functorial geometry

[[!redirects functor of points]]

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###Context###
#### Geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic geometry]], there are two equivalent ways of looking at a [[scheme]]: it may be viewed

1. as a [[petit topos]] with a [[structure sheaf]] of [[commutative rings]], hence as a [[locally ringed space]],

1. as an object of the [[gros topos]] of [[sheaves]] on the [[site]] of [[commutative rings]] (with [[étale topology]] or [[Zariski topology]]) satisfying the condition that it is covered by [[representable functor|representables]] via [[open maps]].

In other words, according to viewpoint (2), a [[scheme]] may be identified with the [[sheaf]] it [[representable functor|represents]]; this sheaf is often called the _functor of points_ of the scheme.

To see this, note that by the [[Yoneda lemma]] a [[scheme]] may be identified with the [[sheaf]] it [[representable functor|represents]] on the gros [[Zariski site]] of [[schemes]]; and since any scheme admits an affine open cover, the [[comparison lemma]] says that [[sheaves]] on the [[site]] of all schemes may be identified with [[sheaves]] on the [[site]] of [[affine schemes]].

The functor of points approach has the advantage of making certain constructions much simpler (e.g. the [[fibered product]] in the category of schemes), and eliminating the need for certain constructions like the [[Zariski spectrum]]. In his famous [1973 Buffalo Colloquium talk](#Grothendieck73), [[Alexander Grothendieck]] urged that his earlier definition of scheme via [[locally ringed spaces]] should be abandoned in favour of the functorial point of view. This is recalled in [Lawvere 03](#Lawvere03):

> {#Lawvere16Quote} The 1973 Buffalo Colloquium talk by Alexander Grothendieck had as its main theme that the 1960 definition of scheme (which had required as a prerequisite the baggage of prime ideals and the spectral space, sheaves of local rings, coverings and patchings, etc.), should be abandoned AS the FUNDAMENTAL one and replaced by the simple idea of a good functor from rings to sets. The needed restrictions could be more intuitively and more geometrically stated directly in terms of the topos of such functors, and of course the ingredients from the "baggage" could be extracted when needed as auxiliary explanations of already existing objects, rather than being carried always as core elements of the very definition.

and in [Lawvere 16](#Lawvere16):

> [[Peter Gabriel]] $[$...$]$ had explained some of the same ideas at Oberwolfach in 1965-66, providing a context in which Grothendieck’s proposal seemed natural. For example, he emphasized the traditional view that the points of an algebraic space form a covariant functor on the category of field extensions of the base.

> Grothendieck’s advice in his Colloquium talk was that 1960 ingredients (like Zariski opens etc.) are easily extracted from the category of functors, when needed.

Of course, this functorial perspective generalizes to other kinds of [[geometry]] and even [[higher geometry]], the general perspective being known as _[[synthetic differential geometry]]_ or similar.  For discussion of functorial ([[higher differential geometry|higher]]) [[differential geometry]] see for instance at _[[smooth set]]_ ([[smooth ∞-groupoid]]), for discussion of functorial [[supergeometry]] see at _[[super formal smooth set]]_ ([[super formal smooth ∞-groupoid]]).

##Example

The functor from commutative rings to sets which sends a ring, $R$, to the set of simultaneous solutions in $R^n$ of a set of polynomials, $f_1, \ldots, f_k$ in $\mathbb{Z}[t_1, \ldots,t_n]$ corresponds to the affine scheme $X = Spec(\mathbb{Z}[t_1, \ldots,t_n]/(f_1, \ldots,f_k))$. These $R$-points are then equivalently the hom-space

$$
Hom_{schemes}(Spec(R), X).
$$

When $X = Spec(\mathbb{Z}[t])$, the functor of points is the [[forgetful functor]].

The functor which sends $R$ to the $R$-points of the projective space $\mathbb{P}^n$ corresponds to a non-affine scheme.

## Value added by the internal language of toposes

Typically, only field-valued points of a scheme are easy to describe. For                   instance, the functor $F$ describing projective $n$-space is given on fields by

$$
  F(K)
  \;=\;
  \text{the set of lines through the origin in}
  \, K^{n+1}
   \cong
  (K^{n+1} \setminus \{0\})/K^\times,                                                       $$

whereas on general rings it is given by

$$
  F(R)
  \;=\;
  \text{the set of linear surjections}
  \, R^{n+1}
  \twoheadrightarrow P,\text{where}
  \;P\,
  \text{is projective, modulo isomorphism.}
$$

On the other hand, it is these more general kinds of points which impart a meaningful sense of cohesion on the field-valued points, so they can't simply be dropped from consideration.

