$ and $\rho$ on $S$ is the equivalence relation $\pi \wedge \rho$ defined by $x(\pi\wedge \rho)y$ iff $x\pi y$ and $x\rho y$
* the [[join]] of two equivalence relations $\pi$ and $\rho$ is the meet of all equivalence relations $\sigma$ such that $\pi \subseteq \sigma$ and $\rho \subseteq \sigma$ (this is well-defined since $\Pi_n$ is finite)

Dually, viewing partitions as surjections from $S$, the lattice structure may be described as follows:

* the [[bottom]] element is the identity function $S \to S$
* the [[top]] element is the unique function $S \to 1$ to the 1-element set
* the [[join]] of two surjections $\pi : S \to B$ and $\rho : S \to C$ is their [[pushout]]
* the [[meet]] of two surjections $\pi : S \to B$ and $\rho : S \to C$ is the join of all surjections $\sigma : S \to A$ for which there exist functions $f : A \to B$ and $g : A \to C$ such that $\pi = f \circ \sigma$ and $\rho = g \circ \sigma$

## References

* {#Birkhoff} [[Garrett Birkhoff]], _Lattice Theory_, AMS, third edition, 1973.

* [[Marcelo Aguiar]] and [[Swapneel Mahajan]]. [[Monoidal Functors, Species and Hopf Algebras]], 2010.

[[!redirects partition]]
[[!redirects partitions]]

[[!redirects tagged partition]]
[[!redirects tagged partitions]]
[[!redirects composition]]


partition function

> This entry is about *partition functions* in the sense of [[statistical mechanics]] and [[quantum field theory]]. For the function in [[number theory]]/[[combinatorics]] that assigns to a [[natural number]] the number of its [[partitions]] see at *[[partition function (number theory)]]*.

***

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## Idea

A _partition function_ is a certain assignment that may be extracted from a system in [[statistical mechanics]], or in [[quantum field theory]].

If the quantum field theory $Z$ is presented as an [[FQFT]], that is as a [[functor]] on a category of $d$-dimensional [[cobordisms]], then the partition function is the assignment to $d$-dimensional [[torus|tori]] $T$ of the values $Z(T)$ assigned to these by the QFT.

By the axioms of functoriality and symmetric monoidalness of a QFT, this means that the partition function is the [[trace]] over the value of the QFT in the cylinder obtained by cutting the torus open.

This is where the partition function originally derives its name from: typically for QFTs on _[[Riemannian metric|Riemannian]]_ [[cobordisms]] the  value of the QFT on a cylinder of _length_ $t$ is a [[linear operator]] of the form $\exp(- t H)$ for some operator $H$.

## Origin of the term

When one thinks of the QFT --- under [[Wick rotation]] --- as describing a physical system in [[statistical mechanics]], then the vector space that $H$ acts on is the vector space of all states of the system and $H$ is the operator whose eigenstates are the states of definite energy. The expression

$$
  tr(exp(-t H))
$$

then is interpreted as

> sum over all states $\Psi$ of the system and weigh each one by its energy $E_\Psi$.

This involves, conversely, counting for each fixed energy $E_\Psi$ the number of states of that energy. This will typically be a sum over certain partitions of various particles of an ensemble into various "bins" of partial energies. Therefore the term _partition_ function.

In fact, the common letter $Z$ uses to denote QFTs (or at least [[TQFT]]s) also derives from this: in German the partition function is called _Zustandssumme_ --- from German _Zustand_ for "state" .

The [[Mellin transform]] of the partition function is known in [[quantum field theory]] as the [[Schwinger parameter]]-formulation which takes the [[worldline theory]] to its [[zeta function regularization|zeta regulated]] [[Feynman propagator]].

## Examples

Partition function for the [[superparticle]]: [[K-theory]] [[index]].

Partition function for the [[type II superstring]]: [[elliptic genus]].

Partition function for the [[heterotic string]]: [[Witten genus]].

For some discussion of partition functions of 1-dimensional QFTs see [[(1,1)-dimensional Euclidean field theories and K-theory]].

