fundamental groupoid of a cubical set and the cubical nerve of a groupoid

[[!redirects fundamental groupoid of a cubical set]]
[[!redirects cubical nerve of a groupoid]]
# Contents
* table of contents
{: toc}

## Introduction

We construct an adjunction between the category of [[groupoid | groupoids]] and the category of [[cubical set | cubical sets]], the left adjoint of which is the _fundamental groupoid_ of a cubical set, and the right adjoint of which is the (cubical) _nerve_ of a groupoid.

## Preliminaries

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

We denote the category of [[groupoid | groupoids]] by $\mathsf{Grpd}$.

=--

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

We make use throughout of the notation of [[category of cubes]], [[cubical set]], and [[cubical truncation, skeleton, and co-skeleton]].

=--

## Fundamental groupoid for 2-truncated cubical sets

+-- {: .num_defn #NotationFundamentalGroupoid}
###### Notation

We denote by $\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd}$ the functor defined as follows.

1) To a 2-truncated cubical set $X$, we associate the groupoid $\Pi^{\leq 2}_{1}(X)$ defined as follows.

a) The objects of $\Pi^{\leq 2}_{1}(X)$ are the 0-cubes of $X$.

b) The arrows of $\Pi^{\leq 2}_{1}(X)$ are zig-zags of 1-cubes of $X$ up to the notion of equivalence defined below, where by a zig-zag of 1-cubes of $X$ we mean, for some integer $n \geq 0$, a set of 1-cubes of $X$ whose faces match up as follows.

$$
   \array{
      x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
$$

We identify a pair of zig-zags if one can be obtained from the other by a sequence of the following manipulations.

i) We may remove or add a pair of arrows (anywhere in the zig-zag) of the form

$$
   \array{
      x_{0} & \overset{f}{\rightarrow} & x_{1} & \overset{f}{\leftarrow} & x_{0}
   }
$$

or of the following form.

$$
   \array{
      x_{0} & \overset{f}{\leftarrow} & x_{1} & \overset{f}{\rightarrow} & x_{0}
   }
$$

ii) We may replace an entire zig-zag

$$
   \array{
      x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
$$

with a zig-zag

$$
   \array{
      x'_{0} & \overset{f'_{1}}{\rightarrow} & x'_{1} \overset{g'_{1}}{\leftarrow} & x'_{2} & \cdots & x'_{2n-2} & \overset{f'_{n}}{\rightarrow} & x'_{2n-1} \overset{g'_{n}}{\leftarrow} & x'_{2n}
   }
$$

if there is, for every $1 \leq i \leq n$, a 2-cube of $X$ whose horizontal 1-cubes are as follows

$$
   \array{
      x_{2i-2}   & \overset{f_{i}}{\to}   & x'_{2i-1} \\
      \downarrow &                        & \downarrow \\
      x'_{2i-2}  & \underset{f'_{i}}{\to} & x'_{2i-1}
   }
$$

and there is, for every $1 \leq i \leq n$, a 2-cube of $X$ whose horizontal 1-cubes are as follows.

$$
   \array{
      x_{2i-1}   & \overset{g_{i}}{\leftarrow}   & x'_{2i} \\
      \downarrow &                        & \downarrow \\
      x'_{2i-1}  & \underset{g'_{i}}{\leftarrow} & x'_{2i}
   }
$$

c) The source of a zig-zag as at the beginning of b) is $x_{0}$, and the target is $x_{2n}$.

d) Composition of arrows is given by concatenation of zig-zags (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).

e) The identity arrow on an object $x$ of $\Pi^{\leq 2}_{1}(X)$ is the zig-zag with $n = 0$ consisting simply of $x$.

f) The inverse of an arrow

$$
   \array{
      x_{0} & \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
$$

of $\Pi_{1}(X)$ is the following arrow (it is immediately verified that this is well-defined with respect to the equivalence relation of b)).

$$
   \array{
      x_{2n} & \overset{g_{n}}{\rightarrow} & x_{2n-1} \overset{f_{n}}{\leftarrow} & x_{2n-2} & \cdots & x_{2} & \overset{g_{1}}{\rightarrow} & x_{1} \overset{f_{1}}{\leftarrow} & x_{0}
   }
$$

2) To a morphism of 2-truncated cubical sets $F : X \rightarrow Y$, we associate the functor $\Pi^{\leq 2}_{1}(F) : \Pi^{\leq 2}_{1}(X) \rightarrow \Pi^{\leq 2}_{1}(Y)$ defined as follows.

a) On objects, $\Pi^{\leq 2}_{1}(F)$ is the same as $F$.

b) To a zig-zag as follows

$$
   \array{
      x_{0}  &  \overset{f_{1}}{\rightarrow} & x_{1} \overset{g_{1}}{\leftarrow} & x_{2} & \cdots & x_{2n-2} & \overset{f_{n}}{\rightarrow} & x_{2n-1} \overset{g_{n}}{\leftarrow} & x_{2n}
   }
$$

we associate the following zig-zag.

$$
   \array{
      F(x_{0}) &  \overset{F(f_{1})}{\rightarrow} & F(x_{1}) \overset{F(g_{1})}{\leftarrow} & F(x_{2}) & \cdots & F(x_{2n-2}) & \overset{F(f_{n})}{\rightarrow} & F(x_{2n-1}) \overset{F(g_{n})}{\leftarrow} & F(x_{2n})
   }
$$

It is immediately verified that this is well-defined with respect to the equivalence relation of 1) b).

=--

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

We refer to $\Pi^{\leq 2}_{1} : \mathsf{Set}^{\square_{\leq 2}^{op}} \rightarrow \mathsf{Grpd}$ as the _fundamental groupoid_ functor for 2-truncated cubical sets.

=--

## 2-truncated nerve functor

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

We denote by $N^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}}$ the functor defined as follows.

1) To a groupoid $\mathcal{A}$, we associate the 2-truncated cubical set $N^{\leq 2}(\mathcal{A})$ defined as follows.

a) The 0-cubes of $N^{\leq 2}(\mathcal{A})$ are the objects of $\mathcal{A}$.

b) The 1-cubes $f : a_{0} \rightarrow a_{1}$ of $N^{\leq 2}(\mathcal{A})$ are the arrows $f$ of $\mathcal{A}$ with source $a_{0}$ and target $a_{1}$.

c) The 2-cubes

$$
   \array{
      a_{0}            & \overset{f_{0}}{\to}  & a_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      a_{2}            & \underset{f_{3}}{\to} & a_{3}
   }
$$

of $N^{\leq 2}(\mathcal{A})$ are the commutative squares $\sigma$ of $\mathcal{A}$ whose boundary looks the same as this.

d) The degenerate 1-cubes of $N^{\leq 2}(\mathcal{A})$ are the identity arrows of $\mathcal{A}$.

e) The degenerate 2-cubes of $N^{\leq 2}(\mathcal{A})$ are the commutative squares of $\Pi^{\leq 2}_{1}(\mathcal{A})$ which look as follows

$$
   \array{
      x_{0}         & \overset{f}{\to}  & x_{1} \\
      id \downarrow & \sigma            & \downarrow id \\
      x_{0}         & \underset{f}{\to} & x_{1}
   }
$$

or as follows.

$$
   \array{
      x_{0}        & \overset{id}{\to}  & x_{0} \\
      f \downarrow & \sigma             & \downarrow f \\
      x_{1}        & \underset{id}{\to} & x_{1}
   }
$$

2) To a functor $F : \mathcal{A} \rightarrow \mathcal{B}$, we associate the morphism of 2-truncated cubical sets $N^{\leq 2}(F) : N^{\leq 2}(\mathcal{A}) \rightarrow N^{\leq 2}(\mathcal{B})$ defined as follows.

a) On 0-cubes, $N^{\leq 2}(\mathcal{A})$ is the same as $F$.

b) On 1-cubes, $N^{\leq 2}(F)$ is the same as $F$.

c) On 2-cubes, $N^{\leq 2}(F)$ sends a commutative square

$$
   \array{
      a_{0}            & \overset{f_{0}}{\to}  & a_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      a_{2}            & \underset{f_{3}}{\to} & a_{3}
   }
$$

of $\mathcal{A}$ to the commutative square

$$
   \array{
      F(a_{0})            & \overset{F(f_{0})}{\to}  & F(a_{1}) \\
      F(f_{2}) \downarrow & F(\sigma)                & \downarrow F(f_{1}) \\
      F(a_{2})            & \underset{F(f_{3})}{\to} & F(a_{3})
   }
$$

of $\mathcal{A}$.

=--

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

We refer to $N^{\leq 2} : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square_{\leq 2}^{op}}$ as the 2-truncated _nerve_ functor.

=--

## Adjunction between the fundamental groupoid functor for 2-truncated cubical sets and the 2-truncated cubical nerve functor

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

We denote by $\epsilon : \Pi^{\leq 2}_{1} \circ N^{\leq 2} \rightarrow id$ the natural transformation which to a groupoid $\mathcal{A}$ associates the functor $\epsilon(\mathcal{A}) : \Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A}) \rightarrow \mathcal{A}$ defined as follows.

1) On objects it is the identity.

2) To an arrow of $\Pi^{\leq 2}_{1} \circ N^{\leq 2}(\mathcal{A})$, given by a zig-zag of arrows

$$
   \array{
      a_{0}  &  \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}}{\leftarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}}{\leftarrow} & a_{2n}
   }
$$

of $\mathcal{A}$, we associate the arrow of $\mathcal{A}$ given by the composition in $\mathcal{A}$ of the arrows

$$
   \array{
      a_{0}  &  \overset{f_{1}}{\rightarrow} & a_{1} \overset{g_{1}^{-1}}{\rightarrow} & a_{2} & \cdots & a_{2n-2} & \overset{f_{n}}{\rightarrow} & a_{2n-1} \overset{g_{n}^{-1}}{\rightarrow} & a_{2n}
   }
$$

of $\mathcal{A}$.

It is straightforward to check that this is well-defined with respect to the equivalence relation of 1 b) of Notation
\ref{NotationFundamentalGroupoid}, and that we indeed have a functor.

=--

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

We denote by $\eta : id \rightarrow N^{\leq 2} \circ \Pi^{\leq 2}_{1}$ the natural transformation which to a 2-truncated cubical set $X$ associates the morphism of 2-truncated cubical sets $\eta(X) : X \rightarrow N^{\leq 2} \circ \Pi^{\leq 2}_{1}(X)$ defined as follows.

1) On objects it is the identity.

2) To a 1-cube $f : x_{0} \rightarrow x_{1}$ of $X$ we associate the following zig-zag of 1-cubes of $X$, where the right arrow is the degeneracy on $x_{1}$.

$$
   \array{
      x_{0}  &  \overset{f}{\rightarrow} & x_{1} \overset{id}{\leftarrow} & x_{1} }
$$

3) To a 2-cube

$$
   \array{
      x_{0}            & \overset{f_{0}}{\to}  & x_{1} \\
      f_{2} \downarrow & \sigma                & \downarrow f_{1} \\
      x_{2}            & \underset{f_{3}}{\to} & x_{3}
   }
$$

of $X$ we associate the commutative square in $\Pi^{\leq 2}_{1}(X)$ given as follows.

$$
   \array{
      x_{0}            & \overset{f_{0}}{\to}  & x_{1} & \overset{id}{\leftarrow}  & x_{1} \\
      f_{2} \downarrow &                       &       &                           & \downarrow f_{1} \\
      x_{2}            &                       &       &                           & x_{3} \\
      id    \downarrow &                       &       &                           & \downarrow id \\
      x_{2}            & \underset{f_{3}}{\to} & x_{2} & \underset{id}{\leftarrow} & x_{3} \\
   }
$$

The following diagram of commutative squares in $\Pi^{\leq 2}_{1}(X)$ illustrates that the above square does indeed commute in $\Pi^{\leq 2}_{1}(X)$.

$$
   \array{
      x_{0}         & \overset{f_{0}}{\rightarrow}  & x_{1}        & \overset{id}{\leftarrow} & x_{1}        & \overset{f_{1}}{\rightarrow}   & x_{2}         & \overset{id}{\leftarrow} & x_{2} \\
      id \downarrow &                               & \downarrow A &                          & \downarrow A &                            & \downarrow id &                          & \downarrow id \\
      x_{0}         & \underset{f_{2}}{\rightarrow} & x_{2}        & \underset{id}{\leftarrow} & x_{2}        & \underset{f_{3}}{\rightarrow} & x_{3}         & \underset{id}{\leftarrow} & x_{3}
   }
$$

Here A is $f_{2} \circ f_{0}^{-1}$. That the square

 $$
   \array{
      x_{1}        & \overset{f_{1}}{\to}     & x_{3} \\
      A \downarrow &                       & \downarrow id \\
      x_{2}        & \underset{f_{3}}{\to} & x_{3}
   }
$$

commutes follows easily from the commutativity of the square arising from the 2-cube $\sigma$ of $X$ above.