We can resolve this tension by observing that the category of functors from rings to sets is a [[topos]] and therefore has an [[internal language]]. We can use this language to describe such functors (and later study their properties) in a naive, element-based way. For instance, the functor $F$ describing projective $n$-space can be given by either of the internal expressions
$$
  \text{the set of lines through the origin in}
  \;
  U^{n+1}
  \quad\text{or}\quad
  (U^{n+1} \setminus \{0\})/U^\times,
$$
where $U$ is the forgetful functor (representing the [[affine line]]). Details on this approach are in Part III of [Blechschmidt 17](#Blechschmidt17).

## Related concepts

* [[gros topos]]

* [[space and quantity]]

## References

The original idea is advertised in

* {#Grothendieck65} [[Alexander Grothendieck]], _Introduction au langage fonctoriel_, course in Algiers in November 1965, lecture notes by [[Max Karoubi]], ([pdf scan](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GrothAlgiers.pdf), [[GrothendieckIntroductionLangageFonctoriel1965.pdf:file]])

* {#Grothendieck73} [[Alexander Grothendieck]], _Introduction to functorial algebraic geometry, part 1: affine algebraic geometry_, summer school in Buffalo, 1973, lecture notes by Federico Gaeta ([pdf scan](http://matematicas.unex.es/~navarro/res/ifag.pdf), [[GrothendieckIntrodFunctorialGeometryI1973.pdf:file]])

and has been re-emphasized in various forms in the writing of [[William Lawvere]], notably:

* {#Lawvere03} [[William Lawvere]], _Grothendieck's 1973 Buffalo Colloquium_, [Categories Mailing List](https://www.mta.ca/~cat-dist/), March 2003, ([cat-dist:03-3](https://www.mta.ca/~cat-dist/archive/2003/03-3)).

* {#Lawvere16} [[William Lawvere]], email to [[Adeel Khan]], March 2016 ([web excerpt](https://nforum.ncatlab.org/discussion/7521/functorial-geometry/?Focus=69419#Comment_69419))

For a pedagogical discussion of the advantages and disadvantages of teaching the functor of points approach, see

* Secret blogging seminar, _Algebraic geometry without prime ideals_, ([blog discussion](https://sbseminar.wordpress.com/2009/08/06/algebraic-geometry-without-prime-ideals/))

Discussion for [[supergeometry]]:

* [[Claudio Carmeli]], Lauren Caston, [[Rita Fioresi]], _Mathematical Foundations of Supersymmetry_, EMS Series of Lectures in Mathematics Volume: 15; 2011; 263 pp; ( [ISBN:978-3-03719-097-5](https://bookstore.ams.org/emsserlec-15/), [arXiv:0710.5742](https://arxiv.org/abs/0710.5742))

See also

* {#Blechschmidt17}[[Ingo Blechschmidt]], _Using the internal language of toposes in algebraic geometry_, doctoral thesis (2017) ([web](https://rawgit.com/iblech/internal-methods/master/notes.pdf))

* [[Michel Demazure]], [[Pierre Gabriel]], _Introduction to algebraic geometry and algebraic groups_, North-Holland Mathematics Studies Volume 39 (1980) ([ISBN:9780080871509](https://www.elsevier.com/books/introduction-to-algebraic-geometry-and-algebraic-groups/demazure/978-0-444-85443-8))

* [[Bertrand Toen]], _[[Master course on algebraic stacks]]_.

* {#Low16} [[Zhen Lin Low]], _Categories of spaces built from local models_, doctoral thesis (2016) ([web](https://www.repository.cam.ac.uk/handle/1810/256998),[doi.org/10.17863/CAM.384](https://doi.org/10.17863/CAM.384))

[[!redirects functors of points]]


functorial quantum field theory - contents

**[[FQFT|functorial quantum field theory]]**

## Contents

* [[cobordism category]]

  * [[cobordism]]

  * [[extended cobordism]]

  * [[(∞,n)-category of cobordisms]]

  * [[bordism categories following Stolz-Teichner|Riemannian bordism category]]

* [[cobordism hypothesis]]

  * [[generalized tangle hypothesis]]

  * [[On the Classification of Topological Field Theories|classification of TQFTs]]

* [[functorial field theory]]

  * [[unitary functorial field theory]]

  * [[extended functorial field theory]]

  * [[conformal field theory|CFT]]

    * [[vertex operator algebra]]

  * [[TQFT]]

    * [[Reshetikhin-Turaev model]] / [[Chern-Simons theory]]

    * [[HQFT]]

    * [[TCFT]]

      * [[A-model]], [[B-model]], [[Gromov-Witten theory]]

      * [[homological mirror symmetry]]

* FQFT and [[cohomology]]

  * [[(1,1)-dimensional Euclidean field theories and K-theory]]

  * [[(2,1)-dimensional Euclidean field theory]]

  * [[geometric models for tmf]]

* [[holographic principle of higher category theory]]

  * [[holographic principle]]

  * [[AdS/CFT correspondence]]

  * [[quantization via the A-model]]


This page is about [[William Lawvere]]'s Ph.D. thesis:

* F. W. Lawvere, _Functorial Semantics of Algebraic Theories_, Ph.D. thesis, Columbia University, 1963.

