Nelson arithmetic

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#### Arithmetic
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# Contents #
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## Idea

An extension of the [[first-order theory]] of [[Peano arithmetic]] as described in [Edward Nelson's paper](#Nelson), by introducing a set of "counting numbers" that we can't use induction on.

## Definition

Nelson arithmetic ($NA$) is expressed in the language with a constant $0$, unary predicate $\mathcal{C}$, unary function symbol $s$ and binary functions $+$,$\cdot$; it extends [[Peano arithmetic]], which consists of the following [[axioms]]:

1. $\forall_x \neg s(x) = 0$;

2. $\forall_{x, y} (s(x) = s(y)) \Rightarrow (x = y)$;

3. $\forall_x x + 0 = x$,

4. $\forall_{x, y} x + s(y) = s(x+y)$;

5. $\forall_x x \cdot 0 = 0$;

6. $\forall_{x, y} x \cdot s(y) = x \cdot y + x$;

7. For any predicate $\varphi$ in first-order logic in the language $0$, $s$, $+$, $\cdot$ (but not containing $\mathcal{C}$), the axiom $\varphi(0) \wedge (\forall_x  \varphi(x) \Rightarrow \varphi(Sx)) \Rightarrow \forall_x \varphi(x)$;

with the two additional axioms regarding $\mathcal{C}$:

8. $\mathcal{C}(0)$;

9. $\forall_{x} \mathcal{C}(x) \Rightarrow \mathcal{C}(s(x))$;

We may choose to omit the axiom schema (7), in which case we refer to Nelson arithmetic without induction ($NA-Ind$). Without induction, it is not possible to prove that addition or multiplication are commutative or associative. Optionally, we could add the axiom $\forall_x x = 0 \vee \exists_y x = s(y)$ from [[Robinson arithmetic]] so that the resulting theory extends Robinson arithmetic, however we don't in this page.

### Counting numbers

The predicate $\mathcal{C}$ is supposed to refer to "counting numbers", in the familiar intuitive sense that the [[natural numbers]] are supposed to be for counting. Nelson states in his paper that

> It follows (...) that $0$, $S0$, $SS0$, and so forth, are counting numbers. But we cannot prove that all numbers are counting numbers.

And indeed, there are many [[nonstandard model of arithmetic|non-standard models of Nelson arithmetic]] where not every number in the model is a "counting number"/"natural number". Unfortunately, nonstandard models of Peano arithmetic, which Nelson arithmetic extends, are hard to describe, and cannot have computable arithmetic operations.

Nonstandard models of $NA-Ind$ are more common though. If one has any [[ordered ring]] R (meaning $R$ is partially ordered with $+$ monotone, $0 \leq 1$, and the elements $\geq 0$ are closed under multiplication), then the subset $R_{\geq 0} = \{r \in R \mid r \geq 0\}$ is a model of $NA-Ind$, where we may take the predicate $C(a)$ to be $a \in \{0, 1, 1+1, 1+1+1, \cdots\}$ (the image of $\mathbb{N}$ under the unique ring homomorphism $\mathbb{Z} \to R$). For example, the non-negative [[rational numbers]] $\mathbb{Q}_{\geq 0}$ and nonnegative [[real numbers]] $\mathbb{R}_{\geq 0}$ both produce models in this way.

On the other extreme, one could define a model where $\mathcal{C}(a)$ is true for all numbers $a$ in the model, and thus all number trivially are "counting numbers", though in that case the "counting numbers" interpretation of the predicate doesn't make much intuitive sense anymore.

### Additionable, multiplicable, and exponentiable numbers

We define the unary predicate $\mathcal{A}$, meaning "additionable number", as $\mathcal{A}(x)$ if for all counting numbers $y$, $y + x$ is also a counting number. Similarly, we define the unary predicate $\mathcal{M}$, meaning "multiplicable number", as $\mathcal{M}(x)$ if for all additionable numbers $y$, $y \cdot x$ is also an additionable number. Without induction, we don't know that addition of "additionable numbers" or multiplication of "multiplicable numbers" is associative.

Let $(-)\uparrow(-)$ denote the exponentiation function; if we aren't using induction, instead suppose that there is a [[binary function]] $(-)\uparrow(-):N \times N \to N$ called "exponentiation" such that

* for all $x \in N$, $x \uparrow 0 = s(0)$

* for all $x, y \in N$, $x \uparrow s(y) = x \cdot x \uparrow y$

Nelson states that

> If we attempt to define “exponentiable number” in the same spirit, we are unable to prove that if $x_1$ and $x_2$ are exponentiable numbers then so is $x_1 \uparrow x_2$. There is a radical difference between addition and multiplication on the one hand and exponentiation, superexponentiation, and so forth, on the other hand. The obstacle is that exponentiation is not associative; for example, $(2 \uparrow 2) \uparrow 3 = 4 \uparrow 3 = 64$ whereas $2 \uparrow (2 \uparrow 3) = 2 \uparrow 8 = 256$. For any specific numeral SSS. . . 0 we can indeed prove that it is an exponentiable number, but we cannot prove that the world of exponentiable numbers is closed under exponentiation.

#### Categorification

The categorification of Nelson's argument would proceed as follows: Suppose we have the subcategory $CountSet \hookrightarrow FinSet$ of the category of finite sets whose cardinality is a counting number. (Here "finite" refers to the internal notion, not the external notion. Although our language only contains numbers, not sets, we can encode this category in $PA$ using a [[skeleton]] of $FinSet$ where objects are (internal) natural numbers and a morphism $m \to n$ is encoded as a natural number less than $n^m$.)

Then we can form a further subcategory $AddSet \hookrightarrow CountSet$, consisting of the objects $A$ such that the coproduct $A + C$ exists in $ContSet$ for all $C$ in $ContSet$. Although $CountSet$ may not have had binary coproducts, $AddSet$ has binary coproducts.

Next, we can form a further subcategory $MultSet \hookrightarrow CountSet$, consisting of the objects $M$ such that the product $M \times A$ exists in $AddSet$ for all $A$ in $AddSet$. Although $AddSet$ may not have had binary products, $MultSet$ has binary products.

Now, we can form a further subcategory $ExpSet \hookrightarrow MultSet$, consisting of the objects $E$ such that the exponent $M^E$ exists in $MultSet$ for all $M$ in $MultSet$ i.e. the [[exponentiable object]]s of $MultSet$. However, in constrast to the previous cases, we cannot prove that $ExpSet$ has exponents (hence can't prove that it's [[cartesian closed]], an [[elementary topos]], a [[Π-pretopos]], etc.).

### Ultrafinitism

The theory $NA$, or a categorification thereof, provides an option to study mathematics using ultrafinitism, by viewing the counting numbers to be the correct notion of natural numbers and the other numbers to be an idealised abstraction. (If one wants to work without these "idealised abstractions", induction would have to be substituted for weaker acceptable axioms.)

Categorifying the current axioms to the category of sets, the counting sets (represented by the sets $S$ in which $\mathcal{C}(S)$ is [[inhabited]]) could not be proven to be [[symmetric monoidal]] with respect to either the [[disjoint union]] or the [[Cartesian product]]. Thus, Nelson's [[CountSet]] forms a framework of mathematics, which is sometimes called "ultrafinite mathematics", that is significantly weaker than [[predicative mathematics|strongly predicative]] [[constructive mathematics|constructive]] [[finite mathematics]]. However, the [[subcategories]] of additionable finite sets $AddSet$ and multiplicable finite sets $MultSet$ can be proven to be symmetric monoidal.

## Models of Nelson arithmetic

A model of Nelson arithmetic is a model of Peano arithmetic $(N,0,S,+,\cdot)$, equipped with a subset $C \subset N$ such that $0 \in C$ and $a \in C$ implies $S(a) \in C$.

A model of Nelson arithmetic without induction is a [[set]] $N$ with an element $0 \in N$, a function $s:N \to N$, binary operations $(-)+(-):N \times N \to N$ and $(-)\cdot(-):N \times N \to N$, and a predicate $\mathcal{C}:N \to \mathrm{Prop}$, such that the axioms 1-6, 8, 9 above are all satisfied. Explicitly, this means:

* for all $x \in N$, $\neg s(x) = 0$

* for all $x, y \in N$, $s(x) = s(y)$ implies $x = y$

* for all $x \in N$, $x + 0 = x$

* for all $x, y \in N$, $x + s(y) = s(x+y)$

* for all $x \in N$, $x \cdot 0 = 0$

* for all $x, y \in N$, $x \cdot s(y) = x \cdot y + x$

* $\mathcal{C}(0)$

* for all $x \in N$, $\mathcal{C}(x)$ implies $\mathcal{C}(s(x))$;

A [[homomorphism]] of models of Nelson arithmetic without induction $N$ and $N^{'}$ is a function $f:N \to N^{'}$ such that

* $f(0_N) = 0_{N^{'}}$

* for all $x \in N$, $\mathcal{C}_N(x)$ implies $\mathcal{C}_{N^{'}}(f(x))$

* for all $x \in N$, $f(s_N(x)) = s_{N^{'}}(f(x))$

* for all $x, y \in N$, $f(x +_N y) = f(x) +_{N^{'}} f(y)$

* for all $x, y \in N$, $f(x \cdot_N y) = f(x) \cdot_{N^{'}} f(y)$

Homomorphisms of models of Nelson arithmetic with induction are defined in the same way, but as with $PA$, the notion of [[elementary embedding]]s may be more useful.

The [[category]] of models of Nelson arithmetic without induction in a [[universe]] $\mathcal{U}$ is the category $\mathrm{NelsonArithmetics}_\mathcal{U}$ whose objects $Ob(\mathrm{NelsonArithmetics}_\mathcal{U})$ are the $\mathcal{U}$-small models of Nelson arithmetic, and for any two objects $A \in Ob(\mathrm{NelsonArithmetics}_\mathcal{U})$ and $B \in Ob(\mathrm{NelsonArithmetics}_\mathcal{U})$ the [[morphisms]] $Mor_{\mathrm{NelsonArithmetics}_\mathcal{U}}(A, B)$ are the homomorphisms of models of Nelson arithmetic as defined above.

This category is an [[accessible category]]. If $\mathcal{U}$ satisfies the [[axiom of finiteness]] (meaning it thinks every set is finite), then (because there are no finite models of $NA-Ind$) the category $\mathrm{NelsonArithmetics}_\mathcal{U}$ is [[empty category|empty]].

### Example models

* The set of [[natural numbers]] $\mathbb{N}$ is the standard model of Nelson arithmetic, and is the initial model of Nelson arithmetic in the [[category]] of models of Nelson arithmetic.

* If $R$ is any [[rig]] such that $+1 : R \to R\setminus\{0\}$ is (total and) injective, then $R$ is a model of Nelson arithmetic without induction, where we may take $S(r) = r + 1$ and $\mathcal{C}$ to be any of: $R$, the image of the function $i : \mathbb{N} \to R$ defined by $i(0) = 0$, $i(n+1) = i(n)+1$, or any other subset $C$ with $0 \in C$ and $a \in C \Rightarrow a + 1 \in C$. Examples include:

- The set of non-negative [[rational numbers]] $\mathbb{Q}_{\geq 0}$ with $s(q) \coloneqq q + 1$ and $\mathcal{C}(q)$ for all $q \in \mathbb{Q}_{\geq 0}$ is a model of Nelson arithmetic without induction.

- The set of non-negative [[real numbers]] $\mathbb{R}_{\geq 0}$ with $s(r) \coloneqq r + 1$ and $\mathcal{C}(r)$ for all $r \in \mathbb{R}_{\geq 0}$ is a model of Nelson arithmetic without induction.

- The subset $P \subset \mathbb{Z}[x]$ of polynomials with leading coefficient nonnegative

- Any [[free object|free]] [[commutative algebra|commutative $\mathbb{N}$-algebra]] on a generating set $S$ with $s(r) \coloneqq r + 1$ and $\mathcal{C}(r)$ for all $r \in \mathbb{N}[S]$, is a model of Nelson arithmetic without induction.

* The [[disjoint union]] set $\mathbb{N} + \mathbb{1}$, where $*:\mathbb{1}$ is the unique element of the [[singleton]], extends the model $\mathbb{N}$ with:

  1. $s(*) = *$

  1. $\forall a \in \mathbb{N}. * + a = * = a + *$ and $* + * = *$

  1. $\forall a \in \mathbb{N}. * \cdot a = a$

  1. $\forall a \in \mathbb{N} + \mathbb{1}. a \cdot * = *$

  1. $\mathcal{C}(*)$

  This defines a model of Nelson arithmetic without induction. Note that multiplication is not commutative and that $0 \cdot * = *$, so this is not an instance of the previous examples.

* Given any nonstandard model $N$ of Peano arithmetic, and any subset $C \subset N$ with $0 \in C$ and $a \in C \Rightarrow a+1 \in C$, $N$ and $C$ together give a model of Nelson arithmetic.