=--

+-- {: .num_defn #PropositionAdjunctionBetweenPi1TwoTruncAndNTwoTrunc}
###### Proposition

The natural transformations $\eta$ and $\epsilon$ define an adjunction between $\Pi^{\leq 2}_{1}$ and $N^{\leq 2}$.

=--

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

Straightforward verification that the triangle identities hold.

=--

## Fundamental groupoid functor

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

Adopting the notation of [[cubical truncation, skeleton, and co-skeleton]], we denote by $\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ the functor

$$
   \array{
      \mathsf{Set}^{\square^{op}} & \overset{tr_{2}}{\rightarrow}  & \mathsf{Set}^{\square_{2}^{op}} & \overset{\Pi_{1}^{\leq 2}}{\rightarrow} & \mathsf{Grpd}.
   }
$$

=--

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

We refer to the functor $\Pi_{1} : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ as the _fundamental groupoid_ functor.

=--

## Nerve functor

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

Adopting the notation of [[cubical truncation, skeleton, and co-skeleton]], we denote by $N : \mathsf{Grpd} \rightarrow \mathsf{Set}^{\square^{op}}$ the functor

$$
   \array{
      \mathsf{Grpd} & \overset{N^{\leq 2}}{\rightarrow} & \mathsf{Set}^{\square_{\leq 2} ^{op}} & \overset{cosk_{2}}{\rightarrow}  & \mathsf{Set}^{\square^{op}}.
   }
$$

=--

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

We refer to the functor $N : \mathsf{Set}^{\square^{op}} \rightarrow \mathsf{Grpd}$ as the _nerve_ functor.

=--

## Adjunction between the fundamental groupoid functor and the nerve functor

Since $\Pi^{\leq 2}_{1}$ is left adjoint to $N^{\leq 2}$ by Proposition \ref{#PropositionAdjunctionBetweenPi1TwoTruncAndNTwoTrunc} above, and since $tr_{2}$ is left adjoint to $cosk_{2}$, we have that $\Pi_{1}$ is left adjoint to $N$.


fundamental infinity-groupoid

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#### Homotopy theory
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#Contents#
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## Idea

The _fundamental $\infty$-groupoid_  $\Pi_\infty(X)$ of a [[topological space]] $X$ is the [[∞-groupoid]] whose [[k-morphisms]] are the $k$-dimensional paths in $X$. This is the higher refinement of the [[fundamental groupoid]] $\Pi_1(X)$.

It is also sometimes called the _$\infty$-Poincaré groupoid_ of the space, in analogy to the term _[[Poincaré groupoid]]_ for the fundamental groupoid.

## Definition

### General version

The following definition is appropriate if we take a [[Kan complex]] as the definition of $\infty$-groupoid.

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

The **fundamental $\infty$-groupoid** $\Pi(X)$ of a [[topological space]] $X$ is given by the [[Kan complex]]

$$
  Sing X : [k] \mapsto Hom_{Top}(\Delta^k, X)
$$

which is the [[singular simplicial complex]] of $X$.

=--

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

This construction is [[right adjoint]] to [[geometric realization]].

=--

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

By choosing [[horn]]-fillers this  becomes an [[algebraic Kan complex]]. In terms of these the [[homotopy hypothesis]] has a direct proof, exhibited by a [[Quillen equivalence]]

$$
  Alg Kan \stackrel{\overset{\Pi}{\leftarrow}}{\to} Top
$$

due to ([Nikolaus](#Nikolaus)).

=--

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

One may regard the singular simplicial complex functor as the instance of the general abstract notion of [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] by regarding [[Top]] as a [[cohesive (∞,1)-topos]]. See [[discrete ∞-groupoid]] for more on this.

=--

For other models of [[∞Grpd]] there are correspondingly other constructions:

* The definition of [[Trimble n-category]] has the concept of fundamental $n$-groupoid built right into it.

### Strict versions

One can consider [[strict ∞-groupoid]] versions of the fundamental $\infty$-groupoid. These lose information about the [[homotopy type]] of the space, though, but are more tractable and may give in some applications all the information that one is interested in.

The study of strict fundamental $\infty$-groupoids have been pursued by [[Ronnie Brown]] and his school.

There is a [[strict 2-groupoid|strict]] homotopy 2-groupoid for a [[Hausdorff space]] defined spring , and a weak homotopy 2-groupoid for a general space (by the same authors). They later introduced  a homotopy double groupoid. There is no $n$-dimensional version of these ideas on offer.

A strict cubical omega-groupoid $\rho X_*$ for a [[filtered space]] $X_*$ was defined by Brown and Higgins in 1981. Form the filtered cubical complex $R X_*$ which in dimension $n$ consists of  filtered maps $I^n_* \to X_*$ and take filter homotopy classes of these _relative to the vertices_. The proof that the compositions in $RX_*$ are inherited by $\rho X_*$ is one of the key points of the development.

It turns out that $\rho X_*$ is equivalent in a clear sense to the crossed complex $\Pi X_*$ defined using relative homotopy groups by Blakers in 1948 (with other terminology) and that the homotopy types modelled by crossed complexes, or by the corresponding globular or cubical gadget, are restricted, essentially to the _linear_ homotopy types, with no quadratic information.  Nonetheless, it is well known in mathematics that linear approximations can be useful.

Loday's paper of 1982 on _Spaces with finitely many homotopy groups_ introduced the  entirely new idea of a _cubical resolution_ of a space. Some details were completed by [[Richard Steiner]]. [[Jean-Louis Loday|Loday]] also introduced the fundamental [[cat-n-group]] of an $n$-cube of spaces. In this way we get a model of a space $X$ by a multiple groupoid in which the $r$-dimensional homotopy of $X$ occurs in the right place in the model. Also you can calculate something with this model, and it has led to new algebraic constructions, such as a nonabelian tensor product of groups, with homotopical applications.

These strict groupoid models do satisfy the dimension condition.

## Properties

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

The 1-[[truncated|truncation]] of $\Pi X$ is the [[fundamental groupoid]] of $X$:

$$
  \tau_{\leq 1} \Pi X \simeq \Pi_1(X)
  \,.
$$

=--

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

For $X$ a [[locally contractible topological space]], the fundamental $\infty$-groupoid $Sing X$ is equivalent to the [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]] of the [[(∞,1)-sheaf (∞,1)-topos]] $(\infty,1)Sh(X)$.

=--

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

Details on this are at [[geometric homotopy groups in an (∞,1)-topos]].

=--

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

This perspective suggests that when $X$ is not locally contractible, a better replacement for its fundamental $\infty$-groupoid (as usually defined) is the [[shape of an (∞,1)-topos|shape]] of $(\infty,1)Sh(X)$. As discussed there, this coincides with the traditional [[shape theory]] of $X$.

=--

## Related concepts

* [[fundamental groupoid]]

* fundamental $\infty$-groupoids incarnated as [[sSet]]-[[enriched groupoids]] aka "Dwyer-Kan [[simplicial groupoids]]" are known as *[[Dwyer-Kan loop groupoids]]*.

* [[fundamental category]], [[fundamental (∞,1)-category]]

* [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] / [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|of a locally ∞-connected (∞,1)-topos]]

* [[shape via cohesive path ∞-groupoid]]

## References

The [[fundamental 2-groupoid]]:

as a [[strict 2-groupoid]]:

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A homotopy 2-groupoid of a Hausdorff space*. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures **8** (2000) 209-234 [[doi:10.1023/A:1008758412196](https://doi.org/10.1023/A:1008758412196)]

as a [[weak 2-groupoid]]:

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A Homotopy Bigroupoid of a Topological Space*, Applied Categorical Structures **9** (2001) 311-327 [[doi:10.1023/A:1011270417127](https://doi.org/10.1023/A:1011270417127)]

See also:

* [[J.-L. Loday]], *Spaces with finitely many homotopy groups*, J. Pure Appl.  Alg., 24 (1982) 179--202.

* J. M. Casas, G. Ellis, M. Ladra, T. Pirashvili, _Derived functors and the homology of $n$-types_, J. Algebra __256__, 583--598 (2002).

The direct proof of the [[homotopy hypothesis]] for the algebraic version of the fundamental $\infty$-groupoid is in

* {#Nikolaus} [[Thomas Nikolaus]], _Algebraic models for higher categories_ ([arXiv](http://arxiv.org/abs/1003.1342))

Strict versions of fundamental $\infty$-groupoids are discussed in

* [[Ronnie Brown]] et al. _[[Nonabelian Algebraic Topology]]_

See also

* [[Ronnie Brown]] and [[Philip Higgins]], Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11-41.

*  [[Ronnie Brown]], A new higher homotopy groupoid: the fundamental  globular $\omega$-groupoid of a filtered space, Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.

* [[Richard Steiner]], Resolutions of spaces by  $n$-cubes of fibrations, J. London Math. Soc.(2), 34, 169-176, 1986.

[[!redirects fundamental infinity-groupoids]]
[[!redirects fundamental ∞-groupoid]]
[[!redirects fundamental ∞-groupoids]]

[[!redirects ∞-Poincaré groupoid]]
[[!redirects ∞-Poincaré groupoids]]
[[!redirects infinity-Poincaré groupoid]]
[[!redirects infinity-Poincaré groupoids]]


fundamental infinity-groupoid in a locally infinity-connected (infinity,1)-topos

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#Contents#
* table of contents
{:toc}

## Definition

For $(\Pi \dashv \Gamma \dashv LConst) :  \mathbf{H} \to \infty Grpd$ a [[locally ∞-connected (∞,1)-topos]] and $X \in \mathbf{H}$ an [[object]], we say that $\Pi(X)$ is the **fundamental $\infty$-groupoid** of $X$ in $\mathbf{H}$.

## Properties

* This is the object that encodes the [[geometric homotopy groups in an (∞,1)-topos]].

* For any $X \in \mathbf{H}$ the $\infty$-groupoid $\Pi(X)$ coincides with the [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]] (see there) of the [[over-(∞,1)-topos]] $\mathbf{H}/X$.

## Examples

* In the [[cohesive (∞,1)-topos]] [[Top]] the intrinsic fundamental $\infty$-groupoid functor coincides with the ordinary [[fundamental ∞-groupoid]] of a [[topological space]]. See [[discrete ∞-groupoid]] for details.

* In [[ETop∞Grpd]] the intrinsic fundamental $\infty$-groupoid is the generalization of that on [[Top]] to [[∞-groupoid]]s in [[paracompact space]]s.

## Related concepts

* [[geometric realization of cohesive ∞-groupoids]]

## References

* [[schreiber:differential cohomology in a cohesive topos]]

See also

* {#Carchedi15} [[David Carchedi]], _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_ ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))

for further discussion of the smooth [[shape modality]] of cohesion (the [[etale homotopy type]] operation in the context of [[smooth infinity-stacks]]) as applied to [[nLab:orbifolds]] and [[étale groupoids]] and generally [[étale ∞-groupoids]].

[[!redirects fundamental infinity-groupoid in a locally ∞-connected (infinity,1)-topos]]
[[!redirects fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]


fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos

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###Context###
#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Every [[(∞,1)-topos]] $E$ has a [[shape of an (∞,1)-topos|shape]] $Shape(E) \in Pro\infty Grpd$. When $E$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] then this is a genuine [[∞-groupoid]] $\Pi(E) \in $ [[∞Grpd]]. We may think of this as the [[fundamental ∞-groupoid]] of the $(\infty,1)$-topos regarded as a generalized [[space]].

