apartness relation

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# Apartness relations
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## Idea

An _apartness relation_ is a [[binary relation]] that, instead of saying when two things are the same (as an [[equivalence relation]]), states when two things are different -- an [[inequality relation]].

Apartness relations are most used in [[constructive mathematics]]; in [[classical mathematics]], equivalence relations can take their place (mediated by [[negation]]).

The apartness relations that we discuss here are sometimes called __point--point apartness__, to distinguish this from the related concepts of _set--set_ or _point--set_ apartness relations; see [[proximity space]] and [[apartness space]] (respectively) for these.

## Definitions

### Abstract

A [[set]] $S$ equipped with an __apartness relation__ is a [[groupoid]] (with $S$ as the set of [[objects]]) [[enriched category|enriched]] over the [[cartesian monoidal category]] $TV^\op$, that is the [[opposite category|opposite]] of the [[partial order|poset]] of [[truth value|truth values]], made into a [[monoidal category]] using [[disjunction]]. By the law of [[excluded middle]] (which says that $TV$ is self-dual under [[negation]]), this is equivalent to equipping $S$ with an [[equivalence relation]] (which makes $S$ a groupoid [[enriched category|enriched]] over the cartesian category $TV$ *itself*). But in [[constructive mathematics]] (or interpreted [[internalization|internally]]), it is a richer concept with a topological flavour, as $TV^\op$ is a [[co-Heyting algebra]].

### Concrete

Of course, nobody but a [[category theory|category-theorist]] would use the above as a *definition* of an apartness relation. Normally, one defines an apartness relation on $S$ as a [[binary relation]] $\#$ satisfying these three properties:

* [[irreflexive relation|irreflexivity]]: for all $x: S$, $x \# x$ is false;
* [[symmetric relation|symmetry]]: for all $x, y: S$, if $y # x$, then $x # y$;
* [[comparison]]: for all $x, y, z: S$, if $x # z$, then $x # y$ or $y # z$.

(Notice that these are dual to the axioms for an [[equivalence relation]]; like those axioms, these correspond to [[identity morphisms]], [[inverses]], and [[composition]] in a groupoid.)

### Related notions

The [[negation]] of an apartness relation is an equivalence relation. (On the other hand, the statement that every equivalence relation is the negation of some apartness relation is equivalent to [[excluded middle]], and the statement that the negation of an equivalence relation is always an apartness relation is equivalent to the nonconstructive [[de Morgan law]].) An apartness relation is a [[tight apartness relation]] if this equivalence relation is [[equality]]; any apartness relation defines a [[tight apartness relation]] on the [[quotient set]].

If $S$ and $T$ are both sets equipped with apartness relations, then a [[function]] $f\colon S \to T$ is __[[strongly extensional function|strongly extensional]]__ if $x \# y$ whenever $f(x) \# f(y)$; that is, $f$ reflects apartness. The strongly extensional functions are precisely the [[enriched functors]] between $TV^\op$-enriched groupoids, so they are the correct morphisms. (Note that there is no nontrivial notion of enriched [[natural isomorphism]], at least not when the apartness in $T$ is tight.)

## (2,1)-category of sets with apartness relations

Sets with apartness relations, strongly extensional functions, and equivalences of strongly extensional functions, which serve to identify unequal but equivalent (that is, not apart) elements of a set, form a locally thin [[(2,1)-category]]; i.e. a [[bicategory]] [[enriched bicategory|enriched]] in [[thin groupoids]]. This bicategory is [[locally small category|locally small]] and a [[univalent bicategory]].

## Topological aspects {#topology}

### The apartness topology

Let $S$ be a set equipped with an apartness relation $\ne$. Using $\ne$, many [[topology|topological]] notions may be defined on $S$. (Often one assumes that the apartness is tight; this corresponds to the $T_0$ [[separation axiom]] in topology.)

If $U$ is a [[subset]] of $S$ and $x$ is an element, then $U$ is a $\ne$-**[[neighbourhood]]** (or $\ne$-**neighborhood**) of $x$ if, given any $y\colon S$, $x \ne y$ or $y \in U$; note that $x \in U$ by irreflexivity. The neighbourhoods of $x$ form a [[filter]]: a superset of a neighbourhood is a neighbourhood, and the intersection of $0$ or $2$ (hence of any finite number) of neighbourhoods is a neighbourhood.

A subset $G$ is $\ne$-**[[open subset|open]]** if it\'s a neighbourhood of all of its members. The open subsets form a [[topological space|topology]] (in the sense of Bourbaki): any union of open subsets is open, and the intersection of $0$ or $2$ (hence of any finite number) of open subsets is open.

The $\ne$-**complement** of $x$ is the subset $\{y\colon S \;|\; x \ne y\}$; this is open by comparison. More generally, the $\ne$-**[[complement]]** of any subset $A$ is the set $\tilde{A}$, defined as:
$$ \tilde{A} \coloneqq \{y \;|\; \forall{x} \in A,\; x \ne y\} .$$
This is not in general open, but you would use it where you would classically use the set-theoretic complement. However, if $A$ is open to begin with, then $\tilde{A}$ equals the set-theoretic complement.

If $x \ne y$, then $x \in \tilde{y}$ and $y \in \tilde{x}$.  Thus, if $\ne$ is tight, then $(S, \ne)$ satisfies the $T_1$ [[separation axiom]]. Symmetry is important here; if we removed symmetry from the axioms of apartness (obtaining a [[quasi-apartness]]) but retained tightness, then we would still get a $T_0$ topology, but it would not be $T_1$. This is a version of the fact that failure of $T_1$ is given by a [[partial order]] (or a [[preorder]] if $T_0$ might also fail).

The $\ne$-**[[closure]]** $\bar{A}$ of a subset $A$ is the complement of its complement. This closure is a [[closure operator]]: $A \subset \bar{A}$, $\overline{\bar{A}} = \bar{A}$ (in fact, $\overline{\tilde{A}} = \tilde{A}$), $\bar{A} \subset \bar{B}$ whenever $A \subset B$, $\overline{S} = S$ (in fact, $\bar{\empty} = \empty$ too), and $\overline{A \cap B} = \bar{A} \cap \bar{B}$ (but *not* $\overline{A \cup B} = \bar{A} \cup \bar{B}$).

The __antigraph__ of a function $f\colon S \to T$ is
$$ \{(x,y) \;|\; x\colon S, y\colon T \;|\; f(x) \ne y\} .$$
Recall that in ordinary topology, a function between [[Hausdorff spaces]] is continuous iff its [[graph of a function|graph]] is closed. Similarly, a function $f\colon S \to T$ is strongly extensional iff its antigraph is open. (Then the [[graph of a function|graph]] of $f$ is the complement of the antigraph.)

One important topological concept that doesn\'t appear classically is locatedness; in an inequality space, a subset $A$ is __[[located subspace|located]]__ if, given any point $x$ and any neighbourhood $U$ of $x$, either $U \cap A$ is [[inhabited set|inhabited]] (that is, it has a point) or some neighbourhood of $x$ (not necessarily $U$) is contained in $\tilde A$. Note that every point is located. (For an example of a set that need not be located, consider $\{x\colon S \;|\; p\}$, where $p$ is an arbitrary [[truth value]]. In an inhabited space, this set is located iff $p$ is true or false.)

### Relation to metric spaces

Recall that, as [[Bill Lawvere]] taught us, a [[metric space]] is a groupoid (or $\dagger$-[[dagger-category|category]]) enriched over the category $([0,\infty[^\op,+)$ of nonnegative [[real numbers]], ordered in reverse, and made monoidal under addition. (Actually, you get a metric only if you impose a tightness condition, although again you can recover this up to equivalence from the $2$-morphisms. Furthermore, Lawvere advocated using $[0,\infty]$ instead of $[0,\infty[$, and also dropping the symmetry requirement to get enriched categories instead of groupoids. Thus, he dealt with extended quasipseudometric spaces. These details are not really important here.)

There is a [[monoidal functor]] from $([0,\infty[^\op,+)$ to $TV^\op$ that maps a nonnegative real number $x$ to the truth value of the statement that $x \gt 0$. Accordingly, any (symmetric) metric space becomes an inequality space, and any function satisfying $d(f(x),f(y)) \leq d(x,y)$) is strongly extensional.

The topological properties of metric spaces fit well with those of inequality spaces if you always work in this direction. For example, a set which is $d$-open will also be $\ne$-open, but not necessarily the other way around. Similarly, a (merely) [[continuous function]] between metric spaces is (still) strongly extensional.

### Relation to gauge spaces and uniform spaces

In analysis, many spaces are given as [[gauge spaces]], that is by families of pseudometrics; these also become inequality spaces by declaring that $x \ne y$ iff $d(x,y) \gt 0$ for some pseudometric $d$ in the family. (This will actually be a *tight* apartness iff the family of pseudometrics is separating.)

Classically, any [[uniform space]] may be given by a family of pseudometrics, but this doesn\'t hold constructively. In particular, a topological group may not be an inequality group (as in the next section).  However, we can generalize a bit beyond gauge spaces: any [[uniformly regular uniform space]] becomes an inequality space by declaring that $x \ne y$ iff there is an entourage $U$ with $(x,y)\notin U$.  (If the uniform space is not uniformly regular, the result is merely an [[inequality relation]], not an apartness.)

### Relation to proximity spaces

The constructive theory of [[proximity spaces]] is based on a generalisation of apartness relations (which here go between *points*) to an apartness relation between *sets*.  These are called _[[apartness space|apartness spaces]]_; just as apartness relations (between points) are classically equivalent to equivalence relations, so apartness spaces are classically equivalent to proximity spaces, with two sets being proximate if and only if they are not apart.

Of course, any apartness space has an apartness relation between points: $x$ and $y$ are apart iff $\{x\}$ and $\{y\}$ are apart.

### Relation to locales

Let $X$ be a set, regarded as a [[discrete locale]], whose [[frame]] of opens is $O(X) = P(X)$, the [[power set]] of $X$.  That is, the opens in the locale $X$ are precisely the subsets of the set $X$.  Since discrete locales are [[locally compact locale|locally compact]] (every set is the union of its [[K-finite set|K-finite]] subsets), the locale product $X\times X$ agrees with the spatial product, so that $X\times X$ is also discrete and every subset of $X\times X$ is open.  Thus, the opens in the locale $X\times X$ are precisely the subsets of $X\times X$.  In particular, an [[equivalence relation]] on the set $X$ can be identified with an *open* equivalence relation (in [[Loc]]) on the discrete locale $X$.

Thus, the following theorem gives a different precise sense in which apartness relations are dual to equivalence relations.