## Groupoidal categorification

The [[groupoidal categorification]] of Nelson arithmetic without induction is a [[groupoid]] $\mathcal{G}$ with

* an object $0 \in \mathcal{G}$,

* a [[functor]] $S:\mathcal{G} \to \mathcal{G}$,

* a functor $(-)+(-):\mathcal{G} \times \mathcal{G} \to \mathcal{G}$

* a functor $(-)\cdot(-):\mathcal{G} \times \mathcal{G} \to \mathcal{G}$

* for all $A \in \mathcal{G}$, a function set $(S(A) \cong 0) \to \emptyset$ from the [[set]] of [[isomorphisms]] $S(A) \cong 0$ to the [[empty set]]

* for all $A, B \in \mathcal{G}$, a function set $(S(A) \cong S(B)) \to (A \cong B)$ from the set of isomorphism $S(A) \cong S(B)$ to the set of isomorphisms $A \cong B$

* for all $A \in \mathcal{G}$, a set of isomorphisms $A + 0 \cong A$

* for all $A, B \in \mathcal{G}$, a set of isomorphisms $A + S(B) \cong S(A+B)$

* for all $A \in \mathcal{G}$, a set of isomorphisms $A \cdot 0 \cong A$

* for all $A, B \in \mathcal{G}$, a set of isomorphisms $A \cdot S(B) \cong A \cdot B + A$

* a [[Set]]-valued functor $\mathcal{C}:\mathcal{G} \to \mathrm{Set}$

* an element $p \in \mathcal{C}(0)$

* for all $A \in \mathcal{G}$, a function $f_A:\mathcal{C}(A) \to \mathcal{C}(S(A))$

## See also

* [[Peano arithmetic]]
* [[finite mathematics]]

## References

* {#Nelson} [[Edward Nelson]], _Warning Signs of a Possible Collapse of Contemporary Mathematics_ ([pdf](https://web.math.princeton.edu/~nelson/papers/warn.pdf))


Nelson Niu

* [personal page](https://nelsonniu.com)

## Selected writing

On [[polynomial functors]] in view of [[categorical systems theory]]:

* {#SpivakPoly} [[David Spivak]], [[Nelson Niu]], _Polynomial Functors: A General Theory of Interaction_ [[webpage](https://topos.site/poly-course/), [pdf](https://topos.site/poly-book.pdf)]

category: people


Nengkun Yu

* [personal page](https://sites.google.com/site/nengkunyu/)

* [institute page](https://www.cs.stonybrook.edu/people/faculty/NengkunYu)

* [GoogleScholar page](https://scholar.google.com/citations?user=i3s2MPwAAAAJ&hl=en)

## Selected writings

Understanding the [[linear logic]]-sector of [[bunched logic]] as a kind of [[quantum logic]]:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

category: people


Nersés Aramyan

* [webpage](http://math.illinois.edu/~aramyan2/)

## related $n$Lab entries

* [[higher holonomy]]

* [[Adams spectral sequence]]

category: people

[[!redirects Nersés Aramian]]

[[!redirects Nerses Aramian]]
[[!redirects Nerses Aramyan]]


nerve

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#Contents#
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The _nerve_ is the right adjoint of a pair of [[adjoint functors]] that exists in many situations.  For the general abstract theory behind this see

* [[nerve and realization]].

## Idea ##

As soon as any [[locally small category]] $C$ comes equipped with a [[simplicial object|cosimplicial object]]

$$
  \Delta_C : \Delta \to C
$$

that we may think of as determining a [[geometric realization|realization]] of the standard $n$-[[simplex]] in $C$, we make every [[object]] of $C$ [[space and quantity|probeable]] by [[simplex|simplices]] in that there is now a way to find the set

$$
  N(A)_n := Hom_C(\Delta_C[n],A)
$$

of ways to map the $n$-[[simplex]] into a given object $A$.

These collections of sets evidently organize into a [[simplicial set]]

$$
  N(A) : \Delta^{op} \to Set
  \,.
$$

This [[simplicial set]] is called the _nerve_ of $A$ (with respect to the chosen [[geometric realization|realization]] of the standard simplices in $C$).  Typically the nerve defines a functor $N \colon C \to Set^{\Delta^op}$ that has a left adjoint $|\cdot| \colon Set^{\Delta^op} \to C$ called [[realization]].

There are many generalizations of this procedure, some of which are described below.

## Definition ##

> (notice that for the moment the following gives just one particular case of the more general notion of nerve)

Let $S$ be one of the categories of [[geometric shapes for higher structures]], such as the [[globe category]] $G$, the [[simplex category]] $\Delta$, the [[cube category]] $\Box$, the [[cycle category]] $\Lambda$ of Connes, or certain category $\Omega$ related to trees whose corresponding presheaves are [[dendroidal set|dendroidal sets]].

If $C$ is any [[locally small category|locally small]] category or, more generally, a $V$-[[enriched category]] equipped with a [[functor]]
$$
  i : S \to C
$$
we obtain a functor
$$
  N : C \to V^{S^{op}}
$$
from $C$ to [[globular sets]] or [[simplicial sets]] or [[cubical sets]], respectively, (or the corresponding $V$-objects) given on an [[object]] $c \in C$ by the [[restricted Yoneda embedding]]
$$
  N_i(c) : S^{op} \stackrel{i}\to C^{op}
    \stackrel{C(-,c)}{\to}
    V
  \,.
$$

This $N_i(c)$ is the **nerve** of $c$ with respect to the chosen $i : S \to C$. In other words, $N = i^* \circ Y$ where $Y: C \to [C^{op}, V]$ is the curried Hom functor; if $V=\mathsf{Sets}$ then $Y$ is the [[Yoneda embedding]].

Typically, one wants that $i$ is [[dense functor]], i.e. that every object $c$ of $C$ is canonically a colimit of a diagram of objects in $M$, more precisely,
$$
\mathrm{colim}((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C) = c,
$$
which is equivalent to the requirement that the corresponding nerve functor is [[full and faithful functor|fully faithful]] (in other words, if $i$ is inclusion then $S$ is a left adequate subcategory of $C$ in terminology of [Isbell 60](#Isbell60)).
The nerve functor may be viewed as a [[singular functor]] of the functor $i$.

## Examples ##

### Nerve of a 1-category
 {#NerveOfACategory}

For fixing notation, recall that the source and target maps of a small  [[category#OneCollectionOfMorphisms|category]] form a [[span]] in the category $Span(Set)$ where composition [[span#categories_of_spans|is given by a pullback]] (fiber product). The pairs of composable morphisms of a category are then obtained composing its source/target span with itself.

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

A _[[small category]]_ $\mathcal{C}_\bullet$ is

* a pair of [[sets]] $\mathcal{C}_0 \in Set $ (the set of [[objects]]) and $\mathcal{C}_1 \in Set$ (the set of [[morphisms]])

* equipped with [[functions]]

  $$
    \array{
      \mathcal{C}_1 \times_{\mathcal{C}_0} \mathcal{C}_1
      &\stackrel{\circ}{\to}&
      \mathcal{C}_1
      & \stackrel{\overset{t}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\to}}}&
      \mathcal{C}_0
    }\,,
  $$

  where the [[fiber product]] on the left is that over $\mathcal{C}_1 \stackrel{t}{\to} \mathcal{C}_0 \stackrel{s}{\leftarrow} \mathcal{C}_1$,

such that

* $i$ takes values in [[endomorphisms]];

  $$
    t \circ i = s \circ i =   id_{\mathcal{G}_0}, \;\;\;
  $$

* $\circ$ defines a partial [[composition]] operation which is [[associativity|associative]] and [[unitality|unital]] for $i(\mathcal{C}_0)$ the [[identities]]; in particular

  $s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$.

=--

#### Definition

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

For $\mathcal{C}_\bullet$ a [[small category]], def. \ref{SmallCategory},
its _simplicial nerve_ $N(\mathcal{C}_\bullet)_\bullet$ is
the [[simplicial set]] with

$$
  N(\mathcal{C}_\bullet)_n \coloneqq \mathcal{C}_1^{\times_{\mathcal{C}_0}^n}
$$

the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$;

with face maps

$$
  d_k \colon N(\mathcal{C}_\bullet)_{n+1} \to N(\mathcal{C}_\bullet)_{n}
$$

being

* for $n = 0$, $d_0= target:arr(\mathcal{C})\to ob(\mathcal{C})$, whilst $d_1$ is similarly the domain /  source function;

* for $n \geq 1$

  * the two outer face maps $d_0$ and $d_{n+1}$ are given by forgetting the first and the last morphism in such a sequence, respectively;

  * the $n$ inner face maps $d_{0 \lt k \lt n+1}$ are given by composing the $k$th morphism with the $k+1$st in the sequence.

The degeneracy maps

$$
  s_k \colon N(\mathcal{C}_\bullet)_{n} \to N(\mathcal{C}_\bullet)_{n+1}
  \,.
$$

are given by inserting an [[identity]] morphism on $x_k$.

=--

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

Spelling this out in more detail: write

$$
  \mathcal{C}_n =
  \left\{
    x_0
      \stackrel{f_{0,1}}{\to}
    x_1
      \stackrel{f_{1,2}}{\to}
    x_2
      \stackrel{f_{2,3}}{\to}
    \cdots
      \stackrel{f_{n-1,n}}{\to}
    x_n
  \right\}
$$

for the set of sequences of $n$ composable morphisms. Given any element of
this set and $0 \lt k \lt n $, write

$$
  f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}
$$

for the composition of the two morphism that share the $i$th vertex.

With this, face map $d_k$ acts simply by "removing the index $k$":

$$
  d_0
  \colon
  (x_0 \stackrel{f_{0,1}}{\to}  x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )
  \mapsto
  (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )
$$

$$
  d_{0\lt k \lt n}
   \colon
  (
    x_0
    \cdots
    \stackrel{}{\to}
    x_{k-1}
    \stackrel{f_{k-1,k}}{\to}
    x_k
    \stackrel{f_{k,k+1}}{\to}
    x_{k+1}
    \stackrel{}{\to}
    \cdots
    x_n
  )
  \mapsto
  (
    x_0
    \cdots
    \stackrel{}{\to}
    x_{k-1}
    \stackrel{f_{k-1,k+1}}{\to}
    x_{k+1}
    \stackrel{}{\to}
    \cdots
    x_n
  )
$$

$$
  d_n
  \colon
  (
    x_0 \stackrel{f_{0,1}}{\to}
    \cdots
    \stackrel{f_{n-2,n-1}}{\to}
    x_{n-1}
    \stackrel{f_{n-1,n}}{\to}
    x_n
  )
  \mapsto
  (
    x_0 \stackrel{f_{0,1}}{\to}
    \cdots
    \stackrel{f_{n-2,n-1}}{\to}
    x_{n-1}
  )
  \,.
$$

Similarly, writing

$$
  f_{k,k} \coloneqq id_{x_k}
$$

for the identity morphism on the object $x_k$,
then the degeneracy map acts by "repeating the $k$th index"

$$
  s_k
  \colon
  (
    x_0 \stackrel{}{\to}
    \cdots
    \to
    x_k
    \stackrel{f_{k,k+1}}{\to}
    x_{k+1}
    \to
    \cdots
  )
  \mapsto
  (
    x_0 \stackrel{}{\to}
    \cdots
    \to
    x_k
     \stackrel{f_{k,k}}{\to}
    x_k
    \stackrel{f_{k,k+1}}{\to}
    x_{k+1}
    \to
    \cdots
  )
  \,.
$$

This makes it manifest that these functions organise into a [[simplicial set]].

=--

More abstractly, this construction is described as follows. Recall that

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

The [[nLab:simplex category|simplex category]] $\Delta$ is equivalent to the [[full subcategory]]
$$
  i \colon \Delta \hookrightarrow Cat
$$
of [[Cat]] on non-empty finite [[linear orders]] regarded as categories, meaning that the object $[n] \in Obj(\Delta)$ may be identified with the category $[n] = \{0 \to 1 \to 2 \to \cdots \to n\}$. The morphisms of $\Delta$ are all functors between these total linear categories.

=--

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

For $\mathcal{C}$ a [[small category|small]] [[strict category]] its _nerve_ $N(\mathcal{C})$ is the [[simplicial set]] given by

$$
  N(\mathcal{C})
  \colon
  \Delta^{op}
     \hookrightarrow
     Cat^{op}
    \stackrel{Cat(-,\mathcal{C})}{\to}
    Set
  \,,
$$

where [[Cat]] is regarded as a [[1-category]] with objects locally small strict categories, and morphisms being [[functors]] between these.

=--

So the set $N(\mathcal{C})_n$ of $n$-[[simplices]] of the nerve is the set of functors $\{0 \to 1 \to \cdots \to n\} \to \mathcal{C}$. This is clearly the same as the set of sequences of composable morphisms in $\mathcal{C}$ of length $n$ obtained by iterated fiber product (as [above](#NerveOfACategory) for pairs of composables):

$$
  N(\mathcal{C})_n =
  \underbrace{
     Mor(\mathcal{C}) \times_{Obj(\mathcal{C})} Mor(\mathcal{C})
     \times_{Obj(\mathcal{C})} \cdots \times_{Obj(\mathcal{C})} Mor(\mathcal{C})
  }_{n \medspace factors}
$$

The collection of all functors between linear orders
$$
  \{
    0 \to 1 \to \cdots \to n
  \}
  \to
  \{
    0 \to 1 \to \cdots \to m
  \}
$$
is generated from those that map almost all generating morphisms $k \to k+1$ to another generating morphism, except at one position, where they

* map a single generating morphism to the composite of two generating morphisms
  $$
   \delta^n_i : [n-1] \to [n]
  $$
  $$
    \delta^n_i : ((i-1) \to i) \mapsto ((i-1) \to i \to (i+1))
  $$

* map one generating morphism to an identity morphism
  $$
   \sigma^n_i : [n+1] \to [n]
  $$
  $$
    \sigma^n_i : (i \to i+1) \mapsto Id_i
  $$

It follows that, for instance

* for $(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{f_2}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(D)_3$ the image under $d_1 := N(\mathcal{C})(\delta_1) : N(\mathcal{C})_3 \to N(\mathcal{C})_2$ is obtained by composing the first two morphisms in this sequence: $(d_0 \stackrel{f_2 \circ f_1}{\to} d_2, d_2 \stackrel{f_3}{\to} d_3) \in N(\mathcal{C})_2$

* for $(d_0 \stackrel{f_1}{\to} d_1) \in N(\mathcal{C})_1$ the image under $s_1 := N(\mathcal{C})(\sigma_1) : N(\mathcal{C})_1 \to N(\mathcal{C})_2$ is obtained by inserting an identity morphism:
$(d_0 \stackrel{f_1}{\to} d_1, d_1 \stackrel{Id_{d_1}}{\to} d_1) \in N(\mathcal{C})_2$.