For some discussion of partition functions of 2-dimensional QFTs see [[(2,1)-dimensional Euclidean field theories and tmf]]

## Related concepts

* [[functional determinant]], [[zeta function of an elliptic differential operator]]

* [[vacuum amplitude]], [[vacuum energy]]

* [[Witten index]]

* [[Riemann hypothesis and physics]]

[[!include genera and partition functions - table]]

## References

### General

* Addison Ault, "The partition function: If that's what it is _Why don't they say so!_" ([pdf](http://people.cornellcollege.edu/aault/Chemistry/PartitionFunction1.pdf))

[[!include elliptic genera as partition functions -- references]]

[[!redirects partition functions]]


partition function (number theory)

> This entry is about the notion in [[number theory]]/[[combinatorics]]. For partition functions in the sense of [[statistical mechanics]] and [[quantum field theory]] see at *[[partition function]]*.

***

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###Context###
#### Combinatorics
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[[!include combinatorics - contents]]
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#### Arithmetic
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[[!include arithmetic geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[number theory]]/[[combinatorics]], the [[function]] which assigns to a [[natural number]] $n \in \mathbb{N}$ the [[number]] $p(n)$ of its [[partitions]] (into [[multisets]] of [[positive number|positive]] [[natural numbers]]) is often called the *paritition function*.

## Properties

### Generating function

Its (ordinary) [[generating function]] is

$$
  \sum_{n=0}^\infty p(n) x^n = \prod_{k=1}^\infty (1 - x^k)^{-1}
  \,.
$$

### Asymptotic expansion
 {#AsymptoticExpansion}

An [[asymptotic expansion]] of the partition function is ([Hardy & Ramanujan 1918](#HardyRamanujan1918), see [DeSalvo 20](#DeSalvo20) for discussion)

$$
 p(n)
  \;\sim\;
 \frac
   { e^{ \sqrt{2/3}\pi \cdot \sqrt{n} } }
   { 4 \sqrt{3} \cdot n}
 \,.
$$

## References

* On-Line Encyclopedia of Integer Sequences, *[OEIS A000041](https://oeis.org/A000041)*

See also:

* Wikipedia, *Partition function (number theory)*

Asymptotic behaviour:

* {#HardyRamanujan1918} [[Godfrey Hardy]], [[Srinivasa Ramanujan]], *Asymptotic formulaæ in combinatory analysis*, Proceedings of the London Mathematical Society, 2(1):75–115, 1918 ([doi:10.1112/plms/s2-17.1.75](https://doi.org/10.1112/plms/s2-17.1.75), [[HardyRamanujanAsymptoticFormulae.pdf:file]])

* {#DeSalvo20} Stephen DeSalvo, *Will the real Hardy-Ramanujan formula please stand up?* ([arXiv:2003.06908](https://arxiv.org/abs/2003.06908))


partition of unity

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### Context
#### Topology
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[[!include topology - contents]]
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# Contents
* table of contents
{: toc}

## Idea

A _partition of unity_ is a [[partition]] of the unit function on a [[topological space]] into a sum of continuous functions that are each non-zero only on small parts of the space.

## Definition

Let $X$ be a [[topological space]]. A **partition of unity** on $X$ is a collection $\{u_j\}_{j \in J}$ of [[continuous functions]] $u_j \colon X \to [0,1]$, $j\in J$ to the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]] such that $\sum_{j \in J} u_j(x) = 1$ for all $x\in X$.

A partition of unity defines an [[open cover]] of $X$, consisting of the open sets $u_j^{-1}(0,1]$. Call this the **induced cover**.

Given a [[cover]] $\mathcal{U} = \{U_j\}_{j\in J}$ of a [[topological space]] ([[open cover]] or closed or neither), the partition of unity $\{u_j\}_J$ is **subordinate** to $\mathcal{U}$ if for all $j\in J$,
$$\overline{u_j^{-1}(0,1]} \subset U_j.$$

What this means is that the open sets $u_j^{-1}(0,1]$ form an open cover _refining_ the cover $\mathcal{U}$.

A partition of unity is __point finite__ if for every $x\in X$ there is only a [[finite number]] of $j\in J$ such that $u_j(x) \neq 0$.