But also every [[locally ∞-connected (∞,1)-topos]] has an _internal_ notion of [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] for objects of $E$, denoted $\Pi_E : E \to \infty Grpd$. (If in addition $\Delta \colon \infty Grpd \to E$ is [[fully faithful (∞,1)-functor|fully faithful]] then we call $\Pi$ the _[[shape modality]]_.) Applied to its [[terminal object]] this does agree with the fundamental ∞-groupoid of the topos:

$$
  \Pi(E) \simeq \Pi_E(*)
  \,.
$$

Conversely, for an object $X\in E$, the fundamental ∞-groupoid $\Pi_E(X)$ internal to $E$ can be identified with the fundamental ∞-groupoid of the locally ∞-connected (∞,1)-topos $E/X$.

## Definition

+-- {: .num_def}

###### Definition

For $(\Pi_E \dashv \Gamma_E \dashv LConst_E) :  E \to \infty Grpd$ a [[locally ∞-connected (∞,1)-topos]] we say its **fundamental $\infty$-groupoid** is

$$
  \Pi(E) := \Pi_E(*)
  \,,
$$

where $*$ is the [[terminal object]] of $E$.

In other words, it is the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos|internal fundamental ∞-groupoid]] of the terminal object of $E$.

=--

## Properties

Let $\mathbf{H}$ be a locally $\infty$-connected $(\infty,1)$-topos and $X \in \mathbf{H}$ an [[object]]. Then also the [[over-(∞,1)-topos]] $\mathbf{H}/X$ is locally $\infty$-connected (as discussed there).

We have then two different definitions of the fundamental $\infty$-groupoid of $X$: once as $\Pi_{\mathbf{H}}(X)$ -- the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]  -- and once as $\Pi(\mathbf{H}/X)$.

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

These agree:

$$
  \Pi_{\mathbf{H}}(X) \simeq \Pi(\mathbf{H}/X)
  \,.
$$

=--

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

Since $X \stackrel{Id}{\to} X$ is the [[terminal object]] in $\mathbf{H}/X$ we have by definition

$$
  \Pi(\mathbf{H}/X) = \Pi_{\mathbf{H}/X}(Id_X)
  \,.
$$

Now observe that $\Pi_{\mathbf{H}/X} = \Pi_{\mathbf{H}} \circ X_!$ since the terminal [[global section]] geometric morphism of the over-topos is

$$
  \mathbf{H}/X
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
  \mathbf{H}
  \stackrel{\overset{\Pi_{\mathbf{H}}}{\to}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\leftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\to}}}
  \infty Grpd
$$

and that $X_!$ in the [[etale geometric morphism]] is the projection map that sends $Y \stackrel{}{\to} X$ to $Y$.
=--

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

Let $LC(\infty,1)Topos$ denote the full [[sub-(∞,1)-category]] of [[(∞,1)Topos]] determined by the locally ∞-connected objects.

=--

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

The $(\infty,1)$-category $\infty Gpd$ (as the category of local homeomorphisms over $\infty Gpd$) is [[reflective sub-(∞,1)-category|reflective]] in $LC(\infty,1)Topos$,

$$
  \infty Grpd
    \stackrel{\overset{\Pi}{\leftarrow}}{\hookrightarrow}
  LC(\infty,1)Topos
$$

with the reflector given by forming the fundamental $\infty$-groupoid.

=--

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

Any [[∞-groupoid]] $G$ gives rise to an [[(∞,1)-presheaf (∞,1)-topos]] $PSh(G) = [G^{op},\infty Gpd]$, which by the [[(∞,1)-Grothendieck construction]] is equivalent to the [[over-(∞,1)-topos]] $\infty Gpd / G$.  The $(\infty,1)$-toposes of this form are, by definition, those for which the unique [[(∞,1)-geometric morphism]] to $\infty Gpd$ is a [[local homeomorphism of toposes]].  This construction embeds $\infty Gpd$ as a full [[sub-(∞,1)-category]] of [[(∞,1)Topos]]:

$$
  PSh(-) : \infty Grpd \hookrightarrow LC(\infty,1)Topos
  \,.
$$

since in particular the $(\infty,1)$-toposes $PSh(G)$ are [[locally ∞-connected (∞,1)-topos|locally ∞-connected]].

To show that $\Pi$ is a [[left adjoint|left]] [[adjoint (∞,1)-functor]] to $PSh(-)$ we demonstrate a natural hom-equivalence

$$
  LC(\infty,1)Topos(E,(\infty,1)PSh(A)) \simeq \infty Grpd(\Pi_E(*), A)
$$

for $E\in LC(\infty,1)Topos$ and $A \in \infty Grpd$.

At [[shape of an (∞,1)-topos]] it is shown that we have a natural equivalence

$$
  (\infty,1)Topos(E, PSh(A)) \simeq \Gamma_E LConst_E G =: Shape(E)(A)
  \,.
$$

Now observe that furthermore we have a sequence of natural equivalences

$$
\begin{aligned}
Shape(E)(A) &= \Gamma(LConst(A))\\
&\simeq \infty Grpd(*, \Gamma(LConst(A)))\\
&\simeq E(LConst(*), LConst(A)) \\
&\simeq E(*, LConst(A)) \\
&\simeq \infty Grpd(\Pi_E(*),A).
\end{aligned}
  \,.
$$

=--

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

So equivalently, one may say that a locally ∞-connected (∞,1)-topos $E$ has a [[shape of an (∞,1)-topos|shape]] which is [[representable functor|representable]], and its [[fundamental ∞-groupoid]] $\Pi(\mathbf{H})$ is the representing object.

=--

## Examples

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

For $X$ a [[locally contractible topological space]], we have an equivalence

$$
  \Pi ((\infty,1)Sh(X)) \simeq Sing X
$$

between the ordinary [[fundamental ∞-groupoid]] of $X$ defined by the [[singular simplicial complex]] and the topos-theoretic fundamental $\infty$-groupoid of the [[(∞,1)-sheaf (∞,1)-topos]] $(\infty,1)Sh(X)$ over $X$.

=--

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

Details are at [[geometric homotopy groups in an (∞,1)-topos]].

=--

More generally the [[shape of an (∞,1)-topos]] of $(\infty,1)Sh(X)$ reproduces the [[shape theory]] of $X$.

## Related concepts

* [[fundamental groupoid]], [[fundamental ∞-groupoid]],

* [[homotopy group]], [[homotopy groups in an (∞,1)-topos]]

* [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] / **of a locally $\infty$-connected $(\infty,1)$-topos**

* [[étale homotopy theory]], [[pro-left adjoint]]

[[!redirects fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]]
[[!redirects fundamental ∞-groupoid of an (∞,1)-topos]]


fundamental M2-brane sigma-model -- references

### Super-membrane/M2-brane as a sigma model
 {#SuperMembraneAsSigmaModelReferences}

The [[Green-Schwarz sigma-model]]-type formulation of the [[super-membrane]] in 11d (as in the [[brane scan]] and in contrast to the [[black brane]]-solutions of [[11d supergravity]]) first appears in:

* {#BergshoeffSezginTownsend87} [[Eric Bergshoeff]], [[Ergin Sezgin]], and [[Paul Townsend]], _Supermembranes and eleven-dimensional supergravity_, Phys. Lett. B189 (1987) 75–78. ([spire:248230](http://inspirehep.net/record/248230/))

The [[equations of motion]] of the super membrane are derived via the [[superembedding approach]] in

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], Chapter 3 of _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B446:79-118, 1995 ([arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113))

and the [[Lagrangian density]] for the super membrane is derived via the [[superembedding approach]] in

* {#HoweSezgin05} [[Paul Howe]], [[Ergin Sezgin]], _The supermembrane revisited_, Class. Quant. Grav. 22 (2005) 2167-2200 ([arXiv:hep-th/0412245](https://arxiv.org/abs/hep-th/0412245))

Discussion from the point of view of [[Green-Schwarz action functional]]-[[schreiber:∞-Wess-Zumino-Witten theory]] is in

* {#FSS2013} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ (2013)

The [[double dimensional reduction]] of the M2-brane to the [[Green-Schwarz superstring]] was observed in

* {#DuffHoweInamiStelle87} [[Michael Duff]], [[Paul Howe]], T. Inami, [[Kellogg Stelle]], _Superstrings in $D=10$ from Supermembranes in $D=11$_, Phys. Lett. B191 (1987) 70 and in [[Michael Duff]] (ed.) _[[The World in Eleven Dimensions]]_ 205-206 (1987) ([spire](http://inspirehep.net/record/245249))

The interpretation of the [[super-membrane]] as an object related to [[string theory]] via [[double dimensional reduction]], hence as the _[[M2-brane]]_ was proposed in

* [[Paul Townsend]], _The eleven-dimensional supermembrane revisited_, Phys.Lett.B350:184-187, 1995 ([arXiv:hep-th/9501068](http://arxiv.org/abs/hep-th/9501068))

around the time when [[M-theory]] became accepted due to

* {#Witten95} [[Edward Witten]], _[[String Theory Dynamics In Various Dimensions]]_ ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

See also

* [[Paul Howe]], [[Ergin Sezgin]], _The supermembrane revisited_, ([arXiv:hep-th/0412245](http://arxiv.org/abs/hep-th/0412245))

* [[Igor Bandos]], [[Paul Townsend]], _SDiff Gauge Theory and the M2 Condensate_ ([arXiv:0808.1583](http://arxiv.org/abs/0808.1583))

* [[Maria P. Garcia del Moral]], C. Las Heras, P. Leon, J. M. Pena, [[Alvaro Restuccia]], _Fluxes, Twisted tori, Monodromy and $U(1)$ Supermembranes_,  J. High Energ. Phys. **2020** 97 (2020) [[arXiv:2005.06397](https://arxiv.org/abs/2005.06397), doi:10.1007/JHEP09(2020)097]

On possible structures in M2-brane dynamics and [[M2-M5-brane bound states]] which could be [[M-theory]]-lifts of the familiar [[integrable system|integrability]] of the [[Green-Schwarz superstring]] on [[AdS spacetime|$AdS_5$]] $\times$ [[5-sphere|$S^5$]]:

* [[Kirill Gubarev]], [[Edvard Musaev]], *Integrability structures in string theory* [[arXiv:2301.06486](https://arxiv.org/abs/2301.06486)]


fundamental particle

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

#Contents#
* table of contents
{:toc}

## Examples

[[!include fields and quanta - table]]

## Related concepts

* [[particle]], [[string]], [[brane]]

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

* [[preon]]

* [[unitary representation of the Poincaré group]]

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

## Scales

[[length scales]] in the [[observable universe]] (from [[cosmological constant|cosmic]] scales, over [[fundamental particle]]-[[masses]] around the [[electroweak symmetry breaking]] to [[GUT scale]] and [[Planck scale]]):

> graphics grabbed from [Zupan 19](flavour+in+particle+physics#Zupan19)

[[!redirects fundamental particles]]

[[!redirects elementary particle]]
[[!redirects elementary particles]]

[[!redirects family of elementary particles]]
[[!redirects families of elementary particles]]


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

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[compact Hausdorff space]], the _fundamental product theorem in topological K-theory_ identifies

1. the [[topological K-theory]]-[[commutative ring|ring]] $K(X \times S^2)$ of the [[product topological space]] $X \times S^2$ with the [[2-sphere]] $S^2 $;

1. the K-theory ring $K(X)$ of the original space $X$ with a generator $H$ for the [[basic line bundle on the 2-sphere]] adjoined:

$$
  K(X) \otimes_{\mathbb{Z}} \mathbb{Z}[H]/(H-1)^2
    \overset{\simeq}{\longrightarrow}
  K(X \times S^2)
  \,.
$$

This theorem in particular serves as a substantial step in a [[proof]] of [[Bott periodicity]] for [[topological K-theory]] (cor. \ref{BottPeriodicity} below).