In this way, generally the face and degeneracy maps of the nerve of a category come from composition of morphisms and from inserting identity morphisms.

In particular in light of their generalization to nerves of higher categories, discussed below, the cells in the nerve $N(\mathcal{C})$ have the following interpretation:

* $N(\mathcal{C})_0 = \{d | d \in Obj(\mathcal{C})\} $ is the collection of [[objects]] of $\mathcal{C}$;

* $N(\mathcal{C})_1 = Mor(\mathcal{C}) = \{d \stackrel{f}{\to} d' | f \in Mor(D)\}$ is the collection of [[morphisms]] of $D$;

* $N(\mathcal{C})_2 = \left\{
    \left.
    \array{
      && d_1
      \\
      & {}^{f_1}\nearrow &\Downarrow^{\exists !}& \searrow^{f_2}
      \\
      d_0 &&\stackrel{f_2 \circ f_1}{\to}&& d_2
    }
    \right|
    (f_1, f_2) \in Mor(D) {}_t \times_s Mor(D)
  \right\}$ is the collection of composable morphisms in $\mathcal{C}$ as in the diagram
  \begin{imagefromfile}
      "file_name": "2-simplex-1-cat-nerve.svg",
      "width": 150
  \end{imagefromfile}
  The 2-cell itself is to be read as the _composition operation_, which is unique for an  ordinary category (there is just one way to compose two morphisms);

* $N(\mathcal{C})_3 = \left\{
      \left.
        \array{
          d_1 &\stackrel{f_2}{\to}& d_2
          \\
          {}^{f_1}\uparrow &
            {}^{f_2 \circ f_1}\nearrow
          & \downarrow^{f_3}
          \\
          d_0 &\stackrel{f_3\circ (f_2\circ f_1)}{\to}&
          d_3
        }
        \;\;\;\;\;\stackrel{\exists !}{\Rightarrow}
        \;\;\;\;\;
        \array{
          d_1 &\stackrel{f_2}{\to}& d_2
          \\
          {}^{f_1}\uparrow &
            \searrow^{f_3\circ f_2}
          & \downarrow^{f_3}
          \\
          d_0 &\stackrel{(f_3\circ f_2) \circ f_1}{\to}&
          d_3
        }
      \right|
      (f_3,f_2, f_1) \in
      Mor(D) {}_t \times_s Mor(D) {}_t \times_s Mor(D)
    \right\}
  $ is the collection of triples of composable morphisms as in the diagram
  \begin{imagefromfile}
      "file_name": "3-simplex-1-cat-nerve.svg",
      "width": 450
  \end{imagefromfile}
  to be read as the unique associators that relate one way to compose three morphisms using the above 2-cells to the other way.

#### Examples

+-- {: .num_example}
###### Example
**(bar construction)**

Let $A$ be a [[monoid]] (for instance a [[group]]) with multiplication $m$, and write $\mathbf{B} A$ for the corresponding one-object [[category]] with $Mor(\mathbf{B} A) = A$. Then the nerve $N(\mathbf{B} A)$ of $\mathbf{B}A$ is the simplicial set which is given by a [[two-sided bar construction]] of $A$, namely $B(1, A, 1)$:
  $$
    N(\mathbf{B}A)
    =
    \left(
       \cdots
       A \times A \stackrel{\to}{\stackrel{\to}{\to}}
       A \stackrel{\to}{\to} {*}
    \right)
  $$
where for example the three parallel face maps on display are $\pi_1, m, \pi_2: A \times A \to A$.

  In particular, when $A = G$ is a discrete group, then the [[geometric realization]] $|N(\mathbf{B} G)|$ of the nerve of $\mathbf{B}G$ is the [[classifying space|classifying]] [[topological space]] $ \cdots \simeq B G$ for $G$-[[principal bundles]].

=--

#### Properties
 {#PropNerveCat}

The following lists some characteristic properties of simplicial sets that are nerves of categories.

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

A simplicial set is the nerve of a category precisely if it satisfies the [[Segal condition]].

=--

See at _[[Segal condition]]_ for more on this.

\begin{proposition}
\label{sSetIsNerveOfCategoryIffAllInnerHornsHaveUniqueFillers}
\linebreak
A [[simplicial set]] is the nerve of a [[small category]] precisely if all [[inner horn|*inner* horns]] have _unique_ fillers.
\end{proposition}

(e.g. [[Kerodon]], [Prop. 1.2.3.1](https://kerodon.net/tag/0031); see also at *[[inner fibration]]*.)

\begin{proposition}
\label{sSetIsNerveOfGroupoidIffAllHornsHaveUniqueFillers}
\linebreak
A [[simplicial set]] is the nerve of a [[groupoid]] precisely if _all_ [[horns]] of dimension $\gt 1$ have _unique_ fillers.
\end{proposition}

(cf. e.g. [[Kerodon]], [Prop. 1.2.4.2](https://kerodon.net/tag/0037))

Here the point as compared to the previous statements is that in particular all the outer horns have fillers for $n \gt 3$.

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

The nerve $N(C)$ of a [[small category]]
is a [[Kan complex]] precisely if $C$ is a [[groupoid]].

=--

The existence of [[inverse]] morphisms in $C$ corresponds to the fact that in the [[Kan complex]] $N(C)$ the "outer" [[horns]]

$$
  \array{
    && d_0
    \\
    & && \searrow^{f}
    \\
    d_1 &&\stackrel{Id_{d_1}}{\to} && d_1
  }
  \;\;\;
  \;\;\;
  and
  \;\;\;
  \;\;\;
  \array{
    && d_1
    \\
    & {}^f\nearrow &&
    \\
    d_0 &&\stackrel{Id_{d_0}}{\to} && d_1
  }
$$

have fillers

$$
  \array{
    && d_0
    \\
    & {}^{f^{-1}}\nearrow&& \searrow^{f}
    \\
    d_1 &&\stackrel{Id_{d_1}}{\to} && d_1
  }
  \;\;\;
  \;\;\;
  and
  \;\;\;
  \;\;\;
  \array{
    && d_1
    \\
    & {}^f\nearrow && \searrow^{f^{-1}}
    \\
    d_0 &&\stackrel{Id_{d_0}}{\to} && d_0
  }
$$

(even unique fillers, due to the above).

It suggests the sense that  a Kan complex  models an [[∞-groupoid]]. The possible lack of uniqueness of fillers in general gives the 'weakness' needed, whilst the lack of a [[coskeletal]] property requirement means that the homotopy type it represents has enough generality, not being constrained to be a 1-type.

\begin{proposition}
\label{NerveOfCategoriesIsFullyFaithfulFunctor}
The nerve functor
$
  N \colon Cat \longrightarrow SSet
$
(on [[small category|small]] [[strict categories]]) is a [[fully faithful functor]].
\end{proposition}

(e.g [[Kerodon]], [Prop. 1.2.2.1](https://kerodon.net/tag/002Z); [Rezk 2022, Prop. 4.10](#Rezk22))

So  [[functors]] between [[locally small category|locally small categories]] are in [[bijection]] with morphisms of [[simplicial sets]] between their nerves.

\begin{proposition}
\label{NerveOfCategoriesPreservesFiniteProducts}
The nerve functor
$
  N \colon Cat \longrightarrow SSet
$
(on [[small category|small]] [[strict categories]]) [[preserved limit|preserves]] [[finite products]], in that it sends:

1. the [[terminal category]] to the terminal simplicial set,

1. any [[product category]] to the [[product of simplicial sets]] of the nerves of the factors:

   $$
     N \;\colon\;
     \mathcal{C} \times \mathcal{D}
     \;\mapsto\;
     N(\mathcal{C})
       \times
     N(\mathcal{D})
   $$

\end{proposition}
\begin{proof}
  By direct inspection, using that the [[morphisms]] in a [[product category]] are just [[pairs]] of morphisms of the two factor categories.
\end{proof}

\begin{proposition}
\label{NerveOfCategoriesPreservesMappingObjects}
The nerve functor
$
  N \colon Cat \longrightarrow SSet
$
sends [[functor categories]] to the [[function complexes]] between the separate nerves:
$$
  N
  \big(
    Maps(\mathcal{X},\,\mathcal{A})
  \big)
  \;\simeq\;
  Maps
  \big(
    N(\mathcal{X})
    ,\,
    N(\mathcal{A})
  \big)
  \,.
$$

\end{proposition}
\begin{proof}
For $n \in \mathbb{N}$ we have the following sequence of [[natural isomorphisms]]:
$$
  \begin{array}{l}
  \Big(
  N
  \big(
    Maps(\mathcal{C}, \mathcal{D})
  \big)
  \Big)_n
  \\
  \;\simeq\;
  Hom_{Cat}
  \big(
    \mathcal{C} \times [n]
    ,\,
    \mathcal{D}
  \big)
  \\
  \;\simeq\;
  Hom_{sSet}
  \big(
    N(\mathcal{C} \times [n])
    ,\,
    N(\mathcal{D})
  \big)
  \\
  \;\simeq\;
  Hom_{sSet}
  \big(
    N(\mathcal{C}) \times N([n])
    ,\,
    N(\mathcal{D})
  \big)
  \\
  \;\simeq\;
  Hom_{sSet}
  \big(
    N(\mathcal{C}) \times \Delta[n]
    ,\,
    N(\mathcal{D})
  \big)
  \\
  \;\simeq\;
  Maps
  \big(
    N(\mathcal{C})
    ,\,
    N(\mathcal{D})
  \big)_n
  \end{array}
$$

Here

* the first step follows as discussed at *[[natural transformation]]* ([here](natural+transformation#InTermsOfCartMon));

* the second step follow by Prop. \ref{NerveOfCategoriesIsFullyFaithfulFunctor};

* the third step follows by Prop. \ref{NerveOfCategoriesPreservesFiniteProducts}.

\end{proof}

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

A [[simplicial set]] $S$ is the nerve of a locally small category $C$ precisely if it satisfies the [[Segal conditions]]: precisely if all the [[commuting squares]]

$$
  \array{
    S_{n+m}
    &
     \overset
       {\cdots \circ d_0 \circ d_0}
       {\longrightarrow}
    &
    S_m
    \\
    \mathllap{
      ^{
       \cdots d_{n+m-1}\circ d_{n+m}
      }
    }
    \big\downarrow && \big\downarrow
    \\
    S_n
    &\underset{d_0 \circ \cdots d_0}{\longrightarrow}&
    S_0
  }
$$

are [[pullback]] diagrams.

=--

Unwrapping this definition inductively in $(n+m)$, this says that a simplicial set is the nerve of a category if and only if all its cells in degree $\geq 2$ are unique compositors, associators, pentagonators, etc of composition of 1-morphisms. No non-trivial such structure cells appear and no further higher cells appear.

This characterization of categories in terms of nerves directly leads to the model of [[(∞,1)-category]] in terms of [[complete Segal spaces]] by replacing in the above discussion sets by [[topological spaces]] (or something similar, like [[Kan complexes]]) and pullbacks by [[homotopy pullback|homotopy pullbacks]].

\begin{proposition}
\label{NerveOfACategoryIsTwoCoskeletal}
The nerve $N(C)$ of a category is [[coskeleton|2-coskeletal]].
\end{proposition}
(e.g. [Duskin 1975, §0.18(b)](monadic+cohomology#Duskin75), [Joyal 2008, Cor. 1.2](quasi-category#Joyal08))

Hence in the nerve of a category, all [[horn]] inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$ have unique fillers for $n \gt 3$, and all [[boundary of a simplex|boundary]] inclusions $\partial \Delta[n] \hookrightarrow \Delta[n]$ have unique fillers for $n \geq 3$.

In summary:

\begin{example}
\label{CoskeletalityOfSimplicialNervesOfCategories}
**(coskeletality of simplicial nerves of categories)**
\linebreak
The [[simplicial nerve]] of a [[category]] (i.e. of a [[1-category]]) is a [[coskeleton|2-coskeletal]] [[simplicial set]] (Prop. \ref{NerveOfACategoryIsTwoCoskeletal}): The unique filler of the [[boundary of a simplex|boundary of an $n \geq 3$-simplex]] encodes the [[associativity]]-condition on [[n-tuple|$n$-tuples]] of composable morphisms.

Of course there is more to a category than its associativity condition, and hence the converse fails: Not every [[coskeleton|2-coskeletal]] [[simplicial set]] is the [[nerve of a category]]. For example the [[boundary of a simplex|boundary of the 2-simplex]], $\partial \Delta^2$, is 2-coskeletal but not the nerve of a category, since it is missing a [[composition]] of the edges $0 \to 1 \to 2$, namely it is missing a filler of this [[inner horn]].

