simple root

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#### Algebra
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[[!include higher algebra - contents]]
=--
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=--

\tableofcontents

## Definition

\begin{definition}
Given a [[polynomial function]] with [[real number|real]] or [[complex number|complex]] [[coefficients]] $x \mapsto f(x)$, a **simple root** is a [[root]] of $x \mapsto f(x)$ such that the [[derivative]] of $x \mapsto f(x)$ evaluated at the root is [[invertible]].
\end{definition}

More generally, let $R$ be a [[commutative ring]]. The [[polynomial ring]] $R[x]$ is a [[differential algebra]] and thus has a derivative function $(-)':R[x] \to R[x]$. There is a canonical ring homomorphism $h:R[x] \to (R \to R)$ from the [[polynomial ring]] $R[x]$ to the [[function algebra]] $R \to R$ which assigns constant polynomials to [[constant functions]] in $R$ and the bare indeterminant to the [[identity function]] of $R$, which when [[uncurried]] leads to the evaluation of polynomials $(-)((-)):R[x] \times R \to R$. Since the set of polynomial functions in a commutative ring is a polynomial ring generated by the identity function, the situation for [[polynomial functions]] is simply a special case of the situation for polynomials. Thus, we have the following definition for general polynomials over [[fields]]:

\begin{definition}
Given a [[Heyting field]] $K$ and a [[polynomial]] $f \in K[x]$ with [[coefficients]] in $K$, a **simple root** is a [[root]] $a \in K$ of $f$ such that $f'(a)$, the evaluation of the [[derivative]] of $f$ at $a$, is [[invertible]].
\end{definition}

In [[classical mathematics]] (implying acceptance of the [[law of excluded middle]]), a simple root $a$ is a root that is not a multiple root: it is not the case that $(x-a)^2$ divides $f(x)$, or "not $f'(a) = 0$". But for the purposes of [[constructive mathematics]], it is preferable to state the condition as "$f'(a)$ is [[apartness|apart]] from $0$", or that $f'(a)$ is invertible. Hence the definitions above.

## Related concepts

* [[root]]
* [[elementary symmetric polynomial]]
* [[unramifiable polynomial]]
* [[fundamental theorem of algebra]]

## References

* {#Ruitenberg91} [[Wim Ruitenberg]], *Constructing Roots of Polynomials over the Complex Numbers*, Computational Aspects of Lie Group Representations and Related Topics, CWI Tract, **84** Centre for Mathematics and Computer Science, Amsterdam (1991) 107–128 [[pdf](https://www.mscsnet.mu.edu/~wim/publica/roots_new.pdf), [[Ruitenberg-Roots.pdf:file]]]

[[!redirects simple roots]]


simple theory

* [[Saharon Shelah]], _Simple unstable theories_, Ann. Math. Logic, 19, 177–203 (1980)
* Rami Grossberg, José Iovino, Olivier Lessmann, _A primer of simple theories_, Archive for Mathematical Logic 41:6 (2002) 541-580, [doi](http://dx.doi.org/10.1007/s001530100126)
* F. O. Wagner, _Simple theories_, Kluwer Academic Publishers, 260 pages (2000)
* J.T. Baldwin, _Fundamentals of stability theory_, Springer 1988


simple topological space

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

## Idea

A [[topological space]] $X$, or rather its [[homotopy type]], is called _simple_ if

1. the [[fundamental group]] $\pi_1(X)$ is [[abelian group|abelian]];

1. its canonical [[action]] on all higher [[homotopy groups]] $\pi_{\bullet \geq 2}(X)$ is trivial.

## Examples

* Every [[connected topological space|connected]] [[H-space]] is nilpotent

and thus, in particular,

* every [[loop space]] is nilpotent

  (since all its [[connected components]] are [[homotopy equivalence|homotopy equivalent]] to the unit component, which is a connected H-space).

(See [May-Ponto 12, p. 49 (77 of 542)](#MayPonto12))

## Properties

Simple spaces are also [[nilpotent spaces]] ([May-Ponto 12, p. 49 (77 of 542)](#MayPonto12)).

## Related concepts

* [[simply connected topological space]]

* [[nilpotent space]]

* [[H-space]]

## References

* {#MayPonto12} [[Peter May]], [[Kate Ponto]], _More Concise Algebraic Topology_, University of Chicago Press (2012) ([pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf))

[[!redirects simple topological spaces]]

[[!redirects simple space]]
[[!redirects simple spaces]]

[[!redirects simple homotopy type]]
[[!redirects simple homotopy types]]


simple type theory

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#### Type theory
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[[!include type theory - contents]]
=--
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\tableofcontents

## Idea

Originally, "*simple type theory*" was the name of the [[type theory]] introduced by [Church (1940)](#Church40) (therefore often and more informatively: "*Church's type theory*" or similar). This [[type theory]] allowed (only) [[function type]]-[[type formation]] (therefore often: "simply typed [[lambda-calculus]]") based on two elementary types (a kind of [[type of natural numbers]] and a [[type of propositions]]).

In mild generalization, if one admits in addition [[product type]]-[[type formation]] then [e.g. [Gunter (1992)](#Gunter92)] these are the type theories whose [[categorical semantics]] is in [[cartesian closed categories]] [[Lambek & Scott (1986), Part I](#LambekScott86)], see also at *[[relation between category theory and type theory]]*.

More generally, the term "simple type theory" has come to refer to any [[type theory]] whose [[type formations]] are *not indexed*, in that the [[judgment]] that a [[type]] $A$ is well-formed has no other inputs, understood *in contrast* to:

* [[polymorphism|polymorphic type theory]], where types depend on a [[context]] of type variables,

* [[dependent type theory]], where types depend on a [[context]] of more general variables.

## Related Pages

* [[pure type system]]

## References

Original articles:

* {#Russell08} [[Bertrand Russell]], _Mathematical Logic as Based on the Theory of Types_, American Journal of Mathematics, Vol. 30, No. 3 (Jul., 1908), pp. 222-262

* {#Church40} [[Alonzo Church]], §5 of: *A Formulation of the Simple Theory of Types*, The Journal of Symbolic Logic **5** 2 (1940) 56-68  [[doi:10.2307/2266170](https://doi.org/10.2307/2266170)]

See also

* Stanford Encyclopedia of Philosophy, *[Church’s Type Theory](https://plato.stanford.edu/entries/type-theory-church/)*

Establishing the [[syntax-semantics duality|syntax/semantics]] [[relation between category theory and type theory|relation between]] [[cartesian closed categories]] and simply-typed lambda calculi:

* {#LambekScott86} [[Joachim Lambek]], [[Philip J. Scott]], Part I of: *Introduction to higher order categorical logic*, Cambridge Studies in Advanced Mathematics **7** (1986) [[ISBN: 0-521-24665-2](https://www.cambridge.org/ae/academic/subjects/mathematics/logic-categories-and-sets/introduction-higher-order-categorical-logic?format=PB&isbn=9780521356534), [pdf](https://raw.githubusercontent.com/Mzk-Levi/texts/master/Lambek%20J.%2C%20Scott%20P.J.%20Introduction%20to%20Higher%20Order%20Categorical%20Logic.pdf)]

Lecture notes:

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

Textbook accounts:

* [[J. Roger Hindley]], *Basic Simple Type Theory*, Cambridge University Press (1997, 2009) [[doi:10.1017/CBO9780511608865](https://doi.org/10.1017/CBO9780511608865)]

* {#Gunter92} [[Carl A. Gunter]], §2-3 in: *Semantics of Programming Languages -- Structures and Techniques*, MIT Press (1992) [[ISBN:9780262570954](https://mitpress.mit.edu/9780262570954/semantics-of-programming-languages/)]

  > (in a context of [[programming languages]])

See also:

* W. Farmer, _The seven virtues of simple type theory_, Journal of Applied Logic, Vol. 6, No. 3. (September 2008), pp. 267-286.

* [[Nathanael Arkor]], [[Marcelo Fiore]], *Algebraic models of simple type theories: a polynomial approach*, Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science (2020) 88-101 [[doi:10.1145/3373718.3394771](https://doi.org/10.1145/3373718.3394771), [arXiv:2006.16949](https://arxiv.org/abs/2006.16949)[

[[!redirects simple type theories]]

[[!redirects simply typed lambda calculus]]
[[!redirects simply typed lambda-calculus]]
[[!redirects simply typed λ-calculus]]

[[!redirects simply-typed lambda calculus]]
[[!redirects simply-typed lambda calculi]]

[[!redirects simply-typed lambda-calculus]]
[[!redirects simply-typed lambda-calculi]]

[[!redirects simply-typed λ-calculus]]
[[!redirects simply-typed λ-calculi]]


simplex

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

## Idea

The _cellular simplex_ is one of the basic [[geometric shapes for higher structures]]. Variants of the same "shape archetype" exist in several settings, e.g., that of a [[simplicial set]], the topological /cellular one, and categorical contexts, plus others.

## Definitions
 {#Definition}

### Simplicial simplices
 {#SimplicialSimplex}

For $n \in \mathbb{N}$, the standard _simplicial $n$-simplex_ $\Delta[n]$ is the [[simplicial set]] which is represented (as a [[presheaf]]) by the object $[n]$ in the [[simplex category]], so $\Delta[n]= \Delta(-,[n])$.

### Cellular (simplicial) simplex
 {#CellularSimplex}

Likewise, there is a standard topological $n$-simplex, which is (more or less by definition) the [[geometric realization]] of the standard simplicial $n$-simplex.

### Topological simplex
 {#TopologicalSimplex}

The _topological $n$-simplex_ $\Delta^n$ is a generalization of the standard filled _[[triangle]]_ in the plane, from [[dimension]] 2 to arbitrary dimensions. Each $\Delta^n$ is [[homeomorphism|homeomorphic]] to the closed $n$-[[ball]] $D^n$, but its defining [[embedding]] into a [[Cartesian space]] equips its [[boundary]] with its cellular decomposition into _[[faces]]_, generalizing the way that the triangle has three [[edges]] (which are 1-simplices) as [[faces]], and three points (which are [[vertices]] or 0-simplices) as corners.

The topological $n$-simplex is naturally defined as a [[subspace]] of a
[[Cartesian space]] given by some relation on its canonical
[[coordinates]]. There are two standard choices for such coordinate
presentation, which of course define [[homeomorphism|homeomorphic]]
$n$-simplices:

* [Barycentric coordinates](#BarycentricCoordinates)

* [Cartesian coordinates](#CartesianCoordinates)

Each of these has its advantages and disadvantages, depending on application, but of course there is a simple coordinate transformation that exhibits an explicit [[homeomorphism]] between the two:

* [Transformation between Barycentric and Cartesian coordinates](#CoordinateTransformation).

#### Barycentric coordinates
 {#BarycentricCoordinates}

In the following, for $n \in \mathbb{N}$ we regard the [[Cartesian space]] $\mathbb{R}^n$ as equipped with the canonical [[coordinates]] labeled $x_0, x_1, \cdots, x_{n-1}$.

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

For $n \in \mathbb{N}$, the **topological $n$-simplex** is,
up to [[homeomorphism]], the [[topological space]] whose underlying set is
the [[subset]]

$$
  \Delta^n
    \;\coloneqq\;
  \Big\{
    \vec x \in (\mathbb{R}_{\geq 0})^{n+1}
    \,\big|\,
    \textstyle{\sum}_{i = 0 }^n x^i = 1
  \Big\}
  \;\subset\;
  \mathbb{R}^{n+1}
$$

of the [[Cartesian space]] $\mathbb{R}^{n+1}$, and whose [[topological space|topology]] is the  [[subspace topology]] inherited from the [[Euclidean topology]] on $\mathbb{R}^{n+1}$.

=--

\linebreak
\linebreak

\[\label{IllustrationOfBarycentricTopologicalSimplices}\]
\begin{imagefromfile}
    "file_name": "TopologicalSimplices.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -100,
        "bottom": 20,
        "right": 0,
        "left": 0
    }
\end{imagefromfile}

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

For $n \in \mathbb{N}$, $\n \geq 1$ and $0 \leq k \leq n$, the
**$k$th $(n-1)$-face (inclusion)**  of the topological $n$-simplex is the subspace inclusion

$$
  \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n
$$

induced under the barycentric coordinates of def. \ref{TopologicalInBarycentricCoords},
by the inclusion

$$
  \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}
$$

which omits the $k$th coordinate

$$
  (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_{n-1})
  \,.
$$

=--

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

The inclusion

$$
  \delta_0 : \Delta^0 \to \Delta^1
$$

is the inclusion

$$
  \{1\} \hookrightarrow [0,1]
$$

of the "right" end of the standard interval. The other inclusion

$$
  \delta_1 : \Delta^0 \to \Delta^1
$$

is that of the "left" end $\{0\} \hookrightarrow [0,1]$.

=--

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

For $n \in \mathbb{N}$ and $0 \leq k \lt n$ the **$k$th degenerate $n$-simplex (projection)** is the surjective map

$$
  \sigma_k : \Delta^{n} \to \Delta^{n-1}
$$

induced under the barycentric coordinates of def. \ref{TopologicalInBarycentricCoords} under the surjection

$$
  \mathbb{R}^{n+1} \to \mathbb{R}^n
$$

which sends

$$
  (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n)
  \,.
$$

=--

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

The collection of face inclusions, def. \ref{FaceInclusionInBarycentricCoords}
and degeneracy projections, def. \ref{DegeneracyProjectionsInBarycentricCoords}
satisfy the (dual) [[simplicial identities]]. Equivalently, they constitute the components of a [[functor]]

$$
  \Delta^\bullet : \Delta \to Top
$$

from the [[simplex category]] $\Delta$ to the category [[Top]] of [[topological spaces]]. This is, up to [[isomorphism]], the standard [[cosimplicial object]] in $Top$.