The usual [[proof]] proceeds by

1.  realizing all [[vector bundles]] on $X \times S^2$ via an $X$-parameterized [[clutching construction]];

1. showing that all the clutching functions are [[homotopy|homotopic]] to those that are Laurent polynomials as functions on $S^1$, hence products of a polynomial clutching $p$ functions with a monomial $z^{-n}$ of negative power;

1. observing that the bundle corresponding to a clutching function of the form $f z^n$ is equivalent to the bundle corresponding to $f$ and tensored with the $n$th [[tensor product of vector bundles]]-power of the [[basic complex line bundle on the 2-sphere]];

1. showing that some [[direct sum of vector bundles]] of the vector bundle corresponding to a polynomial clutching function with one coming from a trivial clutching function is given by a linear clutching function;

1. showing that bundles coming from linear clutching functions are [[direct sums of vector bundles|direct sums]] of one coming from a trivial clutching function with the one coming from the homogeneously linear part;

Applying these steps to a vector bundle on $X \times S^2$ yields a virtual sum of [[external tensor products of vector bundles]] of bundles on $X$ with powers of the [[basic complex line bundle on the 2-sphere]]. This means that the function in the fundamental product theorem is surjective.
By similar means one shows that it is also injective.

## Statement

For $S^2 \subset \mathbb{R}^3$ the [[2-sphere]] with its [[Euclidean space|Euclidean]] [[subspace topology]],
write $h$ for the [[basic line bundle on the 2-sphere]]. Its image in the [[topological K-theory]] ring $K(S^2)$ satisfies the relation

$$
  2 h = h^2 + 1
  \;\;\Leftrightarrow\;\;
  (h-1)^2 = 0
$$

(by [this prop.](basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere)).

Notice that $h-1$ is the image of $h$ in the [[reduced K-theory]] $\tilde K(X)$ of $S^2$ under the splitting $K(X) \simeq \tilde K(X) \oplus \mathbb{Z}$ (by [this prop.](topological+K-theory#KGrupDirectSummandReducedKGroup)). This element

$$
  h - 1 \in \tilde K_{\mathbb{C}}(S^2)
$$

is called the _[[Bott element]]_ of complex [[topological K-theory]].

It follows that there is a [[ring homomorphism]] of the form

$$
  \array{
    \mathbb{Z}[h]/\left( (h-1)^2 \right)
      &\overset{}{\longrightarrow}&
    K(S^2)
    \\
    h &\overset{\phantom{AAA}}{\mapsto}& h
  }
$$

from the [[polynomial ring]] in one abstract generator, [[quotient ring|quotiented]] by this relation, to the [[topological K-theory]] ring.

More generally, for $X$ a [[topological space]] this induces the composite ring homomorphism

$$
  \array{
    \Phi \colon
    &
    K(X) \otimes \mathbb{Z}[h]/((h-1)^2)
      & \longrightarrow &
    K(X) \otimes K(S^2)
      & \overset{\boxtimes}{\longrightarrow} &
    K(X \times S^2)
    \\
    &
    (E, h)
      &\overset{\phantom{AAA} }{\mapsto}&
    (E,H)
      &\overset{\phantom{AAA}}{\mapsto}&
    (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H)
  }
$$

to the topological K-theory ring of the [[product topological space]] $X \times S^2$, where the second map $\boxtimes$ is the [[external tensor product of vector bundles]].

+-- {: .num_prop #FundamentalProductTheorem}
###### Proposition
**(fundamental product theorem in topological K-theory)**

For $X$ a [[compact Hausdorff space]], then ring homomorphism $\Phi \colon     K(X) \otimes \mathbb{Z}[h]/((h-1)^2) \longrightarrow K(X \times S^2)$ is an [[isomorphism]].

=--

(e.g. [Hatcher, theorem 2.2](#Hatcher))

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

More generally, for $L\to X$ a [[complex line bundle]] with class $l \in K(X)$ and with $P(1 \oplus L)$ denoting its [[projective bundle]] then

$$
  K(X)[h]/((h-1)(l \cdot h -1))
   \simeq
  K(P(1 \oplus L))
$$

=--

(e.g. [Wirthmuller 12, p. 17](#Wirthmuller12))

As a special case this implies the first statement above:

For $X = \ast$ the product theorem prop. \ref{FundamentalProductTheorem} says in particular that the first of the two morphisms in the composite is an [[isomorphism]] (example \ref{TopologicalKTheoryRingOfThe2Sphere} below)  and hence by the [[two-out-of-three]]-property for [[isomorphisms]] it follows that

+-- {: .num_cor #ExternalProductTheorem}
###### Corollary
**(external product theorem)**

For $X$ a [[compact Hausdorff space]] we have that the [[external tensor product of vector bundles]] with vector bundles on the [[2-sphere]]

$$
  \boxtimes
    \;\colon\;
  K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2)
$$

is an [[isomorphism]] in [[topological K-theory]].

=--

## Bott periodicity

When restricted to [[reduced K-theory]] then the external product theorem (cor. \ref{ExternalProductTheorem}) yields the statement of [[Bott periodicity]] of topological K-theory:

+-- {: .num_cor #BottPeriodicity}
###### Corollary
**([[Bott periodicity]])

Let $X$ be a  [[pointed topological space|pointed]] [[compact Hausdorff space]].

Then there is an [[isomorphism]] of [[reduced K-theory]]

$$
  (h-1)
    \widetilde \boxtimes
  (-)
    \;\colon\;
  \tilde K(X) \overset{\simeq}{\longrightarrow} \tilde K(\Sigma^2 X)
$$

from that of $X$ to that of its double [[suspension]] $\Sigma^2 X$.

=--

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

By [this example](topological+K-theory#ReducedKTheoryOfProductSpace)
there is for any two pointed compact Hausdorff spaces $X$ and $Y$ an [[isomorphism]]

$$
  \tilde K(Y \times X)
    \simeq
  \tilde K(Y \wedge X)
    \oplus
  \tilde K(Y)
    \oplus
  \tilde K(X)
$$

relating the reduced K-theory of the [[product topological space]]
with that of the [[smash product]].

Using this and the fact that for any pointed compact Hausdorff space $Z$ we have
$K(Z) \simeq \tilde K(Z) \oplus \mathbb{Z}$ ([this prop.](topological+K-theory#KGrupDirectSummandReducedKGroup)) the isomorphism of the external product theorem (cor. \ref{ExternalProductTheorem})

$$
  K(S^2) \otimes K(X)
   \underoverset{\simeq}{\boxtimes}{\longrightarrow}
  K(S^2 \times X)
$$

becomes

$$
  \left(
    \tilde K(S^2) \oplus \mathbb{Z}
  \right)
    \otimes
  \left(
     \tilde K(X) \oplus \mathbb{Z}
  \right)
   \;\simeq\;
  \left(
    \tilde K(S^2 \times X)
     \oplus
     \mathbb{Z}
  \right)
   \simeq
  \left(
    \tilde K(S^2 \wedge X)
      \oplus
    \tilde K(S^2)
      \oplus
    \tilde K(X)
      \oplus
    \mathbb{Z}
  \right)
  \,.
$$

Multiplying out and chasing through the constructions to see that this reduces to an isomorphism on the common summand $\tilde K(S^2) \oplus \tilde K(X) \oplus \mathbb{Z}$, this yields an isomorphism of the form

$$
  \tilde K(S^2) \otimes \tilde K(X)
    \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow}
  \tilde K(S^2 \wedge X)
    =
  \tilde K(\Sigma^2 X)
  \,,
$$

where on the right we used that [[smash product]] with the 2-sphere is the same as double [[suspension]].

Finally there is an [[isomorphism]]

$$
  \array{
    \mathbb{Z}
      &\underoverset{\simeq}{ \beta }{\longrightarrow}&
    \tilde K_{\mathbb{C}}(S^2)
    \\
    1 &\overset{\phantom{AAA}}{\mapsto}& (h-1)
  }
$$

(example \ref{TopologicalKTheoryRingOfThe2Sphere}). The composite

$$
  \array{
    \tilde K_{\mathbb{C}}(X)
      &
      \simeq
    \mathbb{Z} \otimes \tilde K_{\mathbb{C}}(X)
      \overset{ \beta \otimes id }{\longrightarrow}
    \tilde K_{\mathbb{C}}(S^2) \otimes \tilde K_{\mathbb{C}}(X)
      \underoverset{\simeq}{\widetilde \boxtimes}{\longrightarrow}
      &
    \tilde K_{\mathbb{C}}(S^2 \wedge X)
      =
    \tilde K_{\mathbb{C}}(\Sigma^2 X)
    \\
    E - rk_x(E) &\overset{\phantom{AAAA}}{\mapsto}& (h-1) \widetilde \boxtimes (E - rk_x(E))
  }
$$

is the isomorphism to be established.

=--

## Examples

+-- {: .num_example #TopologicalKTheoryRingOfThe2Sphere}
###### Example
**([[topological K-theory]] ring of the [[2-sphere]])

For $X = \ast$ the [[point space]], the fundamental product theorem \ref{FundamentalProductTheorem} states that the homomorphism

$$
  \array{
    \mathbb{Z}[h]/((h-1)^2)
      &\longrightarrow&
    K(S^2)
    \\
    h &\mapsto& h
  }
$$

is an [[isomorphism]].

This means that the relation $(h-1)^2 = 0$ satisfied by the [[basic line bundle on the 2-sphere]] ([this prop.](basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere)) is the _only_ relation is satisfies in topological K-theory.

Notice that the underlying [[abelian group]] of $\mathbb{Z}[h]/((h-1)^2)$ is two [[direct sum]] copies of the [[integers]],

$$
  K(S^2) \simeq \mathbb{Z} \oplus \mathbb{Z} = \langle 1, h\rangle
$$

one copy spanned by the [[trivial vector bundle|trivial]] [[complex line bundle]] on the 2-sphere, the other spanned by the [[basic complex line bundle on the 2-sphere]]. (In contrast, the underlying abelian group of the [[polynomial ring]] $\mathbb{R}[h]$ has infinitely many copies of $\mathbb{Z}$, one for each $h^n$, for $n \in \mathbb{N}$).

It follows (by [this prop.](topological+K-theory#KGrupDirectSummandReducedKGroup)) that the [[reduced K-theory]] group of the 2-sphere is

$$
  \tilde K(S^2) \simeq \mathbb{Z}
  \,.
$$

=--

## Related concepts

* [[equivariant K-theory of projective G-space]]

## References

Review:

* {#Wirthmuller12} [[Klaus Wirthmüller]], section 6 (from p. 19 on) in: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

* {#Hatcher} [[Allen Hatcher]], section 2.1 (from p. 45 on) in: _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* [[Varvara Karpova]], Section 5.2 in: _Complex Topological K-Theory_, 2009 ([pdf](http://infoscience.epfl.ch/record/162450/files/karpova.semestre.hess2.pdf), [[KarpovaTopologicalKTheory.pdf:file]])

[[!redirects external product theorem]]
[[!redirects external product theorem in topological K-theory]]

[[!redirects fundamental product theorem in K-theory]]


fundamental representation

+-- {: .rightHandSide}
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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[finite number|finite]] [[dimension|dimensional]] [[irreducible representation]] of a [[semisimple Lie group]] or [[Lie algebra]] is called the _fundamental representation_ if its highest [[weight]] is a [[fundamental weight]].

## References

See also:

* Wikipedia, _[Fundamental representation](http://en.wikipedia.org/wiki/Fundamental_representation)_

[[!redirects fundamental representations]]


fundamental scales -- contents

**fundamental [[scales]]** (fundamental/[[natural unit|natural]] [[physical units]])

* [[speed of light]] $\,$ $c$

* [[Planck's constant]] $\,$ $\hbar$

* [[gravitational constant]] $\,$ $G_N = \kappa^2/8\pi$

* [[Planck scale]]

  * [[Planck length]] $\,$ $\ell_p = \sqrt{ \hbar G / c^3 } $

  * [[Planck mass]] $\,$ $m_p = \sqrt{\hbar c / G}$

* depending on a given [[mass]] $m$

  * [[Compton wavelength]] $\,$ $\lambda_m = \hbar / m c$

  * [[Schwarzschild radius]] $\,$ $2 m G / c^2$

  * depending also on a given [[charge]] $e$

    * [[Schwinger limit]] $\,$ $E_{crit} = m^2 c^3 /  e \hbar$

* [[GUT scale]]

* [string scale](string+theory#Scales)

  * [[string tension]] $\,$ $T = 1/(2\pi \alpha^\prime)$

  * [[string length scale]] $\,$ $\ell_s = \sqrt{\alpha'}$

  * [[string coupling constant]] $\,$ $g_s = e^\lambda$


fundamental solution

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a linear [[differential operator]] (ordinary or partial) $P$ on a [[domain]] $M\subset\mathbb{R}^n$ or a [[manifold]] $M$, one can consider both the homogeneous [[differential equation]] $P f = 0$ and the nonhomogeneous equation of the form $P f = g$ where $g$ is a given nonhomogeneous term. If $g$ is a [[delta distribution]] and the boundary conditions are given, then the [[generalized solution of a PDE|generalized solution]] of the nonhomogenous equation

$$
  P f = \delta
$$

is called the **fundamental solution** for $P$; alternative names like __Green function__ and __function of influence__ are also used. A particular solution of the nonhomogeneous equation for some other $g$ can be obtained by calculating the [[convolution]] with the fundamental solution. (Compare the fact that the delta distribution is the [[identity element]] for convolution.)