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###### Theorem
An apartness relation on a set $X$ is the same as a (strongly) [[closed subspace|closed]] equivalence relation on the discrete locale $X$.  Moreover, the apartness topology defined above is, as a locale, the quotient of this equivalence relation.
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+--{: .proof}
###### Proof
By definition, a (strongly) closed sublocale of a locale $Y$ is one of the form $\mathsf{C}U$, for some open $U\in O(Y)$.  Thus, when $X$ is a discrete locale, a closed sublocale of $X\times X$ is of the form $\mathsf{C}U$ for some subset $U$ of $X\times X$.  This subset is the extension of the apartness relation, i.e. $U = \{ (x,y) \mid x\#y \}$.

For the first claim, therefore, it remains to show that the three axioms of an equivalence relation for $\mathsf{C}U$ correspond to the apartness axioms for $\#$.  Note that pullback along locale maps respects closed complements, i.e. $f^*(\mathsf{C}U) = \mathsf{C}(f^*U)$.  Thus, the pullback of $\mathsf{C}U$ along the twist map $X\times X \to X\times X$ is the closed sublocale corresponding to the twist of $U$, i.e. the set $\{ (x,y) \mid y\#x \}$.  Since $\mathsf{C}$ is a contravariant order-isomorphism between the posets of open and closed sublocales, symmetry for $\mathsf{C}U$ is equivalent to symmetry for $\#$.  Similarly, pulling $\mathsf{C}U$ back to $X\times X\times X$ along one of the three canonical projections gives the closed sublocale dual to the corresponding pullback of $U$ itself, and $\mathsf{C}$ transforms unions to intersections; thus transitivity for $\mathsf{C}U$ is equivalent to comparison for $\#$.  Finally, the pullback of $\mathsf{C}U$ along the diagonal is the closed sublocale dual to the similar pullback of $U$, so to say that the former is all of $X$ is equivalent to saying that the latter is $\emptyset$; thus reflexivity for $\mathsf{C}U$ is equivalent to irreflexivity for $\#$.

Now, the quotient in $Loc$ of such an an equivalence relation in particular comes equipped with a surjective locale map from $X$.  Thus, it is a spatial locale and can be regarded as a topology on the set $X$.  Moreover, quotients in $Loc$ are constructed as [[equalizers]] in $Frm$, so we have to compute the equalizer of the two maps $O(X) = P(X) \to O(\mathsf{C}U)$, where $O(\mathsf{C}U)$ is the frame of opens of $\mathsf{C}U$ regarded as a locale in its own right.  Equivalently, this means the equalizer of the two maps $P(X) \xrightarrow{\pi_i} P(X\times X) \xrightarrow{j_{\mathsf{C}U}} P(X\times X)$, where $j_{\mathsf{C}U}$ is the [[nucleus]] corresponding to $\mathsf{C}U$.

Now by definition, $j_{\mathsf{C}U}(V) = V\cup U$.  Thus, the elements of this equalizer --- which is to say, the opens in the locale quotient --- are subsets $V$ of $X$ such that $(V\times X) \cup U = (X\times V) \cup U$.  Reexpressed in terms of $\#$, that means that for any $x,y\in X$ we have $(x\in V \vee x\#y) \iff (y\in V \vee x\#y)$.  But since $\#$ is symmetric, this is equivalent to the unidirectional implication $(x\in V \vee x\#y) \to (y\in V \vee x\#y)$, and since $x\#y$ always implies itself, this is equivalent to $x\in V \to (y\in V \vee x\#y)$, which is precisely the condition defining the open sets in the apartness topology above.
=--

Recall that the negation of an apartness relation on $X$ is an equivalence relation on the *set* $X$.  This is the spatial part of the above closed localic equivalence relation, which in general (constructively) need not be itself spatial.  The apartness relation is tight just when this spatial part is the diagonal.  (By contrast, to say that the closed localic equivalence relation is *itself* the diagonal is to say that the discrete locale $X$ is [[Hausdorff space|Hausdorff]], which is only true if $X$ has [[decidable equality]].)

Another characterization of the $\#$-open sets is that $U$ is $\#$-open if $U\times X \subseteq (X\times U) \cup W_\#$, where $W_\#$ is $\#$ regarded as a subset of $X\times X$.  Rephrased in terms of complementary closed sublocales, this says that $\mathsf{C}U$ is "closed under the equivalence relation" dual to $\#$.  Thus, the closed sublocales of $X$ with its $\#$-topology (i.e. the formal complements of $\#$-open sets) correspond precisely to the closed sublocales of $X$ (the formal complements of arbitrary subsets of $X$) that respect this equivalence relation.

As a partial converse to the above theorem, if $X$ is a [[Hausdorff space|localically strongly Hausdorff]] topological space, meaning that its diagonal is a strongly closed sublocale, then the pullback of this diagonal to the discrete locale on the set of points of $X$ is a closed localic equivalence relation, hence an apartness, whose $\ne$-topology refines the given topology.  See [this theorem](/nlab/show/Hausdorff+space#Apartness).  If we are given an apartness relation $\ne$, it is unclear whether the $\ne$-topology is localically strongly Hausdorff; but if it is, then the apartness relation resulting from this topology is stronger than the given $\ne$.

## Related concepts

* [[inequality relation]]

  * [[denial inequality]]

* [[antisubalgebra]]

* [[inequality space]]

[[!include logic symbols -- table]]

## References

According to [Troelstra and van Dalen](#TvD):

> Brouwer ([1919](#Brouwer1919)) introduced the notion of apartness (örtlich verschieden, Entfernung).... The axioms of the theory of apartness were formulated by Heyting ([1925](#Heyting1925)).

* {#Brouwer1919} [[L.E.J. Brouwer]], _Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten II: Theorie der Punktmengen.  1919

* {#Heyting1925} [[Arend Heyting]], _Intuïtionistische Axiomatiek der Projectieve Meetkunde_ (Dutch), Ph.D. Thesis, 1925

* [[Errett Bishop]]'s _Foundations of Constructive Analysis_ (1967) uses apartness for the real numbers and more general metric spaces.

* The 1985 edition with Douglas Bridges, _Constructive Analysis_, includes the general definition of apartness relation, there called an "inequality relation" (though in many other sources, as here, an [[inequality relation]] need not satisfy comparison).

* {#TvD} Anne Troelstra\'s and Dirk van Dalen\'s _Constructivism in Mathematics_ (1988) uses apartness for the reals (volume 1), and general apartness relations in algebra (volume 2, chapter 8).  They say "preapartness" and "apartness" instead of "apartness" and "tight apartness" respectively.

* Apartness plays a minimal role in _A Course in Constructive Algebra_ (also 1988), by Ray Mines, Fred Richman, and Wim Ruitenburg.

* A great reference for point-set topology in constructive mathematics is the Ph.D. thesis of Frank Waaldijk, _[Modern Intuitionist Topology](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf)_ (1996).

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apartness space

# Apartness spaces
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## Idea

An *apartness space* is a set equipped with an "apartness relation" that distinguishes between pairs of points or sets.  They are particularly interesting in [[constructive mathematics]]; in [[classical mathematics]] they tend to be equivalent to better-known definitions expressed in terms of "closeness" rather than "apartness".

## Definitions

There are actually several different notions of "apartness space" depending on whether the objects being compared on each side are points or sets.

* A _point--point apartness space_ is a set $X$ equipped with an _[[apartness relation]]_, usually written $x # y$ on elements $x,y\in X$.  Sometimes it is required to be [[tight relation|tight]], or to be only an [[inequality relation]].

* A __point--set apartness space__ is a set $X$ equipped with a relation $x \bowtie A$ between points $x\in X$ and subsets $A\subseteq X$, satisfying appropriate axioms.  In [[classical mathematics]], these axioms are obtained by [[contrapositive|contraposition]] from the definition of a [[topological space]] in terms of a [[exact functor|right exact]] [[Moore closure]] operator, so that point--set apartness spaces are equivalent to topological spaces.  In [[constructive mathematics]] ... well, keep reading.

* A _set--set apartness space_ is a set $X$ equipped with a relation $A\bowtie B$ between [[subsets]] $A,B\subseteq X$, satisfying appropriate axioms.  A set--set apartness space is one of the ways to describe the [[classical mathematics|classical]] notion of a _[[proximity space]]_.  In [[constructive mathematics]], the definition of a proximity space in terms of $\bowtie$ can be taken as a definition of a set--set apartness space.

* A _uniform apartness space_ is a set $X$ equipped with a collection of "co-[[entourages]]", each giving a different notion of when two points are apart, satisfying appropriate axioms.  In [[classical mathematics]], the co-entourages are exactly the [[complements]] of the entourages of a [[uniform space]], and the same is true constructively if the space is [[uniformly regular]].

Since point--point apartness spaces are described at [[apartness relation]], set--set apartness spaces at [[proximity space]], and uniform apartness spaces at [[uniformly regular space]], the rest of this page will be about point--set apartness spaces.

## Point--set apartness spaces

+-- {: .un_defn}
###### Definition
An __apartness space__ is a set $X$ equipped with a relation $\bowtie$ between points $x\in X$ and subsets $A\subseteq X$ such that

1. $x\bowtie \emptyset$ for any $x$.
2. if $x\bowtie A$, then $x\neq y$ for all $y\in A$.
3. $x\bowtie (A\cup B)$ iff $x\bowtie A$ and $x\bowtie B$.
4. If $x\bowtie A$, then $x \bowtie \{ y \mid \forall z, (z\bowtie A \to z\neq y) \}$.
=--

The relation $x\bowtie A$ should be regarded as a "positive" way of saying that $x$ does not belong to the [[closure]] of $A$, i.e. $x\notin \overline{A}$.  Under this interpretation, the above axioms contrapose to become

1. $\overline{\emptyset} = \emptyset$.
2. If $x\in A$, then $x\in \overline{A}$, i.e. $A\subseteq \overline{A}$.
3. $\overline{A} \cup \overline{B} = \overline{A\cup B}$ (and in particular $(A\subseteq B) \to (\overline{A}\subseteq \overline{B})$).
4. If $B\subseteq \overline{A}$ and $x\in \overline{B}$, then $x\in\overline{A}$, i.e. $\overline{\overline{A}} = \overline{A}$.

which are precisely the axioms of a [[topology]] expressed in terms of a closure operator.  In constructive mathematics, of course, the law of contraposition does not hold.

The axiom $x\bowtie \emptyset$ is almost unnecessary, since the last axiom ensures that if $x\bowtie A$ for any set $A$ then $x\bowtie\emptyset$.  In particular, this is the case if $X$ is $T_1$ (see below) and for any $x\in X$ there is a $y\in X$ with $x\neq y$.