Written under [[S. Eilenberg]] and highly influential, the dissertation was not published at the time, and only two short notices appeared:

* F. W. Lawvere, _Functorial Semantics of Algebraic Theories_, Proc. Nat.  Acad. Sci. **50** (1963) pp. 869--872. ([pdf](http://www.pnas.org/content/50/5/869.full.pdf))

* F. W. Lawvere, _Algebraic theories, algebraic categories, and algebraic functors_, pp. 413--418 in Addison, Henkin, Tarski (eds.), _The Theory of Models_, North-Holland Amsterdam 1965.

It was finally published together with an author's comment and a supplement here:

* Reprints in Theory and Applications of Categories, No. 5 (2004) pp. 1--121 ([tac](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html)).

In his thesis, Lawvere introduces [[Lawvere theories]] and the [[categorical semantics|functorial perspective]] into model theory. He takes steps towards axiomatizing [[ETCC|the category of categories as a foundation for mathematics]]. This  includes introducing the notion of [[comma category]] as an auxiliary to a definition of [[adjoint functors|adjunction]] implicitly involving an isomorphism of [[2-sided discrete fibrations]] (cf. Lawvere 2004 pp.12--13).

Generalization to relational theories:

* [[Filippo Bonchi]], [[Dusko Pavlovic]], [[Pawel Sobocinski]], _Functorial Semantics for Relational Theories_, ([arXiv:1711.08699](https://arxiv.org/abs/1711.08699))

Generalization to partial algebraic theories:

* [[Ivan Di Liberti]], [[Fosco Loregian]], [[Pawel Sobocinski]], [[Chad Nester]] _Functorial semantics for partial theories_, ([arXiv:2011.06644](https://arxiv.org/abs/2011.06644))

## Related concepts

* [[list of theses in category theory]]

category:reference


functoriality of categories of presheaves

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###Context###
#### Topos Theory
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#### Category Theory
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[[!include category theory - contents]]
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#Contents#
* table of contents
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## Idea

### Functoriality w.r.t. functors

Given a [[small category]] $C$, one can consider the [[category of presheaves]] $PSh(C, D)$ valued in some [[category]] $D$.  Given some assumptions on $D$, any [[functor]] of small categories $F : C \to C'$ induces two [[adjunction|adjoint pairs]]

  $$ F_! : PSh(C, D) \rightleftarrows PSh(C', D) : F^* $$

  $$ F^* : PSh(C', D) \rightleftarrows PSh(C, D) : F_* $$

Here, $F^*$ is given by precomposition with $F$, whereas $F_!$ and $F_*$ are the left and right (global) [[Kan extension]]s along $F$.

### Functoriality w.r.t. profunctors

In fact, more generally, any [[profunctor]] $\mathcal{P} : C \nrightarrow C'$ (i.e. $\mathcal{P} : C \times {C'}^{op} \to D$ or, after [[currying]], $P : C \to PSh(C', D)$) gives rise to a single [[adjunction|adjoint pair]]

  $$ P_\odot : PSh(C, D) \rightleftarrows PSh(C', D) : P^\odot $$

($P_\odot$ and $P^\odot$ are not standard notations) where $P_\odot$ is the [[Yoneda extension]] of $P$.

A functor $F : C \to C'$ gives rise to two profunctors:

* a [[companion]] profunctor $\hat F : C \nrightarrow C'$, given by

  $$ \hat F : C \times {C'}^{op} \to D : (c, c') \mapsto Hom_{C'}(c', Fc). $$

  After currying, this amounts to the functor $y \circ F : C \to PSh(C')$.

* a [[conjoint]] profunctor $\check F : C' \nrightarrow C$, given by

  $$ \check F : C' \times {C}^{op} \to D : (c', c) \mapsto Hom_{C'}(Fc, c'). $$

  After currying, this amounts to the functor $F^* \circ y : C' \to PSh(C)$.

The former profunctor produces the adjoint pair $F_! \dashv F^*$, i.e. $F_! = (y \circ F)_\odot$ and $F^* \cong (y \circ F)^\odot$.
The latter profunctor produces the adjoint pair $F^* \dashv F_*$, i.e. $F^* \cong (F^* \circ y)_\odot$ and $F_* = (F^* \circ y)^\odot$.