=--

#### Cartesian coordinates
 {#CartesianCoordinates}

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

The standard **topological $n$-simplex** is, up to [[homeomorphism]],
the [[subset]]

$$
  \Delta^n
  \coloneqq
  \{
     \vec x \in \mathbb{R}^n | 0 \leq x_1 \leq \cdots \leq x_n \leq 1
  \}
  \hookrightarrow \mathbb{R}^n
$$

equipped with the [[subspace topology]] of the standard topology on the [[Cartesian space]] $\mathbb{R}^n$.

=--

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

This definition identifies the topological $n$-simplex with the space of [[interval]] maps (preserving top and bottom) $\{0 \lt 1 \lt \ldots \lt n+1\} \to I$ into the topological interval. This point of view takes advantage of the [duality](http://ncatlab.org/nlab/show/simplex+category#duality_with_intervals_23) between the [[simplex category]] $\Delta$ and the category $\nabla$ of finite [[intervals]] with distinct top and bottom. Indeed, it follows from the duality that we obtain a functor

$$\Delta \simeq \nabla^{op} \stackrel{Int(-, I)}{\to} Top.$$

=--

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

* For $n = 0$ this is the [[point]], $\Delta^0 = *$.

* For $n = 1$ this is the standard [[interval object]] $\Delta^1 = [0,1]$.

* For $n = 2$ this is a triangle sitting in the plane like this:

  $$
    \left\{
      (x_0,x_1) | 0 \leq x_0 \leq x_1 \leq 1
    \right\}
    =
    \left\{
    \array{
       && && (1,1)
       \\
       && & \nearrow & \downarrow
       \\
       && (\tfrac{1}{2}, \tfrac{1}{2}) && (\tfrac{1}{2},1)
       \\
       & \nearrow & && \downarrow
       \\
       (0,0) &\stackrel{}{\to}& (0,\tfrac{1}{2}) & \to & (0,1)
    }
    \right\}
  $$

=--

#### Transformation between Barycentric and Cartesian coordinates
 {#CoordinateTransformation}

For $n \in \mathbb{N}$,  write now explicitly

$$
 \Delta^n_{bar} \hookrightarrow \mathbb{R}^{n+1}
$$

for the topological $n$-simplex in barycentric coordinate presentation, def. \ref{TopologicalInBarycentricCoords}, and

$$
 \Delta^n_{cart} \hookrightarrow \mathbb{R}^{n}
$$

for the topological $n$-simplex in Cartesian coordinate presentation, def. \ref{TopologicalInCartesianCoordinates}.

Write

$$
  S_n : \mathbb{R}^{n+1} \to \mathbb{R}^n
$$

for the [[continuous function]] given in the standard coordinates by

$$
  (x_0, \cdots, x_{n})
  \mapsto
  (x_0, x_0 + x_1, \cdots, \sum_{i = 0}^k x_i, \cdots, \sum_{i = 0}^{n-1} x_i)
  \,.
$$

By restriction, this induces a continuous function on the topological $n$-simplices

$$
  \array{
     \Delta^n_{bar} &\hookrightarrow& \mathbb{R}^{n+1}
     \\
     \downarrow^{\mathrlap{S_n|_{\Delta^n_{bar}}}} && \downarrow^{p_n}
     \\
     \Delta^n_{cart} &\hookrightarrow& \mathbb{R}^n
  }
  \,.
$$

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

For every $n \in \mathbb{N}$ the function $S_n$ is a [[homeomorphism]]
and respects the face and degeneracy maps.

Equivalently, $S_\bullet$ is a [[natural isomorphism]] of [[functors]]
$\Delta^n \to Top$, hence an [[isomorphism]] of [[cosimplicial objects]]

$$
  S_\bullet : \Delta^\bullet_{bar} \stackrel{\simeq}{\to} \Delta^\bullet_{cart}
  \,.
$$

=--

### Singular simplex
 {#SingularSimplex}

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

For $X \in $ [[nLab:Top]]  and $n \in \mathbb{N}$, a **singular $n$-simplex** in $X$ is a [[nLab:continuous map]]

$$
  \sigma : \Delta^n \to X
  \,.
$$

Write

$$
  (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)
$$

for the set of singular $n$-simplices of $X$.

=--

As $n$ varies, this forms the [[singular simplicial complex]] of $X$.

## Properties

### Relation to globes

The [[orientals]] relate simplices to [[globes]].

## Related concepts

* [[vertex]], [[edge]]

* [[triangle]], [[tetrahedron]]

* [[horn]], [[boundary of a simplex]], [[spine]]

* [[differential forms on simplices]]

* [[simplicial set]]

  * [[simplicial complex]]

  * [[Kan complex]]

  * [[weak Kan complex]]

* [[simplicial category]]

* [[globe]],

* [[tree]], [[dendrex]]

* [[Bloch region]]

* [[Delta-generated space]]

## References

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Exp. 1.7 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)]

[[!redirects simplices]]

[[!redirects simplicial simplex]]
[[!redirects cellular simplex]]
[[!redirects topological simplex]]

[[!redirects simplicial simplices]]
[[!redirects cellular simplices]]
[[!redirects topological simplices]]

[[!redirects singular simplex]]
[[!redirects singular simplices]]

[[!redirects k-simplex]]
[[!redirects k-simplices]]

[[!redirects 0-simplex]]
[[!redirects 0-simplices]]

[[!redirects 1-simplex]]
[[!redirects 1-simplices]]

[[!redirects 2-simplex]]
[[!redirects 2-simplices]]

[[!redirects 3-simplex]]
[[!redirects 3-simplices]]

[[!redirects n-simplex]]
[[!redirects n-simplices]]

[[!redirects topological n-simplex]]
[[!redirects topological n-simplices]]

[[!redirects topological k-simplex]]
[[!redirects topological k-simplices]]

[[!redirects geometric n-simplex]]
[[!redirects geometric n-simplices]]


simplex category

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

#Contents#
* table of contents
{:toc}

##Idea##

The **simplex category** $\Delta$ encodes one of the main [[geometric shapes for higher structures]]. Its objects are the standard cellular $n$-[[simplices]]. It is also called the _[[simplicial category]]_, but that term is ambiguous.

## Definition ##
 {#Definition}

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

The **augmented simplex category** $\Delta_a$ is the [[full subcategory]] of [[Cat]] on the [[free categories]] of [[finite set|finite]] linear [[directed graphs]]

$$
  \{c_0 \to c_1 \to \cdots \to c_n\}
  \,.
$$

Equivalently, this is the category whose [[objects]] are finite [[total order|totally ordered sets]], or finite ordinals, and whose [[morphisms]] are order-preserving functions between them.

=--

\begin{rmk} As a consequence of the equivalent description, $\Delta_a$ can be viewed as a full sub-[[2-poset]] of [[Pos]]. \end{rmk}

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

The **simplex category** $\Delta$ is the [[full subcategory]] of $\Delta_a$ (and hence of $Cat$) consisting of the free categories on finite and **[[inhabited set|inhabited]]** linear directed graphs, hence of  _non-empty_ finite linear orders or _non-zero_ ordinals.

=--

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

It is common, convenient and without risk to use a [[skeleton]] of $\Delta$ or $\Delta_a$, where we pick a fixed representative in each [[isomorphism class]] of objects. Since isomorphisms of finite linearly ordered sets are _unique_ this step is so trivial that it is often not even mentioned explicitly.

With this the objects of $\Delta$ are in bijection with [[natural numbers]] $n \in \mathbb{N}$ and one usually writes

$$
  [n] = \{0 \to 1 \to \cdots \to n\}
$$

for the object of $\Delta$ given by the category with $(n+1)$ [[object]]s. Geometrically one may think of this as the [[spine]] of the standard cellular $n$-[[simplex]], see the discussion of [simplicial sets](#SimplicialSets) below. In this context one also writes $\Delta[n]$ or $\Delta^n$ for the [[simplicial set]] [[representable functor|represented]] by the object $[n]$: the simplicial $n$-[[simplex]]. By the [[Yoneda lemma]] one may identify the subcategory of simplicial sets on the $\Delta[n]$ with $\Delta$.

With this convention the first few objects of $\Delta$ are

$$
  [0] = \{0\}
$$

$$
  [1] = \{0 \to 1\}
$$

$$
  [2] = \{0 \to 1 \to 2\}
$$

etc.

The category $\Delta_a$ contains one more object, corresponding to the empty category $\emptyset$. When sticking to the above standard notation for the objects of $\Delta$, that extra object is naturally often denoted

$$
  [-1] = \emptyset
  \,.
$$

However, in contexts where only $\Delta_a$ and not $\Delta$ plays a role, some authors prefer to start counting with 0 instead of with $-1$. Then for instance the notation

$$
  \mathbf{0} = \emptyset
$$

$$
  \mathbf{1} = [0] =  \{0\}
$$

$$
  \mathbf{2} = [1]  = \{0 \to 1\}
$$

and generally

$$
  \mathbf{n} = [n-1]
$$

may be used.

=--

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

The [[skeleton|skeletal]] version of the **augmented simplex category** $\Delta_a$ can be presented as follows:

* objects are the finite totally ordered sets $\mathbf{n} \coloneqq \{0 \lt 1 \lt  \cdots \lt n-1\}$ for all $n \in \mathbb{N}$;

* morphisms  _generated_ by (are all expressible as finite compositions of) the following two elementary kinds of maps

  1. **face maps**: $\;\delta_i^n :\: \mathbf{n-1} \hookrightarrow \mathbf{n}\;$ is the injection whose image leaves out $i \in [n]\quad $ ($n \gt 0$ and $0 \leq i \lt n$);

  1. **degeneracy maps**: $\;\sigma_i^n :\: \mathbf{n+1} \to  \mathbf{n}\;$ is the surjection such that $\sigma_i(i) = \sigma_i(i+1) = i\quad $ ($n \gt 0$ and $0 \leq i \lt n$);

subject to the following relations, called the **simplicial relations** or **[[simplicial identities]]**:

$$
  \array{
     \delta_i^{n+1} \circ \delta_j^n
     =
     \delta_{j+1}^{n+1}\circ \delta_i^n
     &
     \qquad i \leq j
     \\
     \sigma_j^n \circ \sigma_i^{n+1}
     =
     \sigma_i^n \circ \sigma_{j+1}^{n+1}
     &
     \qquad i \leq j
  }
$$
$$
  \sigma_j^n \circ \delta_i^{n+1}
  =
  \left\lbrace
    \array{
      \delta_i^n \circ \sigma_{j-1}^{n-1}
      &
      \qquad i \lt j
      \\
      Id_n
      &
      \qquad i = j \;or\; i = j+1
      \\
      \delta^n_{i-1} \circ \sigma_{j}^{n-1}
      &
      \qquad j+1 \lt i
    }
  \right.
$$
whenever $i, j$ are chosen so that the maps are defined. For example, in
$$\delta_i^{n+1} \circ \delta_j^n
     =
     \delta_{j+1}^{n+1}\circ \delta_i^n
$$
we implicitly require      $0 \leq i, j \leq n$.

=--

## Properties

### Monoidal structure
 {#ordinal sum}

The addition of [[natural number]]s extends to a [[functor]] $\oplus : \Delta_a \times \Delta_a \to \Delta_a$  and $\oplus : \Delta \times \Delta \to \Delta$, by taking $\mathbf{m} \oplus \mathbf{n}$ to be the [[disjoint union]] of the underlying sets of $\mathbf{m}$ and $\mathbf{n}$, with the linear order that extends those on $\mathbf{m}$ and $\mathbf{n}$ by putting every element of $\mathbf{m}$ below every element of $\mathbf{n}$. This is called the _[[ordinal sum]]_ functor. If we visualise $\mathbf{n}$ as a totally ordered set $\{0 \lt 1 \lt \cdots \lt n-1\}$, and similarly for $\mathbf{m}$, then $\mathbf{m} \oplus \mathbf{n}$ looks like

$$
\mathbf{m} \oplus \mathbf{n} = \{0 \lt 1 \lt  \cdots \lt m-1 \lt 0^*\lt 1^* \lt \cdots \lt (n-1)^*\}
$$

where $k^*$ denotes $k$ considered as an element of $\mathbf{n}$.

Clearly $\oplus : \Delta_a \times \Delta_a \to \Delta_a$ acts on objects as
$$
  \mathbf{n} \oplus \mathbf{m} = \mathbf{n+m},
$$

On morphisms, given $f : \mathbf{m} \to \mathbf{m}'$ and $g : \mathbf{n} \to \mathbf{n}'$, we have
$$
  (f\oplus g)(i) = \left\lbrace
     \array{
       f(i) & if \; 0 \leq i  \leq m - 1
       \\
       m' + g(i-m) & if \; m \leq i \leq (m+n-1)
     }
  \right.
  \,.
$$
so that $f \oplus g$ can be visualised as $f$ and $g$ placed side by side.