The above definition is almost exactly that of a "pre-apartness" from [BV11](#BV11).  The differences are (1) they require $X$ to be [[inhabited set|inhabited]] (which category-theoretically is wrong-headed, since it excludes the [[initial object]]), and (2) they assume that $X$ is given with an ambient [[inequality relation]] that is referred to by the symbol $\neq$ in axioms 2 and 4.  (If $\neq$ is the [[denial inequality]], then these axioms can be written more simply as "if $x\bowtie A$ then $x\notin A$" and "if $x\bowtie A$ then $x\bowtie \{ y \mid \neg(y\bowtie A) \}$".)  Note that if the space is $T_1$ (see below) then this ambient inequality is definable in terms of $\bowtie$ as $x\bowtie \{y\}$.  For an "apartness", [BV11](#BV11) also require comparability (see below).

Axiom 4 is phrased in [BV11](#BV11) as "if $\forall x, (x\bowtie A \Rightarrow (\forall y\in B, x\neq y))$, then $\forall x, (x\bowtie A \Rightarrow x\bowtie B)$.  This is equivalent to our version, since $B = \{ y \mid \forall z, (z\bowtie A \to z\neq y) \}$ is the largest set $B$ satisfying $\forall x, (x\bowtie A \Rightarrow (\forall y\in B, x\neq y))$.

The earlier paper [BSV02](#BSV02) omits the axiom $x\bowtie \emptyset$, and phrases axiom 2 with the denial inequality but axiom 4 with an ambient inequality --- although these are probably oversights --- and requires $T_1$ as part of the definition too (see below).  An earlier, more predicative presentation of "apartness spaces" can be found in [Waaldijk](#FW96).

## Separation properties

Any apartness space comes with an [[irreflexive relation]] $\nleq$ defined by $x \nleq y$ iff $x \bowtie \{y\}$.  This is a positive version of the negation of the [[specialization order]].  A topological space is called $T_1$ (see [[separation axioms]]) if its specialization order is discrete, i.e. $(x\le y) \to (x=y)$; thus an apartness space is called **$T_1$** if $\neg(x\nleq y) \to (x=y)$.

We could contrapose this to obtain $\neg(x=y) \to (x\nleq y)$, but it is usually too strong constructively to have a denial statement imply a positive one.  However, if we replace $\neg(x=y)$ with a given [[apartness relation]] or [[inequality relation]] other than the [[denial inequality]], then we obtain an axiom $(x\neq y) \to (x\nleq y)$ that is a purely positive version of $T_1$.  Note that if $\neq$ is [[tight relation|tight]], then this statement implies the negative version of $T_1$, while conversely if $\nleq$ is symmetric (see below) then the negative version of $T_1$ says precisely that $\nleq$ is tight.

Note that axiom 2 implies the converse $(x\bowtie \{y\}) \to (x\neq y)$, so that if the axioms and the $T_1$ property are stated with reference to some ambient inequality $\neq$, then $\neq$ can be defined in terms of $\bowtie$.  If we *define* $\neq$ in this way, then axiom 2 should be stated in terms of the denial inequality (thereby asserting that this relation $x\bowtie \{y\}$ is [[irreflexive]]).

If one wants the relation $x\bowtie \{y\}$ to be [[symmetric relation|symmetric]] and thus an "[[inequality relation]]" one needs to assert this separately.  An apartness space with this property is naturally called **$R_0$**, or perhaps **strongly $R_0$**, since it implies (and classically is equivalent to) the topological property called $R_0$ that the specialization order is symmetric (see [[separation axioms]]).

However, stating axiom 4 in terms of the $\bowtie$-derived inequality is weaker, even in [[classical mathematics]], than stating it in terms of the denial inequality.  For instance, if $X = \{x,y,z\}$ with the only nonempty apartness relations $x\bowtie \{z\}$ and $z\bowtie \{x\}$, then axiom 4 for the denial inequality fails for $A=\{z\}$, since $\{ y \mid \neg(y\bowtie A)\} = \{y,z\}$ which is not apart from $x$; but stated in terms of the $\bowtie$-derived inequality it holds, since $\{ v \mid \forall u, (u\bowtie A \Rightarrow u \bowtie \{v\})\} = \{z\}$.  This is a [[pretopological space]] that is not a topological space; thus only axiom 4 with the denial inequality (or at least a [[tight inequality]], which classically becomes equivalent to denial) gives a notion of point-set apartness space that reduces classically to ordinary toplogical spaces.  In [BV11](#BV11), axiom 4 for the denial inequality is called the **reverse Kolmogorov property**.

An apartness space may be called **comparable** (nonce definition on this page) if $x\bowtie A$ implies $(x\neq y) \vee (y\bowtie A)$ for any $y$, where $\neq$ might also be a given apartness on $X$.  This condition is classically trivial, and generalizes the [[comparison]] axiom on a point--point [[apartness relation]].  In particular, if $\neq$ denotes the relation $\nleq$ defined above, then this property implies that $\nleq$ is a [[comparison]], and hence (if the space is also $R_0$, so it is symmetric) an [[apartness relation]].  It is also related topologically to Penon's definition of intrinsic [[open subset]] in [[synthetic topology]] and to the natural topology on a [[point-point apartness space]], and can be viewed as a very weak version of [[regular space|regularity]].

An apartness space is **[[locally decomposable space|locally decomposable]]** if whenever $x\bowtie A$, there exists a set $B$ such that $x\bowtie B$ and every $y$ satisfies either $y\bowtie A$ or $y\in B$.  This condition is also classically trivial: take $B = \{ y \mid \neg(y\bowtie A) \}$.  It implies comparability (for $\neq$ the [[denial inequality]]).

## Relation to topological spaces

If $X$ is a topological space, we define $x\bowtie A$ if there is a neighborhood of $x$ disjoint from $A$, or equivalently if the complement of $A$ is a neighborhood of $x$.  This makes $X$ an apartness space.  Only the last axiom is somewhat nontrivial: if $x\in U$ with $U$ open and $U\cap A = \emptyset$, and $\forall y, (y\bowtie A \to y\notin B)$, then since $(y\in U) \Rightarrow (y\bowtie A)$ we have $U\cap B = \emptyset$ too, so $x\bowtie B$.

Conversely, if $X$ is an apartness space, define $U\subseteq X$ to be a neighborhood of $x\in U$ if there is a set $A$ such that $x\bowtie A$ and $\forall y, (y\bowtie A \Rightarrow y\in U)$.  This makes $X$ a topological space.  Again the nontrivial part is the "transitivity" of neighborhoods, i.e. that if $U$ is a neighborhood of $x$ then so is $\{ y \mid U$ is a neighborhood of $y \}$.  To see this, if $x\bowtie A$ and $\forall y, (y\bowtie A \Rightarrow y\in U)$, then $U$ is a neighborhood of any point $y$ with $y\bowtie A$, so it suffices to show that $\{ y \mid y\bowtie A\}$ is a neighborhood of $x$; but this is obvious.

If we order the topologies on $X$ by $\tau_1 \le \tau_2$ if $\tau_2 \subseteq \tau_1$ (i.e. $\tau_1$ is finer than $\tau_2$), and the apartnesses by $\bowtie_1 \le \bowtie_2$ if $(x\bowtie_2 A) \Rightarrow (x\bowtie_1 A)$, then these constructions define [[monotone functions]] $\tau \mapsto \bowtie_\tau$ and $\bowtie \mapsto \tau_\bowtie$ respectively.  Moreover, we have:

+-- {: .un_theorem}
###### Theorem
The above functions exhibit the poset of apartnesses on $X$ as a [[reflective subcategory|reflective]] sub-poset of the poset of topologies on $X$: we have $\tau \le \tau_{\bowtie_\tau}$ and $\bowtie_{\tau_\bowtie} = \bowtie$.
=--
+-- {: .proof}
###### Proof
For the former inequality, if $U$ is open in $\tau_{\bowtie_\tau}$, then for every $x\in U$ there is a set $A$ such that $x\bowtie_\tau A$ and $\forall y, (y\bowtie_\tau A \Rightarrow y\in U)$.  Since $x\bowtie_\tau A$, there is an open set $V$ with $x\in V$ and $V\cap A = \emptyset$.  But then every $y\in V$ satisfies $y\bowtie_\tau A$, hence $y\in U$; so $V\subseteq U$.  Thus $U$ is open.

For the latter equation, if $x\bowtie A$, then to show $x \bowtie_{\tau_\bowtie} A$ we must construct an open set $U\in \tau_\bowtie$ with $x\in U$ and $U\cap A = \emptyset$; but it suffices to take $U = \{ y \mid y\bowtie A \}$.  Conversely, if $x \bowtie_{\tau_\bowtie} A$, then there is an open set $U\in \tau_\bowtie$ with $x\in U$ and $U\cap A = \emptyset$.  By definition, $U\in \tau_\bowtie$ and $x\in U$ mean there is a set $B$ with $x\bowtie B$ and $\forall y, (y\bowtie B \Rightarrow y\in U)$.  And by the last axiom of an apartness space, to show $x\bowtie A$ it suffices to show $\forall y, (y\bowtie B \to y\notin A)$; but this follows since $y\bowtie B \Rightarrow y\in U$ and $U\cap A = \emptyset$.
=--

In other words, an apartness space can be regarded as a well-behaved kind of topological space: one in which for every open neighborhood $U$ of a point $x$ there is a set $A$ and an open neighborhood $V$ of $x$ such that $V\cap A = \emptyset$ and $U$ contains the interior of the complement of $A$.  Note that $V$ is then in the interior of the complement of $A$, and if $x$ is in the interior of the complement of $A$ then the latter is such a $V$.  Thus, the condition can equivalently be phrased as: for every open neighborhood $U$ of $x$, there is a set $A$ such that $x \in int(\neg A) \subseteq U$.  In other words, the interiors of complements form a base for the topology.  In [Bridges et. al.](#BSV) this condition is called being **topologically consistent**.

A sufficient condition for topological consistency is local decomposability.  This was defined above for apartness spaces; more generally, a topological space is said to be [[locally decomposable space|locally decomposable]] if for every open neighborhood $U$ of $x$ there is an open neighborhood $V$ of $x$ such that $U \cup \neg V = X$, i.e. every $y\in X$ satisfies $(y\in U)\vee (y\notin V)$.  This implies that $x\in int(\neg\neg V) \subseteq \neg\neg V \subseteq U$, giving topological consistency.  (Of course, in classical mathematics every space is locally decomposable.)