In fact, a [[simplicial set]] is the [[nerve]] of a [[category]] iff it has *unique* [[inner horn|inner $n$-horn]]-fillers for $n \geq 2$ (Prop. \ref{sSetIsNerveOfCategoryIffAllInnerHornsHaveUniqueFillers}).
But [[coskeleton|2-coskeletality]] already implies that all $k \geq 4$-horns have unique filles (first uniquely fill the missing $k-1$-face then the interior $k$)-cell. Together this implies that:

A [[simplicial set]] is the [[nerve]] of a [[category]] iff

1. it is [[coskeleton|2-coskeletal]],

1. all *[[inner horn|inner]]* 2- and 3-[[horns]] have unique fillers (encoding [[composition]] and [[associativity]]).

Similarly for [[groupoids]] (byProp. \ref{sSetIsNerveOfGroupoidIffAllHornsHaveUniqueFillers}):

A [[simplicial set]] is the [[nerve]] of a [[groupoid]] iff

1. it is [[coskeleton|2-coskeletal]],

1. all 2- and 3-[[horns]] have unique fillers.

For better or worse, such a simplicial set has at times also been called a *[[1-hypergroupoid]]*, pointing to the fact that this is the first non-trivial stage in a pattern that recognizes $n+1$-coskeletal [[Kan complexes]] with unique horn fillers as models for [[n-groupoids|$n$-groupoids]]

Notice that a [[Kan complex]] which is [[coskeleton|2-coskeletal]] but with possibly non-unique 2-horn fillers is still a [[homotopy 1-type]] and may still be called a *[[1-groupoid]]* in the sense of [[homotopy theory]], but need not be the [[nerve]] of a [[groupoid]]. It may be thought of as a [[bigroupoid]] (2-[[hypergroupoid]]) which happens to be just a [[homotopy 1-type]].
\end{example}

\linebreak

### Nerve of a 2-category ###

For [[2-categories]] modeled as [[bicategories]] the nerve operation is called the [[Duskin nerve]].

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

A simplicial set is the [[Duskin nerve]] of a [[bigroupoid]] precisely if it is a 2-[[hypergroupoid]]: a [[Kan complex]] such that the horn fillers in dimension $\geq 3$ are _unique_.

=--

This is theorem 8.6 in ([Duskin](http://www.tac.mta.ca/tac/volumes/9/n10/9-10abs.html))

For a [[2-category]], regarded as a [[Cat]]-[[internal category]] one can apply the nerve operation for categories in stages, to obtain the [[double nerve]].

### Nerve of a 3-category ###

One also has a nerve operation for [[3-categories]] modeled as [[tricategories]]: the [[Street nerve]].

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

A simplicial set is the [[Street nerve]] of a [[trigroupoid]] precisely if it is a 3-[[hypergroupoid]]: a [[Kan complex]] such that the horn fillers in dimension $\geq 4$ are _unique_.

=--

This is the main result of ([Carrasco, 2014](#Carrasco2014)).

### Nerve of an $\omega$-category ###

* For [[strict omega-category|strict omega-categories]] there is a nerve induced by the [[orientals]]; see [[omega-nerve]].

### Nerve of chain complexes ###

Let $Ch_+$ be the [[category of chain complexes]]
of abelian groups, then there is a [[simplicial object|cosimplicial chain complex]]

$$
  C_\bullet : \Delta \to Ch_+
$$

given by sending the standard $n$-simplex $\Delta[n]$ first to the free [[simplicial group]] $F(\Delta[n])$ over it and then that to the normalized [[Moore complex]]. This is a small version of the ordinary [[homology]] [[chain complex]] of the standard $n$-[[simplex]].

The nerve induced by this cosimplicial object was first considered in

* D. Kan, _Functors involving c.s.s complexes_, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330--346 ([jstor](http://www.jstor.org/stable/1993103))

The nerve/[[geometric realization|realization]] adjunction induced from this is the [[Dold?Kan correspondence]]. See there for more details.

## Remarks ##

###Geometric realization###
Often the operation of taking the nerve of a (higher) category is followed by forming the [[geometric realization]] of the corresponding cellular set.

###Nerves and higher categories###
For many purposes it is convenient to conceive categories and especially [[∞-categories]] entirely in terms of their nerves: those simplicial sets that arise as certain nerves are usually characterized by certain properties. So one can turn this around and _define_ an [[∞-category]] as a simplicial set with certain properties. This is the strategy of a [[geometric definition of higher category]]. Examples for this are [[complicial set|complicial sets]], [[Kan complex|Kan complexes]], [[quasi-category|quasi-categories]], [[simplicial T-complex|simplicial T-complexes]],...

###Internal nerve###
A variant of the nerve construction can also be applied _internally_ within a category, to any internal category, see the discussion at [[internal category]].

###Direct categories versus finite simplicial sets###
If a [[direct category]] has finitely many objects then its nerve  is a finite [[simplicial set]]. Conversely, if a finite simplicial set is the nerve of a category then the category is a direct category with finitely many objects.

## Properties

### (Non-)Preservation of colimits
 {#PreservationOfColimits}

While the nerve operation is a [[right adjoint]] ([this Prop.](nerve+and+realization#NerveAndRealizationAreAdjoint)) and [[right adjoints preserve limits|hence]] [[preserved limit|preserves]] all [[limits]], the nerve operation does not [[preserved colimit|preserve]] all [[colimits]] (Exp. \ref{NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup}), [[left adjoints preserve colimits|hence]] is not a [[left adjoint]].

However, it does preserve *some* colimits (Exp. \ref{NervePreservesLeftQuotientsOnRightActionGroupoids}); rather special ones, but of central importance in the theory of [[classifying spaces]] constructed via [[geometric realization of simplicial topological spaces]] (Exp. \ref{NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction}).

(In the following Exp. \ref{NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup} we use "card" instead of the more common notation "${\vert - \vert}$" for [[cardinality]] (of [[underlying]] [[sets]]) in order not to clash with the notation for [[geometric realization]], even if the latter is not directly involved in the following examples.)

\begin{example}\label{NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup}
**(Nerve does not preserve quotients of delooping groupoids by normal subgroups)** \linebreak
  Let $H \hookrightarrow G \twoheadrightarrow G/N$ be the inclusion of a [[trivial group|non-trivial]] [[normal subgroup]] $H$ of a [[finite group]] $G$, with its [[quotient group]] denoted $G/H$.

Then the [[cardinalities]] of the $n$th component sets of the [[nerves]]

$$
  N \;\colon\; Grpd \longrightarrow sSet
$$

of their [[delooping groupoids]]

$$
  \mathbf{B}(-)
  \;\colon\;
  Grp \longrightarrow Grpd
$$

satisfy, from degree $n \geq 2$ on, an [[inequality relation]]:

$$
  n \geq 2
  \;\;\;\;\;\;\;
    \Rightarrow
  \;\;\;\;\;\;\;
  card
  \Big(
    \big( N \mathbf{B} G \big)_n / H
  \Big)
  \;=\;
  \frac{
    card(G)^n
  }
  {
    card(H)
  }
  \;\;
    \gt
  \;\;
  \frac{
    card(G)^n
  }
  {
    card(H)^n
  }
  \;=\;
  card
  \Big(
    \big( N \mathbf{B} (G/H) \big)_n
  \Big)
  \,.
$$

But this means that it is impossible for there to be an [[isomorphism]] (namely a degree-wise [[bijection]]) from $N(\mathbf{B}G)/H$ to $N\big(\mathbf{B}(G/H)\big)$, and hence that it is impossible for the nerve operation to preserve the [[colimit]] which is the [[quotient]] by the $H$-[[group action|action]].
\end{example}

\begin{example}\label{NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction}
**(nerve does preserve canonical quotients of chaotic groupoids of groups)**
\linebreak
For $G \,\in\, Grp(Set)$ a ([[discrete group|discrete]]) [[group]], write

* $\mathbf{B}G \;\coloneqq\; \big( G \rightrightarrows \ast\big)$ for its [[delooping groupoid]];

* $\mathbf{E}G \;\coloneqq\; \big( G \times G \rightrightarrows G  \big)$
  for its [[pair groupoid]] equipped with the usual left $G$-[[group action|action]] (discussed [there](codiscrete+groupoid#UniversalGPrincipalBundle)),

so that the [[quotient]] [[coprojection]] of this action is

\[
  \label{QuotientCoprojectionFromPairGroupoidOfGroupToDeloopingGroupoid}
  \mathbf{E}G \xrightarrow{\;\;} (\mathbf{E}G)/G \;=\; \mathbf{B}G
  \,.
\]

Noticing that the [[nerve]] of $\mathbf{E}G$ (which is the [[universal principal simplicial complex]] $N(\mathbf{E}G) \,=\, W G$) has component sets

$$
  N(\mathbf{E}G)_n
  \;=\;
  \big\{
    (g_n, g_{n-1}, \cdots, g_0)
    \;\in\;
    G^{\times_{n+1}}
  \big\}
$$

with the $G$ action given degreewise by left-multiplication on *just the leftmost factor* (see also [this exp.](simplicial+classifying+space#SimplicialClassifyingSpaceOfAnOrdinaryGroup)), we have

$$
 \big(  N(\mathbf{E}G)_n \big)/G
  \;\simeq\;
  \big\{
    (g_{n-1}, \cdots, g_0)
    \;\in\;
    G^{\times_{n+}}
  \big\}
  \;=\;
  N(\mathbf{B}G)_n
$$

and hence here the nerve operation does preserve the [[quotient]] [[coprojection]] (eq:QuotientCoprojectionFromPairGroupoidOfGroupToDeloopingGroupoid):

$$
  W G
  \;=\;
  N(\mathbf{E}G)
    \xrightarrow{\;\;}
  \big(N(\mathbf{E}G)\big)/G
    \simeq
  N\big(
    (\mathbf{E}G)/G
  \big)
  \;=\;
  N\big(
    \mathbf{B}G
  \big)
  \;=\;
  \overline{W} G
  \,.
$$

The result is the [[universal simplicial principal bundle]] of $G \,\in\, Grp(Set) \xhookrightarrow{Grp(Disc)} Grp(sSet)$ regarded as a [[simplicial group]].

\end{example}

\begin{remark}
  The joint relevance of
  Exp. \ref{NervesDoNotPreserveQuotientOfDeloopingByNormalSubgroup} and
  Exp. \ref{NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction}
  has been highlighted in [Guillou, May & Merling 2017 ](equivariant+bundle#GuillouMayMerling17)
(corresponding there to Exp. 2.9 and Lem. 2.10 -- but Exp. 2.9 seems a little broken (?) while Lem. 2.10 does not quite get around to discussing the quotienting, for which it seems to be quoted later on).
\end{remark}

The principle behind Exp. \ref{NerveDoesPreserveQuotientOfPairGroupoidOfGroupByGroupAction} is readily seen to be, more generally, the following:

\begin{example}
  \label{NervePreservesLeftQuotientsOnRightActionGroupoids}
**(nerve preserves left quotients of right action groupoids)**
\linebreak
  For $G_L, G_R \,\in\, Grp(Set)$ a [[pair]] of [[groups]], let $X \in (G_L \times G^{op}_R) Act(Set)$ be a [[set]] equipped with a [[group action|left action]] of $G_L$ and a commuting right action of $G_R$.

Then the [[action groupoid]] of the right $G_R$-action inherits the residual $G_L$-action

$$
  \big(
    X \times G_R
    \underoverset
      {(-)\cdot(-)}
      {pr_1}
      {\rightrightarrows}
    X
  \big)
  \;\;
    \in
  \;\;
  G_L Act\big( Grpd \big)
$$

and the [[quotient]] by this left action is preserved by the [[nerve]] operation:

$$
  \Big(
    N
    \big(
      X \times G_R
      \underoverset
        {(-)\cdot(-)}
        {pr_1}
        {\rightrightarrows}
      X
    \big)
  \Big)
  /G_L
  \;\simeq\;
  N
  \Big(
    \big(
    X \times G_R
    \underoverset
      {(-)\cdot(-)}
      {pr_1}
      {\rightrightarrows}
    X
    \big)/ G_L
  \Big)
  \,.
$$

\end{example}

## Related concepts

* [[monad with arities]]

* [[Duskin nerve]]

* [[Street nerve]]

* [[∞-nerve]]

* [[homotopy coherent nerve]]

* [[dg-nerve]]

## References

### For covers

The notion of the nerve of a [[cover]] (in modern parlance: of its [[Cech groupoid]]) appears in:

* [[Paul Alexandroff]], Section 9 of: _Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung_, Mathematische Annalen 98 (1928), 617–635 ([doi:10.1007/BF01451612](https://doi.org/10.1007/BF01451612)).