As a first step one may also considers searching for $f$ such that $P f - \delta$ is a smooth function, the solution to the latter problem is called a [[parametrix]].

## Examples

### Propagators for free fields

The Green functions for the [[wave operator]]/[[Klein-Gordon operator]] are known as the _[[propagators]]_ for [[free fields]] in [[field theory]]:

[[!include propagators - table]]

## Properties and results

(Malgrange-Ehrenpreis theorem) If $D$ is a differential operator with constant coefficients in $n$-dimensional real space $\mathbf{R}^n$ then the fundamental solution exists in the Schwarz space $\mathcal{S}'(\mathbf{R}^n)$. The proof is using Fourier transform which sends the equation for the fundamental solution into an algebraic equation for the Fourier transform of the solution seeked; the algebraic equation requires inversion and one checks that the conditions for finding the inverse in the Schwarz space are satisfied.

## Related concepts

* [[Feynman propagator]]

* [[Dirac propagator]]

* [[parametrix]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 3.3 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* [[Sergiu Klainerman]], chapter 4 of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], section 2 and 3 of _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_,  Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009

* {#Khavkine14} [[Igor Khavkine]], section 2 of _Covariant phase space, constraints, gauge and the Peierls formula_, Int. J. Mod. Phys. A, 29, 1430009 (2014) ([arXiv.1402.1282](https://arxiv.org/abs/1402.1282))

* Wikipedia, _[Fundamental solution](http://en.wikipedia.org/wiki/Fundamental_solution)_

* Wikipedia, _[Green's functions](http://en.wikipedia.org/wiki/Green%27s_function)_

[[!redirects fundamental solutions]]

[[!redirects Green function]]
[[!redirects Green functions]]

[[!redirects Green's function]]
[[!redirects Green's functions]]

[[!redirects function of influence]]
[[!redirects functions of influence]]

[[!redirects Green operator]]
[[!redirects Green operators]]


fundamental theorem of algebra

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

\tableofcontents

## Statement

Classically, the **fundamental theorem of algebra** states that

* The [[field]] of [[complex number]]s $\mathbb{C}$ is [[algebraically closed field|algebraically closed]]. In other words, every nonconstant [[polynomial function]] with [[coefficients]] in $\mathbb{C}$ has a [[root]] in $\mathbb{C}$.

Since every non-zero polynomial could be made a [[monic polynomial]] by dividing by the leading coefficient, it could also be expressed as

* The [[field]] of [[complex number]]s $\mathbb{C}$ is [[integrally closed field|integrally closed]]. In other words, every [[monic polynomial|monic]] [[polynomial function]] with [[coefficients]] in $\mathbb{C}$ has a [[root]] in $\mathbb{C}$.

Many proofs of this theorem are known (see the [references](#References) below); some use [[complex analysis]] (the reciprocal of a [[polynomial function]] cannot be bounded), some use [[algebraic topology]] (the degree of a map is invariant with respect to homotopy), and some use advanced calculus (polynomial functions on the complex numbers are [[open map|open mappings]]). All of these proofs involve, at some level, the fact that the real numbers are [[Dedekind real number|Dedekind complete]], which has as a consequence the fact that the real numbers are [[archimedean field|archimedean]].

## Algebraic proof via real closed fields

Despite its name, the fundamental theorem of algebra makes reference to a concept from [[analysis]] (the field of complex numbers).  However, the analytic part may be reduced to a minimum: that the field of [[real numbers]] is [[real closed field|real closed]].  This has been known essentially forever, and is easily proved using (for example) the [[intermediate value theorem]].

The rest of the proof is algebraic and, unlike the other proof methods, applies to all [[real closed field]]s, which need not be archimedean. It is due to [[Emil Artin]], and forms a basic chapter in the Artin--Schreier theory of real closed fields.

We recall that a _real closed field_ is an [[ordered field]] such that every positive element has a [[square root]], and every polynomial function of odd degree has a root.  Note that the polynomial function $x^2 + 1$ cannot have any root in a real closed field --- or in fact in any ordered field, since we always have $x^2\ge 0$ and hence $x^2+1 \ge 1$.

+-- {: .num_theorem}
###### Theorem
If $F$ is real closed, then $K = F[\sqrt{-1}]$ is algebraically closed.
=--

+-- {: .proof}
###### Proof
We must show that any [[irreducible polynomial]] $p$ of degree greater than $0$ with coefficients in $K$ has a root in $K$. Since $F$ has characteristic $0$, it is a [[perfect field]].

Thus the [[splitting field]] of $p$ is a finite [[Galois extension]] $L$ of $F$, with [[Galois group]] $G$. If $G(2)$ is the [Sylow 2-group](http://ncatlab.org/nlab/show/class+equation#sylow_theorems) of $G$, then the [[fixed field]] $E$ of $G(2)$ is an odd degree extension of $F$. Any $\alpha \in E$ must then have an irreducible polynomial function $q \in F[x]$ of odd degree. But since $F$ is real closed, $q$ has a root in $F$; by irreducibility, $\deg(q) = 1$ and $\alpha \in F$, forcing $E = F$ and $G = G(2)$. We have ${|G|} \gt 1$ since the splitting field contains $K$.

So $G$ is a $2$-[[primary group]]. But for any [[prime number]] $p$, a nontrivial finite $p$-group has nontrivial [[center]] (see [here](/nlab/show/class+equation#pgroup)), and is therefore [[solvable group|solvable]] by an inductive argument. Therefore the extension $L/F$ arises from a tower of non-trivial [[quadratic extension]]s

$$F \subseteq L_1 \subseteq \ldots \subseteq L_n = L$$

By the [[quadratic formula]], the first field $L_1$ arises by adjoining roots to $F$ of a polynomial function $x^2 + a x + b$,

$$\frac{-a \pm \sqrt{a^2 - 4b}}{2},$$

where $a^2 - 4b$ is negative. Since $F$ is real closed, the positive element $4b - a^2$ has a square root in $F$, so that the roots displayed above belong to $K = F[\sqrt{-1}]$. So $L_1 = K$. But $K$ has no nontrivial quadratic extensions by the lemma that follows, so in fact $L_1 = L_n = K$ and the theorem is proved.
=--

+-- {: .num_lemma #sqrt}
###### Lemma
Every element of $K = F[\sqrt{-1}]$ has a square root in $K$.
=--

+-- {: .proof}
###### Proof
The proof is most easily apprehended by analogy with polar coordinate representations of complex numbers and half-angle formulas, where a square root of $r e^{i\theta}$ is given by $r^{1/2}e^{i\theta /2}$. Let $i$ be a fixed square root of $-1$, and let $a + b i$ be an arbitrary element of $K$, with $a, b \in F$. We must solve $(x + y i)^2 = a + b i$, i.e., find $x, y \in F$ that solve

$$x^2 - y^2 = a, \qquad 2x y = b$$

Since $a^2 + b^2$ has a square root in $F$, we may assume by homogeneity in $x, y$ that $(a, b)$ is on the unit circle: $a^2 + b^2 = 1$. By interchanging $x$ and $y$ if need be, we may assume $0 \leq a \leq 1$; replacing $y$ by $-y$ if need be, we may assume $b \geq 0$. Taking $x, y \geq 0$ such that

$$x^2 = \frac{1+a}{2}, \qquad y^2 = \frac{1-a}{2},$$

we obtain a solution (since $x^2 - y^2 = a$ and $4 x^2 y^2 = b^2$).
=--

## Classical FTA via advanced calculus

As noted above, many proofs of the fundamental theorem are known. The following proof, ultimately rooted in the fact that [[polynomial mappings]] on $\mathbb{C}$ are [[open mappings]], has the advantage that it requires very little machinery. From what I ([[Todd Trimble]]) understand, it is close to the method used by Argand to give his proof (1814)[^1].

[^1]: Despite the credit given to Gauss for his demonstration of 1799, Argand's proof is often credited as the first one that is *fully rigorous*. The proof given here also uses the Bolzano-Weierstrass theorem, first proven by Bolzano in 1817, making it somewhat contemporaneous. Argand is also widely credited as the one who introduced the cutting-edge idea of viewing complex numbers and their operations *geometrically*, which the proof here also uses (the complex plane $\mathbb{C}$ being also known as the Argand plane).

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a nonconstant polynomial mapping, and suppose $f$ has no zero.

1. Let $s$ be the [[infimum]] of values ${|f(z)|}$; choose a [[sequence]] $z_1, z_2, z_3, \ldots$ such that ${|f(z_n)|} \to s$. Since $\lim_{z \to \infty} f(z) = \infty$, the sequence $z_n$ must be bounded; by the Bolzano-Weierstrass theorem it has a [[sequence|subsequence]] $z_{n_k}$ that converges to some point $z_0$. Then ${|f(z_{n_k})|}$ converges to ${|f(z_0)|}$ by continuity, and converges to $s$ as well, so ${|f(z)|}$ attains an absolute minimum $s$ at $z = z_0$. By supposition, $f(z_0) \neq 0$.

2. The polynomial function $f$ may be uniquely written in the form
$$f(z) = f(z_0) + g(z)(z - z_0)^n$$
where $g$ is polynomial function and $g(z_0) \neq 0$. Put
$$F(z) = f(z_0) + g(z_0)(z - z_0)^n$$
and choose $\delta \gt 0$ small so that
$${|z - z_0|} = \delta \Rightarrow {|g(z) - g(z_0)|} \lt {|g(z_0)|}.$$

3. $F$ maps the circle $C = \{z : {|z - z_0|} = \delta\}$ _onto_ a circle of radius $r = {|g(z_0)|}\delta^n$ centered at $f(z_0)$. (This uses the fact that any complex number has an $n^{th}$ root, which one can prove using polar coordinate representations. We omit the details.) Choose $z' \in C$ so that $F(z')$ is on the line segment between the origin and $f(z_0)$ (we can always choose $\delta$ so that also $r \lt {|f(z_0)|}$). Then
$${|F(z')|} = {|f(z_0)|} - r$$
We also have
$${|f(z') - F(z')|} = {|g(z') - g(z_0)|} {|z' - z_0|^n} \lt {|g(z_0)|} \delta^n = r$$
according to how we chose $\delta$ in 2. We conclude by observing the strict inequality
$${|f(z')|} \leq {|F(z')|} + {|f(z') - F(z')|} \lt {|f(z_0)|} - r + r = {|f(z_0)|},$$
which contradicts the fact that ${|f(z)|}$ attains an absolute minimum at $z = z_0$.

## In weak foundations

Many proofs rely explicitly on the [[double negation]] rule by first supposing that a polynomial function $p$ has no root and deriving a contradiction.

### In constructive mathematics

In [[constructive mathematics]], [[integral closure]] and [[algebraic closure]] of a [[field]] are not the same, because not every nonconstant polynomial function has a well-defined degree. This is still the case even if we take "nonconstant polynomial function", we mean "[[tight apartness relation|apart from]] every constant polynomial function". In general, every nonconstant polynomial function has a well-defined degree if and only if the field is a [[discrete field]].

Thus, the fundamental theorem of algebra is usually expressed in terms of integral closure, which classically is the same as algebraic closure for any field:

* The [[field]] of [[complex number]]s $\mathbb{C}$ is [[integrally closed field|integrally closed]]. In other words, every [[monic polynomial|monic]] [[polynomial function]] with [[coefficients]] in $\mathbb{C}$ has a [[root]] in $\mathbb{C}$.