## Definitions

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

Let $F : C \to C'$ be a functor of small categories and $D$ some category.  The **[[restriction of scalars]]** functor $F^* : PSh(C', D) \to PSh(C, D)$ is given by the formula $H \mapsto H \circ f$, i.e. mapping a [[presheaf]] $H : C'^{op} \to D$ to the composite
  $$C^{op} \stackrel{F}{\to} C'^{op} \stackrel{H}{\to} D.$$
=--

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

Suppose $D$ admits small [[colimits]] (resp. small [[limits]]).  Then the functor $F^*$ admits a [[left adjoint]] $F_!$ (resp. [[right adjoint]] $F_*$).
=--

## Properties

+-- {: .num_lemma}
###### Lemma

Let $F : C \rightleftarrows C' : G$ be an [[adjoint pair]] and consider the induced functors $(F_!, F^*, F_*)$ and $(G_!, G^*, G_*)$.  One has

* $F_!$ is left adjoint to $G_!$,
* $F^*$ is left adjoint to $G^*$,
* $F_*$ is left adjoint to $G_*$,
* $F_* \cong G^*$,
* $F^* \cong G_!$.
=--

Note that all these claims are in fact equivalent.

+-- {: .num_lemma}
###### Lemma
If $F$ is fully faithful, then so are $F_!$ and $F_*$.
=--

+-- {: .proof}
###### Proof
A left adjoint functor $L \dashv R$ is fully faithful precisely if $R L$ is naturally isomorphic to the identity functor (by the unit). Dually, $R$ is fully faithful precisely if $L R$ is naturally isomorphic to the identity (by the co-unit). Hence, it suffices to prove $F^* F_! \cong Id$, which by uniqueness of the right adjoint immediately implies that $F^* F_* \cong Id$ and thus proves both claims.

Being a left adjoint, $F^* F_!$ preserves colimits. Because every presheaf is a colimit of representable objects, it is sufficient to show that $F^* F_! y \cong y$ where $y$ is the Yoneda-embedding. We have
\[
(F^* F_! y I) J \cong (F^* y F I) J = (y F I) (F J) = Hom(F J, F I) \cong Hom(J, I) = (y I) J.
\]
=--

## See also

For functoriality of [[sheaves]], see

* [[restriction and extension of sheaves]]
* [[direct image]]
* [[inverse image]]

The pseudofunctor $PSh$ (with $-_!$ as its action on morphisms) takes a category to its [[free cocompletion]]. As such, it has the structure of a weak [[2-monad]], and more specifically it is a prototypical example of a [[lax-idempotent 2-monad]]. It factors over its Eilenberg-Moore 2-category, the 2-category of ([[total category|total]]? [[cocomplete]]?) categories, as a [[lax-idempotent 2-adjunction]].
The bind operation for $PSh$ is given by [[Yoneda extension]] and the [[Kleisli 2-category]] of $PSh$ is [[Prof]].

## References

* [[SGA 4]], Exp. I, no. 5.
* [[Stacks Project]], tags [00VC](http://stacks.math.columbia.edu/tag/00VC) and [00XF](http://stacks.math.columbia.edu/tag/00XF).
 * {#Nuyts24} [[Andreas Nuyts]], _Lifting Profunctors to Presheaf Categories_, [note](https://anuyts.github.io/files/lifting-profunctors.pdf)


functorially finite subcategory

[[!redirects functorially finite]]
##Functorially finite subcategories of abelian categories##

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a full subcategory. Given an object $X$ of $\mathcal{A}$, a morphism $f: C \to X$ from $C \in \mathrm{Ob}(\mathcal{C})$ is called a **right $\mathcal{C}$ approximation of X** if, for any $C' \in \mathrm{Ob}(\mathcal{C})$ and any morphism $f': C' \to X$, we can factor $f'$ and $f \circ g$.

For example, suppose that we have finitely many objects $C_1$, $C_2$, ..., $C_n$ and $\mathcal{C}$ is the full subcategory on direct sums of $C_j$'s. Suppose also that our category is $k$ linear for some field $k$ and that $\mathrm{Hom}(C_j, X)$ is finite dimensional over $k$. Then $\bigoplus_j \Hom(C_j, X) \otimes_k C_j \longrightarrow X$ will be a right approximation of $X$.

As an example where we do NOT have a left approximation, let $\mathcal{A}$ be finitely generated abelian groups and let $\mathcal{C}$ be finite abelian groups. Take $X = \mathbb{Z}$. For any finite abelian group $C$ and map $\mathbb{Z} \to C$, we can choose a prime $p$ not dividing $|C|$, then $\mathbb{Z} \to \mathbb{Z}/p \mathbb{Z}$ will not factor through $\mathbb{Z} \to C$. (On the other hand, all finitely generated abelian groups do have right approximations in this setting -- the inclusion of the torsion subgroup is a right approximation.)