A partition of unity is __locally finite__ if for every $x\in X$ there is an open neighborhood $U$ of $x$ such that for only a [[finite number]] of $j\in J$ there is $x\in U$ such that $u_j(x) \neq 0$.

{#Harmless} Often, the property of local finiteness is included in the definition of a partition of unity. This is harmless, since a result due to [[Michael R. Mather]] (Prop. \ref{MatherTheorem} below) says that for any partition of unity we can find a locally finite partition of unity with the same indexing set and whose induced cover refines the original induced cover.

## Examples

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

Consider $\mathbb{R}$ with its [[Euclidean space|Euclidean]] [[metric topology]].

Let $\epsilon \in (0,\infty)$  and consider the [[open cover]]
$$
  \{
    (n-1-\epsilon , n+1 +  \epsilon) \subset \mathbb{R}
  \}_{n \in \mathbb{Z} \subset \mathbb{R} }.
$$

Then a partition of unity $\{ f_n \colon \mathbb{R} \to [0,1] \}_{n \in \mathbb{N}}$ subordinate to this cover is given by
$$
  f_n(x)
    \coloneqq
  \left\{
    \array{
      x - (n - 1)  &\vert& n - 1 \leq x \leq n
      \\
      1- (x-n) &\vert& n \leq x \leq n+1
      \\
      0 &\vert& \text{otherwise}
    }
  \right\}.
$$

=--

## Properties

### Existence on paracompact topological spaces

+-- {: .num_prop #ParacompactHausdorffEquivalentToexistenceOfParititionsOfUnity}
###### Proposition
**([[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]])**

Assuming the [[axiom of choice]] then:

Let $(X,\tau)$ be a [[topological space]]. Then the following are equivalent:

1. $(X,\tau)$ is a [[paracompact Hausdorff space]].

1. Every [[open cover]] of $(X,\tau)$ admits a subordinate partition of unity.

=--

Similarly [[normal spaces]] are equivalently those such that every [[locally finite cover]] has a subordinate partition of unity (reference Bourbaki, Topology Generale - find this!)

### The case of non-Hausdorff spaces

Slightly more generally,
a [[topological space]] (not necessarily Hausdorff)
is [[fully normal topological space|fully normal]] if and only every [[open cover]] admits a subordinate partition of unity.

A [[T1-space]] is [[fully normal topological space|fully normal]] if and only if it is [[paracompact]],
in which case it is also Hausdorff.

For [[topological spaces]] that are not [[T1-spaces]],
the condition of being [[fully normal topological space|fully normal]] is strictly stronger than [[paracompactness]].

### The case of locales

A [[regular locale]] is [[fully normal topological space|fully normal]] if and only if it is [[paracompact]].

The usual proof of the existence of partitions of unity
goes through for such locales since it does not make any use of points.

### Existence on smooth manifolds
 {#ExistenceOnSmoothManifolds}

Paracompact [[smooth manifolds]] admit locally finite _smooth_ partitions of unity subordinate to any open cover (this follows from the existence of a smooth [[bump function]] on $[-1,1]$). It is not true, however, that [[analytic manifolds]] have analytic partitions of unity - the aforementioned [[bump function]] is smooth but not analytic:

+-- {: .num_lemma #SmoothManifoldClosedBallRefinementOfCover}
###### Lemma
**([[open cover]] of [[smooth manifold]] admits [[locally finite cover|locally finite]] [[refinement]] by [[closed balls]])**

Let $X$ be a [[smooth manifold]] and let
$\{U_i \subset X\}_{i \in I}$ be an [[open cover]].
Then there exists cover

$$
  \left\{
    B_0(\epsilon_j)
      \underoverset{\simeq}{\psi_j}{\to}
    V_j
      \subset
    X
  \right\}_{i \in J}
$$

which is a [[locally finite cover|locally finite]] [[refinement]] of $\{U_i \subset X\}_{i \in I}$ with each patch [[diffeomorphism|diffeomorphic]] to a [[closed ball]] in [[Euclidean space]].

=--

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

First consider the special case that $X$ is [[compact topological space]].