It is easy to see now that $(\Delta_a,\oplus,\mathbf{0})$ is a [[strict monoidal category]].

It is important to note that this tensor does _not_ give a monoidal structure to $\Delta$, as that category does not contain the unit $\mathbf{0} = [-1] = \emptyset$.

Also note that this [[monoidal structure]] is _not_ [[braided monoidal category|braided]]!  One does have isomorphisms $\mathbf{m} \oplus \mathbf{n} \simeq \mathbf{n+m} \simeq \mathbf{n} \oplus \mathbf{m}$ for all $m,n$, but it has to be the identity on $\mathbf{n+m}$, and that is not [[bifunctorial]].

Under [[Day convolution]] this monoidal structure induces the [[join of simplicial sets]].

### $\Delta$ and $\Delta_a$ as 2-categories
 {#As2Categories}

Being full subcategories of the [[2-category]] $Cat$, $\Delta$ and $\Delta_a$ are themselves 2-categories: their [[2-cell|2-cells]] $f \Rightarrow g$ are given by the pointwise order on monotone functions.  Equivalently, they are generated under (vertical and [[horizontal composition|horizontal]]) composition by the inequalities
$$ \delta^n_{i+1} \leq \delta^n_i \qquad \qquad \sigma^n_i \leq \sigma^n_{i+1} \, .$$
Of course, the ordinal sum functor $\oplus$ extends to a [[2-functor]] in the obvious way.

For each $n$ there is [a string of adjunctions](https://ncatlab.org/nlab/show/adjoint+string)
$$
  \delta^n_{n-1} \dashv \sigma^n_{n-2} \dashv \delta^n_{n-2} \dashv
  \cdots \dashv \delta^n_1 \dashv \sigma^n_0 \dashv \delta^n_0
$$
where the counit of $\sigma_i \dashv \delta_i$ and the unit of $\delta_{i+1} \dashv \sigma_i$ are identities.

For each $n \geq 2$, the object $\mathbf{n+1}$ is given by the [[pushout]]
$$
\array{
\mathbf{n-1} & \overset{\delta_0}{\to} & \mathbf{n} \\
\mathllap{\scriptsize{\delta_{n-1}}} \downarrow & & \downarrow \mathrlap{\scriptsize{\delta_n}} \\
\mathbf{n} & \underset{\delta_0}{\to} & \mathbf{n+1}
}
$$

This means that $\Delta_a$ is generated as a 2-category by these pushouts and by taking adjoints of morphisms.  Its monoidal structure is also determined in this way: for each $n$, write $\bot_n = \delta_{n-1}\cdots\delta_2\delta_1$ for the (morphism $\mathbf{1} \to \mathbf{n}$ corresponding to the) least element $0$ of $\mathbf{n}$, and $\top_n = \delta_0\cdots\delta_0\delta_0$ for the greatest.  Then there are [[cospans]] $\mathbf{1} \to \mathbf{n} \leftarrow \mathbf{1}$ given by $\top_n$ and $\bot_n$, and each such is equivalent to the $(n-1)$ fold cospan composite (i.e. pushout) of $\mathbf{1} \to \mathbf{2} \leftarrow \mathbf{1}$ with itself.  The ordinal sum $\mathbf{n} \oplus \mathbf{m}$ is given by the composite
$$
\array{
  & & & & \mathbf{n} \oplus \mathbf{m} & & & & \\
  & & & \nearrow & & \nwarrow & & & \\
  & & \mathbf{n} & & & & \mathbf{m} & & \\
  & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\
  \mathbf{1} & & & & \mathbf{1} & & & & \mathbf{1}
}
$$
The universal property of pushouts, together with those of the initial and terminal objects $\mathbf{0},\mathbf{1}$, then suffices to define $\oplus$ as a 2-functor.

### Universal properties

The morphisms $\mathbf{0} \overset{\delta_0}{\to} \mathbf{1} \overset{\sigma_0}{\leftarrow} \mathbf{2}$ in $\Delta_a$ make $\mathbf{1}$ into a [[monoid object]].  Indeed, it is easy to see that
$$
\begin{aligned}
\delta^n_i & = \mathbf{i} \oplus \delta^0_0 \oplus \mathbf{n-i} \\
\sigma^n_i & = \mathbf{i} \oplus \sigma^1_0 \oplus \mathbf{n-i-1}
\end{aligned}
$$
so that the morphisms of $\Delta_a$ are generated under $\circ$ and $\oplus$ by $\delta^0_0$ and $\sigma^1_0$, together with exactly the equations needed to make them the structure maps of the monoid $[1]$.  The objects of $\Delta_a$ are the elements of the free monoid generated by $\mathbf{1}$ and $\oplus$.

$\Delta_a$ thus becomes the universal category-equipped-with-a-monoid, in the sense that for any strict monoidal category $B$, there is a bijection between monoids $(M,m,e)$ in $B$ and strict [[monoidal functors]] $\Delta_a \to B$ such that $\mathbf{1} \mapsto M$, $\sigma_0 \mapsto m$ and $\delta_0 \mapsto e$.

In particular, for $K$ a 2-category, [[monads]] in $K$ correspond to 2-functors $\mathbf{B}\Delta_a \to K$, where $\mathbf{B}\Delta_a$ is $\Delta_a$ considered as a one-object 2-category.  Because monads in $K$ are also the same as [[lax functors]] $1 \to K$, this correspondence exhibits $\mathbf{B}\Delta_a$ as the [[lax morphism classifier]] for the terminal category $1$.

When $\Delta_a$ is considered as a 2-category, a similar argument to the above shows that the one-object [[3-category]] $\mathbf{B}\Delta_a$ classifies [[lax-idempotent monads]]: given a 3-category $M$ and a lax-idempotent monad $t$ therein, there is a unique 3-functor $\mathbf{B}\Delta_a \to M$ sending $[1]$ to $t$, essentially because $\sigma^1_0 \dashv \delta^1_0 = \delta^0_0 \oplus \mathbf{1}$ with identity counit.

### Duality with intervals
 {#DualityWithIntervals}

Recall that an [[interval]] is a [[linearly ordered set]] with a top and bottom element; interval maps are [[monotone functions]] which preserve top and bottom.

Parallel to the categories $\Delta$ and $\Delta_a$, let $\nabla$ denote the category of finite intervals where the top and bottom elements are _distinct_, and let $\nabla_a$ denote the category of all finite intervals, including the terminal one where top and bottom coincide. Then we have [[duality|concrete dualities]], or equivalences of the form

$$\Delta_a^{op} \simeq \nabla_a; \qquad \Delta^{op} \simeq \nabla,$$

both induced by the [[duality|ambimorphic object]] $\mathbf{2}$, seen as both an ordinal and an interval. In other words, we have in each case an adjoint equivalence

$$Int(-, \mathbf{2})^{op} \dashv Ord(-, \mathbf{2})$$

inducing the first equivalence $Ord(-, \mathbf{2}): \Delta_a^{op} \to \nabla_a$, and the second equivalence by restriction.

This fact is mentioned in ([Joyal](#Joyal)), to help give some intuition for his [[Theta category|category]] $\Theta$ as dual to a category of disks. See also _[Interval -- Relation to simplices](interval#RelationToSimplices)_, and the section on dualities in ([Wraith](#Wraith)).

### Combinatorics

The homogenous symmetric function $h_m(x_1,x_2,{\dots},x_n)$ is the generating function for the set of all morphisms from $[m-1]$ to $[n-1]$.

As an order-preserving function between finite ordinals, any morphism $f : \mathbf{m} \to \mathbf{n}$ in $\Delta_a$ is completely specified by fixing $k$ elements of $\mathbf{n}$ as the [[image]] of $f$, together with a [[partition#of_numbers|composition]] of $\mathbf{m}$ into $k$ parts, each part denoting a non-empty, contiguous subset of elements of $\mathbf{m}$ sharing their value of $f$. That is, each such composition is given by a collection of $k$ interval parts $[0,i_1], [i_1 + 1, i_2], \ldots, [i_{k-1}+1, m-1]$, determined by a $(k-1)$-element subset $\{i_1, \ldots, i_{k-1}\}$ of an $(m-1)$-element set $\{0, \ldots, m-2\}$. Hence, there are a total of

$$
\sum_k \binom{n}{k} \binom{m-1}{k-1} = \sum_k \binom{n}{k} \binom{m-1}{m-k} = \binom{n+m-1}{m}
$$

different morphisms of type $\mathbf{m} \to \mathbf{n}$ in $\Delta_a$, where we obtain the expression on the right by applying the [[Chu–Vandermonde identity]].  For example, there are

$$
\binom{2}{1}\binom{2}{0} + \binom{2}{2}\binom{2}{1} = 2+2 = 4 = \binom{4}{3}
$$

different morphisms $\mathbf{3} \to \mathbf{2}$, corresponding to the four functions

1. $f(0) = f(1) = 0, f(2) = 1$
1. $f(0) = 0, f(1) = f(2) = 1$
1. $f(0) = f(1) = f(2) = 0$
1. $f(0) = f(1) = f(2) = 1$

The [stars and bars argument](https://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29) gives a direct [[bijective proof]] of the identity $|\Delta_a(\mathbf{m},\mathbf{n})| = \binom{n+m-1}{m}$: see [this comment](http://nforum.ncatlab.org/discussion/3636/number-of-morphisms-in-the-simplex-category/?Focus=29835#Comment_29835) on the nForum.

As some interesting special cases, taking $m=n$ gives the number of monotone endofunctions on $\mathbf{n}$ (OEIS sequence [A088218](https://oeis.org/A088218), or [A001700](https://oeis.org/A001700) if we consider endomorphisms $[n] \to [n] \in \Delta$), while taking $m=2$ gives the triangular numbers (OEIS sequence [A000217](https://oeis.org/A000217)).

### Extra Degeneracies

In some applications, in addition to the usual maps in the simplex category, one has "extra degeneracies" whose key property can be described as constructing an [[absolute limit]] of shape $\Delta$.

Let $\Delta_\bot \subseteq \Delta$ be the [[wide subcategory]] spanned by the functors that preserve minima. Equivalently, $\Delta_\bot \subseteq \Delta_*$ is the full subcategory of pointed objects of the form $(S, \bot_S)$ where $\bot_S$ is the minimum of $S$.

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

$1 \oplus - : \Delta_a \to \Delta_\bot$ is an [[absolute limit]] cone in the sense of [[(∞,1)-categories]] (and thus in the sense of ordinary categories).

=--

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

By the characterization of absolute limits by representable functors, we need to show every $\Delta_\bot(1 \oplus -, m) : \Delta_a^{\op} \to \infty Grpd$ is a colimit cocone. Observe $\Delta_\bot(1 \oplus -, m) \cong \Delta_a(-, m)$. Since the cone point is $\Delta_a(0, m) = *$, we need to show that the colimit of the simplicial [[∞-groupoid]] $\Delta(-, m)$ is contractible. Since the colimit (in simplicial ∞-gropuoids) of a simplicial set  is the precisely its homotopy type, the theorem follows from the fact that $\Delta^{m-1}$ is a contractible simplicial set.

=--

Since the retract of an absolute limit is again an absolute limit, we can generalize. Define $\Delta_x$ by the pushout in $(\infty,1)Cat$:

$$
\begin{matrix}
\Delta_a^{\leq 0} \times \Delta_a &\xrightarrow{\oplus}& \Delta_a
\\ \!\!\!\!\!\!\!\!\!\!\!\!(1 \oplus -) \times id \downarrow
& & \downarrow \iota\!
\\ \Delta_\bot^{\leq 1} \times \Delta_a &\xrightarrow{\rho}& \Delta_x
\end{matrix}
$$

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

$\iota : \Delta_a \to \Delta_x$ is an [[absolute limit]] cone in the sense of [[(∞,1)-categories]] (and thus in the sense of ordinary categories).

=--

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

$\rho$ transposes to a retraction diagram in the category of augmented simplicial objects of $\Delta_x$. This retraction expresses $\iota$ as a retract of the functor $\Delta_a \xrightarrow{1 \oplus -} \Delta_\bot \subseteq \Delta_a \xrightarrow{\iota} \Delta_x$. Since the first map is an absolute limit, so is the composite.

=--

The homotopy category $h \Delta_x$ has a similar property.

To mesh with the usual description of extra degeneracies, the categories $\Delta_\bot$ and $h \Delta_x$ can be given by

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

Let $\mathbf{C}$ be the category constructed from $\Delta_a$ by adjoining arrows $s_{-1} : m \to (m+1)$ for all $m \geq 0$, and by imposing the cosimplicial identities

* $s_{-1} d_0 = 1$
* $s_{-1} d_{k+1} = d_k s_{-1}$ for $k \geq 0$
* $s_{-1} s_{k+1} = s_k s_{-1}$ for $k \geq 0$

Then the monomorphism $1 \oplus - : \Delta_a \to h \Delta_x$ extends to an equivalence $\mathbf{C} \to h \Delta_x$.

Let $\mathbf{C}'$ be constructed by imposing the remaining cosimplicial identity
$s_{-1} s_0 = s_{-1} s_{-1}$. Then $1 \oplus - : \Delta_a \to \Delta_\bot$
extends to an equivalence $\mathbf{C}' \to \Delta_\bot$.