The full subcategory $\mathcal{C}$ is called **right functorially finite** if every object in $\mathcal{A}$ has a **right $\mathcal{C}$ approximation**. We define **left $\mathcal{C}$ approximation** and **left functorially finite** in a dual manner. A full subcategory which is both right and left functorially finite is called functorially finite.


functors and comma categories

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### Context
#### Category theory
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[[!include category theory - contents]]
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This entry is about special properties of [[functor]]s on [[comma category|comma categories]]. See also [[category of presheaves]].

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

## Presheaves on over-categories and over-categories of presheaves

Let $C$ be a [[category]], $c$ an [[object]] of $C$ and let $C/c$ be the [[over category]] of $C$ over $c$. Write
$PSh(C/c) = [(C/c)^{op}, Set]$ for the category of [[presheaf|presheaves]] on $C/c$ and write
$PSh(C)/Y(c)$ for the [[over category]] of [[presheaf|presheaves]] on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the [[Yoneda embedding]].

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

There is an [[equivalence]] of categories

$$
  e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c)
  \,.
$$

=--

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

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf
$F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map

$$
  \eta_d : \sqcup_{f \in C(d,c)} F(f) \to C(d,c)
  : ((f \in C(d,c), \theta \in F(f)) \mapsto f
  \,.
$$

A weak inverse of $e$ is given by the functor
$$
  \bar e : PSh(C)/Y(c) \to PSh(C/c)
$$
which sends $\eta : F' \to Y(c)$ to $F \in PSh(C/c)$ given by
$$
  F : (f : d \to c) \mapsto F'(d)|_c
  \,,
$$
where $F'(d)|_c$ is the [[pullback]]
$$
  \array{
     F'(d)|_c &\to& F'(d)
     \\
     \downarrow && \downarrow^{\eta_d}
     \\
     pt &\stackrel{f}{\to}& C(d,c)
  }
  \,.
$$

=--

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

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphsims to $c$, i.e. suppose that it factors through the forgetful functor from the [[over category]] to $C$:
$$
  F : (C/c)^{op} \to C^{op} \to Set
  \,.
$$

Then
$
  F'(d) = \sqcup_{f \in C(d,c)} F(f)
  = \sqcup_{f \in C(d,c)} F(d)
  \simeq
  C(d,c) \times F(d)
$
and hence $F ' = Y(c) \times F$ with respect to the [[closed monoidal structure on presheaves]].

=--

See [[over-topos]] for more.

## Over-categories of presheaf categories and presheaves on categories of elements

Generalizing the above,

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

For every $P \colon \mathbf{Set}^D$, there is an [[equivalence]] of categories

$$
  \varphi : \mathbf{Set}^{el(P)} \stackrel{\simeq}{\to} \mathbf{Set}^D / P
  \,.
$$

where $el(P) = \ast / P$ is the [[category of elements]] of $P$.
=--

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

The construction is completely analogous to the above; Given $F \colon \mathbf{Set}^{el(P)}$, $\varphi(F)$ is defined pointwise as a coproduct:
$$
  \varphi(F)(c) = \coprod_{(x,Pc)} F(x,Pc)
$$
where $(x,Pc) = (\ast \stackrel{x}{\to} Pc)$ is an object of $el(P)$. The action on morphisms is defined analogously. This comes equipped with a natural transformation $\alpha \colon \varphi(F) \to P$, with component
$$
  \array{
    \alpha_c \colon \varphi(F)(c) \to Pc          \\
    \alpha_c(F(x,Pc)) = x \in Pc                  \\
  }
$$
Given an object $(Q \stackrel{\alpha}{\to} P)$ of $\mathbf{Set}^D / P$ the action of a weak inverse $\bar \varphi$ can be specified as $ \bar{\varphi}(\alpha)(x,Pc) = \alpha_c^{-1}(x)$, that is, the wedge of the pullback:
$$
  \array{
     \bar{\varphi}(\alpha)(x,Pc) &\to& Qc
     \\
     \downarrow && \downarrow^{\alpha_c}
     \\
     pt &\stackrel{x}{\to}& Pc
  }
  \,.
$$
The action of $\bar{\varphi}(\alpha)$ on arrows of $el(P)$, functoriality, etc is derived from its definition as a pullback and the def of morphisms in $el(P)$.