There is a second problem, namely with Lemma \ref{sqrt}. It is usually impossible to prove that the monic quadratic function $x \mapsto x^2 + c$ has a [[root]] for all complex numbers $c \in \mathbb{C}$. The problem here is once again that any constructive notion of complex numbers is only a [[Heyting field]], rather than a [[discrete field]]. The function $x \mapsto x^2 + c$ has a root for $c = 0$ and for invertible $c$, but it is not provable that every complex number $c \in \mathbb{C}$ is either invertible or equal to zero. In fact, this fails in certain [[topoi]], such as [[sheaves]] over $\mathbb{C}$, because that the complex numbers have a continuous square root over $\mathbb{C}$ can only be proven in the presence of [[weak countable choice]].

However, the following weaker theorem is still true in constructive mathematics for the [[modulated Cantor real number|modulated Cantor complex numbers]]:

\begin{theorem}
Every [[unramifiable polynomial|unramifiable]] [[monic polynomial|monic]] [[polynomial function]] with coefficients in $\mathbb{C}$ has a [[simple root]] in $\mathbb{C}$.
\end{theorem}

A fully choice-free constructive proof of this theorem by Wim Ruitenberg ([Ruitenberg 1991](#Ruitenberg91)) exists for the [[modulated Cantor real number|modulated Cantor complex numbers]] (which agree with the [[Dedekind real number|Dedekind complex numbers]] by [[weak countable choice]] or the [[limited principle of omniscience]]). It is unknown if this weaker FTA is true of the [[Dedekind real number|Dedekind complex numbers]] as well.

[Richmann 2000](#Richman00) has proposed that, in the absence of $WCC$, the FTA should be interpreted as a statement about sets of roots rather than about individual roots.  He constructs a [[complete metric space]] $\hat{M}_n(\mathbb{C})$ which, classically, is the space of $n$-element [[multisets]] of complex numbers (and constructively is the completion of that space) and proves that every complex polynomial function $p$ of degree $n$ may be associated with a point in this space in such a way that the $n$ elements of that point (when viewed as a multiset, if possible, and morally in any case) are the $n$ roots of $p$.

### In RCA_0

Assuming classical logic, but weak foundations, it can be shown that FTA is true in the [[reverse mathematics]] system $RCA_0$ ([Tanaka-Yamazaki 2005](#TY2005)).

## History
 {#History}

The proof was attempted many times before Gauss gave what is accepted as the first proof in his dissertation ([Gauss 1799](#Gauss1799)), although this was not without issues (Gauss 'fixed' this proof almost 50 years later, but the last gap was not filled until the 20th century).

All proofs of this fact (of which there are many) require something analytic, in the sense that ordinary algebra will not suffice: one needs to know that the real numbers (or the complex numbers) 'have no algebraic gaps'. For instance, the rational numbers famously don't contain the square root of $2$. The cleanest proof I know, due to Artin, that isolates this analytic germ, uses the step-ladder result that the real numbers form what is called a [[real closed field]]. This is essentially saying that non-negative real numbers have square roots, and odd degree polynomial functions have roots (anyone who has plotted a cubic can appreciate this fact). Alternatively, one can characterise real closed fields as those for whom the Intermediate Value Theorem (IVT) holds for polynomial functions. Accepting this result (which does need proof), the FTA follows using pure algebra (although not of the high-school sort).

However, it is of interest, partly theoretical, partly for the sake of finding the bare minimum needed to prove the FTA, to know an elementary proof, namely one that minimises the use of analytic techniques (for instance, the IVT for polynomial functions follows from the IVT for continuous functions, but that is like killing a mosquito with a bazooka). Gauss' second proof ([Gauss 1866](#Gauss1866)) is elementary (and predates Artin's by a long time). Since Gauss lacked modern algebraic techniques, some of his proof is laborious, but ([Taylor 85](#Taylor85)) gives a modern gloss. (With some amusing side notes: as Taylor puts it -- 'Gauss takes the opportunity [to] be rude to his inferior contemporaries'.) Gauss' proof, in modern language, takes up less than a page and a half, but this presupposes familiarity with some of the theory of fields (but which is pure algebra). Artin's proof, by comparison, drawing on major theorems can be given in half a page.

It should be noted, in the context of the last statement, that proofs of the FTA can be given, relying on analytic 'bazooka' theorems, that are one sentence. However, to spell out the proofs of the necessary theorems, one needs a course in analysis, of some variety, so one is merely sweeping a lot under a very small rug.

## References
 {#References}

* {#Gauss1799} [[Carl Gauss]], _Demonstratio nova theorematis functionem algebraicam rationalem integramunius variabilis in factores reales primi vel secundi gradus resolvi posse_, Dissertation, Helmstedt (1799); Werke 3, 1–30 (1866) ([English transl. pdf](http://www.jon-arny.com/httpdocs/Gauss/Gauss%20Fundamental%20Theorem.pdf)))

* {#Gauss1866} [[Carl Gauss]], _Demonstratio nova altera theorematis omnem functionem algebraicamrationalem integram unius variabilis in factores reales primi vel secundi gradus resolviposse_, Comm. Soc. Reg. Sci. Göttingen 3, 107–142 (1816); Werke 3, 33–56 (1866)

* Michael Eisermann. _An Elementary Real-Algebraic Proof via Sturm Chains_. [pdf](http://www.jon-arny.com/httpdocs/Gauss/Constructive%20roots-annotated.pdf)

  _Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree_ translated by [[Paul Taylor]] and B. Leak (1983) ([web](http://www.paultaylor.eu/misc/gauss-web))

* {#Taylor85} [[Paul Taylor]], _Gauss' Second Proof_, Eureka 45 (1985) 42-47 ([pdf](http://www.jon-arny.com/httpdocs/Gauss/Eureka-2-aug24.pdf))

A proof of a variant of the fundamental theorem of algebra using finite multisets of complex numbers:

* {#Richman00} [[Fred Richman]], *The fundamental theorem of algebra: a constructive development without choice*, Pacific Journal of Mathematics **196** 1 (2000) 213–230 [[doi:10.2140/pjm.2000.196.213](http://dx.doi.org/10.2140/pjm.2000.196.213), [pdf](https://msp.org/pjm/2000/196-1/pjm-v196-n1-p10-p.pdf)]

see also:

* [[Fred Richman]], *Constructive Mathematics without Choice*,
in: *Reuniting the Antipodes -- Constructive and Nonstandard Views of the Continuum* Synthese Library **306**, Springer (2001) 199-206 [[doi:10.1007/978-94-015-9757-9_17](https://doi.org/10.1007/978-94-015-9757-9_17)]

The Reverse Mathematical treatment is given in

* {#TK2005} Kazuyuki Tanaka and Takeshi Yamazaki, _Manipulating the reals in $RCA_0$_ in _Reverse Mathematics 2001_, Lecture Notes in Logic **21** (2005)

A [[constructive mathematics|constructive]] algebraic proof of the fundamental theorem of algebra for the [[modulated Cauchy real numbers]] without choice princples such as [[weak countable choice]]

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

A full formalization in the [[Coq]] [[proof assistant]] is in

* Herman Geuvers, [[Freek Wiedijk]], Jan Zwanenburg, A Constructive Proof of the Fundamental Theorem of Algebra without Using the Rationals ([web](http://dl.acm.org/citation.cfm?id=696038))

See also

* MathOverflow, _[Ways to prove the fundamental theorem of algebra](http://mathoverflow.net/questions/10535/ways-to-prove-the-fundamental-theorem-of-algebra)_

[[!redirects fundamental theorem of algebra]]
[[!redirects Fundamental Theorem of Algebra]]
[[!redirects FTA]]


fundamental theorem of arithmetic

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

# The Fundamental Theorem of Arithmetic
* table of contents
{: toc}

## Idea

The so-called _Fundamental Theorem of Arithmetic_ (sometimes _FTA_, although this can also mean the [[Fundamental Theorem of Algebra]]) is the statement that the [[ring of integers]] is a [[unique factorization domain]] (UFD).  The theorem is much older than the abstract concepts of [[ring theory]], and indeed the concept of UFD is an abstraction of this theorem.  Inasumuch as '[[arithmetic]]' can refer to [[number theory]], this is a pretty basic theorem in that field.

## Statements

Let $\mathbb{N}_+$ be the positive natural numbers, let $\mathbb{N}_+ \times \mathbb{N}_+$ be the [[cartesian product]] of $\mathbb{N}_+$, with canonical product projection functions $\pi_1:\mathbb{N}_+ \times \mathbb{N}_+ \to \mathbb{N}_+$ and $\pi_2:\mathbb{N}_+ \times \mathbb{N}_+ \to \mathbb{N}_+$, and let $A^*$ be the [[free monoid]] on the set $A$. Let $p:\mathbb{N}_+ \to \mathbb{N}_+$ be the function which takes a positive natural number $n$ to the $n$-th [[prime number]].

Since the positive natural numbers form a [[monoid]] with respect to multiplication, finite products exist, given by the [[monoid homomorphism]]

$$\prod_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):\mathbb{N}_+^* \to \mathbb{N}_+$$

inductively defined on $\mathbb{N}_+^*$ by

$$\prod_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 1$$

$$\prod_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a$$

$$\left(\prod_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) \cdot \left(\prod_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \prod_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)$$

+-- {: .num_theorem}
###### Theorem
There is a [[bijection]] $\mathrm{uf}:\mathbb{N}_+ \cong (\mathbb{N}_+ \times \mathbb{N}_+)^*$ such that for every positive natural number $n$,

$$n = \prod_{i = 0}^{\mathrm{len}(\mathrm{uf}(n)) - 1} p(\pi_1(\mathrm{uf}(n)(i)))^{\pi_2(\mathrm{uf}(n)(i))}$$

equivalently, for every list of pairs of positive natural numbers $a:(\mathbb{N} \times \mathbb{N})^*$

$$\mathrm{uf}^{-1}(a) = \prod_{i = 0}^{\mathrm{len}(a) - 1} p(\pi_1(a(i)))^{\pi_2(a(i))}$$
=--

Equivalently, instead of using lists of pairs of positive natural numbers, one could just use [[finite multisets]] of positive natural numbers, which are lists of lists of positive natural numbers.

+-- {: .num_theorem}
###### Theorem
There is a [[bijection]] $\mathrm{uf}:\mathbb{N}_+ \cong (\mathbb{N}_+^*)^*$ such that for every positive natural number $n$,

$$n = \prod_{i = 0}^{\mathrm{len}(\mathrm{uf}(n))} \prod_{j = 0}^{\mathrm{len}(\mathrm{uf}(n)(i)) - 1} p(\mathrm{uf}(n)(i)(j))$$

equivalently, for every finite multiset of positive natural numbers $a:(\mathbb{N}_+^*)^*$

$$\mathrm{uf}^{-1}(a) = \prod_{i = 0}^{\mathrm{len}(a) - 1} \prod_{j = 0}^{\mathrm{len}(a(i)) - 1} p(a(i)(j))$$
=--

+-- {: .num_theorem}
###### Theorem
The [[free commutative monoid]] on the [[prime numbers]] is [[isomorphic]] to the [[multiplicative subset]] of [[positive number|positive]] [[natural numbers]].
=--

+-- {: .num_cor}
###### Corollary

The [[ring of integers]] is a [[unique factorization domain]].
=--

[[!redirects fundamental theorem of arithmetic]]
[[!redirects Fundamental Theorem of Arithmetic]]


fundamental theorem of calculus

# The fundamental theorem of calculus
* table of contents
{: toc}

## Idea

The fundamental theorem of the [[infinitesimal calculus]] (FTC) states that the [[antiderivatives]] and [[indefinite integrals]] of a [[function]] (typically a [[real number|real]]-valued function on a closed [[interval]] in the [[real line]]) are the same.  It is now such a truism that calculus textbooks tend to use 'indefinite integral' to simply mean an antiderivative, but there is a different concept, definable from the [[definite integral]] and sometimes called the *semidefinite* integral, that is really in play here.

The FTC is usually split into two parts.  The first part (sometimes called the second part) states that every indefinite integral is an antiderivative.  The second part (sometimes called the first part) states that every antiderivative is an indefinite integral.  Somewhere in here (usually in the first part) we also want to state that such antiderivatives and indefinite integrals actually exist.  Their uniqueness (such as it is) may also be included.