### For categories

The notion of the nerve of a general category already appears in

* {#Grothendieck61} [[Alexander Grothendieck]], above Proposition 4.1 of: _Techniques de construction et théorèmes d'existence en géométrie algébrique III : préschémas quotients_, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf))

Another early appearance in print is:

* [[Graeme Segal]], Section 2 of: _Classifying spaces and spectral sequences_,  Publications Mathématiques de l'IHÉS, Volume 34  (1968), p. 105-112  ([numdam:PMIHES_1968__34__105_0](http://www.numdam.org/item/PMIHES_1968__34__105_0/))

  > (in the context of constructing [[classifying spaces]] for [[principal bundles]] in [[algebraic topology]])

Review and exposition:

* [[Saunders MacLane]], §XII.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Kerodon]], *The Nerve of a Category* ([002L](https://kerodon.net/tag/002L))

* [[Kerodon]], *The Nerve of a Groupoid* ([0035](https://kerodon.net/tag/0035))

* {#Rezk22} [[Charles Rezk]], Part 1 of: *Introduction to quasicategories* (2022) [[pdf](https://faculty.math.illinois.edu/~rezk/quasicats.pdf), [[Rezk-IntroToQuasicategories.pdf:file]]]

See also:

* [[Tom Leinster]], p. 117 onwards in: _Higher operads, higher categories_ , London Mathematical Society Lecture Note Series, 298. Cambridge Univ. Press 2004. xiv+433 pp. ISBN: 0-521-53215-9 ([arXiv:math.CT/0305049](https://arxiv.org/abs/math/0305049))

* [[Tom Leinster]], *[How I learned to love the nerve construction](https://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html)*, $n$-Category Café, January 6, 2008.

  > (an explanation of how the simplex category and the nerve construction arise canonically from the free category monad)

See also:

* {#Isbell60} [[John Isbell]], *Adequate subcategories*, Illinois J. Math. 4, 541--552 (1960) ([doi:10.1215/ijm/1255456274](https://www.projecteuclid.org/journals/illinois-journal-of-mathematics/volume-4/issue-4/Adequate-subcategories/10.1215/ijm/1255456274.full))

* [[W. G. Dwyer]], [[D. M. Kan]], Singular functors and realization functors, Nederl. Akad. Wetensch. Indag. Math. 46 (1984), no. 2, 147--153. [pdf](http://www.nd.edu/~wgd/Dvi/SingularAndRealization.pdf)

### For higher categories

For [[strict omega-categories]]:

* [[Ross Street]], _The algebra of oriented simplexes_, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 ([pdf](http://www.math.mq.edu.au/~street/aos.pdf), <a href="https://doi.org/10.1016/0022-4049(87)90137-X">doi:10.1016/0022-4049(87)90137-X</a>).

For [[2-categories]] and [[bicategories]]:

* {#BullejosCegarra03} [[Manuel Bullejos]], [[Antonio M. Cegarra]],  *On the geometry of 2-categories and their classifying spaces*, K-Theory **29** 3 (2003) 211-229  [[doi:10.1023/B:KTHE.0000006921.50151.00](http://dx.doi.org/10.1023/B:KTHE.0000006921.50151.00), [pdf](http://www.ugr.es/\%7Ebullejos/geometryampl.pdf)]

* [[Stephen Lack]], [[Simona Paoli]], _2-nerves for bicategories_, K-Theory  **38** (2008) [[arXiv:math/0607271](https://arxiv.org/abs/math/0607271), [doi:10.1007/s10977-007-9013-2](http://dx.doi.org/10.1007/s10977-007-9013-2)]

* [[Paul Bressler]], Alexander Gorokhovsky, [[Ryszard Nest]], [[Boris Tsygan]], _Formality for algebroids I: Nerves of two-groupoids_ [[arxiv:1211.6603v3](https://arxiv.org/abs/1211.6603v3)]

For [[3-categories]]:

* {#Carrasco2014} Pilar Carrasco, _Nerves of Trigroupoids as Duskin-Glenn’s $3$-Hypergroupoids_, Applied Categorical Structures 23.5 (2015): 673-707 ([doi:10.1007/s10485-014-9374-7](https://doi.org/10.1007/s10485-014-9374-7))

[[!redirects nerves]]
[[!redirects nerve functor]]
[[!redirects nerve functors]]
[[!redirects simplicial nerve]]
[[!redirects simplicial nerves]]
[[!redirects simplicial nerve functor]]
[[!redirects simplicial nerve functors]]

[[!redirects nerve of a category]]
[[!redirects nerves of a categories]]


nerve and realization

[[!redirects realization and nerve]]

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

#Contents#
* table of contents
{:toc}

## Idea

Under rather general conditions a [[functor]]

$$
  S_C \;\colon\; S \to C
$$

into a [[cocomplete category]] $C$ (possibly a $V$-[[enriched category]] with $V$ some [[complete category|complete]] [[symmetric monoidal category]]) induces a pair of [[adjoint functors]]
$$
   C \stackrel{\xleftarrow{|-|}}{\underset{N}{\rightarrow}} [S^{op}, V],
$$
where $|-| \dashv N$, between $C$ and the [[category of presheaves]] $PSh(S) = [S^{op}, V]$ on $S$ (here $V$ = [[Set]] for the unenriched case)

where

* $N$ behaves like a [[nerve]] operation;

* $|-|$ behaves like [[geometric realization]].

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

Here "$S$" is supposed to be suggestive of a category of certain "[[geometric shapes for higher structures|geometric Shapes]]".  The canonical example is $S = \Delta$, the [[simplex category]], and the reader may find it helpful to keep that example in mind.
=--

## Definition

We place ourselves in the context of $V$-[[enriched category theory]]. The reader wishing to stick to the ordinary notions in [[locally small category|locally small categories]] takes $V$= [[Set]].

The _realization_ operation is the left [[Kan extension]] of $S_C : S \to C$ along the [[Yoneda embedding]] $S \hookrightarrow [S^{op},V]$ (i.e. the [[Yoneda extension]]):

$$
  \array{
    S &\stackrel{S_C}{\to}&& C
    \\
    \downarrow^{Y} &\Downarrow& \nearrow_{|-|}
    \\
    [S^{op},V]
  }
  \,.
$$

If we assume that $C$ is [[copower|tensored]] over $V$, then by the general [[coend]] formula for left [[Kan extension]] we find that for $X \in [S^{op}, V]$ we have

$$
  |X| \simeq
   \int^{s \in S}
   S_C(s) \cdot X_s
  \,.
$$

For instance when $S = \Delta$ is the [[simplex category]] this reads more recognizably

$$
  |X| \simeq
   \int^{[n] \in \Delta}
   \Delta_C[n] \cdot X_n
  \,.
$$

The corresponding [[nerve]] operation

$$
  N :
  C \stackrel{}{\to} [S^{op},V]
$$

is given by the [[restricted Yoneda embedding]]

$$
  N(c) : S^{op} \stackrel{S_C^{op}}{\to} C^{op}
  \stackrel{C(-,c)}{\to}
  V
  \,.
$$

\begin{theorem}\label{NerveAndRealizationAreAdjoint}
Nerve and realization are a pair of [[adjoint functors]]

$$
    {|-|} \,\dashv\, N
$$

with $N$ the [[right adjoint]].

\end{theorem}
([Kan 1958, Sec. 3](#Kan58))
+-- {: .proof}
###### Proof

Using the fact that the Hom in its first argument sends [[coends]] to [[ends]]
and then using the definition of [[copower|tensoring]]
over $V$, we check the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism):

$$
  \begin{aligned}
    Hom_C(|X|, c)
    &
    \coloneqq
    Hom_C( \int^{s} S_C(s) \cdot X_s, c)
    \\
    & \simeq
    \int_{s} Hom_C( S_C(s) \cdot X_s, c)
    \\
    & \simeq
    \int_{s} Hom_V( X_s , C(S_C(s), c))
    \\
    & =
    \int_{s} Hom_V( X_s , N(c)_s)
    \\
    & \simeq
    Hom_{[S^{op},V]}(X, N(c))
    \,,
  \end{aligned}
$$

where in the last step we used the definition of the
[[enriched functor category]] in terms of an [[end]].

=--

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

In many cases we have $V = $ [[Set]] and the [[copower|tensoring]] of an object $c$ over a set $I$ is given by coproducts as
  $$
    c \cdot I = \coprod_{i \in I} c
    \,.
  $$
  This is the case for instance for the below examples of realization of [[simplicial sets]], nerves of categories and the [[Dold-Kan correspondence]].

=--

## Examples

### Topological realization of simplicial sets
 {#TopologicalRealizationOfSimplicialSets}

A classical example is given by the [[simplicial topological space|cosimplicial topological space]]

$$
  \Delta_{Top} : \Delta \to Top
$$

that sends the abstract $n$-[[simplex]] $[n]$ to the standard topological $n$-simplex $\Delta_{Top}[n] \subset \mathbb{R}^n$.

* The corresponding realization is what is traditionally called _[[geometric realization]]_ of simplicial sets.

  By restricting this to simplicial sets which are themselves simplicial nerves of [[categories]] (see [below](NervesOfCategories)) or more generally are [[quasi-categories]], this also induces the notion of geometric realization of categorical structures.

  The construction generalizes also to a notion of _[[geometric realization of simplicial topological spaces]]_.

* The corresponding [[nerve]] is the [[singular simplicial complex]] functor, producing the [[fundamental ∞-groupoid]] of a topological space.

This topological nerve and realization adjunction plays a central role as a presentation of the [[Quillen equivalence]] between the [[model structure on simplicial sets]] and the [[model structure on topological spaces]]. This is discussed in detail at _[[homotopy hypothesis]]_.

### Nerve and realization of categories
 {#NervesOfCategories}

Pretty much every notion of [[category]] and [[higher category theory|higher category]] comes, or should come, with its canonical notion of simplicial nerve, induced from a functor
  $$
    \Delta_C : \Delta \to n Cat
  $$

  that sends the standard $n$-[[simplex]] to something like the free $n$-category on the $n$-[[directed n-graph|directed graph]] underlying that simplex.

For ordinary categories see the discussion at _[[nerve]]_ and at _[[geometric realization of categories]]_.

One formalization of this for $n = \infty$ in the context of [[strict ∞-categories]] is the cosimplicial $\omega$-category called the [[orientals]]
  $$
    \Delta_{\omega} : \Delta \to \omega Cat
    \,.
  $$

* The induced [[nerve]] is the [[∞-nerve]].

* The induced realization operation is the operation of forming the free $\omega$-category on a simplicial set. See [[oriental]] for more details.

### Dold--Kan correspondence

The [[Dold-Kan correspondence]] is the nerve/realization adjunction for the [[homology]] functor
  $$
    \Delta_{C_\bullet} : \Delta \to Ch_+
  $$
  to the [[category of chain complexes]] of abelian groups, which sends the standard $n$-[[simplex]] to its [[homology]] chain complex, more precisely to its normalized [[Moore complex]].

* The induced realization is the normalized [[Moore complex]] functor extended from $\Delta$ to all [[simplicial sets]].

### Higher Lie integration / smooth Sullivan construction

see at _[[Lie integration]]_ [this Prop.](Lie+integration#LieIntegrationIsRightQuillenFunctor)

### Simplicial models for $(\infty,1)$-categories

The canonical cosimplicial [[simplicially enriched category]]

$$
  \Delta \to SSet\text{-}Cat
$$

induces the [[homotopy coherent nerve]] of [[SSet]]-[[enriched category|enriched categories]] and establishes the relation between the [[quasi-category]]
and the simplicially enriched model for [[(infinity,1)-category|(infinity,1)-categories]]. See

* [[relation between quasi-categories and simplicial categories]].

## Properties

### Full and faithfulness
 {#FullAndFaithfulness}

When the nerve functor $N$ is a [[full and faithful functor]], the functor $S_C : S \to C$ is said to be a [[dense functor]].

See also at *[[monad with arities]]*.

## Related concepts

* [[nerve]], [[monad with arities]]

* [[geometric realization]]

* [[totalization]]

## References

The notion of nerve and realization (not with these names yet) was introduced and proven to be an [[adjunction]] in section 3 of

* {#Kan58} [[Daniel Kan]], _Functors involving c.s.s complexes_, Transactions of the American Mathematical Society, Vol. 87, No. 2 (Mar., 1958), pp. 330--346 ([jstor:1993103](http://www.jstor.org/stable/1993103)).

In fact, in that very article apparently what is now called [[Kan extension]] is first discussed.

Also, in that article, as an example of the general mechanism, also the [[Dold-Kan correspondence]] was found and discussed, independently of the work by Dold and Puppe shortly before, who used a much less general-nonsense approach.

In an article in 1984, Dwyer and Kan look at this 'nerve-realization' context from a different viewpoint, using the term 'singular functor' where the above has used 'nerve'. Their motivation example is that in which $S$ is the [[orbit category]] of a group $G$, and the realisation starts with a functor on that category with values in spaces and returns a $G$-space:

* [[W. G. Dwyer]] and [[D. M. Kan]], _Singular functors and realization functors_, Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 – 153.

We should also mention the treatment in Leinster's book and the relation to the notions of [[dense subcategory]] or adequate subcategory in the sense of Isbell.

In a [blog post](https://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html)  on the n-Category Café, Tom Leinster illustrates that "sections of a bundle" is a nerve operation, and its corresponding geometric realization is the construction of the étalé space of a presheaf.

[[!redirects realization]]
[[!redirects realizations]]


nerve theorem

> This page is about a property of [[Cech nerves]] in [[homotopy theory]]. For the "nerve theorem" in [[category theory]] see at _[[Segal conditions]]_. For the "nerve theorem" for [[monads with arities]] see [there](monad+with+arities#NerveTheorem).

***

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

#Contents#
* table of contents
{:toc}

## Idea

The **nerve theorem** asserts that the [[homotopy type]] of a sufficiently nice [[topological space]] is encoded in the [Čech nerve](Čech+nerve#FromACover) of a [[good open cover]] (as used in [[Čech homology]]).

This can be seen as a special case of some aspects of [[étale homotopy]] as the étale homotopy type of nice spaces coincides with the homotopy type of its Cech nerve.