=--

### Relationship with the over-categories statement

Putting $D = C^{op}, P = Y(c)$ in the above yields:
$$
  \mathbf{Set}^{el(Y(c))} \simeq \mathbf{Set}^D / Y(c)
$$
Now it is easy to see that $el(Y(c)) \simeq (C / c)^{op}$; we get then:
$$
  \mathbf{Set}^{(C / c)^{op}} \simeq \mathbf{Set}^{C^{op}} / Y(c)
$$

## In higher category theory

For the analogous result in the context of [[(∞,1)-category]] theory see
(∞,1)-Category of (∞,1)-presheaves -- Interaction with overcategories

## Related concepts

* [[comma category]]

* [[presheaf]]


fundamental (infinity,1)-category

[[!redirects fundamental (∞,1)-category]]

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

## Idea

In analogy to how a [[topological space]] has a [[fundamental ∞-groupoid]], a [[directed space]] has a fundamental [[(∞,1)-category]]

## Definition

### Directed topological spaces

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

By a **[[directed space|directed topological space]]** we here mean specifically a pair $(X,D)$ consisting of

* a [[topological space]] $X$;

* a subset $D \subset X^{[0,1]}$ of [[continuous map]]s $\gamma : [0,1] \to X$ that are are labeled as being _directed_ ,

* such that

  * for every orientation preserving [[homeomorphism]] $\phi : [0,1] \to [0,1]$ and every directed map $\gamma$ also $\gamma \circ \phi$ is directed;

  *  for any two directed paths $\gamma_1, \gamma_2$ with $\gamma_1(1) = \gamma_2(0)$ also the concatenation

     $$
       [0,1] \stackrel{\times 2}{\to} [0,2]
       \simeq [0,1] \coprod_{*} [0,1]
       \stackrel{(\gamma_1, \gamma_2)}{\to}
       X
     $$

     is a directed path.

A morphism $(X_1,D_1) \to (X_2, D_2)$ of directed spaces is a continuous map $X_1 \to X_2$ that takes elements in $D_1$ to elements in $D_2$.

This defines the [[category]] $DTop$ of directed topological spaces.

=--

+-- {: .un_example}
###### Example
**(topological poset)**

For $X$ a topological space equipped with the structure of a [[poset]] on its underlying set, say that $\gamma : [0,1] \to X$ is directed if for all $x \leq y$ in $[0,1]$ we have $\gamma(x) \leq \gamma(y)$ in $X$.

=--

+-- {: .un_example}
###### Example
**(directed geometric simplex)**
{#DirectedSimplex}

For $\Delta^k_{Top}$ the standard topological $k$-[[simplex]] its _standard directed paths_ are order-preserving maps into its 1-[[simplicial skeleton|skeleton]] (the union of the 1-faces equipped with the evident [[poset]]-structure induced from that on the vertices), such that the endpoints land on vertices.

=--

+-- {: .un_example}
###### Example
**(directed geometric realization)**

For $C$ a [[quasi-category]] its [[geometric realization]] $|C|$ becomes a directed topological space by taking the directed paths to be all maps that factor through its 1-[[simplicial skeleton|skeleton]]:

$$
  \int^{[k] \in \Delta_{\leq 1}} \Delta^k_{Top} \times C_k
  \hookrightarrow
  \int^{[k] \in \Delta}
  \Delta^k_{Top} \times C_k
  =
  |C|
$$

while preserving the canonical order and so that the endpoints land on vertices.

=--

The previous example of the directed topological simplex is the special case of this for $C = \Delta[k]$.

### Fundamental $(\infty,1)$-category

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

The **fundamental $(\infty,1)$-category** of a directed topological space $(X,D)$ is given by the [[quasi-category]] $\Pi(X,D)$ whose [[k-morphism]]s are those continuous maps

$$
  \sigma : \Delta^k_{Top} \to X
$$

that are morphisms of directed spaces with respect to the [standard directed paths](#DirectedSimplex) in $\Delta^k$.

$$
  \Pi(X,D)_k := Hom_{DTop}(\Delta^k_{Top}, (X,D))
 \,.
$$

=--

+-- {: .un_lemma}
###### Lemma

This does indeed define a quasi-category

=--

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

Use the standard [[retract]]s $\Delta^k_{Top} \to \Lambda^k_i$ of the topological [[horn]] inclusions $\Lambda^k_i \hookrightarrow \Delta^k$ to fill horns in $\Pi(X,D)$.

For $k \geq 3$ these retracts are the identity on 1-faces, hence trivially preserve the directed paths. For $k = 2$ it is precisely the retract of the _inner_ horn $\Lambda^2_1$ that preserves the directed paths. This is sufficient to satisfy the inner horn filler condition of quasi-categories.

=--

## Properties

+-- {: .un_lemma}
###### Observation

If $X$ is undirected in that _all_ paths are labeled as directed -- $D = X^{[0,1]}$ -- then $\Pi(X,D)$ coincides with the [[fundamental ∞-groupoid]] $\Pi(X)$ of $X$.

=--

## Fundamental $\infty$-categories induced from intervals

The interest in interval objects is that various further structures of interest may be built up from them. In particular, since picking an interval object $I$ is like picking a notion of _path_, in a category with interval object there is, under mild assumptions, for each object $X$ an [[infinity-category]] $\Pi_I(X)$ -- the fundamental $\infty$-category of $X$ with respect to $I$ -- whose [[k-morphism]]s are $k$-fold $I$-paths in $X$.