=--

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

The case of $\mathbf{C}'$ follows by the same argument showing that $\Delta$ is presented by the face and degeneracy maps modulo the cosimplicial identities by an algorithm reducing arrows to a normal form. The key observation is that a map of $\Delta$ is contained in $\Delta_\bot$ iff its normal form does not contain any copies of $d_0$.

The case of $\mathbf{C}$ can be determined by working through the construction of $h \Delta_x$ as a pushout of ordinary categories.

=--

The construction of $h \Delta_x$ from $\Delta_a$ consists of the usual definition of "extra degeneracies". The construction of $\Delta_\bot$ from $\Delta_a$ can be described as having "strong extra degeneracies".

## Related constructions ##

### Simplicial sets
 {#SimplicialSets}

[[presheaf|Presheaves]] on $\Delta$ are [[simplicial set|simplicial sets]]. Presheaves on $\Delta_a$ are **[[augmented simplicial sets]].

Under the [[Yoneda embedding]] $Y : \Delta \to $ [[SSet]] the object $[n]$ induces the standard simplicial $n$-[[simplex]] $Y([n]) =: \Delta^n$. So
in particular we have $(\Delta^n)[m] = Hom_{\Delta}([m],[n])$ and hence
$\Delta^n[m]$ is a finite set with $\binom{n+m+1}{n}$ elements.

The face and degeneracy maps and the relation they satisfy are geometrically best understood in terms of the [[full and faithful functor|full and faithful]] image under $Y$ in [[SSet]]:

* the face map $Y(\delta_i) : \Delta^{n-1} \to \Delta^{n}$ injects the standard simplicial $(n-1)$-simplex as the $i$th face into the standard simplicial $n$-simplex;

* the degeneracy map $Y(\sigma_i) : \Delta^{n+1} \to \Delta^{n}$ projects the standard simplicial $(n+1)$-simplex onto the standard simplicial $n$-simplex by collapsing its vertex number $i$ onto the face opposite to it.

#### Extra degeneracies

Presheaves on $\Delta_x$ are _(augmented) simplicial sets with extra degeneracies_,
and presheaves on $\Delta_a$ are _(augmented) simplicial sets with strong extra degeneracies_. The key feature is

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

If an augmented simplicial object $P : \Delta_a^{op} \to C$ can be extended to $\Delta_x^{op} \to C$, then $P$ is a colimit diagram.

=--

### Realization and nerve
 {#RealizationAndNerve}

There are important standard functors from $\Delta$ to other categories which _[[nerve and realization|realize]]_ $[n]$ as a concrete model of the standard $n$-[[simplex]].

* The functor $\Delta[-] : \Delta \to $ [[sSet]] (the [[Yoneda embedding]]) realizes $[n]$ as a [[simplicial set]].

* The functor $|\cdot| : \Delta \to $ [[Top]]

  sends $[n]$ to the standard topological $n$-simplex $[n] \mapsto \{(x_0, ... x_n) : 0 \leq x_0 \leq x_1 \leq \cdots \leq x_n \leq 1\}\subset \mathbb{R}^{n}$. This functor induced [[geometric realization]] of [[simplicial sets]].

* The functor $O : \Delta \to Str\omega Cat $
  sends $[n]$ to the $n$th [[oriental]]. This induces simplicial [[nerves]] of [[omega-category|omega-categories]].

  Under the functor $Str \omega Cat \to Cat$ which discards all higher morphisms and identifies all 1-morphisms that are connected by a 2-morphisms, this becomes again the identification of $\Delta$ with the full subbcategory of $Cat$ on linear [[quivers]] that we started the above definition with

  $$
    [n] \mapsto \{0 \to 1 \to \cdots \to n\}
    \,.
  $$

## Related concepts

* [[geometric shapes for higher structures]]

  * **simplex category**

  * [[cube category]]

  * [[globe category]]

  * [[cell category]]

  * [[Segal's category]]

* [[simplicial set]]

* [[augmented simplicial set]]

* [[join of simplicial sets]]

## References

An early description of the simplex category is in

* {#Grothendieck61} [[Alexander Grothendieck]], p. 107 (10 of 21) in: [[FGA]] _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), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-III.pdf), [web version](https://thosgood.com/fga/book/FGA-3-III)).

See also the references at [[simplicial set]].

See also:

* Section VII.5 of [[Categories Work|Categories for the Working Mathematician]]

* Section II.2 of P. Gabriel and M. Zisman, _Calculus of Fractions and Homotopy Theory_. Springer, 1967

* Section 2, Doctrines, of [[Ross Street]], [_Fibrations in Bicategories_](http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0), 1980

The relation to [[intervals]] and the generalization to the [[cell category]] is due to

* [[André Joyal]], _Disks, duality and $\Theta$-categories_, preprint, 1997 ([pdf](http://ncatlab.org/nlab/files/JoyalThetaCategories.pdf))
 {#Joyal}

A discussion of the opposite categories of $\Delta, \Delta_a$ and related categories can be found here:

* [[Gavin C. Wraith]], _Using the generic interval_, Cah. Top. Géom. Diff. Cat. **XXXIV** 4 (1993) pp.259-266. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1993__34_4/CTGDC_1993__34_4_259_0/CTGDC_1993__34_4_259_0.pdf))
 {#Wraith}

[[!redirects augmented simplex category]]

category: category


simplex in a lined topos > history

< [[simplex in a lined topos]]


simplical commutative differential graded algebra

Of course, a simplicial commutative differential graded algebra is a [[simplicial object]] in the category [[CDGA]] of commutative [[differential graded algebra|differential graded algebras]].

+--{: .query}
Note that we already have [[CoDGCA]].  ---[[Toby Bartels]]
=--


simplicial abelian group

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

#Contents#
* table of contents
{:toc}

## Definition

A *simplicial abelian group* is

* an [[abelian group|abelian]] [[group object]] [[internalization|internal to]] [[SimplicialSets]];

* a [[simplicial object]] [[internalization|internal to]] [[AbelianGroups]].

## Properties

### Basic properties

\begin{proposition}\label{FreeSimplicialAbelianGroupAdjunction}
**([[free simplicial abelian group]]-[[adjunction]])** \linebreak
  There is a [[pair]] of [[adjoint functors]]
  (a [[free functor|free]]$\dashv$[[forgetful]]-[[adjunction]])
  $$
    sAb
      \underoverset
        {\underset{ frgt }{\longrightarrow}}
        {\overset{\mathbb{Z}(-)}{\longleftarrow}}
        {\;\;\;\;\;\bot\;\;\;\;\;}
    sSet
  $$

  between [[SimplicialAbelianGroups]] and [[SimplicialSets]], where

  * the [[left adjoint]] is the [[free simplicial abelian group]]-[[functor]];

  * the [[right adjoint]] is the [[functor]] that assigns [[underlying]] [[simplicial sets]].

  This is a [[Quillen adjunction]] with respect to the [[classical model structure on simplicial sets]] and the projective [[model structure on simplicial abelian groups]].
\end{proposition}

### More properties

* [[Dold-Kan correspondence]]

* [[Eilenberg-Zilber theorem]]

* [[model structure on simplicial abelian groups]]

* [[cosimplicial abelian group]]

## Related concepts

* [[Moore complex]]

* [[Eilenberg-Zilber map]]

* [[simplicial group]], [[simplicial topological group]]

* [[free simplicial abelian group]]

[[!redirects simplicial abelian groups]]

[[!redirects sAb]]
[[!redirects sAbGrp]]
[[!redirects SimplicialAbelianGroups]]


simplicial approximation theorem

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

# Contents #
* table of contents
{: toc}

## Idea

The *simplicial approximation theorem* (also: *simplicial approximation lemma*) roughly says that if $X$ and $Y$ are [[simplicial complexes]] and $f \colon {|X|} \to {|Y|}$ is a [[continuous map]] between their [[topological realizations]], then after further [[subdivisions]] $X'$, $Y'$ there is a simplicial map $g: X' \to Y'$ such that ${|g|}$ is [[homotopy|homotopic]] to $f$.

## Statement

Let $X$ be a simplicial complex, with vertex set $X_0$. We recall that the geometric realization of $X$ is a space whose points may be described as functions $\alpha: X_0 \to [0, 1]$ such that $\alpha^{-1}((0, 1])$ is a simplex of $X$ (in particular, finite) and $\sum_{v \in X_0} \alpha(v) = 1$. For a simplex $s$ of $X$, we let ${|s|}$ denote the _closed_ (affine) simplex in the geometric realization ${|X|}$, defined by the formula

$${|s|} = \{\alpha \in {|X|}: \alpha(v) \gt 0 \; implies \; v \in s\}.$$

This may be identified with a standard affine simplex, and we topologize ${|s|}$ so that this identification is a homeomorphism. ${|X|}$ is then given the coherent topology: the largest topology rendering each inclusion ${|s|} \hookrightarrow {|X|}$ is continuous.

If $X, Y$ are simplicial complexes, then a function $f: X_0 \to {|Y|}$ can be extended linearly to a continuous map $\tilde{f}: {|X|} \to {|Y|}$ by the rule

$$\tilde{f}(\alpha) = \sum_{v \in X_0} \alpha(v)f(v),$$

provided that for every simplex $s$ of $X$, all convex combinations of elements of $f(s)$ lie in ${|Y|}$. Naturally this is the case if $f = j \circ \phi$ where $\phi: X_0 \to Y_0$ is a simplicial map and $j: Y_0 \to {|Y|}$ is the natural inclusion.

We may then define a general _subdivision_ (not necessarily an iterated barycentric [[subdivision]]) of a simplicial complex $X$ to be a simplicial complex $X'$ such that

* Vertices of $X'$ are points of ${|X|}$,

* For every simplex $s'$ of $X'$, there is a simplex of $X$ such that $s' \subseteq {|s|}$,

* The linear map $\tilde{i}: {|X'|} \to {|X|}$ induced from the inclusion $i: X' \hookrightarrow {|X|}$ is a homeomorphism.

Often one treats such $\tilde{i}$ as an identity map.

+-- {: .num_defn}
###### Definition
Let $X, Y$ be simplicial complexes, and let $f: {|X|} \to {|Y|}$ be a continuous map. A simplicial map $\phi: X \to Y$ is a **simplicial approximation** to $f$ if $f(\alpha) \in {|s|}$ implies ${|\phi|}(\alpha) \in {|s|}$.
=--

+-- {: .num_prop}
###### Proposition
Suppose $\phi: X \to Y$ is a simplicial approximation to a map $f: {|X|} \to {|Y|}$, and suppose $f = {|\phi|}$ on a subset $A \subseteq {|X|}$. Then there is a homotopy ${|\phi|} \simeq f \; rel \; A$.
=--

+-- {: .proof}
###### Proof
Define the homotopy $H$ by $H(\alpha, t) = t f(\alpha) + (1-t){|\phi|}(\alpha)$. This makes sense since if $f(\alpha) \in {|s|}$, then ${|\phi|}(\alpha) \in {|s|}$ and ${|s|}$ is closed under convex combinations. Clearly $H(\alpha, t) = f(\alpha)$ for all $t$ on a set where $f$ and ${|\phi|}$ agree.
=--

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

Given simplicial pairs $(X, A)$, $(Y, B)$ and a map of pairs $f: ({|X|}, {|A|}) \to ({|Y|}, {|B|})$, there is a subdivision $i: (X', A') \hookrightarrow ({|X|}, {|A|})$ and a simplicial approximation $\phi: (X', A') \to (Y, B)$ to $f \circ \tilde{i}: ({|X'|}, {|A'|}) \to ({|Y|}, {|B|})$.

Moreover, if $(X, A)$ is a finite simplicial pair, we may choose the subdivision to be an iterated barycentric subdivision $(sd^n X, sd^n A)$ for any sufficiently large $n$ (given $f$).

=--

## Related concepts

* [[cellular approximation theorem]]

## References

* {#Switzer75} [[Robert Switzer]], Lemma 6.8, p. 76 in: _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 ([doi:10.1007/978-3-642-61923-6](https://link.springer.com/book/10.1007/978-3-642-61923-6))

* {#FritschPiccini90} [[Rudolf Fritsch]], [[Renzo A. Piccinini]], Section 2.4 in: _Cellular structures in topology_, Cambridge Studies in Advanced Mathematics **19**, Cambridge University Press (1990) [[doi:10.1017/CBO9780511983948](https://doi.org/10.1017/CBO9780511983948), [pdf](https://epub.ub.uni-muenchen.de/4493/1/4493.pdf)]

* {#Jardine04} [[John F. Jardine]], *Simplicial approximation*, Theory and Applications of Categories, **12** 2 (2004) 34-72 [[tac:12-02](https://www.emis.de/journals/TAC/volumes/12/2/12-02abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/12/2/12-02.pdf)]


simplicial bar construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **simplicial bar construction** is a way of building a cofibrant [[resolution]] of a diagram, suitable for computing [[homotopy colimit|homotopy colimits]] in [[simplicial model category|simplicial model categories]].