In contrast to the above theorem, it is not quite justified to say that the *category* of apartness spaces is a subcategory of the category of topological ones.  It is most natural to say that a function $f:X\to Y$ between apartness spaces is **continuous** if for all $x\in X$ and $A\subseteq X$, if $f(x) \bowtie f(A)$ then $x\bowtie A$.  It is true that if $X$ and $Y$ are topological spaces and $f$ is topologically continuous, then it is apartness-continuous in this sense for the induced apartnesses.  For if $f(x)\bowtie f(A)$, then $f(x) \in U$ for some open set $U$ disjoint from $f(A)$; topological continuity of $f$ then gives that $f^{-1}(U)$ is an open set containing $x$ and disjoint from $A$, so that $x\bowtie A$.  Thus we do have a functor $Top \to Apart$.

However, a continuous map between apartness spaces in the above sense apparently need not be continuous for the induced topologies; but this is true if $Y$ is locally decomposable.  For if $U\subseteq Y$ is open and $x\in f^{-1}(U)$, we have a set $A\subseteq Y$ such that $f(x)\bowtie A$ and $y\bowtie A$ implies $y\in U$, and by local decomposability gives a $B\subseteq Y$ such that $f(x)\bowtie B$ and every $y$ satisfies either $y\bowtie A$ or $y\in B$.  Let $C = f^{-1}(B)$; if $x'\bowtie C$, we have $x'\notin C$ and hence $f(x')\notin B$, so $f(x')\bowtie A$ and thus $f(x')\in U$ and $x'\in f^{-1}(U)$.  Moreover, since $f(x)\bowtie B$ we have $x\bowtie C$ by apartness-continuity.  Thus, $f^{-1}(U)$ is open.

Thus, the category of locally decomposable apartness spaces is equivalent to the category of locally decomposable topological spaces.

## References

Notions of "apartness space" based on a presentation in terms of basic opens, somewhat akin to [[formal topology]], were developed in:

* [[Frank Waaldijk]], *Modern intuitionistic topology*, Ph.D. thesis, 1996, [link](http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf)
 {#FW96}

* [[Frank Waaldijk]], *Natural topology*, 2011, 2012, [arxiv](https://arxiv.org/abs/1210.6288)

The above definition in terms of an apartness relation between points and subsets is slightly adapted from the version given in:

* [[Douglas Bridges]], Peter Schuster, and Luminita Vita, *Apartness, Topology, and Uniformity: a Constructive View*, 2002,  [doi](http://dx.doi.org/10.1002/1521-3870%28200210%2948:1%2B%3C16::AID-MALQ16%3E3.0.CO;2-7)
 {#BSV02}

* [[Douglas Bridges]] and Luminita Vita, *Apartness and Uniformity: A Constructive Development*. 2011, [link](http://link.springer.com/book/10.1007%2F978-3-642-22415-7)
 {#BV11}

[[!redirects apartness space]]
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apartness-open predicate

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

\tableofcontents

## Idea

In [[material set theory]], a [[subset]] $S$ of a [[set]] $A$ with a [[apartness relation]] $\#$ is an apartness-[[open subset]] $S \subseteq A$ if for all $x \in A$ and $y \in A$,

$$(x \in S) \implies (y \in S) \vee (x \# y)$$

In [[dependently sorted set theory]], where membership $x \in S$ is not a [[relation]], the above statement that $x \in S$ for every element $x \in A$ in material set theory is equivalently a predicate in the logic $x \in A \vdash P_S(x)$. The given apartness-open subset $S$ is then defined by [[restricted separation]] as the set $S \coloneqq \{x \in A \vert P_S(x)\}$, which in structural set theory automatically comes with an [[injection]]
$$i:\{x \in A \vert P_S(x)\} \hookrightarrow A$$
such that
$$\exists y \in \{x \in A \vert P_S(x)\}.x = i(y) \iff P_S(x)$$
Hence, the notion of apartness-open predicate, a formulation of the notion of apartness-open as a predicate rather than a subset.

## Definition

Given a set $A$ with an [[apartness relation]] $\#$, an $\#$-open predicate is a predicate $x \in A \vdash P_S(x)$ which satisfies

$$x \in A, y \in A \vdash P_S(x) \implies P_S(y) \vee (x \# y)$$

The $\#$-open subset $S$ is then defined by [[restricted separation]] as
$S \coloneqq \{x \in R \vert P_S(x)\}$

## See also

* [[apartness relation]]
* [[predicate]]
* [[anti-ideal predicate]]
* [[restricted separation]]

[[!redirects apartness-open predicate]]
[[!redirects apartness-open predicates]]


application of holographic QCD to B-meson physics -- references

### Application of holographic QCD to B-meson physics and flavour anomalies

Application of [[holographic QCD]] ([[holographic light front QCD]]) to [[B-meson]] physics and [[flavour anomalies]]:

* Ruben Sandapen, [[Mohammad Ahmady]], _Predicting radiative B decays to vector mesons in holographic QCD_ ([arXiv:1306.5352](https://arxiv.org/abs/1306.5352))

* [[Mohammad Ahmady]], R. Campbell, S. Lord, Ruben Sandapen, _Predicting the $B \to \rho$ form factors using AdS/QCD Distribution Amplitudes for the $\rho$ meson_, Phys. Rev. D88 (2013) 074031 ([arXiv:1308.3694](https://arxiv.org/abs/1308.3694))

* [[Mohammad Ahmady]], Dan Hatfield, Sébastien Lord, Ruben Sandapen, _Effect of $c \bar c$ resonances in the branching ratio and forward-backward asymmetry of the decay $B \to K^\ast\mu^+ \mu^-$_

* [[Mohammad Ahmady]], Alexandre Leger, Zoe McIntyre, Alexander Morrison, Ruben Sandapen, _Probing transition form factors in the rare $B \to K^\ast \nu \bar \nu$ decay_, Phys. Rev. D 98, 053002 (2018) ([arXiv:1805.02940](https://arxiv.org/abs/1805.02940))

* [[Mohammad Ahmady]], _Holographic light-front QCD in B meson phenomenology_,  PoS DIS2013 (2013) 182  ([arXiv:2001.00266](https://arxiv.org/abs/2001.00266))


application of topology

# Applications of Topology

There are many applications of [[topology]] as like ...

category: topology

[[!redirects application of topology]]
[[!redirects applications of topology]]


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

# Contents
* table of contents
{: toc}

## Idea

> I can illustrate the second approach with the same image of a nut to be opened. The first analogy which came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise you let time pass. The shell becomes more flexible through weeks and months – when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

> A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance.

> [[Alexander Grothendieck]], _[[Récoltes et semailles]]_, 1985–1987, pp. 552-3-1 ("[[The Rising Sea]]")

> I don't want you to think all this is theory for the sake of it, or rather for the sake of itself. It's theory for the sake of other theory.

> ([[Jacob Lurie|J. Lurie]], [ICM 2010](http://gowers.wordpress.com/2010/08/31/icm2010-spielman-csornyei-lurie/))

The tools of [[category theory]] and [[higher category theory]] serve to organize other structures. There is a plethora of applications that have proven to be much more transparent when employing the [[nPOV]]. Higher category theory has helped foster entire new fields of study that would have been difficult to conceive otherwise. This page lists and discusses examples.

## Examples

The following is a (incomplete) list of examples of topics for which higher category theory has proven to be useful.

### In geometry

The field of [[differential geometry]] has long managed to avoid the change to an $n$-point of view that had been found to be unavoidable, natural and fruitful in algebraic geometry long ago. But more recently -- not the least due to the recognition of differential [[higher geometry|higher geometric]] structures in the physics of [[gauge theory]] and [[supergravity]] (such as that of [[orbifold]]s and [[orientifold]]s, of smooth [[gerbe]]s and smooth [[principal ∞-bundle]]s) -- [[sheaf and topos theory|sheaf and topos theoretic]] concepts, such as [[synthetic differential geometry]], [[diffeological space]]s and [[differentiable stack]]s are gaining wider recognition and appreciation.

For instance the ordinary category [[Diff]] of [[smooth manifold]]s fails to have all [[pullback]]s, it only has pullbacks along [[transversal map]]s.
This observation is usually the starting point for realizing that differential geometry is in need of a bit of [[category theory]] in the form of [[higher geometry]].

In all notions of [[generalized smooth space]]s all pullbacks do exist. But they may still not be the "right" pullbacks. For instance cohomology of pullback objects may not have the expected properties. This is solved by passing to smooth [[derived stack]]s, such as [[derived smooth manifold]]s.

Recent developments in [[higher category theory]], such as the concept of higher [[Structured Spaces]] based on [[Higher Topos Theory]], put all these notions of generalized geometries into a unified picture of [[higher geometry]] that realizes old ideas about how category theory provides a language for [[space and quantity]] in great detail and powerful generality and sheds new light on old [[classical mathematics|classical]] problems such the description of the [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves|derived moduli stack of derived elliptic curves]] and the construction of the [[tmf]] [[spectrum]] from it.
This construction has benefited tremendously from the adoption of the [[nPOV]]. Using this point of view, the general strategy becomes naturally evident.

#### In differential equations {#DiffEqu}

Much of [[topological vector space]] theory, e.g., the theory of [[distribution]]s, [[nuclear space]]s, etc. has its origins in [[partial differential equation]] theory and is intensely conceptual (categorical) in spirit. It is routine these days to accept distributional solutions, but it wasn't always so, and it was the efficacy of the abstract TVS theory which changed people's minds.

Way back Cartan studied differential equations in terms of [[exterior differential system]]s. From the $n$POV, these may be understood naturally as sub [[Lie ∞-algebroid]]s of a [[tangent Lie algebroid]].

[[Bill Lawvere]] noticed in the 1960s that the notion of differential equation makes sense in any [[smooth topos]] (as described [here](http://ncatlab.org/nlab/show/differential+equation#InSynthDiff)). In his highly influential article _Categorical dynamics_ he promoted the point of view that all things [[differential geometry|differential geometric]] can be formulated in abstract category theory internal to a suitable [[topos]]. This is the origin of [[synthetic differential geometry]]. It may be understood as providing the fundamental characterization of the notion of the [[infinitesimal space|infinitesimal]].

Closely related to both these perspectives, a modern point of view on differential equations that is proving to be very fruitful regards them as part of the theory of [[D-module]]s.

### In cohomology

A multitude of notions of cohomology and its variants are unified from the $n$POV when viewed as [[derived hom space|∞-categorical hom-spaces]] in [[(∞,1)-topoi]]. See [[cohomology]].

#### Hochschild (co)homology

Specifically, the subject of [[Hochschild cohomology]], when generalized to _higher order Hochschild cohomology_ effectively merges into the canonical concept of [[(∞,1)-powering]] of an [[(∞,1)-topos]] over [[∞Grpd]]. See [[Hochschild cohomology]] for details.