## Statement

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

Let $X$ be a [[paracompact space]] and $\{U_i \to X\}$ a [[good open cover]]. Write $C(\{U_i\})$ for the [[Cech nerve]] of this cover

$$
  C(\{U_i\}) = \left(
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} U_i \cap U_j \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} U_i
  \right)
  \,,
$$

(a [[simplicial object|simplicial]] [[space]]) and write

$$
  \tilde C(\{U_i\}) = \left(
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}\coprod_{i,j} * \stackrel{\longrightarrow}{\longrightarrow} \coprod_{i} *
  \right)
$$

for the [[simplicial set]] obtained by replacing in $C(\{U_i\})$ each [[direct sum]]mand space by the [[point]]. Let $|\tilde C(\{U_i\})|$ be the [[geometric realization]].

This is [[homotopy equivalence|homotopy equivalent]] to $X$.

=--

The proof relies on the existence of [[partitions of unity]].

This is usually attributed to ([Borsuk 1948](#Borsuk48)). It appears more explicitly as [Weil 52, p. 141](#Weil52) [McCord 67, Thm. 2](#McCord67), review in [Hatcher, prop. 4G.3](#Hatcher).

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

This statement implies that in the [[cohesive (∞,1)-topos]] [[ETop∞Grpd]] the intrinsic [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] coincides with the ordinary [[fundamental ∞-groupoid]] functor of [[paracompact topological spaces]]. See Euclidean-topological ∞-groupoid : Geometric homotopy for details.

=--

## Related concepts

* [[Čech homology]]

## References

Original references:

* {#Borsuk48} [[Karol Borsuk]], _On the imbedding of systems of compacta in simplicial complexes_ , Fund. Math. 35, (1948) 217–234 ([dml:213158](https://eudml.org/doc/213158))

* [[Jean Leray]], _L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue_, J. Math. Pures Appl. (9) 29 (1950), 1–139.

* {#Weil52} [[André Weil]], §5. in: _Sur les theoremes de de Rham_, Comment. Math. Helv. 26 (1952), 119–145 ([dml:139040](https://eudml.org/doc/139040))

* {#McCord67} [[Michael C. McCord]], _Homotopy type comparison of a space with complexes associated with its  open covers_, Proc. Amer. Math. Soc. 18 (1967), 705–708 ([jstor:2035443](https://www.jstor.org/stable/2035443))

* [[Graeme Segal]],  §4 in: _Classifying spaces and spectral sequences_, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.

* [[Armand Borel]] and [[Jean-Pierre Serre]], Theorem 8.2.1. in: _Corners and arithmetic groups_,
Comment. Math. Helv. 48 (1973), 436–491.

A version for hypercovers is discussed in

* [[Daniel Dugger]], [[Daniel C. Isaksen]], _Topological hypercovers and $\mathbb{A}^1$-realizations_, Math. Z. 246 (2004), no. 4, 667–689

A review appears as corollary 4G.3 in the textbook

* {#Hatcher} [[Allen Hatcher]], _Algebraic topology_ ([web](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)) .

Some slightly stronger statements are discussed in

* Anders Björner, _Nerves, fibers and homotopy groups_, Journal of combinatorial theory, series A, 102 (2003), 88-93

* Andrzej Nagórko, _Carrier and nerve theorems in the extension theory_, Proc. Amer. Math. Soc.  135  (2007), 551-558.  ([web](http://www.ams.org/journals/proc/2007-135-02/S0002-9939-06-08477-2/home.html))

A nerve theorem for categories:

* Kohei Tanaka, _Cech complexes for covers of small categories_, Homology, Homotopy and Applications 19(1), (2017), pp. 281-291. [arXiv:1508.03688](https://arxiv.org/abs/1508.03688)

[[!redirects nerve theorems]]

[[!redirects Borsuk's nerve theorem]]


Nesta van der Schaaf

* [GoogleScholar page](https://scholar.google.nl/citations?user=cikMSzMAAAAJ&hl=en)

## Selected writings

On [[axiom|axiomatic]] characterization of [[categories]] of [[Hilbert spaces]]:

* [[Chris Heunen]], [[Andre Kornell]], [[Nesta van der Schaaf]], *Axioms for the category of Hilbert spaces and linear contractions* [[arXiv:2211.02688](https://arxiv.org/abs/2211.02688)]

category: people


net

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

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

A _net_ in a [[set]] $X$ is a [[function]] from a [[directed set]] $D$ to $X$. Special cases of nets are [[sequences]], for which $D = \mathbb{N}_{\leq}$ is the [[natural numbers]]. Regarded as a generalization of sequences, nets are used in [[topology]] for formalization of the concept of [[convergence]].

Nets are also called _Moore--Smith sequences_ and are equivalent (in a certain sense) to [[proper filters]] (def. \ref{Filter} below), their _[[eventuality filters]]_ (def. \ref{EventualityFilter} below).

The concept of nets is motivated from the fact that where plain [[sequences]] detect [[topology|topological]] properties in [[metric spaces]], in generally they fail to do so in more general [[topological spaces]].
For example [[sequentially compact metric spaces are equivalently compact metric spaces]], but for general [[topological spaces]] being [[sequentially compact space|sequentially compact]] neither implies nor is implied by being [[compact space|compact]] (see at _[[sequentially compact space]]_ [Examples and counter-examples](sequentially+compact+topological+space#Examples)).

Inspection of these counter-examples reveals that the problem is that sequences indexed by the [[natural numbers]] may be "too short" in that they cannot go deep enough into uncountable territory, and they are "too slim" in that they proceed to their potential limiting point only from one direction, instead of from many at once. The use of general [[directed sets]] for nets in place of just the [[natural numbers]] for sequences fixes these two issues.

And indeed, as opposed to sequences, nets do detect

1. the topology on general topological spaces (prop. \ref{TopologyDetectedByNets} below),

1. the continuity of functions between them (prop. \ref{ContinuousFunctionsDetectedByNets} below),

1. the [[Hausdorff topological space|Hausdorff]] property (prop. \ref{NetsDetectHausdorff} below),

1. [[compact topological space|compactness]] (prop. \ref{CompactSpacesEquivalentlyHaveConvergetSubnets} below).

While the concept of nets is similar to that of sequences, one gets a cleaner theory still by considering not the nets themselves but their "[[filters]] of [[subsets]] which they eventually meet" (def. \ref{EventuallyAndFrequently} below), called their _[[eventuality filters]]_ (def. \ref{EventualityFilter} below). For example equivalent filters are equal (in contrast to nets) and (unless in [[predicative mathematics]]) the set of filters on a set $X$ is [[small set|small]] (not a proper class).

## Definitions

### Directed sets
 {#DirectedSets}

+-- {: .num_defn #DirectedSet}
###### Definition
**([[directed set]])**

A _[[directed set]]_ is

* a [[preordered set]] $(D, \leq)$, hence a set $D$ equipped with a [[reflexive relation|reflexive]] and [[transitive]] [[relation]] $\leq$

such that

* every [[finite set|finite]] [[subset]] has an [[upper bound]], hence for any $a,b \in D$ there exists $c \in D$ with $a \leq c$ and $b \leq c$.

=--

+-- {: .num_example #DirectedSetOfNaturalNumbers}
###### Example
**([[directed set]] of [[natural numbers]])**

The [[natural numbers]] $\mathbb{N}$ with their canonical
lower-or-equal relation $\leq$ form a [[directed set]] (def. \ref{DirectedSet}).

=--

The key class of examples of nets, underlying their relation to [[topology]] ([below](#RelationToTopology)) is the following:

+-- {: .num_example #DirectedSetOfNeighbourhods}
###### Example
**([[directed set]] of [[neighbourhoods]])**

Let $(X, \tau)$ be a [[topological space]] and let $x \in X$ be an element of the underlying set. Then then set of $(Nbhd_X(x)_{\supset})$ [[neighbourhoods]] of $x$, ordered by _reverse_ inclusion, is a [[directed set]] (def. \ref{DirectedSet}).

=--

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

Let $A_{\geq}$ and $B_{\geq}$ be two [[directed sets]] (def. \ref{DirectedSet}). Then the [[Cartesian product]] $A \times B$ of the underlying sets becomes itself a directed set by setting

$$
  \left(
    (a_1, b_1) \leq (a_2, b_2)
  \right)
  \,\coloneqq\,
  \left(
     \left( a_1 \leq a_2\right)
     \,\text{and}\,
     \left(
        b_1 \leq b_2
     \right)
  \right)
  \,.
$$

=--

### Nets
 {#Nets}

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

For $X$ a [[set]], then a _net_ in $X$ is

1. a [[directed set]] $A$ (def. \ref{DirectedSet}), called the _index set_,

1. a [[function]] $\nu \colon A \to X$ from (the underlying set of) $A$ to $X$.

We say that $A$ _indexes_ the net.

=--

+-- {: .num_example #SequencesAreNets}
###### Example
**([[sequences]] are [[nets]])**

A [[sequence]] is a net (def. \ref{Net}) whose directed set of indices is the [[natural numbers]] $(\mathbb{N}, \leq)$ (example \ref{DirectedSetOfNaturalNumbers}).

=--

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

Although the index set $A$ in def. \ref{Net}, being a [[directed set]], is equipped with a [[preorder]], the function $\nu \colon A \to X$ is not required to preserve this in any way.  This forms an exception to the rule of thumb that a preordered set may be replaced by its quotient [[partial order|poset]].

You can get around this if you instead define a net in $X$ as a [[multi-valued function]] from a partially ordered directed set $A$ to $X$.  Although there is not much point to doing this in general, it can make a difference if you put restrictions on the possibilities for $A$, in particular if you consider the definition of [[sequence]].  In some [[type theory|type-theoretic]] [[foundations]] of mathematics, you can get the same effect by defining a net to be an 'operation' (a [[prefunction]], like a function but not required to preserve [[equality]]).  On the other hand, every net with domain $A$ is equivalent (in the sense of having the same [[eventuality filter]]) to a net with domain $A \times \mathbb{N}$, made into a partial order by defining $(a,m) \leq (b,n)$ iff $a = b$ and $m \leq n$ or $a \leq b$ and $m \lt n$.

=--

+-- {.num_defn #EventuallyAndFrequently}
###### Definition
**([[eventually]] and frequently)**

Consider [[net]] $\nu \colon A \to X$ (def. \ref{Net}), and given a [[subset]] $S \subset X$. We say that

1. $\nu$ is _[[eventually]] in $S$_ if there exists $i \in A$ such that $\nu_j \in S$ for every $j \ge i$.

1. $\nu$ is _frequently in $S$_ if for every index $i \in A$, then $\nu_j \in S$ for some $j \ge i$.

=--

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

Sometimes one says 'infinitely often' in place of 'frequently' in def. \ref{EventuallyAndFrequently} and even 'cofinitely often' in place of 'eventually'; these derive from the special case of sequences, where they may be taken literally.

=--

+-- {: .num_defn #Convergence}
###### Definition
**([[convergence]] of [[nets]])**

Let $(X,\tau)$ be a [[topological space]], and let $\nu \colon A \to X$ be a [[net]] in the underlying set (def. \ref{Net}).

We say that the net $\nu$

1. _[[convergence|converges]]_ to an element $x \in X$ if given any [[neighbourhood]] $U$ of $x$, $\nu$ is eventually in $U$ (def. \ref{EventuallyAndFrequently}); such $x$ is called a _[[limit point]]_ of the net;

1. _clusters_ at $x$ if, for every [[neighbourhood]] $U$ of $x$, $\nu$ is frequently in $U$ (also def. \ref{EventuallyAndFrequently}); such $x$ is called a _[[cluster point]]_ of the net.

=--

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

Beware that [[limit points]] of nets, according to def. \ref{Convergence}, need not be unique. They are guaranteed to be unique in [[Hausdorff topological spaces|Hausdorff spaces]], see prop. \ref{NetsDetectHausdorff} below.

=--

### Subnets

The definition of the concept of _[[sub-nets]]_ of a net requires some care. The point of the definition is to ensure that prop. \ref{CompactSpacesEquivalentlyHaveConvergetSubnets} below becomes true, which states that [[compact spaces equivalently have converging subnet of every net|compact spaces are equivalently those for which every net has a converging subnet]].

There are several different definitions of '[[subnet]]' in the literature, all of which intend to generalise the concept of subsequences.  We state them now in order of increasing generality.  Note that it is Definition \ref{AA} which is correct in that it corresponds precisely to refinement of filters.  However, the other two definitions (def. \ref{Willard}, def. \ref{Kelley}) are sufficient (in a sense made precise by theorem \ref{EquivalenceOfDefinitionsOfSubnets} below) and may be easier to work with.

+-- {.num_defn #Willard}
###### Definition
(Willard, 1970).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a __subnet__ of $x$ if:

We have a [[function]] $f\colon B \to A$ such that

*  $f$ maps $x$ to $y$ (that is, for every $\beta \in B$, $y_{\beta} = x_{f(\beta)}$);
*  $f$ is monotone (that is, for every $\beta_1 \geq \beta_2 \in B$, $f(\beta_1) \geq f(\beta_2)$);
*  $f$ is cofinal (that is, for every $\alpha \in A$ there is a $\beta \in B$ such that $f(\beta) \geq \alpha$).
=--

+-- {.num_defn #Kelley}
###### Definition
(Kelley, 1955).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a __subnet__ of $x$ if:

We have a [[function]] $f\colon B \to A$ such that

*  $f$ maps $x$ to $y$ (that is, for every $\beta \in B$, $y_{\beta} = x_{f(\beta)}$);
*  $f$ is strongly cofinal (that is, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$, $f(\beta_1) \geq \alpha$).