Let

$$
  \left\{
    \mathbb{R}^n
      \underoverset{\simeq}{\phi_j}{\longrightarrow}
    V_j
    \subset X
  \right\}
$$

be a smooth [[atlas]] representing the [[smooth structure]] on $X$. The [[intersections]]

$$
  \left\{
    U_i \cap V_j
  \right\}_{i \in I, j \in J}
$$

still form an open cover of $X$. Hence for each point $x \in X$ there is $i \in I$ and $j \in J$ with $x \in U_i \cap V_j$. By the nature of the [[Euclidean topology]], there exists a [[closed ball]] $B_x$ around $\phi_j^{-1}(x)$ in $\phi_j^{-1}(U_i \cap V_j) \subset \mathbb{R}^n$. Its [[image]] $\phi_j(B_x) \subset X$ is a neighbourhood of  $x \in X$ diffeomorphic to a closed ball.

The [[interiors]] of these balls form an [[open cover]]

$$
  \left\{
    Int(B_x)
     \subset
    X
  \right\}_{x \in X}
$$

of $X$ which, by construction, is a refinement of $\{U_i \subset X\}_{i \in I}$. By the assumption that $X$ is compact, this has a finite subcover

$$
  \left\{
    Int(B_l)
    \subset
    X
  \right\}_{l \in L}
$$

for $L$ a [[finite set]]. Hence

$$
  \left\{
    B_l
    \subset
    X
  \right\}_{l \in L}
$$

is a finite cover by closed balls, hence in particular locally finite, and by construction it is still a refinement of the orignal cover.
This shows the statement for $X$ compact.

Now for general $X$, notice that without restriction we may assume that $X$ is [[connected topological space|connected]], for if it is not, then we obtain the required refinement on all of $X$ by finding one on each [[connected component]].

But if a locally Euclidean paracompact Hausdorff space $X$ is connected, then it is [[sigma-compact topological space|sigma-compact]] and in fact admits a countable increasing exhaustion

$$
  V_0 \subset V_1 \subset V_2 \subset \cdots
$$

by [[open subsets]] whose [[topological closures]]

$$
  K_0 \subset K_1 \subset K_2 \subset \cdots
$$

exhaust $X$ by [[compact topological space|compact]] subspaces $K_n$ (by the proof of [this prop.](topological+manifold#RegularityConditionsForTopologicalManifoldsComparison)).

For $n \in \mathbb{N}$, consider the open subspace

$$
  V_{n+2} \setminus K_{n-1}
    \;\subset\;
  X
$$

which canonically inherits the structure of a smooth manifold ([this prop.](differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds)). As above we find a refinement of the restriction of $\{U_i \subset X\}_{i \in I}$ to this open subset by closed balls and since the further subspace $K_{n+1}\setminus K_n$ is still compact (by [this lemma](compact+space#UnionsAndIntersectionOfCompactSubspaces)) there is a finite set $L_n$ such that

$$
  \{B_{l_n} \subset V_{n+2} \setminus K_{n-1} \subset X \}_{l_n \in L_n}
$$

is a finite cover of $K_{n+1} \setminus K_n$ by closed balls refining the original cover.

It follows that the union of all these

$$
  \left\{
   B_{l_n} \subset X
  \right\}_{n \in \mathbb{N}, l_n \in L_n}
$$

is a refinement by closed balls as required. Its local finiteness follows by the fact that each $B_{l_n}$ is contained in the "strip" $V_{n+2} \setminus K_{n-1}$, each strip contains only a finite set of $B_{l_n}$-s and each strip intersects only a finite number of other strips. (Hence an open subset around a point $x$ which intersects only a finite number of elements of the refined cover is given by any one of the balls $B_{l_n}$ that contain $x$.)

=--

+-- {: .num_prop #SmoothManifoldAdmitsSmoothPartitionsOfUnity}
###### Proposition
**([[smooth manifolds]] admit locally finite smooth partitions of unity)**

Let $X$ be a paracompact [[smooth manifold]]. Then every [[open cover]] $\{U_i \subset X\}_{i \in I}$ has a subordinate partition of unity by functions $\{f_i \colon U_i \to \mathbb{R}\}_{i \in I}$ which are _[[smooth functions]]_.