### In homotopy theory {#InHomotopyTheory}

The study of [[homotopy theory]] originated in the study of categories such as those of [[topological space]]s and other objects such as [[chain complex]]es whose [[morphism]]s were known to admit a notion of [[homotopy]]. Historically, in a sequence of steps formalisms were proposed that would organize the rich interesting structure found in such situations. As a first approximation the notion of [[homotopy category]] and [[derived category]] was introduced in order to deal with structures "up to homotopy". But it was clear that the [[homotopy category]] captured only a very small part of the interesting information. Quillen introduced the notion of [[model category]] as a formalization of the full structure, and this formalization turned out to yield a powerful theory that today provides a powerful toolset for dealing with homotopy theoretic situations.

But also the notion of model category was seen to not be the full answer. For instance a model category in a sense retains _too much_ non-intrinsic information. Equivalence classes of model categories under [[Quillen equivalence]] are a more intrinsic characterization of a given [[homotopy theory]]. But this means that one needs some [[higher category theory|higher categorical]] notion for the collection of all model categories. This problem came to be known as the search for the **homotopy theory of homotopy theories**.

Recently, this problem was fully solved and homotopy theory fully understood as the special case of [[higher category theory]] that deals with [[(∞,1)-category|(∞,1)-categories]]:

* the notion of [[model category]], in particular when refined to that of a [[simplicial model category]] serves as a [[presentable (infinity,1)-category|presentation]] of the notion of [[(∞,1)-category]];

* the "homotopy theory of homotopy theories" is accordingly the [[(∞,1)-category of (∞,1)-categories]] $(\infty,1)Cat$;

  better yet: there is an [[(∞,n)-category|(∞,2)-category]] of all $(\infty,1)$-categories;

* in $(\infty,1)Cat$ two $(\infty,1)$-categories presented by model categories are equivalent precisely if the presenting model categories may be connected by a zig-zag sequence of [[Quillen equivalence]]s;

* all "homotopy"-constructions in model category theory, such as [[homotopy limit]]s, [[mapping cone]]s etc. are _tools for constructing_ the corresponding higher categorical intrinsic notions, such as [[limit in a quasi-category|limit in an (∞,1)-category]].

* all variant notions find their intrinsic higher categorical interpretation this way: for instance [[stable homotopy theory]] is the study of [[stable (∞,1)-category|stable (∞,1)-categories]];

* the [[homotopy category]] of a [[model category]] is simply the [[decategorification]] of the corresponding $(\infty,1)$-category to just a [[1-category]];

* and for instance the notion of homotopy category of a stable $(\infty,1)$-category reproduces the notion of [[triangulated category]], thus incorporating also a large toolset from [[homological algebra]] into the picture.

#### In rational homotopy theory {#RationalHomotopyTheory}

... The study of [[rational homotopy theory]] is naturally understood as the study of the [[localization of an (∞,1)-category|localizations]] of [[(∞,1)-topos]]es at morphism that induce equivalences in [[cohomology]] with certain line-object coefficients. See [[rational homotopy theory in an (∞,1)-topos]]. ...

### In K-theory

In full generality, ([[algebraic K-theory|algebraic]]) [[K-theory]] is a universal assignment of [[spectra]] to [[stable (∞,1)-categories]].

...

### In Tannaka duality

... see [[Tannaka duality]] ...

### In differential geometry

See at

* [[higher differential geometry applied to plain differential geometry]]

### In differential cohomology
 {#InDifferentialCohomology}

[[cohesion]] on [[(∞,1)-toposes]] solves the [Simons-Sullivan question](differential%20cohomology%20diagram#SimonsSullivan07) characterization on the characterization of [[generalized (Eilenberg-Steenrod) cohomology|generalized (Eilenberg-Steenrod-type)]] [[differential cohomology]]. See at _[[differential cohomology hexagon]]_ for details.

### In deformation theory {#DeformationTheory}

In deformation theory it was early on recognized that for a good theory the notion of [[Kähler differential]]s has to be generalized to the notion of [[cotangent complex]]. With the advent of the study of derived [[moduli space]]s, such as the [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves|derived moduli space of derived elliptic curves]], this needed to be further generalized to notions of cotangent complexes not just of [[ring]]s, but of [[E-∞-ring]]s.

It turns out that all these concepts are special cases of a construction obtained from a simple higher categorical notion, that of [[left adjoint]] [[section]]s of a [[tangent (∞,1)-category]].

### In logic and type theory {#Logic}

While it is common to view logic as the study of absolute truth, in fact logic can have many different interpretations, or [[semantics]].  A particular semantics for logic can be useful both to inform the study of logic, and to prove facts logically about the semantics.  One very fruitful semantics of this sort is *categorical semantics* for logic and [[type theory]], according to which every category (and especially every [[topos]]) has an [internal language](/nlab/show/type+theory#CategoricalSemantics) and [[internal logic]].  Interpreting "ordinary" mathematical statements in the internal language of exotic categories can make it much easier to study those categories, while on the other hand it can provide new insight into otherwise mysterious logical notions.

In particular, the internal logic of a category (such as a topos) is, in general, [[constructive mathematics|constructive]], i.e. the principle of [[excluded middle]] (and also stronger statements, such as the [[axiom of choice]]) are generally false.  Thus, in order for a theorem to be interpretable internally in such categories, its proof must be constructive.  So while the original "constructivists" believed that classical mathematics was "wrong," nowadays there are good reasons to care about constructive mathematics even if one believes that excluded middle and the axiom of choice are "true," since regardless of their "global" truth they will *not* be true in the internal logic of many interesting categories.  Conversely, category-theoretic models have provided new insight into the independence of various axioms in constructive mathematics, such as differing forms of the axiom of choice.

As another example, the [[identity types]] in Martin-Löf's original constructive dependent [[type theory]] construct, from any [[type]] $A$ and terms $a, b \in A$, a new type $Id_A(a, b)$.  According to the [[propositions as types]] interpretation, the elements of $Id_A(a,b)$ are proofs that $a$ and $b$ are propositionally equal; thus $Id_A(a,b)$ is a replacement for the [[truth value]] of the [[proposition]] $(a=b)$.   There are type-theoretic [[functions]] $1 \to Id(a, a)$, $Id(b, c) \times Id(a, b) \to Id(a, c)$ and $Id(a, b) \to Id(b, a)$ expressing the [[equivalence relation|reflexivity, transitivity and symmetry]] of this propositional [[equality]], but in general an identity type (even the "reflexive" identity type $Id(a,a)$) can have many distinct elements.  This has long been a source of discomfort to type theorists.  However, from a higher-categorical point of view, it is natural to view the terms of identity types as *isomorphisms* in a [[groupoid]]---or, more precisely, an [[∞-groupoid]], since identity types have their own identity types, and all the laws of associativity, exchange, etc. only hold up to terms of these higher identity types.  This suggests that the nonuniqueness of identity proofs should be embraced rather than denigrated, producing a theory at least related to the "internal logic" of [[(∞,1)-category]] theory and [[homotopy theory]]; see [[identity type]] for more details.

This is now known as _[[homotopy type theory]]_, see there for more.

### In physics {#Physics}

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

#### Classical mechanics and its geometric quantization {#ClassMech}

By the end of the 19th century a fairly complete, powerful and elegant mathematical formulation of [[classical mechanics]]: in terms of [[symplectic geometry]]. By the middle of the 20th century, the passage to the corresponding quantum theory was pretty well modeled by the [[geometric quantization]] of symplectic geometries.

But there were some loose ends. Notably the fully general theory involved [[Poisson manifold]]s, not just symplectic manifolds. And the mechanics of relativistic [[classical field theory]] was realized to be more naturally described by [[multisymplectic geometry]].

Both these generalizations have a natural common higher categorical formulation: that of [[Lie ∞-algebroid]]s: a Poisson geometry is naturally encoded in its corresponding [[Poisson Lie algebroid]]. Its higher categorical versions -- the [[n-symplectic manifold]]s -- encode the corresponding multisymplectic geometry.

Moreover, the quantization step of geometric quantization was understood to be effectively the [[Lie integration]] of these [[Lie ∞-algebroid]]s to the corresponding [[Lie ∞-groupoid]]s (currently this is well understood for low $n$).

#### Quantum mechanics and quantum information {#QuantumMechanics}

The basic structure of [[quantum mechanics]] and [[quantum information theory]] is encoded in the theory of [[dagger-compact categories]].

...

#### Gauge theory {#GaugeTheory}

Maxwell realized that the [[electromagnetic field]] is controlled by a degree 2-cocycle in [[de Rham cohomology]]: the electromagnetic [[field strength]]. Later Dirac noticed that this is one part of a degree 2-cocycle in [[differential cohomology]] that characterize a [[connection on a bundle|connection]] on a [[line bundle]].

Later the [[Yang-Mills field]] was understood to similarly be a [[connection on a bundle]], this time on a $G$-[[principal bundle]] for $G$ some possibly nonabelian [[group]].

While thinking about the mathematical structures possibly underlying [[standard model of particle physics]] and [[gravity]], theoretical physicists considered more general hypothetical [[gauge field]]s, such as the [[Kalb-Ramond field]], the [[RR-field]] or the [[supergravity C-field]]. Today all these gauge fields are understood to be modeled, mathematically, by generalized [[differential cohomology]].

##### Supergravity {#Supergravity}

Theories of [[supergravity]] have been known to require higher [[gauge field]]s in the above sense -- hence the term [[supergravity C-field]]. A powerful formalism for handling these theories is the [[D'Auria-Fre formulation of supergravity]]. As described there, this is secretly (but evidently) nothing but a description of supergravity as a theory of connections on nonabelian $G$-[[principal ∞-bundle]]s for $G$ some super [[Lie ∞-groupoid|Lie ∞-group]]. For instance Cremmer-Scherk 11-dimensional supergravity theory is governed by the super Lie 3-group $G$ whose [[L-∞-algebra]] is the [[supergravity Lie 3-algebra]].

#### BV-BRST formalism {#BVBRST}

The [[BV-BRST formalism]] is secretly a way to talk about the fact that configuration spaces of [[gauge theory|gauge theories]] are not naive spaces such as [[manifold]]s, but are general [[space]]s in the sense of [[higher geometry]]:

the configuration space is really an object $Conf \in Sh_{(\infty,1)}((dgAlg^-)^{op})$ in the [[∞-stack]] [[(∞,1)-topos]] on the [[(∞,1)-site]] $(dgAlg^-)^{op}$ of certain [[algebra in an (∞,1)-category|∞-algebras]] modeled as [[dg-algebra]]s. The BV-BRST-complex of a physical system is the global [[derived geometry|derived]] function algebra

$$
  \mathcal{O}(Conf) \in dgAlg
  \,.
$$

> (many more aspects go here, eventually)...