=--

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

Notice that the function $f$ in definitions \ref{Willard} and \ref{Kelley} is _not_ required to be an [[injection]], and it need not be. As a result, a [[sequence]] regarded as a [[net]] in general has more sub-nets than it has sub-sequences.

=--

+-- {.num_defn #AA}
###### Definition
(Smiley, 1957; Årnes & Andenæs, 1972).

Given a net $(x_{\alpha})$ with index set $A$, and a net $(y_{\beta})$ with an index set $B$, we say that $y$ is a __subnet__ of $x$ if:

The [[eventuality filter]] of $y$ (def. \ref{EventualityFilter}) refines the eventuality filter of $x$.  (Explicitly, for every $\alpha \in A$ there is a $\beta \in B$ such that, for every $\beta_1 \geq \beta \in B$ there is an $\alpha_1 \geq \alpha \in A$ such that $y_{\beta_1} = x_{\alpha_1}$.)
=--

The equivalence between these definitions is as follows:

+-- {.num_theorem #EquivalenceOfDefinitionsOfSubnets}
###### Theorem
([Schechter, 1996](#Schechter96)).

1.  If $y$ is a (\ref{Willard})-subnet of $x$, then $y$ is also a (\ref{Kelley})-subnet of $x$, using the same function $f$.
2.  If $y$ is a (\ref{Kelley})-subnet of $x$, then $y$ is also a (\ref{AA})-subnet of $x$.
3.  If $y$ is a (\ref{AA})-subnet of $x$, then there is some net $z$ such that
    *  $z$ is equivalent to $y$ in the sense that $y$ and $z$ are (\ref{AA})-subnets of each other, and
    *  $z$ is a (\ref{Willard})-subnet of $x$, using some function.
=--

So from the perspective of definition (\ref{AA}), there are enough (\ref{Willard})-subnets and (\ref{Kelley})-subnets, up to equivalence.

### Eventuality filters
 {#NetsAndFilters}

Recall that:

+-- {: .num_defn #Filter}
###### Definition
**([[filter]])**

Given a [[set]] $X$ then a [[set]] of [[subsets]] of $X$, hence a subset of the [[power set]]

$$
  \mathcal{F} \subset P(X)
$$

is called a _[[filter]]_ of subsets if it is closed under [[intersections]] and under taking supersets.

The filter $\mathcal{F}$ is called _proper_ if each set in it is [[inhabited subset|inhabited]].

=--

+-- {: .num_defn #EventualityFilter}
###### Definition
**([[eventuality filter]])**

Let $X$ be a [[set]] and let $\nu \colon D \to X$ be a net in $X$ (def. \ref{Net}).

The _[[eventuality filter]]_ $\mathcal{F}_\nu$ of the net $\nu$ is the [[filter]] (def. \ref{Filter}) onsisting of the subsets that $\nu$ is _eventually in_, according to def. \ref{EventuallyAndFrequently}.

$$
  \left(
    (U \subset X) \in \mathcal{F}_\nu
  \right)
   \,\Leftrightarrow\,
  \left(
    \nu \, \text{is eventually in}\, U
  \right)
  \,.
$$

=--

+-- {: .num_remark}
###### Remark
**(equivalence of nets)**

Two nets are to be considered __equivalent__ if they have the same [[eventuality filter]] according to def. \ref{EventualityFilter}.
By def. \ref{AA} and theorem \ref{EquivalenceOfDefinitionsOfSubnets}, this means equivalently that they are both subnets of each other.

In particular, equivalent nets define the same logical quantifiers (see [below](#LogicOfNets)) and are therefore indeed equivalent for the application to [[topology]] (see [below](#RelationToTopology)).

(Of course, it is possible to distinguish them by using the standard logical quantifiers instead.)

=--

Conversely,  every [[filter]] is the [[eventuality filter]] of some net:

+-- {: .num_defn #FilterNet}
###### Definition
**(nets from filters)**

Let $X$ be a [[set]] and let  $\mathcal{F} \subset P(X)$ be a [[filter]] of subsets of $X$ (def. \ref{Filter}).
Ss
Consider the [[disjoint union]] $\underset{U \in \mathcal{F}}{\sqcup}$ of subsets in  $\mathcal{F}$, hence the set whose elements are [[pairs]] of the form $(U,x)$, where $x \in U \in \mathcal{F}$. Equipped with the ordering

$$
  \left(
    (U,x) \geq (V,y)
  \right)
  \,\Leftrightarrow\,
  \left(
    U \subset V
  \right)
  \phantom{AAA}
  \text{regardless of}\, x\, \text{and} \, y
$$

the fact that $\mathcal{F}$ is a proper filter implies that this is a [[directed set]] according to def. \ref{DirectedSet}. (It is actually enough to use only a base of the filters).

Then the _filter net_ $\nu_F$ of $\mathcal{F}$ is the [[net]] on $X$ (def. \ref{Net}) given by

$$
  \array{
    \left(
      \underset{U \in \mathcal{F}}{\sqcup} U
    \right)_{\supset}
      &\overset{\nu_{\mathcal{F}}}{\longrightarrow}&
    X
    \\
    (U,x) &\overset{\phantom{AAA}}{\mapsto}& x
  }
  \,.
$$

=--

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

Given a [[set]] $X$ and a [[filter]] of subsets $\mathcal{F} \subset P(X)$ (def. \ref{Filter}), then $\mathcal{F}$ is the [[eventuality filter]] (def. \ref{EventualityFilter}) of its filter net (def. \ref{FilterNet}).

=--

## Properties

### Relation to topology
 {#RelationToTopology}

We discuss that nets detect:

1. the topology on general topological spaces (prop. \ref{TopologyDetectedByNets} below),

1. the continuity of functions between them (prop. \ref{ContinuousFunctionsDetectedByNets} below),

1. the [[Hausdorff topological space|Hausdorff]] property (prop. \ref{NetsDetectHausdorff} below),

1. [[compact topological space|compactness]] (prop. \ref{CompactSpacesEquivalentlyHaveConvergetSubnets} below).

+-- {: .num_prop #TopologyDetectedByNets}
###### Proposition
**(topology detected by nets)**

Using the [[axiom of choice]] then:

Let $(X, \tau)$ be a [[topological space]]. Then a [[subset]] $(S \subset X)$ is [[open subset|open]] in $X$ (is an element of $\tau \subset P(X)$) precisely if its [[complement]] $X \backslash S$ is a [[closed subset]] as seen not just by sequences but by nets, in that no [[net]] with elements in $X\backslash S$,  $\nu \colon A \to X\backslash S \hookrightarrow X$, [[converges]] to an element in $S$.

=--

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

In one direction, let $S \subset X$ be open, and consider a net $\nu \colon A \to X \backslash S \subset X$. We need to show that for every point $x \in S$, $x$ is not a limiting point of the net.

But by assumption then $S$ is a [[neighbourhood]] of $x$ which does not contain any element of the net, and so by definition of convergence it is not a limit of this net.

Conversely, let $S \subset X$ be a subset that is not open. We need to show that then there exists a net $\nu \colon A \to X\backslash S \subset X$ that converges to a point in $S$.

For $x \in X$, consider the [[directed set]] $Nbhd_X(x)_{\supset}$ of [[open neighbourhoods]] of this element (example \ref{DirectedSetOfNeighbourhods}).
Now the fact that the set $S$ is not open means that there exists an element $s \in S \subset X$ such that every [[open neighbourhood]] $U$ of $s$ intersects $X \backslash S$. This means that we may [[axiom of choice|choose]] elements $x_U \in U \cap (X \backslash S)$, and hence define a net

$$
  \array{
    Nbhds_X(s) &\overset{\nu}{\longrightarrow}& X \backslash S  \subset X
    \\
    U &\mapsto& x_U
  }
  \,.
$$

But by construction this net has the property that for every neighbourhood $V$ of $s$ there exists $U \in Nbhd_X(s)$ such that for all $U' \subset U$ then $x_{U'} \in V$, namely $U = V$. Hence the net converges to $s$.

=--

+-- {: .num_prop #ContinuousFunctionsDetectedByNets}
###### Proposition
**(continuous functions detected by nets)**

Assuming [[excluded middle]], then

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be two [[topological space]]. Then a [[function]] $f \colon X \to Y$ between their underlying sets is [[continuous function|continuous]] precisely if for every net $\nu \colon A \to X$ that [[convergence|converges]] to some [[limit point]] $x \in X$ (def. \ref{Convergence}), the image net $f\circ \nu$ converges to  $f(x)\in Y$.

=--

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

In one direction, suppose that $f \colon X \to Y$ is continuous, and that $\nu \colon A \to X$ converges to some $x \in X$. We need to show that $f \circ \nu$ converges to $f(x) \in Y$, hence that for every neighbourhood $U_{f(x)} \subset Y$ there exists $i \in A$ such that $f(\nu(j)) \in U_{f(x)}$ for all $j \geq i$.

But since $f$ is continuous, the [[pre-image]] $f^{-1}(U_{f(x)}) \subset X$ is an open neighbourhood of $x$, and so by the assumption that $\nu$ converges there is an $i \in A$ such that $\nu(j) \in f^{-1}(U_{f(x)})$ for all $j \geq i$. By applying $f$, this is the required statement.

We give two proofs of the other direction.

_proof 1_

Conversely, suppose that $f$ is not continuous. We need to find a net $\nu$ that converges to some $x \in X$, and show that $f \circ \nu$ does not converge to $f(x)$. (This is the [[contrapositive]] of the reverse implication, and by [[excluded middle]] equivalent to it.)

Now that $f$ is not continuous means that there exists an open subset $U \subset Y$ such that the pre-image $f^{-1}(U)$ is not open. By prop. \ref{TopologyDetectedByNets} this means that there exists a net $\nu$ in $X \backslash f^{-1}(U)$ that converges to an element $x \in f^{-1}(U)$. But this means that $f \circ \nu$ is a net in the $Y \backslash U$, which is a [[closed subset]] by the assumption that $U$ is open. Again by prop. \ref{TopologyDetectedByNets} this means that $f\circ \nu$ converges to an element in $Y \backslash U$, and hence not to $f(x) \in U$.

_proof 2 (not using excluded middle)_

Assume for every net $\nu \colon A \to X$ that converges to some limit point $x \in X$, the image net $f\circ \nu$ converges to  $f(x)\in Y$. It is sufficient to prove that $f$ is continuous.

Let $V \subset Y$ be open. It is sufficient to prove that $f^{-1}(V)$ is open.

Let $\nu$ be a net with range contained in $X \backslash f^{-1}(V)$. Let $x$ be a limit point of $\nu$. By \ref{TopologyDetectedByNets}, it is sufficient to prove that $x\notin f^{-1}(V)$.

By our assumption, and since $f$ converges to $x$, $f \circ \nu$ converges to $f(x)$. So by \ref{TopologyDetectedByNets}, since $V$ is open, $f(x) \notin V$. So $x \notin f^{-1}(V)$.

=--

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

It is possible to define elementary conditions on this [[convergence]] relation that characterise whether it is topological (that is whether it comes from a topology on $X$), although these are a bit complicated.

By keeping only the simple conditions, one gets the definition of a [[convergence space]]; this is a more general concept than a topological space and includes many non-topological situations where we want to say that a sequence converges to some value (such as convergence in measure).

=--

+-- {: .num_prop #NetsDetectHausdorff}
###### Proposition
**(Hausdorff property detected by nets)**

Assuming [[excluded middle]] and the [[axiom of choice]], then:

A [[topological space]] $(X,\tau)$ is [[Hausdorff topological space]] precisely if no [[net]] in $X$ (def. \ref{Net}) converges to two distinct [[limit points]] (def. \ref{Convergence}).

=--

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

In one direction, assume that $(X,\tau)$ is a Hausdorff space, and that $\nu \colon A \to X$ is a net in $X$ which has limits points $x_1, x_2 \in X$. We need to show that then $x_1 = x_2$.

Assume on the contrary that the two points were different, $x_1 \neq x_2$. By assumption of Hausdorffness, these would then have disjoint open neighbourhoods $U_{x_1}, U_{x_2}$, i.e. $U_1 \cap U_2 = \emptyset$. By definition of convergence, there would thus be $a_1, a_2 \in A$ such that $\nu_{a_1 \leq \bullet} \in U_{x_1}$ and $\nu_{a_2 \leq \bullet} \in U_{x_2}$. Moreover, by the definition of directed set, this would imply $a_3 \in A$ with $a_1, a_2 \leq a_3$, and hence that $x_{a_3 \leq \bullet} \in U_{x_1} \cap U_{x_2}$.  This is in contradiction to the emptiness of the intersection, and hence we have a [[proof by contradiction]].

Conversely, assume that $(X,\tau)$ is not a Hausdorff space. We need to show that then there exists a net $\nu$ in $X$ with two distinct limit points.