The second part is a low-dimensional version of the [[Stokes theorem]].

The FTC may or may not actually be true, depending on what class of functions is considered and what notions of [[derivative]] and [[integral]] are used.

## Statements

Let $f\colon [a, b] \to \mathbb{R}$ be a [[function]].  Recall that an __antiderivative__ of $f$ is a solution $F$ to the [[differential equation]]
$$ F'(x) = f(x) .$$
Also recall (or learn) than an __indefinite integral__ of $f$ is any function $F$ of the form
$$ F(x) = \int_a^x f(t) \,\mathrm{d}t + C $$
for a real constant $C$.

+-- {: .un_theorem #FTC}
###### Theorems (Fundamental Theorem of Calculus)

Suppose that $F$ and $f$ are appropriate functions, and use an appropriate notion of derivative and integral.  Then:

1. First part: If $F$ is an indefinite integral of $f$, then $F$ is an antiderivative of $f$.
2. Second part: If $F$ is an antiderivative of $f$, then $F$ is an indefinite integral of $f$.
3. Existence: $f$ has an indefinite integral / antiderivative.
4. Uniqueness: If $F_1$ and $F_2$ are antiderivatives of $f$, then $F_1 - F_2$ is a [[constant function]].
=--

This holds under any of the following conditions:

1. $F$ is [[continuously differentiable function|continuously differentiable]], $f$ is [[continuous function|continuous]], differentiation is pointwise, and integration is the [[Riemann integral]].
2. $F$ is [[absolutely continuous function|absolutely continuous]], $f$ is [[integrable function|integrable]], differentiation is [[almost everywhere]], and integration is the [[Lebesgue integral]].
3. In pointwise [[constructive mathematics]] with the [[fan theorem]], we may proceed as in (1).
4. In pointwise constructive mathematics with [[countable choice]], $F$ is uniformly continuously differentiable, $f$ is [[uniformly continuous function|uniformly continuous]], differentiation is uniform differentiability, and integration is the [[Riemann integral]].
5. In constructive [[locale theory]], we may proceed as in (1) if the 'functions' involved are locale-theoretic [[continuous maps]] rather than set-theoretic functions.
6. In [[synthetic differential geometry]], $F$ and $f$ are [[smooth map|smooth]], differentiation is axiomatic, and integration is given by the [[integration axiom]].

One may rephrase the first part in various ways as follows.  First, simply incorporating the definitions of 'indefinite integral' and 'antiderivative', we have
$$ \frac{\mathrm{d}}{\mathrm{d}x} \left(\int_a^x f(t) \,\mathrm{d}t + C\right) = f(x) .$$
Taking for granted that adding a constant to a function leaves its derivative unchanged, this simplifies to
$$ \frac{\mathrm{d}}{\mathrm{d}x} \left(\int_a^x f(t) \,\mathrm{d}t\right) = f(x) .$$
When combined with the [[chain rule]], this generalizes to
$$ \frac{\mathrm{d}}{\mathrm{d}x} \left(\int_a^{q(x)} f(t) \,\mathrm{d}t\right) = f(q(x)) q'(x).$$
Taking for granted that $\int_u^v = \int_a^v - \int_a^u$, this generalizes to
$$ \frac{\mathrm{d}}{\mathrm{d}x} \left(\int_{p(x)}^{q(x)} f(t) \,\mathrm{d}t\right) = f(q(x)) q'(x) - f(p(x)) p'(x).$$
Writing $u$ for $p(x)$ and $v$ for $q(x)$ and multiplying both sides by $\mathrm{d}x$, we get the [[differential]] form
$$ \mathrm{d}\left(\int_u^v f(t) \,\mathrm{d}t\right) = f(v) \,\mathrm{d}v - f(u) \,\mathrm{d}u .$$
In other words, the differential and definite integral operators cancel upon performing the [[substitution]] directed by the limits of the definite integral.

One may rephrase the second part in various ways as follows.  First, simply incorporating the definitions of 'indefinite integral' and 'antiderivative', we have
$$ \int_a^x F'(t) \,\mathrm{d}t + C = F(x) $$
for some constant $C$.  Taking for granted that $\int_a^a = 0$, we see that $C = F(a)$, so
$$ F(x) = F(a) + \int_a^x F'(t) \,\mathrm{d}t .$$
It is sufficient to consider the special case where $x = b$:
$$ \int_a^b F'(t) \,\mathrm{d}t = F(b) - F(a) .$$
This can also be written
$$ \int_a^b \mathrm{d}F(t) = F(b) - F(a) $$
to match the last version of the first part; again, the differential and definite integral operators cancel (but now in the opposite order) upon performing the substitution directed by the limits of the definite integral.

## Proofs

If convenient, one may prove the theorem first for the [[unit interval]] $[0, 1]$. By applying a suitable [[affine transformation]], we can deduce the corresponding result for each interval $[a, b]$ where $a$ and $b$ are real numbers with $a \leq b$.

In [[classical mathematics]] and pointwise constructive mathematics, existence of definite integrals must be proved first; existence of indefinite integrals follows.  We can then prove that every indefinite integral is an antiderivative by applying the [[mean value theorem]].  Uniqueness of antiderivatives has a simple direct proof; since at least one antiderivative is an indefinite integral, this shows that every antiderivative is an indefinite integral.  Of course, we have existence of an antiderivative, so now everything is proved.

In the locale-theoretic approach, we essentially *define* an antiderivative to be an indefinite integral.  Formally, the data in a [[continuously differentiable map]] on $[a,b]$ consists of a [[continuous map]] (its derivative) and a real number (its value at $a$).  We must prove the existence of indefinite integrals directly; a definite integral up to each point is not sufficient.  Then there is nothing more to prove (but one might want to prove that differentiation has its usual properties).

Conversely, in synthetic differential geometry, we essentially define an indefinite integral to be an antiderivative.  We must prove uniqueness of antiderivatives; but their existence is given by an [[axiom]], the [[integration axiom]], and then there is nothing more to prove.  Sometimes one studies versions of SDG without this axiom; then there is no FTC (and indeed no integrals).

## Integral formula for antiderivative

In the literature, the first part of the FTC is sometimes stated as merely existence of an antiderivative.  The purpose of this section is to show, independent of foundations, that the second part gives the integral formula for the antiderivative as an indefinite integral.

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

Suppose $x$ is in $[0, 1]$. By assumption, there is $F$ such that $F' = f$. By the other part of the fundamental theorem, we have $\int_0^x F'(t) \, {\mathrm{d}}t = F(x) - F(0)$. Thus, $\int_0^x f(t) \, {\mathrm{d}}t = F(x) - F(0)$. Differentiating gives the required result.
=--

## References

* Wikipedia, _[Fundamental theorem of calculus](http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus)_

* [[Steve Vickers]], *The Fundamental Theorem of Calculus point-free, with applications to exponentials and logarithms*, ([arXiv:2312.05228](https://arxiv.org/abs/2312.05228))

[[!redirects fundamental theorem of calculus]]
[[!redirects fundamental theorems of calculus]]
[[!redirects fundamental theorem of infinitesimal calculus]]
[[!redirects fundamental theorems of infinitesimal calculus]]
[[!redirects fundamental theorem of differential calculus]]
[[!redirects fundamental theorems of differential calculus]]
[[!redirects fundamental theorem of integral calculus]]
[[!redirects fundamental theorems of integral calculus]]
[[!redirects Fundamental Theorem of Calculus]]
[[!redirects FTC]]


fundamental theorem of covering spaces

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

#Contents#
* table of contents
{:toc}

## Idea

In [[topological homotopy theory]], the _fundamental theorem of covering spaces_ says that for a sufficiently well-behaved [[topological space]] $X$, the [[functor]] which sends a [[covering space]] of $X$ to its [[monodromy]] [[Set]]-[[action]] ([[permutation representation]]) of the [[fundamental groupoid]] of $X$ on the [[fibers]] of $E$ is an [[equivalence of categories]].

{#PathLifting} This is a basic instance of the general principle of [[Galois theory]].

<img src="https://ncatlab.org/nlab/files/PathLiftingInCoveringSpace-230214.jpg" width="500">

It follows in particular that for [[connected topological space|connected]] $X$ the [[automorphism group]] of the [[universal covering space]] of $X$ coincides with the [[fundamental group]] $\pi_1(X,x)$ itself (for any basepoint $x$). This often yields a convenient means to determine the [[fundamental group]] of $X$ in the first place.

This is closely related to the [[(∞,1)-Grothendieck construction]]; the equivalence $\infty Grpd_{/\esh X} \simeq \infty Grpd^{\esh X}$ restricts to an equivalence between the subcategories of bundles $Y \to \esh X$ with 0-truncated fibers and of set-valued functors on $\esh X$. Observe in particular that $Set^{\Pi_1(X)} \simeq Set^{\esh X}$.

(Here $\esh X$ denotes the [[shape modality|shape]] of the topological space $X$, [[shape via cohesive path ∞-groupoid|hence]] its [[fundamental infinity-groupoid|fundamental $\infty$-groupoid]]).

## Statement

+-- {: .num_theorem #FundamentalTheoremOfCoveringSpaces}
###### Theorem
**(fundamental theorem of covering spaces)**

Let $X$ be a [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply-connected topological space]]. Then the operations on

1. extracting the [[monodromy]] $Fib_{E}$ of a [[covering space]] $E$ over $X$

1. [[reconstructing a covering space from monodromy]] $\; Rec(\rho)$

constitute an [[equivalence of categories]]

$$
  Cov(X)
    \underoverset
      {\underset{Fib}{\longrightarrow}}
      {\overset{Rec}{\longleftarrow}}
      {\simeq}
  Set^{\Pi_1(X)}
$$

between the [[category of covering spaces]], and the category  of [[permutation representation|permutation]] [[groupoid representations]] of the [[fundamental groupoid]] of $X$.

=--

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

With the standard definitions of the two functors, both are in fact inverse [[isomorphisms]] of categories instead of just [[equivalences of categories]] (meaning that the required [[natural isomorphisms]] from the composites of the two functors to the [[identity functor]] are componentwise [[equalities]]), which establishes the claim right away. For definiteness, we make this explicit:

Given $\rho \in Set^{\Pi_1(X)}$ a [[permutation representation]],
we need to exhibit a [[natural isomorphism]] of permutation representations.

$$
  \eta_{\rho}
    \;\colon\;
  \rho \longrightarrow Fib(Rec(\rho))
$$

First consider what the right hand side is like:
By [this def.](reconstruction+of+covering+spaces+from+monodromy#ElementaryReconstructionCoveringSpace) of $Rec$ and [this def.](monodromy#CoveringSpaceMonodromy) of $Fib$ we have for every $x \in X$ an actual equality

$$
  Fib(Rec(\rho))(x) = \rho(x)
  \,.
$$

To similarly understand the value of $Fib(Rec(\rho))$ on morphisms $[\gamma] \in \Pi_1(X)$, let $\gamma \colon [0,1] \to X$ be a representing [[path]] in $X$. As in the proof of the path lifting lemma for covering spaces ([this lemma](covering+space#CoveringSpacePathLifting)) we find a [[finite number]] of paths $\{\gamma_i\}_{i \in \{1,n\}}$ such that

1. regarded as morphisms $[\gamma_i]$ in $\Pi_1(X)$ they [[composition|compose]] to $[\gamma]$:

   $$
     [\gamma] = [\gamma_n] \circ \cdots \circ [\gamma_2] \circ [\gamma_1]
   $$

1. each $\gamma_i$ factors through an open subset $U_i \subset X$ over which $Rec(\rho)$ trivializes.

Hence by [[functor|functoriality]] of $Fib(Rec(\rho))$ it is sufficient to understand its value on these paths $\gamma_i$. But on these we have again by direct unwinding of the definitions that

$$
  Fib(Rec(\rho))([\gamma_i]) = \rho([\gamma_i])
  \,.
$$

This means that if we take

$$
  \eta_\rho(x) \colon \rho(x) \overset{=}{\longrightarrow} Fib(Rec(\rho))
$$

to be the above identification, then this is a [[natural transformation]] and hence in a particular a natural isomorphism, as required.