#### Quantum field theory {#QFT}

There are essentially two axiomatizations of what [[quantum field theory]] is, both of which are inherently $\infty$-categorical:

* in the [[FQFT]] picture -- the _Schrödinger picture_ -- a quantum field theory is described as an [[(∞,n)-functor]] on an [[(∞,n)-category of cobordisms]]. The [[cobordism hypothesis]] -- now a theorem that characterizes central properties of these [[(∞,n)-categories]], has been a major driving force in the development of [[higher category theory]].

* in the [[AQFT]]/[[factorization algebra]] picture -- the _Heisenberg picture_  -- a quantum field theory is described as an $\infty$-copresheaf of observables on its parameter space.

##### Holography

* [[holographic principle of higher category theory]]

##### 3d TFT and 2d CFT {#3dTFT2dCFT}

3-dimensional [[TFT]] such as [[Chern-Simons theory]] and [[Dijkgraaf-Witten theory]] and the global aspects of 2-dimensional [[conformal field theory]] are inherently governed by the theory of [[modular tensor categories]].

The local aspects of 2-dimensional conformal field theory are governed by [[vertex operator algebra]]s. A [[vertex operator algebra]] is really the [[algebra over an operad]], for the operad of holomorphic pointed spheres (as described there).

...

### In your favorite topic here

...

## Related pages

* [[geometry of physics]]

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[twisted smooth cohomology in string theory]]

* [[motives in physics]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

* [[motivation for cohesion]]


applications of double category theory

A list of works and resources about applications of [[double category]] theory.

See there for introductory material.

## Theory
### [[2-category equipped with proarrows|Equipments]]
* [[David Jaz Myers]], _String diagrams for double categories and (virtual) equipments_, [(arXiv)](https://arxiv.org/abs/1612.02762), 2016
* [[Mike Shulman]], _Equipments_, [(n-Café)](https://golem.ph.utexas.edu/category/2009/11/equipments.html), 2009
* [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, [(arXiv)](https://arxiv.org/abs/0706.1286), 2009
* Max New, Dan Licata, _A Formal Logic for Formal Category Theory_, [(arXiv)](https://arxiv.org/abs/2210.08663), 2022

### Grothendieck constructions
* Evan Patterson, _Grothendieck construction for double categories_, [(Topos Institute)](https://topos.site/blog/2022/05/grothendieck-construction-for-double-categories/), 2022
* Cruttwell, Lambert, Pronk, Szyld, _Double Fibrations_, [(arXiv)](https://arxiv.org/abs/2205.15240), 2022

## Applications
### Lenses & optics
* [[Bryce Clarke]], _The double category of lenses_, [(online)](https://bryceclarke.github.io/The_Double_Category_Of_Lenses_Phd_Thesis.pdf), 2022
* Capucci, _Seeing double through dependent optics_, [(arXiv)](https://arxiv.org/abs/2204.10708), 2022
* Guillaume Boisseau, Chad Nester, Mario Roman, _Cornering optics_,[(arXiv)](https://arxiv.org/abs/2205.00842), 2022

### Open systems theory
* Jared Culbertson, Paul Gustafson, Daniel E. Koditschek, Peter F. Stiller, _Formal composition of hybrid systems_, [(arXiv)](https://arxiv.org/abs/1911.01267)
* Kenny Courser, _Open systems: a double categorical perspective_, [(arXiv)](https://arxiv.org/abs/2008.02394), 2020
See also [[structured cospans]] and [[decorated cospans]]
* Chad Nester, _Situated Transition Systems_, [(arXiv)](https://arxiv.org/abs/2105.04355), 2021
* [[David Jaz Myers]], _Categorical systems theory_, [(online)](http://davidjaz.com/Papers/DynamicalBook.pdf), 2021
* Evan Patterson, _Decorated cospans via the double Grothendieck construction_, [(Topos Institute)](https://topos.site/blog/2022/05/decorated-cospans-via-the-grothendieck-construction/), 2022
* Evan Patterson, _Structured cospans as a cocartesian equipment_, [(Topos Institute)](https://topos.site/blog/2023/03/structured-cospans-as-a-cocartesian-equipment/), 2023
* [[John Baez]], Kenny Courser, [[Christina Vasilakopoulou]], _Structured versus Decorated Cospans_, [(Compositionality)](https://compositionality-journal.org/papers/compositionality-4-3/pdf), 2022
* Chad Nester, _Concurrent Process Histories and Resource Transducers_, [(arXiv)](https://arxiv.org/abs/2010.08233), 2022

### Theory of programming languagues
* Pierre-Evariste Dagand, [[Conor McBride]], _A Categorical Treatment of Ornaments_, [(acm)](https://dl.acm.org/doi/abs/10.5555/2591370.2591396), 2013
* Max New, Dan Licata, _Call-by-name Gradual Type Theory_, [(arXiv)](https://arxiv.org/abs/1802.00061), 2018
* Max New, Dan Licata, _A Formal Logic for Formal Category Theory_, [(arXiv)](https://arxiv.org/abs/2210.08663), 2022
* Clovis Eberhart, Tom Hirschowitz,
_Fibred Pseudo Double Categories for Game Semantics_, [TAC 34(19)](http://www.tac.mta.ca/tac/volumes/34/19/34-19abs.html), 2019. Beware, “fibred” is use there to mean that the codomain functor is a fibration.


applicative functor

#Contents#
* table of contents
{:toc}

##Idea

In [[computer science]], **applicative functors** (also known as **idioms**) are the programming equivalent of [[lax monoidal functors]] with a [[tensorial strength]] in category theory.

A [[monad (in computer science)|monad]] gives rise to an applicative functor, but not all applicative functors result from monads. Unlike monads, applicative functors are closed under composition.

##Related concepts

* [[arrow (in computer science)]]

##References

* Conor Mcbride, Ross Paterson, _Applicative programming with effects_, Journal of Functional Programming. 18 (01): 1–13. ([doi:10.1017/S0956796807006326](https://doi.org/10.1017/S0956796807006326); [author’s version](http://www.staff.city.ac.uk/~ross/papers/Applicative.html))

* Ross Paterson, _Constructing Applicative Functors_, in Mathematics of Program Construction, Madrid, 2012, Lecture Notes in Computer Science vol. 7342, pp. 300-323, Springer, 2012. ([paper](http://www.staff.city.ac.uk/~ross/papers/Constructors.html))

[[!redirects applicative functors]]


applied category theory

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

#Contents#
* table of contents
{:toc}

##Idea##

The whole point of [[category theory]] is to study fundamental [[general abstract]] patterns and phenomena that (re-)appear throughout [[mathematics]]. Hence applications of [[theorems]] of category theory are ubiquituous in [[mathematics]] and in subjects with a mathematical basis, such as [[physics]] and [[computer science]]. Often this goes without saying.

In the introduction to [Bradley 18](#Bradley18) it says:

> ... ideas and results from category theory have found applications
in computer science and quantum physics (not to mention pure mathematics itself), but these are not the only applications to which the word _applied_ in _applied category theory_ is being applied...

> category theory has found applications in a wide range of disciplines outside of pure mathematics—even beyond the closely related fields of computer science and quantum physics. These disciplines include chemistry, neuroscience, systems biology, natural language processing, causality, network theory, dynamical systems, and database theory to name a few. And what do they all have in common? ... In other words, the techniques, tools, and ideas of category theory are being used to identify recurring themes across these various disciplines with the purpose of making them a little more formal.

##References##

* {#Bradley18} [[Tai-Danae Bradley]], _What is applied category theory?_, [arXiv:1809.05923](https://arxiv.org/abs/1809.05923)

* [[Brendan Fong]] and [[David Spivak]], _Seven Sketches in Compositionality: An Invitation to Applied Category Theory_, [arXiv:1803.05316](https://arxiv.org/abs/1803.05316).

* [Applied category theory](https://www.appliedcategorytheory.org/)

[[John Baez]] ran a series of lectures based on this book:

* lectures on chapter 1

  * [Lecture 1 - Introduction](https://forum.azimuthproject.org/discussion/1807/lecture-1-introduction/p1)
  * [Lecture 2 - What is Applied Category Theory?](https://forum.azimuthproject.org/discussion/1808/lecture-2-what-is-applied-category-theory/p1)
  * [Lecture 3 - Chapter 1: Preorders](https://forum.azimuthproject.org/discussion/1812/lecture-3-chapter-1-posets/p1)
  * [Lecture 4  - Chapter 1: Galois Connections](https://forum.azimuthproject.org/discussion/1828/lecture-4-chapter-1-galois-connections/p1)
  * [Lecture 5 - Chapter 1: Galois Connections](https://forum.azimuthproject.org/discussion/1845/lecture-5-chapter-1-galois-connections/p1)
  * [Lecture 6 - Chapter 1: Computing Adjoints](https://forum.azimuthproject.org/discussion/1901/lecture-6-chapter-1-computing-adjoints/p1)
  * [Lecture 7 - Chapter 1: Logic](https://forum.azimuthproject.org/discussion/1909/lecture-7-chapter-1-logic/p1)
  * [Lecture 8 - Chapter 1: The Logic of Subsets](https://forum.azimuthproject.org/discussion/1921/lecture-8-chapter-1-the-logic-of-subsets/p1)
  * [Lecture 9 - Chapter 1: Adjoints and the Logic of Subsets](https://forum.azimuthproject.org/discussion/1931/lecture-9-chapter-1-adjoints-and-the-logic-of-subsets/p1)
  * [Lecture 10 - Chapter 1: The Logic of Partitions](https://forum.azimuthproject.org/discussion/1963/lecture-10-the-logic-of-partitions/p1)
  * [Lecture 11 - Chapter 1: The Poset of Partitions](https://forum.azimuthproject.org/discussion/1991/lecture-11-chapter-1-the-poset-of-partitions/p1)
  * [Lecture 12 - Chapter 1: Generative Effects](https://forum.azimuthproject.org/discussion/1999/lecture-12-chapter-1-generative-effects/p1)
  * [Lecture 13 - Chapter 1: Pulling Back Partitions](https://forum.azimuthproject.org/discussion/2008/lecture-13-chapter-1-pulling-back-partitions/p1)
  * [Lecture 14 - Chapter 1: Adjoints, Joins and Meets](https://forum.azimuthproject.org/discussion/2013/lecture-14-adjoints-joins-and-meets/p1)
  * [Lecture 15 - Chapter 1: Preserving Joins and Meets](https://forum.azimuthproject.org/discussion/2027/lecture-15-chapter-1-preserving-joins-and-meets/p1)
  * [Lecture 16 - Chapter 1: The Adjoint Functor Theorem for Posets](https://forum.azimuthproject.org/discussion/2031/lecture-16-chapter-1-the-adjoint-functor-theorem-for-posets/p1)
  * [Lecture 17 - Chapter 1: The Grand Synthesis](https://forum.azimuthproject.org/discussion/2037/lecture-17-chapter-1-the-grand-synthesis/p1)