## Definition

Let $\mathcal{V}$ be a [[cocomplete category|cocomplete]] [[symmetric monoidal category|symmetric]] [[monoidal closed category]] and let $\mathbb{C}$ be a small $\mathcal{V}$-[[enriched category|category]]. We define a [[simplicial object]] $W_\bullet (c', \mathbb{C}, c)$ for each pair $(c', c)$ of objects in $\mathbb{C}$ by
$$W_n (c', \mathbb{C}, c) = \coprod_{(c_0, \ldots, c_n)} \mathbb{C} (c_n, c') \otimes \mathbb{C} (c_{n-1}, c_n) \otimes \cdots \otimes \mathbb{C} (c_0, c_1) \otimes \mathbb{C} (c, c_0)$$
with the obvious face and degeneracy operators induced by the composition and identities of $\mathbb{C}$. Note that $W_\bullet (\blank, \mathbb{C}, \blank)$ is a $\mathcal{V}$-functor $\mathbb{C}^{op} \otimes \mathbb{C} \to [\mathbf{\Delta}^{op}, \mathcal{V}]$.

Let $\mathcal{M}$ be a [[copower|tensored]] $\mathcal{V}$-category. The **two-sided simplicial bar construction** of a $\mathcal{V}$-diagram $F : \mathbb{C} \to \mathcal{M}$ weighted by a $\mathcal{V}$-functor $G : \mathbb{C}^{op} \to \mathcal{V}$ is a simplicial object $B_\bullet (G, \mathbb{C}, F)$ equipped with isomorphisms
$$\mathcal{M} (B_n (G, \mathbb{C}, F), M) \cong \int_{(c', c) : \mathbb{C}^{op} \otimes \mathbb{C}} \mathcal{V} (W_n (c', \mathbb{C}, c), \mathcal{M}(G c' \odot F c, M))$$
that are natural in $n$ and $\mathcal{V}$-natural in $M$, where the integral sign on the right hand side denotes a $\mathcal{V}$-[[end]].

Now suppose a [[cosimplicial object]] $\Delta^\bullet : \mathbf{\Delta} \to \mathcal{V}$ is given, so that we may define the [[realisation]] of a simplicial object $X_\bullet$ in a $\mathcal{V}$-category as the [[weighted colimit]] $\left| X_\bullet \right| = \Delta^\bullet \star X_\bullet$. The **two-sided bar construction** of $F : \mathbb{C} \to \mathcal{M}$ weighted by $G : \mathbb{C}^{op} \to \mathcal{V}$ is then defined to be the geometric realisation of the two-sided simplicial bar construction:
$$B (G, \mathbb{C}, F) = \left| B_\bullet (G, \mathbb{C}, F) \right|$$

## Properties

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

If $\mathcal{M}$ is a tensored $\mathcal{V}$-category and has coproducts for small families of objects, then $\mathcal{M}$ has two-sided simplicial bar constructions for all diagrams and weights. If $\mathcal{M}$ is moreover $\mathcal{V}$-cocomplete, then $\mathcal{M}$ also has two-sided bar constructions.

=--

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

If $\mathcal{M}$ is a tensored $\mathcal{V}$-category and has coproducts for small families of objects, then:

* For each $\mathcal{V}$-diagram $F : \mathbb{C} \to \mathcal{M}$, the $\mathcal{V}$-functor $B_n (-, \mathbb{C}, F) : [\mathbb{C}^{op}, \mathcal{V}] \to \mathcal{M}$ has a right $\mathcal{V}$-[[adjoint functor|adjoint]].

If $\mathcal{M}$ is also _cotensored_, then:

* For each weight $G : \mathbb{C}^{op} \to \mathcal{V}$, the $\mathcal{V}$-functor $B_n (G, \mathbb{C}, -) : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}$ has a right $\mathcal{V}$-adjoint.

=--

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

Let $W (c', \mathbb{C}, c)$ be the realisation of $W_\bullet (c', \mathbb{C}, c)$. If $\mathcal{M}$ is a cotensored $\mathcal{V}$-cocomplete $\mathcal{V}$-category, then there are isomorphisms
$$\mathcal{M} (B (G, \mathbb{C}, F), M) \cong \int_{(c', c) : \mathbb{C}^{op} \otimes \mathbb{C}} \mathcal{V} (W (c', \mathbb{C}, c), \mathcal{M} (G c' \odot F c, M))$$
that are $\mathcal{V}$-natural in $M$.

=--

The next theorem is essentially a version of the Fubini theorem for coends.

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

Let $F : \mathbb{C} \to \mathcal{M}$, $G : \mathbb{D}^{op} \to \mathcal{V}$, and $H : \mathbb{C}^{op} \times \mathbb{D} \to \mathcal{V}$ be $\mathcal{V}$-functors. If $\mathcal{M}$ is a _cotensored_  $\mathcal{V}$-cocomplete $\mathcal{V}$-category, then we have the following isomorphism:
$$B (B (G, \mathbb{D}, H), \mathbb{C}, F) \cong B (G, \mathbb{D}, B (H, \mathbb{C}, F))$$

=--

## Applications

As stated in the introduction, the two-sided simplicial bar construction can be used to compute homotopy colimits.

Let us write $y c$ for the representable $\mathcal{V}$-functor $\mathbb{C}(-, c)$ and $B (\mathbb{C}, \mathbb{C}, F)$ for the diagram $c \mapsto B (y c, \mathbb{C}, F)$.

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

Let $\mathcal{M}$ be a [[simplicial model category]] and let $\mathbb{C}$ be a small category.

1. For each diagram $F : \mathbb{C} \to \mathcal{M}$, there is a [[weak equivalence]] $B (y c, \mathbb{C}, F) \to F c$ that is natural in $F$ and $c$.
2. The functor $colim : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}$ preserves weak equivalences between diagrams of the form $B (\mathbb{C}, \mathbb{C}, F)$ where $F$ is a diagram such that each $F c$ is a cofibrant object in $\mathcal{M}$.
3. The functor $B (1, \mathbb{C}, Q -) : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}$, where $Q : \mathcal{M} \to \mathcal{M}$ is a cofibrant replacement functor, is a left [[derived functor]] for $colim : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}$, i.e. it computes [[homotopy colimit|homotopy colimits]].

=--

## References

* [[Mike Shulman|Shulman, M.]], _Homotopy limits and colimits and enriched homotopy theory_ ([arXiv:math/0610194](http://arxiv.org/abs/math/0610194)).


simplicial C-infinity-ring

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A **simplicial $C^\infty$-ring** is a [[simplicial object]] in the category of [[generalized smooth algebra|C∞-rings]].

## Applications

When equipped with the [[model structure on simplicial algebras]] over the [[Lawvere theory]] [[CartSp]], simplicial $C^\infty$-rings are a model for [[smooth (∞,1)-algebras]], hence for $(\infty,1)$-algebras over [[CartSp]] regarded as an  [[(∞,1)-algebraic theory]].

Therefore, in higher analogy to how $C^\infty$-rings serve as function algebras on [[smooth loci]] in [[differential geometry]], so simplicial $C^\infty$-rings serve as function rings on [[derived smooth manifold]]s and more general spaces in [[derived differential geometry]].

## References

* [[David Spivak]], _Derived smooth manifolds_ ([arXiv:0810.5174](http://arxiv.org/abs/0810.5174))
{#Spivak}

* Dennis Borisov, Justin Noel, _Simplicial approach to derived differential manifolds_ ([arXiv:1112.0033](http://arxiv.org/abs/1112.0033))

[[!redirects simplicial C∞-ring]]
[[!redirects simplicial C-∞-ring]]
[[!redirects simplicial C? ring]]
[[!redirects simplicial C-∞ ring]]
[[!redirects simplicial C-infinity-ring]]
[[!redirects simplicial C-infinity ring]]
[[!redirects simplicial C∞-rings]]
[[!redirects simplicial C-∞-rings]]
[[!redirects simplicial C? rings]]
[[!redirects simplicial C-∞ rings]]
[[!redirects simplicial C-infinity-rings]]
[[!redirects simplicial C-infinity rings]]


simplicial category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The term **simplicial category** has at least three common meanings.  Thus, to avoid ambiguity, it is perhaps better to avoid it entirely and use an equivalent, unambiguous term for the particular meaning one has in mind.

First of all

* The **simplicial category** $\Delta$ is the domain category for the presheaf category of [[simplicial set|simplicial sets]].  This is the [[full subcategory]] of [[Cat]] on categories which are linear [[quiver]]s, or equivalently the category of finite, non-empty totally ordered sets and order-preserving functions between them.  To avoid ambiguity, $\Delta$ may also be called the **[[simplex category]]** or the **simplicial indexing category**.

More importantly, there are these two uses of the word:

1. A **simplicial category** is a [[simplicial object]] in [[Cat]] (that is, a functor from $\Delta^{op}$ to $Cat$), just like a simplicial set is a simplicial object in [[Set]], a simplicial space is a simplicial object in [[Top]], a simplicial abelian group is a simplicial object in [[Ab]], and so on.  To avoid ambiguity, **[[simplicial objects in Cat]]** may be called exactly that.  They may equivalently be regarded as [[internal categories]] in simplicial sets.

1. A **simplicial category** also frequently means a category [[enriched category|enriched]] over the category of [[simplicial sets]] ([Quillen 67, II.1](#Quillen67)), i.e. an [[sSet]]-[[enriched category]].  Such categories can be identified with simplicial objects in [[Cat]] all of whose face and degeneracy morphisms are [[bo functor|bijective on objects]].  To avoid ambiguity, categories enriched over simplicial sets may be called **[[simplicially enriched category|simplicially enriched categories]]**.

The analogous issue arises for "[[simplicial groupoids]]" etc., see also there.

Of course there are close relations between these three meanings. In particular [[simplicially enriched categories]] may be identified with those [[simplicial objects in Cat]] whose face and degeneracy maps are constant on [[objects]] and only affect the [[morphisms]].

An example of a text that speaks about what elsewhere are mostly regarded as [[simplicially enriched categories]] as [[simplicial objects in Cat]] which are constant on objects is ([Lydakis 98, around def. 3.2, remark 3.6](#Lydakis98)).

## Related concepts

* **simplicial category**

  * [[simplicially enriched category]]

  * [[simplicial object in Cat]]

* [[simplicial groupoid]]

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

## References

* {#Quillen67} [[Dan Quillen]], chapter II, section 1 of _Homotopical algebra_, Lecture Notes in Mathematics __43__, Springer-Verlag 1967, iv+156 pp.

* {#Lydakis98} Lydakis, _Simplicial functors and stable homotopy theory_ Preprint, available via Hopf archive, 1998 ([pdf](http://hopf.math.purdue.edu/Lydakis/s_functors.pdf))

[[!redirects simplicial categories]]


simplicial classifying space

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

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[simplicial group]], there is a [[reduced simplicial set|reduced]] [[simplicial set]], traditionally denoted $\overline W G$ and called the *[[classifying space]]* or *classifying complex* of $G$, which is a model for the [[delooping]] of $G$ and such that the [[functor]] $\overline{W}(-)$ is [[right adjoint]] to the standard [[simplicial loop space]]-construction $G$ (here denoted by $L$ to avoid a clash of notations).

This pair of [[adjoint functors]]
$$
  SimplicialGroups
    \underoverset
      {\;\;\;\underset{\overline{W}}{\longrightarrow}\;\;\;}
      {\;\;\;\overset{L}{\longleftarrow}\;\;\;}
      {\bot}
  SimplicialSets_{red}
$$
is a [[Quillen equivalence]] (Prop. \ref{QuillenEquivalenceBetweenSimplicialGroupsAndReducedSimplicialSets} below) between the [[model structure on simplicial groups]] and the [[model structure on reduced simplicial sets]], modelling [[looping and delooping]] of [[homotopy types]] in [[simplicial homotopy theory]].

The construction generalized from [[simplicial groups]] to [[simplicial groupoids]], where the groupoidal $\overline{W}$ is accompanied by a corresponding [[left adjoint]] known as the *[[Dwyer-Kan loop groupoid]]-construction*, and together they constitute  a [[Quillen equivalence]] between then [[model structure on simplicial groupoids]] and the [[classical model structure on simplicial sets]], exhibiting [[simplicial groupoids]] as an [[equivalence of (infinity,1)-categories|equivalent]] presentation of [[classical homotopy theory]].

## Definition

### In components

\begin{definition}\label{WGInComponents}
**(standard universal principal simplicial complex)** \linebreak
  For $G$ a [[simplicial group]],
  one writes
  $$
    W G
    \;\in\;
    SimplicialSets
  $$
  for the the [[simplicial set]] whose

  * underlying sets are

    $$
      (W G)_n
      \;\coloneqq\;
      G_{n} \times G_{n-1} \times \cdots \times G_0
    $$

  * face maps are given by

    \[
      \label{FaceMapsOfWG}
      \begin{aligned}
      &
      d_i
      \big(
        g_n, g_{n-1}, \cdots, g_0
      \big)
      \\
      &
      \;\coloneqq\;
      \left\{
      \array{
        \big(
          d_i(g_n),
          \,
          d_{i-1}(g_{n-1}),
          \,
          \cdots
          ,\,
          d_0(g_{n-i}) \cdot g_{n-i-1},
          \,
          g_{n -  i - 2},
          \,
          \cdots,
          \,
          g_0
        \big)
        & \text{if} &
        i \lt n
        \\
        \big(
          d_n(g_n),
          \,
          d_{n-1}(g_{n-1}),
          \,
          \cdots,
          \,
          d_1(g_1)
        \big)
        & \text{if} &
        i = n
      }
      \right.
      \end{aligned}
    \]

* degeneracy maps are given by

  \[
    \label{DegeneracyMapsOfWG}
    \begin{aligned}
    &
    s_i(g_n, g_{n-1}, \cdots, g_0)
    \\
    &
    \;\coloneqq\;
    \big(
      s_i(g_n),
      \,
      s_{i - 1}(g_{n-1}),
      \,
      \cdots,
      \,
      s_0(g_{n-i}),
      \,
      e,
      \,
      g_{n-i-1},
      \,
      \cdots,
      \,
      g_0
    \big)
    \,,
    \end{aligned}
  \]

  where $e$ denotes the respective [[neutral element]].