It remains to see that these morphisms $\eta_\rho$ are themselves natural in $\rho$, hence that for each morphism $\phi \colon \rho \to \rho'$ the diagram

$$
  \array{
    \rho &\overset{\phi}{\longrightarrow}& \rho'
    \\
    {}^{\mathllap{eta_\rho}}\downarrow
      &&
    \downarrow^{\mathrlap{\eta_{\rho'}}}
    \\
    Fib(Rec(\rho))
    &\underset{Fib(Rec(\phi))}{\longrightarrow}&
    Fib(Rec(\rho'))
  }
$$

commutes as a diagram in $Rep(\Pi_1(X), Set)$. Since these morphisms are themselves groupoid homotopies (natural isomorphisms) this is the case precisely if for all $x \in X$ the corresponding component diagram commutes. But by the above this is

$$
  \array{
    \rho(x) &\overset{\phi(x)}{\longrightarrow}& \rho'(x)
    \\
    {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{=}}
    \\
    Fib(Rec(\rho))(x)
    &\underset{Fib(Rec(\phi))(x) }{\longrightarrow}&
    Fib(Rec(\rho'))(x)
  }
$$

and hence this means that the top and bottom horizontal morphism are in fact equal. Direct unwinding of the definitions shows that this is indeed the case.

Conversely, given $E \in Cov(X)$ a covering space, we need to exhibit a natural isomorphism of covering spaces of the form

$$
  \epsilon_E
    \;\colon\;
  Rec(Fib(E)) \longrightarrow E
  \,.
$$

Again by [this def.](reconstruction+of+covering+spaces+from+monodromy#ElementaryReconstructionCoveringSpace) of $Rec$ and [this def.](monodromy#CoveringSpaceMonodromy) of $Fib$  the underlying set of $Rec(Fib(E))$ is actually equal to that of $E$, hence it is sufficient to check that this [[identity function]] on underlying sets is a [[homeomorphism]] of [[topological spaces]].

By the assumption that $X$ is [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply connected topological space|semi-locally simply connected]], it is sufficient to check for $U\subset X$ an open path-connected subset and $x \in X$ a point with the property that $\pi_1(U,x) \to \pi_1(X,x)$ lands is constant on the trivial element, that the open subsets of $E$ of the form $U \times \{\hat x\} \subset p^{-1}(U)$ form a basis for the topology of $Rec(Fib(E))$. But this is the case by definition of $Rec$.

It remains to see that $\epsilon_E$ is itself natural in $E$. But as for the converse direction, since the components of $\epsilon_E$ are in fact equalities, this follows by direct unwinding of the definitions.

This establishes an equivalence as required. In fact this is an [[adjoint equivalence]].

=--

## Applications

* [[fundamental group of the circle is the integers]]

## In homotopy type theory

In [[homotopy type theory]], the fundamental theorem of covering spaces is really just a special case of the [[universal property]] of [[n-truncation modality|$1$-truncation]].

Indeed, in HoTT a covering space over a [[type]] $X$ is usually defined as a [[dependent type]] $C \colon X \to \mathsf{Sets}$. We can get the more familiar bundle definition of a covering space by taking the [[dependent sum type]] $\sum_{x \colon X} C(x)$, which comes equipped with a canonical projection $\operatorname{proj}_1 \colon \sum_{x \colon X} C(x) \to X$.

Since $\mathsf{Sets}$ is a $1$-type, then by the universal property of 1-truncation we have an equivalence:

$$
  \big(
    \Vert X \Vert_1 \to \mathsf{Sets}
  \big)
  \;\simeq\;
  \big(
    X \to \mathsf{Sets}
  \big)
  \,.
$$
Since the 1-truncation $\Vert X \Vert_1$ is the [[fundamental groupoid]] of $X$, this really is the fundamental theorem of covering spaces.

\linebreak

{#InCohesiveHomotopyTypeTheory}
Or rather, more faithful to the traditional concepts in topology, it is the [[shape modality]] $\esh$ in [[cohesive homotopy type theory]]
which  turns a [[geometric homotopy type]] into $X$ its [[fundamental infinity-groupoid|fundamental $\infty$-groupoid]] $\esh X$ (see at *[[shape via cohesive path ∞-groupoid]]*) truncating to the actual [[fundamental groupoid]] $\Vert \esh X \Vert_1$.

Then the [[adjunction]] between the [[shape modality]] and the [[flat modality]] $\flat$ says that

$$
  \big(
    \esh X \to \mathsf{Sets}
  \big)
  \;\simeq\;
  \big(
    X \to \flat \mathsf{Sets}
  \big)
  \,,
$$

or, equivalently, that

$$
  \big(
    \Vert \esh X \Vert_1 \to \mathsf{Sets}
  \big)
  \;\simeq\;
  \big(
    X \to \flat \mathsf{Sets}
  \big)
  \,,
$$

where $\flat Sets$ is the actual classifier for covering spaces in the generality of cohesive (e.g. topological) homotopy types. This reflects the fundamental theorem of covering spaces as traditionally understood in [[topology]].

This is the topic of [[schreiber:differential cohomology in a cohesive topos|dcct, sec. 3.8.6]], [p. 358](https://arxiv.org/pdf/1310.7930v1.pdf#page=358), see also [Cherubini & Rijke 2020, Thm. 8.7](#CherubiniRijke20).

## Related concepts

* [[transport]]

* [[fundamental group]], [[fundamental groupoid]]

* [[universal cover]]

* [[G-set]]

* [homotopy of smooth paths relative to their endpoints](smooth+homotopy#HomotopyOfSmoothPathsRelativeToTheirEndpoints)

* [[Grothendieck's Galois theory]]

## References

On the classical theory:

A detailed treatment is available in

* [[Nicolas Bourbaki]], _Topologie Algébrique_, Chapitres 1 à 4, Springer (1998, 2016)  [[doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8), ISBN 978-3-662-49361-8]

Textbook account:

* {#tomDieck2008} [[Tammo tom Dieck]], Thm. 3.3.2 in: _Algebraic topology_, European Mathematical Society, Zürich (2008) [[doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf)]

Lecture notes:

* {#Waldhausen} [[Friedhelm Waldhausen]], around p. 122 of  _Topologie_ ([pdf](https://www.math.uni-bielefeld.de/~fw/ein.pdf))

* {#Moller11} [[Jesper Møller]], Thm. 7.8 in:  _The fundamental group and covering spaces_ (2011) [[arXiv:1106.5650](https://arxiv.org/abs/1106.5650), [pdf](http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf), [[Moller-FundamentalGroup.pdf:file]]]

Discussion in [[cohesive homotopy theory]]:

* [[Urs Schreiber]], [[schreiber:differential cohomology in a cohesive topos|dcct, sec. 3.8.6]], [p. 358](https://arxiv.org/pdf/1310.7930v1.pdf#page=358)

Discussion in [[cohesive homotopy type theory]]:

* {#CherubiniRijke20} [[Felix Cherubini]], [[Egbert Rijke]], Thm. 8.7 in: *Modal Descent*, Mathematical Structures in Computer Science (2020)  ([arXiv:2003.09713](https://arxiv.org/abs/2003.09713), [doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201))


fundamental theorem of dg-algebraic equivariant rational homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The fundamental [[theorem]] of [[equivariant rational homotopy theory]] modeled by [[equivariant dgc-algebras]].

## Preliminaries

Let $G$ be a [[finite group]].

Write

* $\big(G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj}$ for the [[opposite model structure|opposite]] of the [[projective model structure on equivariant connective dgc-algebras]];

* $G SimplicialSets_{Qu}$ for the [[model structure on equivariant simplicial sets]], namely the global projective [[model structure on functors]] from the [[opposite category|opposite]] of the [[orbit category]] to the [[classical model structure on simplicial sets]]).

+-- {: .num_defn #NilpotententFiniteRationalHomotopyTypes}
###### Definition
**(simply connected and finite equivariant rational homotopy types)**

Write

\[
  \label{NilpotentQFiniteHomotopyTypes}
  Ho
  \big(
     G SimplicialSets_{Qu}
  \big)^{fin_{\mathbb{Q}}}_{\geq 2}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
    G SimplicialSets_{Qu}
  \big)
\]

for the [[full subcategory]] of the [[homotopy category of a model category|homotopy category]] of the [[model structure on equivariant simplicial sets]] on those [[equivariant homotopy types]] $X$ which over each $G/H \in G Orbits$ are

* [[connected topological space|connected]]: $\pi_0(X^H) = \ast$

* [[simply connected topological space|simply connected]]: $\pi_1(X^H) = 1$ is the [[trivial group]];

* rationally [[of finite type]]: $dim_{\mathbb{Q}}\big( H^n(X^H;,\mathbb{Q}) \big) \lt \infty$ for all $n \in \mathbb{N}$.

and

\[
  \label{NilpotentFiniteRationalHomotopyTypes}
  Ho
  \big(
     G SimplicialSets_{Qu}
  \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 2}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     SimplicialSets_{Qu}
  \big)
\]

for the futher [[full subcategory]] on those [[equivariant homotopy types]] that are already [[rational spaces|rational]].

Similarly, write

\[
  \label{NilpotentFiniteTypedgcAlgebras}
  Ho
  \big(
     G dgcAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)_{fin}^{\geq 1}
  \overset{
    \phantom{AAA}
  }{\hookrightarrow}
  Ho
  \big(
     G dgcAlgebras^{\geq 0}_{\mathbb{Q}}
  \big)
\]

for the [[full subcategory]] of the [[homotopy category of a model category|homotopy category]] of the [[projective model structure on equivariant connective dgc-algebras]] on those [[equivariant dgc-algebras]] $A$ which for each $G/H \in G Orbits$ are

* connected: $H^0(A^H) \simeq \mathbb{Q}$

* simply connected: $H^1(A^H) \simeq 0$

* [[finite type]]: $dim_{\mathbb{Q}}\big( H^n(A^H) \big) \lt \infty$ for all $n \in \mathbb{N}$.

=--

([Scull 08, p. 12, 14](#Scull08))

## Statement

+-- {: .num_prop #FundamentalTheoremOfdgAlgebraicRationalHomotopyTheory}
###### Proposition
**([[fundamental theorem of equivariant dg-algebraic rational homotopy theory]])**

The [[derived adjunction]]

$$
  Ho
  \left(
    \big(
      G dgcAlgebras^{\geq 0}_{k}
    \big)^{op}_{proj}
  \right)
  \underoverset
    {
      \underset
        {\;\;\; \mathbb{R} exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot}
  Ho
  \big(
    G SimplicialSets_{Qu}
  \big)
$$

of the [[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]] (whose [[left adjoint]] is the [[equivariant PL de Rham complex]]-[[functor]]) has the following properties:

* on connected, simply connected, rationally finite equivariant homotopy types $X$ (eq:NilpotentQFiniteHomotopyTypes) the [[derived adjunction unit]] is [[equivariant rationalization]]

  $$
    \array{
    Ho
    \big(
       G SimplicialSets_{Qu}
    \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil}
    &
      \overset{
      }{\longrightarrow}
    &
    Ho
    \big(
       G SimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
    \\
    X
    &\mapsto&
    \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
    }
  $$

  $$
    X
    \underoverset
      {\eta_X^{der}}
      {rationalization}
      {\longrightarrow}
    \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)
  $$

* on the [[full subcategories]] of connected, simply connected, and finite rational homotopy types from Def. \ref{NilpotententFiniteRationalHomotopyTypes} it restricts to an [[equivalence of categories]]:

  $$
    Ho
    \left(
      \big(
        G dgcAlgebras^{\geq 0}_{k}
      \big)^{op}_{proj}
    \right)^{\geq 1}_{fin}
    \underoverset
      {
        \underset
          {\;\;\; \mathbb{R} exp \;\;\;}
          {\longrightarrow}
      }
      {
        \overset
          {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;}
          {\longleftarrow}
      }
      {\simeq}
    Ho
    \big(
      G SimplicialSets_{Qu}
    \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}
  $$

=--

([Scull 08, Theorem 5.5](#Scull08))

## Related concepts

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

## References

* [[Marek Golasiński]], _Equivariant rational homotopy theory as a closed model category_, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (doi:10.1016/S0022-4049(97)00127-8)

  > (for Hamiltonian groups)

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

[[!redirects fundamental theorem of equivariant dg-algebraic rational homotopy theory]]