There are different ways to make this precise and realize it in detail. The main distinction is whether one uses

* an [[algebraic definition of higher category]]

or

* a [[geometric definition of higher category]].

We describe an algebraic version in terms of [[Trimble n-category|Trimble omega-categories]] and then a geometric version in terms of [[cubical object]]s and [[simplicial object]]s.

### Fundamental algebraic $\infty$-categories

The collection of objects
$\{ {}_{pt}[I, I^{\vee n}]_{pt}\}_{n \in \mathbb{N}}$ in a category with interval object naturally comes equipped with the structure of an [[operad]]: this is the tautological co-[[endomorphism operad]] on the object $I$ in the symmetric closed monoidal category of [[bi-pointed object]]s from $pt$ to $pt$.

This induces in turn for all objects $X \in V$ on the object $[I,X]$ the structure of an operad, which is naturally interpreted as an internal $A_\infty$-[[A-infinity-category|category]] structure on

$$
  (X_0 := [pt,X])
  \stackrel{s := [\sigma,X]}{\leftarrow}
  [I,X]
  \stackrel{t := [\tau, X]}{\to} (X_0 := [pt,X])
  \,.
$$

This internal $A_\infty$-category is denoted

$$
  \Pi_1(X)
$$

and interpreted as the [[fundamental groupoid]] or rather, in general, the [[fundamental category]] of the object $B$ with respect to the interval object $I$ -- all internal to $V$.

Moreover, by iterating this process as described at [[Trimble n-category]] one should obtain, if everything goes through, on $X$ the structure of a [[Trimble n-category|Trimble]] $\omega$-[[infinity-category|category]] and indeed a functor

$$
  \Pi_\omega : V_0 \to Trimble \omega Cat
  \,.
$$

(... to be continued ...)

+-- {: .un_remark }
###### Remarks

* The condition that all ${}_{pt}[I, I^{\vee n}]_{pt}$ are contractible is the _coherence condition_ on all composition operations.

* The above is _not_ demanding that the interval object $I$ itself is is weakly equivalent to the point. If it is, then $\Pi_1(X)$ is indeed a [[fundamental groupoid]]. If it is not, then $\Pi_1(X)$ may just be a [[fundamental category]].

* If $X_0$ has a notion of [[path object]] one may consider imposing the condition that $[I,X]$ is a path object of $X$ for all $X$. Similarly, if $V_0$ has a [[cylinder functor]], one may consider imposing the condition that it is given by $-\otimes I$.

=--

### Fundamental geometric $\infty$-categories {#FundGeomInftCat}

Let $C$ be a category with finite [[limit]]s and (plain) interval object $ {*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*}$, where ${*}$ denotes the [[terminal object]].

This may or may not come with further structures and properties as discussed in the definitions above. For the following however no more than that is neceesray.

Recall that the **cube category** is the initial strict [[monoidal category]] $(M, \otimes, I)$ equipped with an object $int$ together with two maps $i_0, i_1: I \to int$ and a map $p: int \to I$ such that $p i_0 = 1_I = p i_1$.

In a tautological way, $I$ induces a [[cocubical object]] in $C$, a [[functor]]

$$
  \Box_I : \Box \to C
$$

from the [[cube category]] $\Box$ to $C$. This sends the abstract interval object $int$ that the [[cube category]] is freely generated from to the given $I$, and sends $int^{\otimes n}$ to the $n$-cuber $\Box_I^n := I^{\times n}$.

For every object $X \in C$ homming cubes into $X$ thereby produces a [[cubical set]]

$$
  X^{\Box^\bullet_I} : int^n \mapsto C(\Box_I^n,X)
  \,.
$$

One tends to want to regard this as the cubical incarnation of the fundamental $\infty$-category of $X$ with respect to the notion of path given by $I$.

However, while cubes are nice for many purposes, it is a sad fact of life that the [[homotopy theory]] for cubical structures (while certainly it does exist in full beauty in principle) is much less well developed to date (maybe that will change in the future) than that of [[simplicial object|simplicial]] structures. For many purposes in [[higher category theory]], therefore, it will be useful to take a slightly different perspective on $X^{\Box}$, without essentially changing it.