* lectures on chapter 2

  * [Lecture 18 - Chapter 2: Resource Theories](https://forum.azimuthproject.org/discussion/2075/lecture-18-chapter-2-resource-theories/p1)
  * [Lecture 19 - Chapter 2: Chemistry and Scheduling](https://forum.azimuthproject.org/discussion/2079/lecture-19-chapter-2-chemistry-and-scheduling/p1)
  * [Lecture 20 - Chapter 2: Manufacturing](https://forum.azimuthproject.org/discussion/2081/lecture-20-chapter-2-manufacturing/p1)
  * [Lecture 21 - Chapter 2: Monoidal Preorders](https://forum.azimuthproject.org/discussion/2082/lecture-21-chapter-2-monoidal-preorders/p1)
  * [Lecture 22 - Chapter 2: Symmetric Monoidal Preorders](https://forum.azimuthproject.org/discussion/2084/lecture-22-chapter-2-symmetric-monoidal-preorders/p1)
  * [Lecture 23 - Chapter 2: Commutative Monoidal Posets](https://forum.azimuthproject.org/discussion/2086/lecture-23-chapter-2-commutative-monoidal-posets/p1)
  * [Lecture 24 - Chapter 2: Pricing Resources](https://forum.azimuthproject.org/discussion/2089/lecture-24-chapter-2-pricing-resources/p1)
  * [Lecture 25 - Chapter 2: Reaction Networks](https://forum.azimuthproject.org/discussion/2090/lecture-25-chapter-2-reaction-networks/p1)
  * [Lecture 26 - Chapter 2: Monoidal Monotones](https://forum.azimuthproject.org/discussion/2095/lecture-26-chapter-2-monoidal-monotones/p1)
  * [Lecture 27 - Chapter 2: Adjoints of Monoidal Monotones](https://forum.azimuthproject.org/discussion/2098/lecture-27-chapter-2-adjoints-of-monoidal-monotones/p1)
  * [Lecture 28 - Chapter 2: Ignoring Externalities](https://forum.azimuthproject.org/discussion/2105/lecture-28-chapter-2-ignoring-externalities/p1)
  * [Lecture 29 - Chapter 2: Enriched Categories](https://forum.azimuthproject.org/discussion/2121/lecture-29-chapter-2-enriched-categories/p1)
  * [Lecture 30 - Chapter 2: Preorders as Enriched Categories](https://forum.azimuthproject.org/discussion/2124/lecture-30-chapter-1-preorders-as-enriched-categories/p1)
  * [Lecture 31 - Chapter 2: Lawvere Metric Spaces](https://forum.azimuthproject.org/discussion/2128/lecture-31-chapter-2-lawvere-metric-spaces/p1)
  * [Lecture 32 - Chapter 2: Enriched Functors](https://forum.azimuthproject.org/discussion/2169/lecture-32-chapter-2-enriched-functors/p1)
  * [Lecture 33 - Chapter 2: Tying Up Loose Ends](https://forum.azimuthproject.org/discussion/2192/lecture-33-chapter-2-tying-up-loose-ends/p1)

* [[David Spivak]], _Category theory for the sciences._ MIT Press, 2014.

* [[Brandon Coya]], _Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective_, [arXiv:1805.08290](https://arxiv.org/abs/1805.08290)

* [[John Baez]] and [[Brendan Fong]], _A Compositional Framework for Passive Linear Networks_, [arXiv:1504.05625](https://arxiv.org/abs/1504.05625)

* [[Blake Pollard]], _Open Markov processes: A compositional perspective on non-equilibrium steady states in biology_, [arXiv:1601.00711](https://arxiv.org/abs/1601.00711)

* [[Brendan Fong]], _The Algebra of Open and Interconnected Systems_, [arXiv:arXiv:1609.05382](https://arxiv.org/abs/1609.05382)

* [[John Baez]] and [[Blake Pollard]], _A Compositional Framework for Reaction Networks_, [arXiv:1704.02051](https://arxiv.org/abs/1704.02051)

*  [[John C. Baez]], [[Brendan Fong]] and [[Blake Pollard]], _A Compositional Framework for Markov Processes_, [arXiv:1508.06448](https://arxiv.org/abs/1508.06448)

* [[Jules Hedges]] and [[Martha Lewis]], _Towards Functorial Language-Games_, [arXiv:1807.07828](https://arxiv.org/abs/1807.07828)


applied mathematics

The area conventionally called _applied mathematics_ is not the central focus of the $n$Lab: our main focus is in (higher) category theory, [[sheaves]], [[stack]]s, [[homotopy theory|homotopy]] and (co)homological methods, foundations, [[topology]], [[algebra]] and modern geometry as well as mathematical physics and philosophical aspects.

Unfortunately, there is a problem in defining what applied mathematics is, and many mathematicians disagree about the range. Some, including [[Henri Poincaré]] and [[Vladimir Arnol'd]], said that

> there is no applied mathematics, there are only applications of mathematics.

Moreover, in many areas and departments the subdivision into pure and applied mathematics (or any similar variants) appears to be a social designation, belong to a group rather than an imperative to apply anything to real world.

One can argue that what will be applied can not be nearly fully predicted by the area and intention of the creator of particular mathematical result, but by the internal power of the result and by the future of applications themselves. Hence it is a bit presumptuous that people working on some particular mathematics problems, who do not themselves apply mathematics to real-life problems, declare themselves applied just by the common opinion on classification of their result.

For instance it is common to assume that work on [[partial differential equation]]s (especially by analytic and numerical methods) is a subarea of applied mathematics, even if one studies completely unnatural and never applied differential equations, while it is not considered applied mathematics to prove theorems about the homology of spaces, even if they strongly influence [[index theorem]]s widely used in applications to other sciences and "real-life problems".

In any case, to well-spirited people who believe they do applied mathematics we may vaguely recommend our moderately developed pages relating

* [[mathematical physics]],

which is important for us. Unfortunately, in the departments of the applied mathematics world, there is not really any interest in the foundations of [[quantum field theory]] (and siblings like [[superstring theory]], [[statistical field theory]], [[quantum gravity]], ...) which is our central interest in mathematical physics. We have stubs for some other areas intersecting with traditional "applied mathematics" but not many:

* [[hydrodynamics]]
* [[finite element method]]
* [[regular differential operator]]
* [[homological algebra in finite element method]]
* [[Runge-Kutta method]]
* [[symplectic integrator]]
* [[preconditioner]]
* [[numerical analysis]]

Wikipedia: [applied mathematics](http://en.wikipedia.org/wiki/Applied_mathematics)

As, if and when applications of the nPOV to areas traditionally called 'applied mathematics', we cordially invite practitioners to contribute so that those developments may be recorded and developed here. Recent developments of category theoretical methods and insights to areas of Chemistry and Biology are being discussed in [Azimuth](https://johncarlosbaez.wordpress.com/).

category: applications
[[!redirects applied math]]
[[!redirects applied maths]]
[[!redirects applied mathematics]]


applied topology

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

#Contents#
* table of contents
{:toc}

## Idea

The application of [[topology]] ([[general topology]] or [[algebraic topology]]) to other fields.

## Related concepts

* [[computational topology]]

* [[topological data analysis]]

## References

Textbook account:

* [[Robert Ghrist]], *Elementary Applied Topology* (2014) [[web](https://www2.math.upenn.edu/~ghrist/notes.html), [ISBN:978-1502880857]()]

See also:

* [[Dmitry Feichtner-Kozlov]], *[Applied Topology Unit](https://groups.oist.jp/atu)*

On [[combinatorics|combinatorial]] topology

* [[Dmitry N. Kozlov]], *Trends in Topological Combinatorics* [[arXiv:math/0507390](https://arxiv.org/abs/math/0507390)]

* [[Dmitry Kozlov]], *Combinatorial Algebraic Topology*,  Algorithms and Computation in Mathematics **21**, Springer (2008) [[doi:10.1007/978-3-540-71962-5](https://doi.org/10.1007/978-3-540-71962-5)]


approach space

## Idea

The concept of approach space generalized the concept of [[metric space]]. The idea is that the distance describes not only the distance between two points but the distance of a point to a subset. This relatively tangible generalization gives the missing link in the triad of the concepts of [[uniform space|uniformity]], [[topological space|topology]], and [[metric space|metric spaces]].

## Definition

An __approach space__ is a set $X$ together with a __distance__ $d\colon X \times \mathcal{P}(X) \to [0,\infty]$ (where $\mathcal{P}(X)$ denotes the [[power set]]) such that the following axioms hold for all $x \in X$

1. $d(x, \{x\}) = 0$

1. $d(x, \emptyset) = \infty$

1. for all $A, B \in \mathcal{P}$:
   $d(x, A\cup B) = \min\{ d(x, A), d(x, B) \}$

1. for all $A \in \mathcal{P}$ and $\varepsilon \in [0,\infty]$:
   $d(x, A ) \leq d(x, A^{\varepsilon]} ) + \varepsilon $

where $A^{\varepsilon]} \coloneqq \{x \in X \mid d(x, A) \leq \varepsilon \}$.

## Properties

Every approach space $d$ induces a [[topological space|topology]] on $X$ via the [[closed subspace |closure operator]] $Cl_d(A) = \{x \in X \mid d(x, A) = 0 \}$.

## Examples

* Every [[topological space]] is induced by a canonical approach structure given by $d(x, A) = 0$ if $x \in Cl(A)$ and $d(x, A) = \infty$ otherwise.

* The [[one-point compactification]] of a metric space $d$ can be metrised in a canonical way as an approach space by

$$
d^*(x, A) =
\begin{cases}
d(x, A\setminus\{\infty\})  & x \neq \infty \\
0  & x = \infty\, and\, A\, is\, not\, precompact \\
\infty & x = \infty\, and\, A\, is\, precompact.
\end{cases}
$$

* A [[gauge space|gauge]], or more generally a gauge base, $G$ on $X$ gives a distance on $X$ by $d_G(x, A) = \sup_{d \in G} \inf_{y\in A} d(x,y)$.

## Related concepts

* [[metric space]]

* [[gauge space]]

* [[relational beta-module]]

## References

 * Robert Lowen, _Approach spaces: the missing link in the topology-uniformity-metric triad_, Oxford Mathematical Monographs. 1997. ([publisher link](https://global.oup.com/academic/product/approach-spaces-9780198500308)).


approximate fibration

The approximate homotopy lifting property is a weak version of the [[homotopy lifting property]] in the setup of [[metric spaces]].