=--

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

By lemma \ref{SmoothManifoldClosedBallRefinementOfCover} the given cover
has a [[locally finite cover|locally finite]] [[refinement]]
by [[closed subsets]] [[diffeomorphism|diffeomorphic]] to [[closed balls]]:

$$
  \left\{
    B_0(\epsilon_j)
      \underoverset{\simeq}{\psi_j}{\to}
    V_j
      \subset
    X
  \right\}_{j \in J}
  \,.
$$

Given this, let

$$
  h_j \;\colon\; X \longrightarrow \mathbb{R}
$$

be the function which on $V_j$ is given by a smooth [[bump function]]

$$
  b_j \;\colon\; \mathbb{R} \longrightarrow \mathbb{R}
$$

with [[support]]  $supp(b_j) = B_0(\epsilon_j)$:

$$
  h_j
  \;\colon\;
  x
  \mapsto
  \left\{
    \array{
      b_j(\psi_j^{-1}(x)) &\vert& x \in V_j
      \\
      0 &\vert& \text{otherwise}
    }
  \right.
  \,.
$$

By the nature of [[bump functions]] this is indeed a [[smooth function]] on all of $X$. By local finiteness of the cover by closed balls, the function

$$
  h \;\colon\; X \longrightarrow \mathbb{R}
$$

given by

$$
  h(x)
    \coloneqq
  \underset{j \in J}{\sum} h_j(x)
$$

is well defined (the sum involves only a finite number of non-vanishing contributions) and is smooth. Therefore setting

$$
  f_j \;\coloneqq\; \frac{h_j}{h}
$$

then

$$
 \left\{
    f_j
 \right\}_{j \in J}
$$

is a subordinate partition of unity by smooth functions as required.

=--

### From a partition of unity to a locally finite partition of unity

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

A collection of functions $\mathcal{U} = \{u_i : X \to [0,1]\}$ such that every $x\in X$ is in the support of some $u_i$. Then $\mathcal{U}$ is called _locally finite_ if the cover $u_i^{-1}(0,1]$ (i.e. the induced cover) is [[locally finite cover|locally finite]].
=--

+-- {: .num_prop #MatherTheorem}
###### Proposition
([Mather, 1965](#Mather65))

Let $\{u_i\}_J$ be a partition of unity. Then there is a locally finite partition of unity $\{v_i\}_{i\in J}$ such that the induced cover of the latter is a refinement of the induced cover of the former.

=--

The **proof** is Proposition A.2.8 in [Dold 95](#Dold95) and Lemma 5.1.8 on page 301 of [Engelking 89](#Engelking89).

This implies that (locally finite) [[numerable covers]] are cofinal in induced covers arising from collections of functions as in the definition. In particular, given the [[Milnor classifying space]] $\mathcal{B}^M G$ of a [[topological group]] $G$, which comes with a countable family of 'coordinate functions' $\mathcal{B}^M G \to [0,1]$, has a numerable cover. This is shown by Dold to be a trivialising cover for the universal bundle constructed by Milnor, and so the universal bundle is [[numerable bundle|numerable]].

## Applications

### Maps to geometric realizations

Partitions of unity can be used in constructing maps from spaces to [[geometric realization]]s of [[simplicial spaces]] (incl. simplicial sets) - for example a [[classifying map]] for a $G$-[[bundle]] where $G$ is a [[Lie group]].

### Coboundaries for Cech cocycles {#CechCoboundaries}

Partitions of unity can be used to give explicit coboundaries for the cocycles of the complex of functions on a cover.

Let $\{U_i \to X\}$ be a [[open cover]] and $\{\rho_i \in C(X,\mathbb{R})\}$ a collection of functions with

* $(x \notin U_i) \Rightarrow \rho_i(x) = 0$

* $\sum_i \rho_i = const_1$.