This carries a $G$-[[action]] by left multiplication on the top degree component:

\[
  \label{GActionOnWG}
  \array{
    G \times W G &\overset{}{\longrightarrow}& W G
    \\
    \big(h_n, (g_n, g_{n-1}, \cdots, g_0)\big)
     &\mapsto&
    (h_n \cdot g_n, \, g_{n-1}, \cdots, g_0)
    \mathrlap{\,.}
  }
\]

\end{definition}

\begin{remark}
It is the straightforward simplicial incarnation of the left $G$-action (eq:GActionOnWG) that singles out the model $W G$ (Def. \ref{WGInComponents}) for the universal simplicial principal space. For another model with an alternative good property see at *[[groupal model for universal principal simplicial complex]]*.
\end{remark}

\begin{definition}\label{BarWGInComponents}
**(standard simplicial classifying complex)** \linebreak
For $G$ a [[simplicial group]], its *standard simplicial classifying complex* is the
[[quotient]] of $W G$ (Def. \ref{WGInComponents}) by its $G$-action (eq:GActionOnWG)

\[
  \label{OverlineWGAsQuotientOfWG}
  \overline{W} G
  \;\coloneqq\;
  (W G) / G
  \,.
\]

The corresponding quotient [[coprojection]], whose [[fiber]] is, manifestly, $G$

\[
  \label{TheUniversalSimplicialPrincipalBundle}
  \array{
    G &\xhookrightarrow{fib}& W G
    \\
    && \big\downarrow
    \\
    && \overline{W} G \mathrlap{\; \coloneqq (W G)/G}
  }
\]

is known as the standard model for the simplicial $G$-[[universal principal bundle]] (see [below](#ClassificationOfSimplicialPrincipalBundles)).

{#StructureMapsOfSimplicialClassifyingComplex} This means, under the [[isomorphism]]

$$
  \big(
    \overline{W}G
  \big)_{n}
  \;\coloneqq\;
  (W G)_n/G_n
  \;\simeq\;
  G_n/G_n \times G_{n-1} \times \cdots \times G_0
  \;\simeq\;
  G_{n-1} \times \cdots \times G_{0}
  \,,
$$

that the above face maps (eq:FaceMapsOfWG) and degeneracy maps (eq:DegeneracyMapsOfWG) of $W G$ imply the following structure maps on the simplicial classifying complex:

$$
  \overline{W}G
  \;\;\;
  \in
  \;
  SimplicialSets
$$

* underlying sets are

  $$
    (\overline{W}G)_n
    \;\coloneqq\;
    G_{n-1} \times \cdots \times G_1 \times G_0
    \,;
  $$

* face maps are given by:

   \[
      \label{FaceMapsOfbarWG}
      \begin{aligned}
      &
      d_i
      \big(
        g_{n-1}, \cdots, g_0
      \big)
      \\
      &
      \;\coloneqq\;
      \left\{
      \array{
        \big(
          g_{n - 2},
          \,
          \cdots,
          \,
          g_0
        \big)
        & \text{if} &
        i = 0
        \\
        \big(
          d_{i-1}(g_{n-1}),
          \,
          \cdots
          ,\,
          d_0(g_{n-i}) \cdot g_{n-i-1},
          \,
          g_{n -  i - 2},
          \,
          \cdots,
          \,
          g_0
        \big)
        & \text{if} &
        0 \lt i \lt n
        \\
        \big(
          d_{n-1}(g_{n-1}),
          \,
          \cdots,
          \,
          d_1(g_1)
        \big)
        & \text{if} &
        i = n
        \mathrlap{\,;}
      }
      \right.
      \end{aligned}
    \]

* degeneracy maps are given by:

  \[
    \label{DegeneracyMapsOfbarWG}
    \begin{aligned}
    &
    s_i(g_{n-1}, \cdots, g_0)
    \\
    &
    \;\coloneqq\;
    \left\{
    \array{
    \big(
      e,
      \,
      g_{n-1},
      \,
      \cdots,
      \,
      g_0
    \big)
    & \text{if} &
    i = 0
    \\
    \big(
      s_{i - 1}(g_{n-1}),
      \,
      \cdots,
      \,
      s_0(g_{n-i}),
      \,
      e,
      \,
      g_{n-i-1},
      \,
      \cdots,
      \,
      g_0
    \big)
    & \text{if} &
    0 \lt i
    \mathrlap{\,.}
    }
    \right.
    \end{aligned}
  \]

\end{definition}

(This goes back to [MacLane 1954, p. 3](#MacLane54), [Kan 1958, Def. 10.3](#Kan58); the above follows [Goerss & Jardine 1999/2009, p. 269](#GoerssJardine09).)

More generally:
\begin{definition}\label{BarWForSimplicialGroupoid}
 For $\mathcal{G}$ a Dwyer-Kan [[simplicial groupoid]], $\overline{W}\mathcal{G}$ is the [[simplicial set]] with

1. set of vertices (0-cells) equal to the set of [[objects]] of $\mathcal{G}$,

1. set of [[n-simplex|$n$-simplices]] for $n \geq 1$ equal to sequences of morphisms of the form

   $$
     x_n
       \overset{f_{n-1}}{\longrightarrow}
     x_{n-1}
       \overset{f_{n-2}}{\longrightarrow}
     x_{n-2}
       \to \cdots \to
     x_1
       \overset{f_{0}}{\longrightarrow}
     x_0
   $$

   where

   * $x_i \in Obj(\mathcal{G})$

   * $f_k \in Mor(\mathcal{G}_k)$

\end{definition}
[[Dwyer & Kan 1984, §3.2](#DwyerKan84)]

\begin{remark}\label{Decalage}
**([[décalage]])** \linebreak
Conversely, comparison of Def. \ref{BarWGInComponents} with Def. \ref{WGInComponents} shows that $W G$ is obtained from $\overline{W} G$ by shifting down in degree and discarding the 0th face- and degeneracy maps:

$$
  (W G)_{n} \;\simeq\; (\overline{W}G)_{n+1}
$$

$$
  \begin{aligned}
    d^{W G}_i & \;=\; d^{ \overline{W}G }_{i+1}
    \\
    s^{W G}_i & \;=\; s^{\overline{W}G}_{i + 1}
    \mathrlap{\,.}
  \end{aligned}
$$

One refers to this relation as saying that $W G$ is the *[[décalage]]* of $\overline{W}G$, in the form

$$
  W G \;=\; Dec^0\big( \overline{W}G  \big)
  \,.
$$

See the example [there](decalage#ExampleForSimplicialClassifyingSpaces).

\end{remark}

\begin{example}\label{LowDimensionCellsOfWG}
**(low-dimension cells of $W G$)** \linebreak
  Unwinding the definition of the face maps (eq:FaceMapsOfWG), one finds that the  generic [[1-simplex]] in $W G$ (Def. \ref{WGInComponents}) looks as follows:

  \begin{tikzcd}
    d_1(g_1)
    \ar[
      rr,
      "{(g_1, g_0)}"
    ]
    &&
    d_0(g_1)\cdot g_0
    \mathrlap{\,,}
  \end{tikzcd}

while the generic [[2-simplex]] in $W G$ looks as follows:

  \begin{tikzcd}[column sep=-10pt, row sep=large]
    &&
    \scalebox{.7}{$
      {d_0 d_2(g_2) \cdot d_1(g_1)}
      \,=\,
      {d_1 d_0(g_2) \cdot d_1(g_1)}
    $}
    {}
    \ar[
      ddrr,
      "{
        \scalebox{1.3}{$($}
          d_0(g_2) \cdot g_1,
          \,
          g_0
        \scalebox{1.3}{$)$}
      }"{above, sloped, pos=.6},
      "\ "{below, name=s, pos=.02}
    ]
    \\
    \\
    {
      {\phantom{=\,} d_1 d_2(g_2)}
      \atop
      {=\, d_1 d_1(g_2)}
    }
    \ar[
      rrrr,
      "{
        \scalebox{1.3}{$($}
        d_1(g_2),
        \,
        d_0(g_1)\cdot g_0
        \scalebox{1.3}{$)$}
      }"{below},
      "{\ }"{above, name=t}
    ]
    \ar[
      uurr,
      "{
        \scalebox{1.3}{$($}
          d_2(g_2),
          \,
          d_1(g_1)
        \scalebox{1.3}{$)$}
      }"{above, sloped, pos=.4}
    ]
    &&&&
    {
      {\phantom{=,} d_0 d_0(g_2) \cdot d_0(g_1) \cdot g_0}
      \atop
      {=\,  d_0 d_1(g_2)\cdot d_0(g_1) \cdot g_0}
    }
    \ar[
      from=s,
      to=t,
      Rightarrow,
      "{
        (g_2, g_1, g_0)
      }"{description}
    ]
  \end{tikzcd}

\end{example}

\begin{example}\label{SimplicialClassifyingSpaceOfAnOrdinaryGroup}
**(simplicial classifying space of an ordinary group)** \linebreak
  In the special case that
  $$
    G \in Groups \overset{const}{\hookrightarrow} SimplicialGroups
  $$
  is an ordinary [[discrete group]] regarded as a [[simplicial group]] (which is [[constant functor|constant]] as a [[functor]] on the [[opposite category|opposite]] [[simplex category]]) the definitions in Def. \ref{WGInComponents} reduce as follows:

The simplicial set $W G$ is that whose

  * underlying sets are
    $$
      (W G)_n
      \;\coloneqq\;
      G^{\times_{n + 1}}
    $$

  * face maps are given by

    $$
      d_i
      \big(
        g_n, g_{n-1}, \cdots, g_0
      \big)
      \;\coloneqq\;
      \left\{
      \array{
        \big(
          g_n,
          \,
          g_{n-1},
          \,
          \cdots
          ,\,
          g_{n-i} \cdot g_{n-i-1},
          \,
          g_{n -  i - 2},
          \,
          \cdots,
          \,
          g_0
        \big)
        & \text{if} &
        i \lt n
        \\
        \big(
          g_n,
          \,
          g_{n-1},
          \,
          \cdots,
          \,
          g_1
        \big)
        & \text{if} &
        i = n
      }
      \right.
    $$

* degeneracy maps are given by

  $$
    s_i(g_n, g_{n-1}, \cdots, g_1)
    \;\coloneqq\;
    \big(
      g_n,
      \,
      g_{n-1},
      \,
      \cdots,
      \,
      g_{n-i},
      \,
      e,
      \,
      g_{n-i-1},
      \,
      \cdots,
      \,
      g_0
    \big)
    \,,
  $$

This identifies

\[
  \label{SimplicialClassifyingSpaceAsNerve}
  W G
  \;=\;
  N
  \big(
    G \times G
      \rightrightarrows
    G
  \big)
\]

with the [[nerve]] of the [[action groupoid]] of $G$ acting on itself by right multiplication ([[isomorphism|isomorphic]] to the [[pair groupoid]] on the underlying set of $G$):

\begin{tikzcd}
  & g_2 g_1
  \ar[
    dr,
    "{(g_2 g_1, g_0)}"{sloped},
    ""{name=s, below, pos=0.01}
  ]
  \\
  g_2
  \ar[
    ur,
    "{(g_2, g_1)}"{sloped}
  ]
  \ar[
    rr,
    "{(g_2, g_1 g_0)}"{below},
    ""{name=t, above}
  ]
  &
  {}
  &
  g_2 g_1 g_0
  \ar[
    from=s, to=t,
    Rightarrow,
    "{(g_2, g_1, g_0)}"{description}
  ]
\end{tikzcd}

Finally this means that the simplicial classifying complex (eq:OverlineWGAsQuotientOfWG) of an ordinary group is isomorphic to the [[nerve]] of its [[delooping groupoid]]:

$$
  \overline{W}G
  \;\simeq\;
  N
  \big(
    G \rightrightarrows \ast
  \big)
  \,.
$$

\end{example}

### Via total simplicial sets
 {#ViaTotalSimplicialSets}

Equivalently, $\overline{W}(-)$ is the following [[composition|composite]] [[functor]]:

$$
  \overline{W}
  \;\colon\;
  [\Delta^{op}, Groups]
  \overset
    {\;\;[\Delta^{op}, \mathbf{B}]\;\;}
    {\longrightarrow}
  [\Delta^{op}, Groupoids]
  \overset
    {\;\;[\Delta^{op}, N]\;\;}
    {\longrightarrow}
  [\Delta^{op}, SimplicialSets]
  \overset
    {\;\;\sigma_\ast\;\;}
    {\longrightarrow}
  SimplicialSets
  \,.
$$

([Stevenson 11, Lemma 15](#Stevenson11) following [[John Duskin]], see also [NSS 12, Def. 3.26](#NSS12))

Here:

* $\mathbf{B} \;\colon\; Groups \to Groupoids$ forms the one-object groupoid with [[hom-set]] the given group (the *[[delooping groupoid]]*);

* $N \;\colon\; Groupoids \longrightarrow SimplicialSets $ is the [[nerve]] construction;

* $\sigma_\ast$ is the [[total simplicial set]]-functor ([[right adjoint]] to pre-composition with [[ordinal sum]]).