That $(X,\tau)$ is not Hausdorff means that there are two distinct points $x_1, x_2 \in X$ such that every open neighbourhood of $x_1$ intersects every open neighbourhood of $x_2$. Hence we may [[axiom of choice|choose]] elements in these intersections

$$
  x_{U_{x_1}, U_{x_2}} \in U_{x_1}, U_{x_2}
  \,.
$$

Consider the directed neighbourhood sets $Nbhd_X(x_1)_{\supset}$ and $Nbhd_X(x_2)_{\supset}$ of these two points (example \ref{DirectedSetOfNeighbourhods}) and their directed Cartesian product set (example \ref{DirectedProductSet}) $Nbhd_X(x_1)_{\supset} \times Nbhd_X(x_2)_{\supset}$. The above elements then define a net

$$
  \array{
     Nbhd_X(x_1) \times Nbhd_X(x_2)
       &\overset{\nu}{\longrightarrow}&
     X
     \\
     (U_{x_1}, U_{x_2})
       &\overset{\phantom{AAA}}{\mapsto}&
     x_{U_1, U_2}
  }
  \,.
$$

We conclude by claiming that $x_1$ and $x_2$ are both limit points of this net. We show this for $x_1$, the argument for $x_2$ is directly analogous:

Let $U_{x_1}$ be an open neighbourhood of $x_1$. We need to find an element $(V_1, V_2) \in Nbhd_X(x_1) \times Nbhd_X(x_2)$ such that for all $(W_1, W_2) \subset (V_1, V_2)$ then $\nu_{(W_1, W_2)} \in U_{x_1}$.

Take $V_1 \coloneqq U_{x_1}$ and take $V_2 = X$. Then by construction

$$
  \begin{aligned}
    \nu_{(W_1, W_2)}
      & \in W_1 \cap W_2
    \\
      & \subset V_1 \cap V_2
    \\
     & = U_{x_1} \cap X
    \\
     & = U_{x_1}
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop #CompactSpacesEquivalentlyHaveConvergetSubnets}
###### Proposition
**([[compact spaces equivalently have converging subnet of every net|compact spaces are equivalently those for which every net has a converging subnet]])**

Assuming [[excluded middle]] and the [[axiom of choice]], then:

A [[topological space]] $(X,\tau)$ is [[compact topological space|compact]] precisely if every [[net]] in $X$ (def. \ref{Net}) has a [[sub-net]] (def. \ref{Willard}) that [[convergence|converges]] (def. \ref{Convergence}).

=--

We break up the **proof** into that of lemmas \ref{InACompactSpaceEveryNetHasAConvergentSubnet} and \ref{IfEveryNetHasConvergentSubnetThenSpaceIsCompact}:

+-- {: .num_example #InACompactSpaceEveryNetHasAConvergentSubnet}
###### Lemma
**(in a compact space, every net has a convergent subnet)**

Let $(X,\tau)$ be a [[compact topological space]]. Then every net in $X$ has a convergent subnet.

=--

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

Let $\nu \colon A \to X$ be a net. We need to show that there is a subnet which converges.

For $a \in A$ consider the [[topological closures]] $Cl(S_a)$ of the sets $S_a$ of elements of the net beyond some fixed index:

$$
  S_a
    \;\coloneqq\;
  \left\{
    \nu_b  \in X \;\vert\; b \geq a
  \right\}
  \subset X
  \,.
$$

Observe that the set $\{S_a \subset X\}_{a \in A}$ and hence also the set $\{Cl(S_a) \subset X\}_{a \in A}$ has the [[finite intersection property]], by the fact that $A$ is a [[directed set]]. Therefore [this prop. ](finite+intersection+property#CompactnessInTermsOfFiniteIntersectionProperty) implies from the assumption of $X$ being compact that the intersection of _all_ the $Cl(S_a)$ is non-empty, hence that there is an element

$$
  x \in \underset{a \in A}{\cap}  Cl(S_a)
  \,.
$$

In particular every [[neighbourhood]] $U_x$ of $x$ intersects each of the $Cl(S_a)$, and hence also each of the $S_a$. By definition of the $S_a$, this means that for every $a \in A$ there exists $b \geq a$ such that $\nu_b \in U_x$, hence that $x$ is a [[cluster point]] (def \ref{Convergence}) of the net.

We will now produce a [[sub-net]]

$$
  \array{
     B && \overset{f}{\longrightarrow} && A
     \\
     & \searrow && \swarrow_{\nu}
     \\
     && X
  }
$$

that converges to this cluster point. To this end, we first need to build the domain directed set $B$. Take it to be the sub-directed set of the Cartesian product directed set (example \ref{DirectedProductSet}) of $A$ with the directed neighbourhood set $Nbhd_X(x)$ of $x$ (example \ref{DirectedSetOfNeighbourhods})

$$
  B \subset A_{\leq} \times Nbhd_X(x)_{\supset}
$$

on those pairs such that the element of the net indexed by the first component is contained in the second component:

$$
  B
    \;\coloneqq\;
  \left\{
     (a,U_x)  \,\vert \, \nu_a \in U_X
  \right\}
  \,.
$$

It is clear $B$ is a [[preordered set]]. We need to check that it is indeed directed, in that every pair of elements $(a_1, U_1)$, $(a_2, U_2)$ has a common upper bound $(a_{bd}, U_{bd})$. Now since $A$ itself is directed, there is an upper bound $a_3 \geq a_1, a_2$, and since $x$ is a cluster point of the net there is moreover an $a_{bd} \geq a_3 \geq a_1, a_3$ such that $\nu_{a_{bd}} \in U_1 \cap U_2$. Hence with $U_{bd} \coloneqq U_1 \cap U_2$ we have obtained the required pair.

Next take the function $f$ to be given by

$$
  \array{
     B &\overset{f}{\longrightarrow}& A
     \\
     (a, U) &\overset{\phantom{AAA}}{\mapsto}& a
  }
  \,.
$$

This is clearly order preserving, and it is cofinal since it is even a surjection. Hence we have defined a subnet $\nu \circ f$.

It now remains to see that $\nu \circ f$ converges to $x$, hence that for every open neighbourhood $U_x$ of $x$ we may find $(a,U)$ such that for all $(b,V)$ with $a \leq b$ and $U \supset V$ then $\nu(f(b,V)) = \nu(b) \in U_x$.  Now by the nature of $x$ there exists some $a$ with $\nu_a \in U_x$, and hence if we take $U \coloneqq U_x$ then nature of $B$ implies that with $(b, V) \geq (a,U_x)$ then $b \in V \subset U_x$.

=--

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

Assuming [[excluded middle]], then:

Let $(X,\tau)$ be a [[topological space]]. If every [[net]] in $X$ has a [[subnet]] that [[convergence|converges]], then $(X,\tau)$ is a [[compact topological space]].

=--

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

By [[excluded middle]] we may equivalently prove the [[contrapositive]]: If $(X,\tau)$ is not compact, then not every net in $X$ has a convergent subnet.

Hence assume that $(X,\tau)$ is not compact. We need to produce a net without a convergent subnet.

Again by [[excluded middle]], then by [this prop.](finite+intersection+property#CompactnessInTermsOfFiniteIntersectionProperty) $(X,\tau)$ not being compact means equivalently that there exists a set $\{C_i \subset X\}_{i \in I}$ of [[closed subsets]] satisfying the [[finite intersection property]], but such that their intersection is empty: $\underset{i \in I}{\cap} C_i = \emptyset$.

Consider then $P_{fin}(I)$, the set of [[finite set|finite]] [[subsets]] of $I$. By the assumption that $\{C_i \subset X\}_{i \in I}$ satisfies the [[finite intersection property]], we may [[axiom of choice|choose]] for each $J \in P_{fin}(I)$ an element

$$
  x_J \in \underset{i \in J \subset I}{\cap} C_i
  \,.
$$

Now $P_{fin}(X)$ regarded as a [[preordered set]] under inclusion of subsets is clearly a [[directed set]], with an upper bound of two finite subsets given by their [[union]]. Therefore we have defined a net

$$
  \array{
     P_{fin}(X)_{\subset} &\overset{\nu}{\longrightarrow}& X
     \\
     J &\overset{\phantom{AAA}}{\mapsto}& x_J
  }
  \,.
$$

We will show that this net has no converging subnet.

Assume on the contrary that there were a subnet

$$
  \array{
     B && \overset{f}{\longrightarrow} && P_{fin}(X)
     \\
     & \searrow && \swarrow_{\nu}
     \\
     && X
  }
$$

which converges to some $x \in X$.

By the assumption that $\underset{i \in I}{\cap} C_i = \emptyset$, there would exist an $i_x \in I$ such that $x \neq C_{i_x}$, and because $C_i$ is a [[closed subset]], there would exist even an [[open neighbourhood]] $U_x$ of $x$ such that $U_x \cap C_{i_x} = \emptyset$. This would imply that $x_J \neq U_x$ for all $J \supset \{i_x\}$.

Now since the function $f$ defining the subset is cofinal, there would exist $b_1 \in B$ such that $\{i_x\} \subset f(b_1)$. Moreover, by the assumption that the subnet converges, there would also be $b_2 \in B$ such that $\nu_{b_2 \leq \bullet} \in U_x$. Since $B$ is directed, there would then be an upper bound $b \geq b_1, b_2$ of these two elements. This hence satisfies both $\nu_{f(e)} \in U_x$ as well as $\{i_x\} \subset f(b_1) \subset f(b)$. But the latter of these two means that $\nu_{f(b)}$ is not in $U_x$, which is a contradiction to the former. Thus we have a [[proof by contradiction]].

=--

### Logic of nets
 {#LogicOfNets}

A [[property]] of [[elements]] of a [[set]] $X$ (given by the [[subset]] $S \subset X$ of those elements of $X$ satisfying this property) may be applied to nets in $X$.

Being eventually in $S$, def. \ref{EventuallyAndFrequently}, is a weakening of being always in $S$ (given by the [[universal quantifier]] $\forall_\nu$), while being frequently in $S$ is a strengthening of being sometime in $S$ (given by the [[particular quantifier]] $\exists_\nu$).  Indeed we can build a [[formal logic]] out of these.  Use $\ess\forall i, p[\nu_i]$ or $\ess\forall_\nu p$ to mean that a predicate $p$ in $X$ is eventually true, and use $\ess\exists i, p[\nu_i]$ or $\ess\exists_\nu p$ to mean that $p$ is frequently true.  Then we have:
$$\forall_\nu p \;\Rightarrow\; \ess\forall_\nu p \;\Rightarrow\; \ess\exists_\nu p \;\Rightarrow\; \exists_\nu p$$
$$\ess\forall_\nu (p \wedge q) \;\Leftrightarrow\; \ess\forall_\nu p \wedge \ess\forall_\nu q$$
$$\ess\exists_\nu (p \wedge q) \;\Rightarrow\; \ess\exists_\nu p \wedge \ess\exists_\nu q$$
$$\ess\forall_\nu (p \vee q) \;\Leftarrow\; \ess\forall_\nu p \wedge \ess\forall_\nu q$$
$$\ess\exists_\nu (p \vee q) \;\Leftrightarrow\; \ess\exists_\nu p \vee \ess\exists_\nu q$$
$$\ess\forall_\nu \neg{p} \;\Leftrightarrow\; \neg\ess\exists_\nu p$$
and other analogues of theorems from [[predicate logic]].  Note that the last item listed requires [[excluded middle]] even though its analogue from ordinary predicate logic does not.

A similar logic is satisfied by '[[almost everywhere]]' and its dual ('not almost nowhere' or 'somewhere significant') in [[measure spaces]].

## Related concepts

* [[sequential net]]

* [[limit of a net]]

* [Tychonoff theorem -- Proof via net convergence](Tychonoff+theorem#ProofViaNets)

## References

A textbook account is in

* {#Schechter96} [[Eric Schechter]], sections 7.14--7.21 of _[[Handbook of Analysis and its Foundations]]_, Academic Press (1996)

Lecture notes include

* {#Vermeeren10} [[Stijn Vermeeren]], _Sequences and nets in topology_, 2010 ([pdf](http://stijnvermeeren.be/download/mathematics/nets.pdf))

[[!redirects net]]
[[!redirects nets]]


net of C-star-systems

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###Context###
#### AQFT
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[[!include AQFT and operator algebra contents]]
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#### Physics
+--{: .hide}
[[!include physicscontents]]
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=--

#Contents#
* table of contents
{:toc}

## Idea ##

The notion of a _net $C^*$-systems_ combines the notion of a [[C-star system]] with the notion of [[local net of observables]]. In this way, the notion of [[global gauge groups]] is introduced into the [[Haag-Kastler approach]] to [[AQFT]].

## Definition ##

Let $\mathcal{A}_I$ be a [[local net]] of [[C-star algebras]]. Let $G$ be a [[locally compact space|locally compact]] [[topological group]] and $\alpha_G$ a [[representation]] of $G$ on the quasi-local algebra $\mathcal{A}$, that is

$$
\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{i \in I} \mathcal{A}_i \bigr)
$$

so that $(\mathcal{A}, \alpha_G)$ is a [[C-star system]].

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

The tupel $(\mathcal{A}_I, \alpha_G)$ is a **net of $C^*$-systems** if $\alpha_g(\mathcal{A}_i) \subseteq \mathcal{A}_i \; \forall g \in G$.

=--

In the context of [[Haag-Kastler nets]] the group $G$ is called the
**[[global gauge group]]** and every [[automorphism]] $\alpha_g$ is called a **[[gauge transformation|gauge automorphism]]**.

This definition makes sense also if the net consists of [[star-algebras]] only, of course.

## Related concepts

* [[local net of observables]]

* [[field net]]

## References ##

Chapter 6 of:

* [[Hellmut Baumgärtel]], Manfred Wollenberg: _Causal nets of operator algebras._ Berlin: Akademie Verlag 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete))

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[[!redirects net of C-star systems]]
[[!redirects net of C*-systems]]

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