Write $C(\{U_i\}) : \Delta^{op} \to Top$ for the [[Cech nerve]] of the cover and $C(C(\{U_i\}), \mathbb{R})$ for the [[cosimplicial ring]] of functions on this [[simplicial object|simplicial]] topological space; and $(C_\bullet(C(\{U_i\}), \mathbb{R}), \delta)$ for the corresponding (normalized) [[Moore complex|cochain complex]]: its differential is the alternating sum of the pullbacks of functions along the face maps, i.e. along the restriction maps

$$
  \delta = \sum_k (-1)^k \delta_{k}^*
  \,.
$$

For instance for $f = \{f_{i_1, i_2, \cdots, i_n} \in C(U_{i_1} \cap \cdots \cap U_{i_{n+1}})\}$ a collection of functions in degree $n$, we have

$$
  (\delta f)_{i_0 \cdots i_n i_{n+1}} =
  \sum_{k = 0}^{n+1} (-1)^k f_{i_0 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}}
  \,.
$$

This cochain complex has vanishing [[cochain cohomology]] in positive degree. We can explicitly construct corresponding coboundaries using the partion of unity:

assume that with the above notation $f$ is a cocycle in positive degree, in that $\delta f  = 0$. Then define the $(n-1)$-cochain

$$
  \lambda_{i_1 \cdots i_n}
  :=
  \sum_{i_0} \rho_{i_0} f_{i_0 i_1 \cdots i_n}
  \,.
$$

Here in the summands on the right the product is defined on $U_{i_0} \cap U_{i_1} \cap \cdots \cap U_{i_n}$ and extended as 0 to all of $U_{i_1} \cap \cdots \cap U_{i_n}$.

With this definition we have

$$
  \delta \lambda = f
  \,.
$$

To see this we compute

$$
  \begin{aligned}
    (\delta \lambda)_{i_1 \cdots i_{n+1}}
    & :=
    \sum_{i_0} \rho_{i_0} \sum_{k=1}^n (-1)^k f_{i_0 i_1 \cdots i_{k-1} i_{k+1} \cdots i_{n+1}}
    \\
    & =
    \pm
    \sum_{i_0} \rho_{i_0}
    f_{i_1 \cdots i_{n+1}}
    \\
    & = f_{i_1 \cdots i_{n+1}}
  \end{aligned}
  \,,
$$

where in the second step we used the condition $\delta f = 0$ and in the last step we used the property of the partition of unity.

This construction is used a lot in [[Cech cohomology]]. For instance it can be used to show in Chech cocycles that every [[principal bundle]] admits a [[connection on a bundle]] (see there for the details).

## References

* [[Albrecht Dold]], _Partitions of unity in the theory of fibrations_, Ann. of Math. 78. (1963), 223-255.

* {#Mather65} [[Michael R. Mather]], _Paracompactness and partitions of unity_, Mimeographed notes, Cambridge (1964).

* {#Mather66} [[Michael R. Mather]], _Products in spectral sequences and other topics_, PhD dissertation, Cambridge (1966).

* {#Dold95} [[Albrecht Dold]], p. 354 in: *Lectures on Algebraic Topology*, Springer 1995 ([doi:10.1007/978-3-642-67821-9](https://www.springer.com/gp/book/9783540586609), [pdf](https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf), [GoogleBooks](https://books.google.com.au/books?id=4AmzD9XhGDkC&lpg=PA354&vq=locally%20finite%20oartition%20unity&pg=PA354#v=onepage&q&f=false))

* {#Engelking89} [[Ryszard Engelking]], p. 301 in: *General Topology*, Sigma series in pure mathematics **6**, Heldermann 1989 ([ ISBN 388538-006-4](https://www.heldermann.de/SSPM/SSPM06/sspm06.htm))

Discussion of partitions of unity in [[constructive mathematics]] is in

* {#Waaldijk96} [[Frank Waaldijk]], section 3.1 of _modern intuitionistic topology_, 1996 ([pdf](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf))

[[!redirects partition of unity]]
[[!redirects partitions of unity]]
[[!redirects Partition of unity]]
[[!redirects Partitions of unity]]


Pascal Hitzler

[[!redirects P. Hitzler]]
* [homepage](http://www.pascal-hitzler.de/)

category:people