## Properties

In all of the following, $G$ is any simplicial group.

### Basic properties
 {#KanFibrancy}

\begin{prop}\label{SimplicialClassifyingSpaces}
**([[simplicial classifying spaces]] are [[Kan complexes]])** \linebreak
  The underlying simplicial set of any simplicial classifying $\overline{W}G$ (Def. \ref{BarWGInComponents}) is a [[Kan complex]].
\end{prop}
(e.g. [Goerss & Jardine 09, Sec. V Cor. 6.8 (p. 287)](#GoerssJardine09))
\begin{proof}
This follows as the combination of the following facts:

1. every [[simplicial group]] is fibrant in the projective [[model structure on simplicial sets]] ([this Prop.](model+structure+on+simplicial+groups#EveryObjectIsFibrant));

1. $\overline{W}(-)$ is a [[right Quillen functor]] from there to the injective [[model structure on reduced simplicial sets]] (Prop. \ref{QuillenEquivalenceBetweenSimplicialGroupsAndReducedSimplicialSets});

1. every injectively fibrant reduced simplicial set is a Kan complex ([this Prop.](model+structure+on+reduced+simplicial+sets#FibrationsInTermsOfKanFibrations)).

\end{proof}

\begin{prop}\label{CoprojectionOutOfBorelConstructionIsKanFibration}
  For a [[simplicial group action]] of $G$ on a [[Kan complex]] $X$, the canonical coprojection from the [[Borel construction]] to the simplicial classifying space is a [[Kan fibration]]:
$$
  (W G \times X)/G
  \xrightarrow{ \;\in Fib \; }
  \overline{W}G
  \,.
$$
\end{prop}
\begin{proof}
  By the fact ([this Prop.](Borel+model+structure#QuillenEquivalenceToSliceOverSimplicialClassifyingSpace)) that the [[Borel construction]] is a [[right Quillen functor]] from the [[model structure on simplicial group actions]] to the [[slice model structure]] of the [[classical model structure on simplicial sets]] over the simplicial classifying space; see [this Example](Borel+construction#ProjectionToSimplicialClassifyingSpaceIsKanFibration).
\end{proof}

In particular:
\begin{prop}\label{UniversalSimplicialPrincipalBundleIsKanFibration}
  The coprojection $W G \overset{}{\longrightarrow} \overline{W}G$
  (eq:TheUniversalSimplicialPrincipalBundle)
  is a [[Kan fibration]].
\end{prop}
(e.g [Goerss & Jardine 09, Sec. V Lemma 4.1 (p. 270)](#GoerssJardine09))
\begin{proof}
  This is the special case of Prop. \ref{CoprojectionOutOfBorelConstructionIsKanFibration}
for $X = G$ equipped with the left multiplication [[simplicial group action|action]] on itself, using again that
the underlying simplicial set of any simplicial group is a Kan complex
([this Prop.](model+structure+on+simplicial+groups#EveryObjectIsFibrant)):
$$
  \array{
    G &\longrightarrow& W G
    \\
    &&
    \big\downarrow{}^{\mathrlap{\in Fib}}
    \\
    && \overline{W}G
  }
  {\phantom{AAAAAA}}
  \simeq
  {\phantom{AAAA}}
  \left(
  W G \times
  \left(
    \array{
      G
      \\
      \downarrow
      \\
      \ast
    }
  \right)
  \right)
  \big/ G
  \,.
$$
\end{proof}

\begin{prop}\label{UniversalPrincipalSimplicialComplexIsContractible}
  The simplicial set $W G$ is [[contractible]].
\end{prop}
(e.g [Goerss & Jardine 09, Sec. V Lemma 4.6 (p. 270)](#GoerssJardine09), see also the discussion at *[[décalage]]*)

\begin{prop}\label{HomotopyGroupsOfBarWG}
  The [[simplicial homotopy groups]] of $\overline{W} G$ are those of $G$, shifted up in degree by one:

$$
  \pi_n(G) \;\simeq\; \pi_{n+1}\big(\overline{W}(G)\big)
  \,.
$$
\end{prop}
\begin{proof}
  By Prop. \ref{UniversalSimplicialPrincipalBundleIsKanFibration} the [[universal simplicial principal bundle]] (eq:TheUniversalSimplicialPrincipalBundle) is a [[Kan fibration]] between [[Kan complexes]] (by Prop. \ref{SimplicialClassifyingSpaces} and [this Prop.](simplicial+group#EverySimplicialGroupIsAKanComplex)). Therefore this is a [[homotopy fiber sequence]] (by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback))
$$
  G
  \xrightarrow{hofib}
  W G
  \xrightarrow{\;\;q\;\;}
  \overline{W}G
  \,.
$$
This implies a [[long exact sequence of homotopy groups]] of the form
$$
  \cdots
  \xrightarrow{\;}
  \pi_{n+1}(W G)
  \xrightarrow{\;\;}
  \pi_{n+1}(\overline{W}G)
  \xrightarrow{\;\;}
  \pi_n(G)
  \xrightarrow{\;\;}
  \pi_n(W G)
  \xrightarrow{\;}
  \cdots
  \,.
$$
But Prop. \ref{UniversalPrincipalSimplicialComplexIsContractible} says that
$\pi_n(W G)$ is [[trivial group|trivial]] for all $n$, so that this collapses to [[short exact sequences]]:

$$
  0
  \xrightarrow{\;\;}
  \pi_{n+1}(\overline{W}G)
  \xrightarrow{\;\simeq\;}
  \pi_n(G)
  \xrightarrow{\;\;}
  0
$$
which exhibit the claim to be proven.
\end{proof}

\begin{proposition}\label{DeloopedKanFibrationOfHomomorphismOfSimplicialGroups}
  Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a [[homomorphism]] of [[simplicial groups]] which is a [[Kan fibration]].
Then the induced morphism of [[simplicial classifying spaces]]
$\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a [[Kan fibration]] if and only if $\phi$ is a [[surjection]] on [[connected components]]: $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$.
\end{proposition}
([Goerss & Jardine, Ch. V, Cor. 6.9](#GoerssJardine09), see the proof [here](model+structure+on+reduced+simplicial+sets#DeloopedKanFibrationOfHomomorphismOfSimplicialGroups))

### Classification of simplicial principal bundles
 {#ClassificationOfSimplicialPrincipalBundles}

The object $\overline{W}G$ serves as the [[classifying space]] for [[simplicial principal bundles]] ([May 67, §21](#May67), [Goerss & Jardine 09, Section V, Thm. 3.9](#GoerssJardine09), see also [NSS 12, Section 4.1](#NSS12)).

### Quillen equivalence between simplicial groups and reduced simplicial sets

\begin{proposition}\label{QuillenEquivalenceBetweenSimplicialGroupsAndReducedSimplicialSets}
**([[Quillen equivalence between simplicial groups and reduced simplicial sets]])** \linebreak
  The [[simplicial classifying space]]-construction $\overline{W}(-)$ (Def. \ref{BarWGInComponents}) is the [[right adjoint]] in a [[Quillen equivalence]] between the projective [[model structure on simplicial groups]] and the injective [[model structure on reduced simplicial sets]].

$$
  Groups(sSet)_{proj}
  \underoverset
    {\underset{\overline{W}}{\longrightarrow}}
    {\overset{ \Omega }{\longleftarrow}}
    {\;\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;\;}
  (sSet_{\geq 0})_{inj}
  \,.
$$

The [[left adjoint]] $\Omega$ is the [[simplicial loop space]]-construction.
\end{proposition}

(e.g. [Goerss & Jardine 09, V Prop. 6.3](#GoerssJardine09))

### Slice model structure
 {#SliceModelStructure}

The [[slice model category]] of the [[classical model structure on simplicial sets]] over the simplicial classifying complex $\overline{W}G$ is [[Quillen equivalence|Quillen equivalent]] to the [[Borel model structure]] for $G$-[[equivariant homotopy theory]]:

$$
  \big(
    SimplicialSets_{Qu}
  \big)_{/\overline{W} G}
  \;\simeq_{Qu}\;
  G Actions(SimplicialSets)_{proj}
  \,.
$$

([Dror, Dwyer & Kan 1980](#DDK80)) See [there](Borel+model+structure#RelationToSliceOverSimplicialClassifyingSpace) for details. This is a model for the general abstract situation discussed at *[[∞-action]]*.

## Related concepts

* [[simplicial loop functor]]

* [[free loop space of classifying space]]

* [[simplicial homotopy theory]]

* [[Borel model structure]]

* [[simplicial principal bundle]], [[universal simplicial principal bundle]]

## References

The idea of constructing $\overline{W}$ using the [[bar construction]]
is due

* {#EilenbergMacLane53I} [[Samuel Eilenberg]], [[Saunders Mac Lane]], _On the Groups $H(\Pi,n)$, I_, Annals of Mathematics Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106 ([jstor:1969820](https://www.jstor.org/stable/1969820)) (See, in particular, §17.)

who apply it to [[simplicial rings]] with the usual [[tensor product]] operation. This was also later discussed in:

* {#MacLane54} [[Saunders MacLane]], *Constructions simpliciales acycliques*, Colloque Henri Poincaré 1954 ([[MacLaneConstructionsSimplicialesAcycliques.pdf:file]])  (See, in particular, §3.)

The first reference where $\overline{W}$ is defined explicitly for [[simplicial groups]] and the [[adjoint functor|adjunction]] between [[simplicial groups]] and [[reduced simplicial sets]] is explicitly spelled out:

* {#Kan58} [[Daniel Kan]], Sections 10-11 in: _On homotopy theory and c.s.s. groups_, Ann. of Math. 68 (1958), 38-53 ([jstor:1970042](https://www.jstor.org/stable/1970042))

The [[left adjoint]] [[simplicial loop space]] functor $L$ is also discussed by [[Kan]] (there denoted "$G$") in

* [[Daniel M. Kan]], §7 of: _A combinatorial definition of homotopy groups_, Annals of Mathematics 67:2 (1958), 282–312.  [doi](https://doi.org/10.2307/1970006).

The [[Quillen equivalence]] was established in

* {#Quillen69} [[Dan Quillen]], Section 2 of: _Rational homotopy theory_, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 ([jstor:1970725](http://www.jstor.org/stable/1970725))

Textbook accounts:

* {#May67} [[Peter May]], p. 87-88 in: _Simplicial objects in algebraic topology_, University of Chicago Press 1967 ([ISBN:9780226511818](https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html), [djvu](http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu), [[May_SimplicialObjectsInAlgebraicTopology.pdf:file]])

* {#GoerssJardine09} [[Paul Goerss]], [[J. F. Jardine]], Section V.4 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) ([doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/))

Streamlining:

* {#Stevenson11} [[Danny Stevenson]], around Lemma 15 of: *Décalage and Kan's simplicial loop group functor*, Theory and Applications of Categories, Vol. 26, 2012, No. 28, pp 768-787 ([arXiv:1112.0474](https://arxiv.org/abs/1112.0474), [tac:26-28](http://www.tac.mta.ca/tac/volumes/26/28/26-28abs.html))

Identification of the [[slice model structure]] over $\overline{W}G$ with the [[Borel model structure]]:

* {#DDK80} [[Emmanuel Dror]], [[William Dwyer]], [[Daniel Kan]], _Equivariant maps which are self homotopy equivalences_, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 ([jstor:2043448](http://www.jstor.org/stable/2043448))

Generalization to [[simplicial groupoids]], now [[right adjoint]] to the [[Dwyer-Kan loop groupoid]]-construction:

* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], §3.2 of:  *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [<a href="https://doi.org/10.1016/1385-7258(84)90038-6">doi:10.1016/1385-7258(84)90038-6</a>]

Generalization to [[simplicial presheaves]]:

* {#NSS12} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Section 3.5 of: _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]_, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 ([doi:10.1007/s40062-014-0077-4](http://link.springer.com/article/10.1007/s40062-014-0077-4), [arXiv:1207.0249](http://arxiv.org/abs/1207.0249))

[[!redirects simplicial classifying spaces]]

[[!redirects simplicial classifying complex]]
[[!redirects simplicial classifying complexes]]

[[!redirects simplicial delooping]]
[[!redirects simplicial deloopings]]

[[!redirects simplicial delooping functor]]
[[!redirects simplicial delooping functors]]

[[!redirects universal principal simplicial complex]]
[[!redirects universal principal simplicial complexes]]


simplicial cochain

> This entry may need to be merged with _[[cochain on a simplicial set]]_.

\tableofcontents

## Idea

Given a [[simplicial set]] $X$, the simplicial cochains on $X$ form a [[cochain complex]].
The [[cohomology]] of this [[cochain complex]] computes the [[cohomology]]
of the [[simplicial set]] $X$.

As a special case, if $X$ is the [[singular simplicial set]]
of a [[topological space]] $S$, then
the simplicial cochains of $X=Sing(S)$
are precisely the [[singular cochains]] of $S$.