A [[proper map]] $p:E\to B$ between [[locally compact space|locally compact]] metric [[absolute neighborhood retract]]s (ANRs) satisfies the **approximate homotopy lifting property** for a space $X$ if for any [[open covering]] $U$ of B, and any map $h : X\to E$ with a [[homotopy]] $H : X \times I \to B$ such that $p\circ h = H_0$, there exists a homotopy $G : X\times I\to E$ such that $G_0 = h$ and the maps $p\circ G$ and $H$ are $U$-[[close map|close]].

A proper map $p : E\to B$ between locally compact metric ANRs is an **approximate fibration** if $p$ has the approximate homotopy lifting property for all metric spaces.

It is straightfoward to generalize this notion to the level maps of inverse systems of locally compact metric ANRs.

[[!redirects approximate homotopy lifting property]]


approximate integral domain

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

\tableofcontents

## Idea

A weaker notion of [[integral domain]] which allows for some [[zero divisors]], but for which one may [[quotient ring|quotient out]] the zero divisors to obtain an [[integral domain]].

"Approximate integral domain" is a placeholder name for a concept which may or may not have another name in the mathematics literature. The idea however is that approximate integral domains are to integral domains as [[local rings]] are to [[Heyting fields]], and as [[weak local rings]] are to [[weak Heyting fields]].

## Definition

An **approximate integral domain** is a [[commutative ring]] $R$ such that:

* $R$ is nontrivial ($0 \ne 1$); and

* The [[zero divisors]] form an [[ideal]]. (equivalently, the non-[[cancellative elements]] form an [[ideal]]).

Thus, the [[quotient ring|quotient]] of an approximate integral domain by its [[ideal]] of [[zero divisors]] is an [[integral domain]].

Every approximate integral domain has an [[equivalence relation]] $\approx$, defined as $x \approx y$ if and only if $x - y$ is a [[zero divisor]]. Hence the name "approximate" integral domain. Then [[integral domains]] are precisely the approximate integral domains for which $\approx$ implies [[equality]].

## In constructive mathematics

In constructive mathematics, similar to the notion of [[local ring]], [[integral domain]], and [[field]], the notion of approximate integral domain bifurcates into multiple distinct notions:

A **weak approximate integral domain** is an approximate integral domain defined as above.

Recall that a [[cancellative element]] in a commutative ring $R$ is an element $a \in R$ for which both left and right multiplication by $a$ is an injection, and [[zero divisors]] are precisely the elements in $R$ which are not cancellative.

A **strict approximate integral domain** is a weak approximate integral domain for which additionally the cancellative elements form an [[anti-ideal]]:

* $0$ is not cancellative;
* if $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative;
* if $a \cdot b$ is cancellative, then $a$ is cancellative and $b$ is cancellative (this is trivially true in any commutative ring).

The [[quotient object|quotient ring]] of a strict approximate integral domain by its anti-ideal of cancellative elements is a [[Heyting integral domain]].

One can define an [[apartness relation]] in any strict approximate integral domain: $x \# y$ iff $x - y$ is cancellative. Then the local ring is a Heyting field if and only if this apartness relation is [[tight relation|tight]].

+-- {: .num_prop #internal}
###### Proposition
The addition and multiplication operations on a strict approximate integral domain $R$ are strongly extensional with respect to the canonical apartness relation $\#$ defined by $x \# y$ iff $x - y$ is cancellative. In this way a strict approximate integral domain becomes an internal [[ring object]] in the category $Apart$, consisting of sets with apartness relations and maps (strongly extensional functions) between them.
=--

+-- {: .proof}
###### Proof
Recall that products $X \times Y$ in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by $(x, y) \# (x', y')$ iff $x \# x'$ in $X$ *or* $y \# y'$ in $Y$. Recall also that a function $f: X \to Y$ between sets with apartness relations is *strongly extensional* if $f(x) \# f(y)$ implies $x \# y$.

For addition, if $(x + y) \# (x' + y')$, then $x + y - (x' + y') = (x - x') + (y - y')$ is cancellative, so $x - x'$ or $y - y'$ is cancellative since $R$ is a strict approximate integral domain, whence $(x, y) # (x', y')$. Thus addition is strongly extensional.

For multiplication, if $x y # x' y'$, then $x y - x' y'$ is cancellative. Write $x y - x' y' = (x - x')y + x'(y - y')$. Since $R$ is a strict approximate integral domain, either $(x - x')y$ is a cancellative element or $x'(y - y')$ is a cancellative element. From this we easily conclude $x - x'$ is a cancellative element or $y - y'$ is, since cancellative elements are closed under multiplicaiton, whence $(x, y) # (x', y')$. So multiplication is also strongly extensional.
=--

\begin{theorem}
For a strict approximate integral domain, the [[ring of fractions]] obtained by inverting the cancellative elements is a [[local ring]].
\end{theorem}

## Examples and non-examples

* The [[integers]] are an approximate integral domain which are an [[integral domain]].

* The [[dual numbers|dual]] integers $\mathbb{Z}[\epsilon]/\epsilon^2$ is an approximate integral domain where the [[nilpotent]] [[infinitesimal]] $\epsilon \in \mathbb{Z}[\epsilon]/\epsilon^2$ is a non-[[zero]] [[zero divisor]].

* For any [[prime number]] $p$ and any [[positive number|positive]] [[natural number]] $n$, the [[prime power local ring]] $\mathbb{Z}/p^n\mathbb{Z}$ is an approximate integral domain, whose [[ideal]] of [[zero divisors]] is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$. The quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by its ideal of [[zero divisors]] is the [[finite field]] $\mathbb{Z}/p\mathbb{Z}$, indicating that it is also a [[weak local ring]].

* There exist commutative rings which are not approximate integral domains. For example, the [[integers modulo]] 6 $\mathbb{Z}/6\mathbb{Z}$ is not an approximate integral domain, because $3$ and $4$ are both zero divisors, but $3 + 4$ is cancellative. When one tries to quotient out the zero divisors, the resulting ring is [[trivial ring|trivial]].

## See also

* [[integral domain]]

* [[ordered integral domain]]

* [[weak local ring]]

* [[local ring]]

[[!redirects approximate integral domain]]
[[!redirects approximate integral domains]]

[[!redirects weak approximate integral domain]]
[[!redirects weak approximate integral domains]]

[[!redirects strict approximate integral domain]]
[[!redirects strict approximate integral domains]]


approximation

## Related concepts

* [[epsilontic analysis]]

## References

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

[[!redirects approximations]]


approximation of the identity

"Approximation of the identity" is a rubric for a general technique in functional analysis for proving that certain inclusions of [[topological vector space]]s are dense. It refers to the fact that an identity for a convolution product (aka "Dirac distribution") may not literally exist in a particular TVS, but is virtually there in the sense that it can be approximated by elements in the subspace which is being included.

## Examples ##

We illustrate the technique with two examples.

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###### Example

Consider first the problem of showing that restrictions of polynomials to $[-1, 1]$ are dense in $L^p([-1, 1])$ (under Lebesgue [[measure space|measure]]). The idea is to take the formula

$$ f(y) = (f * \delta)(y) = \int_{-1}^1 f(x) \delta(y - x) d x $$

(which literally makes no sense because $\delta$ is not an actual integrable function) and then replace $\delta$ by polynomial functions $p_n$ which "approximate" to it (so each $p_n$ has "mass" 1 and is vanishingly small outside a given neighborhood of 0, if $n$ is sufficiently large).

Put for example $f_n(x) = (1 - x^2)^n$ and "normalize" it, putting

$$p_n = \frac{f_n}{\|f_n\|_1}$$

where $\|g\|_1$ indicates $L^1$ norm. By "differentiating under the integral sign", we have

$$ D^j (f * p_n) = (-1)^j f * D^j(p_n) $$

so that for each $n$, the $j^{th}$ derivative of $f * p_n$ is identically zero for $j$ sufficiently large. Hence $f * p_n$ is polynomial. Next, the claim is that for $f \in L^p$, we have

$$lim_{n \to \infty} \|f - (f * p_n)\|_p = 0$$

Intuitively, the idea is that $ f - (f * p_n) = f * (\delta - p_n) $ and that (because $L^p$ is a module over the [[Banach algebra]] $L^1$)

$$\|f * (\delta - p_n)\|_p \leq \|f\|_p \|\delta - p_n\|_1 \to 0$$

as $n \to \infty$. For a more careful proof, see theorem 9.6 in Wheeden and Zygmund (referenced below).
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+-- {: .num_example}
###### Example

For a second example, consider how to prove that the functions $z^n$, with $n$ ranging over integers, forms an orthonormal basis of the [[Hilbert space]] $L^2(S^1)$ where $S^1$ is the unit circle in the complex plane, where the inner product is given by

$$\langle f, g \rangle = \frac1{2\pi i} \int_{S^1} \overline{f(z)} g(z) \frac{d z}{z}$$

The monomials $z^n$ are clearly orthonormal, so again the idea is to use appropriate linear combinations of the $z^n$ (i.e., Laurent polynomials) to approximate a Dirac mass concentrated at the identity $z = 1$ in $S^1$.
There are various ways of doing that; one of the most useful is by taking the Féjer kernel

$$F_n(z) = \frac1{n+1} (\sum_{-n \leq k \leq n} z^{k/2})^2 = \frac1{n+1}(z^{-n} + 2z^{-n+1} + \ldots + (n+1)z^0 + \ldots + 2z^{n-1} + z^n)$$

Each Laurent polynomial $F_n(z)$ is real-valued, nonnegative, and its $L^1$ norm is 1. Putting $z = e^{i x}$, we have

$$F_n(e^{i x}) = \frac1{n+1} \frac{\sin^2((n+1)x/2)}{\sin^2(x/2)}$$

which makes it clear that $F_n(e^{i x})$ becomes very small outside a neighborhood of 0 (in $\mathbb{R}/2\pi\mathbb{Z}$) as $n$ grows large. Thus $F_n$ approximates the identity; therefore for any $L^2$ function $f$ on $S^1$, we have

$$\lim_{n \to \infty} \|(F_n * f) - f\|_2 = 0$$

Finally, $F_n * f$ is itself a Laurent polynomial; this follows from the fact that for the function $e_n(z) = z^n$, one has

$$ (e_n * f)(w) = \frac1{2\pi i}\int_{S^1} (w/z)^n f(z) \frac{d z}{z} = e_n(w)\langle e_n, f \rangle$$

It follows from all this that the Laurent polynomials on $S^1$ are dense in $L^2(S^1)$.

A similar technique applies to any compact Hausdorff abelian group $G$ equipped with its normalized Haar measure $d\mu$, in place of the measure space $(S^1, \frac1{2\pi i}\frac{d z}{z})$, and shows that the characters on the group span a dense subspace in $L^2$ norm. In other words, the characters form an orthonormal basis of $L^2(G, d\mu)$.
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