## Definition

Given an [[abelian group]] $A$,
the __simplicial cochains__ functor is a functor
$$C^*(-,A)\colon sSet \to coCh.$$
It is defined as the composition
of the [[simplicial chains]] functor (with integer coefficients)
$$C\colon sSet \to sAb \to Ch$$
with the dualization functor
$$Hom(-, A[0])\colon Ch \to coCh.$$

## Additional structures

The simplicial cochains of a [[simplicial set]] $X$
with coefficients in a [[commutative ring]] $A$
admit an action of the [[sequence operad]],
which turns $C^*(X,A)$ into an [[E-infinity algebra]].

In particular, this structure incorporates simplicial [[cup products]]
of cochains, as well as Steenrod's [[generalized cup products]].

## Related concepts

* [[simplicial chains]]

* [[singular cochains]]

* [[singular cohomology]]

[[!redirects simplicial cochains]]


simplicial complex

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

#Contents#
* table of contents
{:toc}

##Idea and definition##

A polyhedron (beware remark \ref{RemarkOnTerminology}) is a [[topological space]] made up of very simple bits 'glued' together. The 'bits' are **[[simplices]]** of different [[dimensions]]. An abstract _simplicial complex_ is a neat combinatorial way of giving the corresponding 'gluing' instructions, a bit like the plan of a construction kit!

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

A **simplicial complex** $K$ (sometimes called an **abstract** simplicial complex) consists of

1. a [[set]] of objects, $V(K)$, called **vertices**

1. a set, $S(K)$, of finite non-empty subsets of $V(K)$, called **[[simplex|simplices]]**.

such that the simplices satisfy the following conditions:

1. if $\sigma \subset V(K)$ is a simplex and $\tau \subset \sigma$, $\tau \ne \emptyset$, then $\tau$ is also a simplex;

1. every singleton $\{v\}$, $v \in V(K)$, is a simplex.

=--

We say $\tau$ is a **face** of $\sigma$.  If $\sigma \in S(K)$ has $p+1$ elements it is said to be a **$p$-simplex**.  The set of $p$-simplices of $K$ is denoted by $K_p$. The **dimension** of $K$ is the largest $p$ such that $K_p$ is non-empty.

A **map** of simplicial complexes $K \to L$ is a function $f: V(K) \to V(L)$ such that whenever $\sigma \subseteq V(K)$ belongs to $S(K)$, the [[image]] $f(\sigma)$ belongs to $S(L)$.

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

The word 'polyhedron' is used here as it is often used by [[algebraic topology|algebraic topologists]], as a space described by a simplicial complex. Elsewhere in mathematics, it might mean a finite union of finite intersections of sets in Euclidean space defined by linear inequalities, usually assumed compact, and often with other assumptions as well (e.g., connected or convex). The various usages have a long history, as recounted in detail in Lakatos's _Proofs and Refutations_.

=--

##Examples##
1. Recall that a 'simple' graph is an undirected [[graph]] with no loops or multiple edges. A simple graph is essentially the same thing as a 1-dimensional simplicial complex, by interpreting edges as simplices and vice versa (see [[graph#undirected_graphs_as_1complexes_barycentric_subdivision|undirected graphs as 1-complexes]]). (On the other hand, a 1-dimensional [[simplicial set]] is essentially the same thing as a 'directed multigraph', i.e., a [[quiver]].)

1. Given a space and an [[open cover]], the nerve of the cover is a simplicial complex (see [[Čech methods]] and the discussion there). The [[Vietoris complex]] is another given by a related method.

1. Given any two sets $X$ and $Y$, and a relation $R\subseteq X\times Y$, there are two simplicial complexes that encode information on the relation. These are generalisations of the nerve and the Vietoris complex. They are studied in detail in [[Dowker's theorem]].

1.  If $(P,\leq)$ is a [[poset]], then the [[nerve]] of the associated category has a simple description. The vertices are the points of $P$ and the simplices are the [[flag|flags]].

1. **Buildings:**
An important class of simplicial complexes is provided by the notion of [[building]], due to Jacques Tits.

##Simplicial complexes v. simplicial sets {#vsSSet}

Simplicial complexes are, in some sense, special cases of [[simplicial set|simplicial sets]], but only 'in some sense'.

To get from a simplicial complex to a fairly small simplicial set, you  pick a [[total order]] on the set of vertices. Without an order on the vertices, you cannot speak of the $k^{th}$ face of a simplex, which is an essential feature of a simplicial set! The degeneracies are obtained by repeating an element when listing the vertices of a simplex. If $\sigma = \{v_0,v_1,\ldots, v_n\}$, with $v_0\lt v_1\lt \ldots \lt v_n$ then, for instance, $s_0(\sigma) = \{v_0,v_0, v_1,\ldots, v_n\}$.

If you do not want to pick an order then you can still form a simplicial set where to each $n$-simplex of the original simplicial complex will correspond to $(n+1)!$ simplices of that associated simplicial set. The result is more unwieldy, but can be useful under some circumstances as it defines a functor from the category of simplicial complexes to that of simplicial sets.  This is very important when discussing group actions on simplicial complexes and how this transfers to the associated simplicial set.

##Simplicial complexes as sheaves on a site##

Simplicial sets are essentially (that is, up to equivalence) [[presheaves]] on the [[simplex category]] of finite nonempty totally ordered sets, whereas simplicial complexes may be regarded as [[concrete sheaf|concrete presheaves]] on the category $Fin_{+}$ of finite nonempty sets and functions between them. This works as follows: given a simplicial complex, $K = (V(K), S(K))$, define a presheaf $K^\sim: Fin_{+}^{op} \to Set$ whose values are sets of functions $\phi$:

$$K^\sim(B) \stackrel{def}{=} \{\phi: B \to V(K)| \phi(B) \in S(K)\}, \qquad K^\sim(f: A \to B)(\phi) \stackrel{def}{=} \phi \circ f$$

This defines an evident [[functor]]

$$SimpComplex \to Set^{Fin_{+}^{op}}: K \mapsto K^\sim$$

that is [[full and faithful functor|full and faithful]]. The essential image is the subcategory of concrete presheaves, where a presheaf $F \colon Fin_{+}^{op} \to Set$ is **concrete** if the canonical map

$$F(B) \to F(1)^{\hom(1, B)}$$

is an injection. The point is that a morphism of concrete presheaves $F \to G$ is uniquely determined from the function $F(1) \to G(1)$ between their underlying sets (i.e., the underlying-set functor on concrete presheaves is [[faithful functor|faithful]], so that a concrete presheaf is a set equipped with extra [[stuff, structure, property|structure]] -- that's what makes it "concrete").

(Equivalently but somewhat more elaborately, the category of concrete presheaves is the same as the full subcategory of [[concrete sheaf|concrete sheaves]] on $Fin_{+}$ with respect to the trivial [[Grothendieck topology|topology]], where the only covering sieve $F \hookrightarrow hom(-, D)$ is the maximal sieve.)

From this point of view, it is immediate that simplicial complexes are the [[separated presheaf|separated objects]] for the [[Lawvere-Tierney topology]] on $Set^{Fin_{+}^{op}}$ whose sheaves are sets, via the sheafification functor

$$\Gamma = \hom(\mathbf{1}, -) = ev_1 \colon Set^{Fin_{+}^{op}} \to Set$$

which has a right adjoint. (See [[local topos]].) It follows from this characterization that the category of simplicial complexes is a [[quasitopos]], and in particular is locally cartesian closed. The category of simplicial sets on the other hand is a [[topos]].

##Geometric realisations and Polyhedra##

An abstract simplicial complex is a combinatorial gadget that models certain aspects of a spatial configuration.  Sometimes it is useful, perhaps even necessary, to produce a topological space from that data in a simplicial complex.

### Idea
To each simplicial complex $K$, one can associate a topological space called the _polyhedron_ of $K$ often also called the  _geometric realisation_ of $K$ and denoted $|K|$.   (This is essentially a special case of the geometric realisation of a simplicial sets.)

This can be constructed by taking a copy $K(\sigma)$ of a standard topological $p$-simplex for each $p$-simplex of $K$ and then 'gluing' them together according to the face relations encoded in $K$.

We therefore first need the definition of a standard $p$-simplex

+--{: .num_defn}
######Definition######
The _standard (topological) $p$-simplex_ is (usually)  taken to be the convex hull of the basis vectors $\mathbf{e}_1, \mathbf{e}_2,\ldots, \mathbf{e}_{p+1}$ in $\mathbb{R}^{p+1}$.
=--

###Intuition
The geometric realisation $|K|$ of a simplicial complex, $K$  is then constructed by taking, for each abstract $p$-simplex, $\sigma\in S(K)$, a copy, $K(\sigma)$ of such a standard topological $p$-simplex, and then 'gluing' faces together, so whenever $\tau$ is a face of $\sigma$ we identify $K(\tau)$ with the corresponding face of $K(\sigma)$. This space is usually denoted $\Delta^p$.

###Canonical construction

As a set, $|K|$ is constructed as follows:

$|K|$ is the set of all functions from $V(K)$ to  the closed interval $[0,1]$ such that

* if $\alpha \in {|K|}$, the set

$$\{v \in V(K) \mid \alpha(v)  \neq  0\}$$

is a simplex of $K$;

 * for each $\alpha \in {|K|}$,
$$\sum_{v \in V(K)} \alpha (v) = 1.$$

There are two commonly used topologies on the set $|K|$. The first is the _metric topology_: we put a metric $d$ on $|K|$ by

$$d(\alpha,\beta) = \Big(\sum_{v\in V(K)} (\alpha(v) - \beta(v))^2\Big)^\frac{1}{2}.$$

$|K|$, when endowed with the metric space topology, will be denoted $|K|_d$. Notice that when $V(K)$ is finite, this gives $|K|_d$ as a subspace of the metric space $\mathbb{R}^{\#(V(K))}$ (which is usually of much higher dimension than might seem geometrically significant in a given context).

The second topology is the _coherent topology_: each geometric simplex $|s|$ consists of all $\alpha \in {|K|}$ supported in $s$, and is given the subspace topology inherited as a subset of $|K|_d$; then the coherent topology on $|K|$ is the largest topology for which all inclusions ${|s|} \hookrightarrow {|K|}$ are continuous. This topological space is normally denoted just $|K|$, reflecting the fact that the coherent topology is regarded as the default topology to put on the set $|K|$.

Note that if $s \subseteq t$ is an inclusion of simplices in $K$, then there is an induced subspace inclusion ${|s|} \hookrightarrow {|t|}$. The space $|K|$ may then be characterized as the colimit in $Top$ of the diagram consisting of geometric simplices $|s|$ and inclusions between them, so that a function $f: {|K|} \to X$ is continuous if and only if its restriction to each simplex $|s|$ is continuous. In particular, the identity function ${|K|} \to {|K|}_d$ is continuous, so that the coherent topology contains the metric topology (and is often strictly larger).

* **Warning:** The geometric realization of a simplicial complex does not preserve products. Indeed, the product of two intervals in the category of simplicial complexes is the tetrahedron!

## Triangulable spaces

If a topological space can be described up to homeomorphism as the geometric realization of a simplicial complex, we say it is **triangulable**, and a **triangulation** of a space $X$ is a simplicial complex $K$ together with a homeomorphism $h: |K| \to X$. (This is discussed in a bit more detail in the entry on [[classical triangulation]].

There is another stronger notion of triangulation used by geometric topologists: a **piecewise-linear (PL) structure** on a [[topological manifold]] $X$ is given by a [[manifold|PL atlas]], where the transition functions are piecewise-linear homeomorphisms. (A homeomorphism $U \to V$ is **piecewise linear** if its graph is the intersection of $U \times V$ with a semilinear set $S$, meaning that $S$ is given by a finite Boolean combination of solution sets of linear inequalities.)

#### Examples from manifold theory

* All [[manifold|smooth manifolds]] are triangulable and, in fact, admit PL structures.

* All topological manifolds in dimensions 2 and 3 admit PL structures, and are in fact smoothable (admit a smooth manifold structure).

* The $E_8$ [[E8 manifold|manifold]] does not admit a triangulation, much less a PL structure.

* In dimensions $n \geq 5$, the $(n-2)$-fold suspension of the [[Poincaré sphere]] is homeomorphic to the $n$-sphere, hence is triangulable, but it does not admit a PL structure.

#### Relation to simplicial sets

The following statement may seem obvious, but it requires careful proof:

* A space is triangulable if and only if it is homeomorphic to the geometric realization of a simplicial set.

As an important step:

* The geometric realization of the nerve of a poset is triangulable.

The basic technique is to use [[subdivision]].

## Related concepts

* [[triangulation]], [[triangulation theorem]]

## References

Textbook account:

* {#Spanier66} [[Edwin Spanier]], _Algebraic topology_, Springer 1966 ([doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1))

Exposition:

* {#Erleben2010} Kenny Erleben, *Simplicial complexes*, 2010 ([[Erleben_SimplicialComplexes.pdf:file]])

On simplicial complexes forming a [[quasitopos]] of [[concrete sheaves]]:

* [[John Baez]] and [[Alex Hoffnung]], _Convenient Categories of Smooth Spaces_, Trans. Amer. Math. Soc. 363 (2011), 5789–5825. [(arXiv)](http://arxiv.org/abs/0807.1704)

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