In fact, there is also naturally the structure of a  [[cosimplicial object]] of the collection $I^\times \bullet$ of $I$-cubes. This differs from the cubical structure only in were precisely one injects the boundaries into an $I^{\times n}$

+-- {: .un_defn}
###### Definition (cosimplicial object induced from interval object)

Given a cartesian interval object $I \in C$, define a [[cosimplicial object]]

$$
  \Delta_I : \Delta \to C
$$

as follows:

* the object in degree $n$ is the $n$-fold [[product]] of $I$ with itself:

  $$
    \Delta_I^n := I^{\times n}
  $$

* the degeneracy map $\sigma_i : \Delta_I^{n+1} \to \Delta_I^n$ is given by
  projecting out the $(i+1)$-factor:

  $$
    \sigma_i := p_1 \times p_2 \times \cdots \times p_i \times p_{i+2} \times p_{i+3}
        \times \cdots \times p_{n+1}
  $$

* the face map $\delta_i : \Delta_I^{n} \to \Delta_I^{n+1}$ is given

  * for $i = n+1$ by inserting $0$ in the $(n+1)$-factor:

    $$
      \delta_{n+1} := p_1 \times \cdots \times p_{n} \times 0
    $$

  * for $i = 0$ by inserting $1$ in the $0$-factor:

    $$
      \delta_0 = 1 \times p_1 \times \cdots \times p_2
    $$

  * for $0 \lt i \lt n+1$ by duplicating the $i$-factor:

    $$
      \delta_i := p_1 \times \cdots \times_{p_{i-1}}
        \times p_i \times p_i \times p_{i+1} \times \cdots \times p_n
    $$

=--

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

The maps defined this way indeed satisfy the [[simplicial identities]].

=--

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

This is straightforward to check, if a little tedious due to the many case distinctions.

=--

+-- {: .un_remark}
###### Remark (unwrapping the definition)

It may be helpful to unpack the above definition a bit.

* The two faces of $\Delta_I^1$ are just the "boundary points" of the interval itself.

  * $(\delta_0 : {*} \to I) = (1 : {*} \to I)$

  * $(\delta_1 : {*} \to I) = (0 : {*} \to I)$

* The face maps of $\Delta^2_I$ may be depicted as follows:

  $$
    \delta_2 :
    \array{
      I &\times& I
      \\
      \uparrow^{Id} && \uparrow^{0}
      \\
      I &\times& {*}
    }
    \,,
    \;\;\;\;\;
    \delta_1 :
    \array{
      I &\times& I
      \\
      & \uparrow^{Id \times Id}
      \\
      & I
    }
    \,,
    \;\;\;\;\;
    \delta_0 :
    \array{
      I &\times& I
      \\
      \uparrow^{1} && \uparrow^{Id}
      \\
      {*} &\times& I
    }
  $$

  Therefore $\delta_1$, being a diagonal morphism (cartesian [[pairing]] of an [[identity]] with itself),
  literally identifies the **diagonal** in the "square" $I \times I$ as the 1st 1-dimensional boundary.

=--

+-- {: .un_remark}
###### Remark
**([[collars]])**

This construction gives "collared simplices" in much the same sense as in [[collar|collared]] [[cobordism]] and in [[A1-homotopy]]:

there is no condition that the morphisms $0,1  : {*} \to I$ "hit a boundary point" -- whatever that may mean
 in $C$ -- of $I$. For instance in a [[lined topos]] $I$ is canonically chosen to be the given line object and will hence typically "extended indefinitely" beyond these points.

An example of this in practice is the [[A1-homotopy theory]] of schemes -- there is no exact analogue of the interval (with boundary points) in an algebraic setting, but the [[affine line]] $A^1$ together with the canonical points 0 and 1 is an interval object.

So $I$ need not "look" much like a 1-simplex, but the _choice of boundary points_ $\delta_1 = 0 : {*} \to I$ and $\delta_0 = 1 : {*} \to I$  allows us to regard
it as an interval for all practical purposes.

Similarly and more generally what the above construction manifestly defines are _cubes_ $I^n$ built from $I$. But then the simplicial choice of  boundaries inside these cubes allows to think of them as just the simplices
 "sitting inside" these cubes.

All these statement become precise for specific typical choices of the ambient category $C$, discussed in the examples below.

An important aspect is that once the cosimplicial object of _collared simplices_ $\Delta_I$ is used to form [[simplicial object]]s $\Pi(X) := [\Delta^\bullet_I,X]$ (discussed below) and when these are interpreted as [[model category|models]] for [[∞-groupoid]]s, then **the collars disappear**: they are part of the model, but, roughly, don't affect the equivalence class of the object that this model models.

For instance with ${*}\amalg {*} \stackrel{0, 1}{\to}\mathbb{R}$ used as the interval in [[Top]], a _path_ in a [[topological space]] $X$ is an entire curve $\gamma : \mathbb{R} \to X$, but two such paths $\gamma_1,\gamma_2$ are composable already when $\gamma_1(1) = \gamma_2(0)$, irrsepctive of how $\gamma_1$ extends $\gt 1$ and irrsepctive of how $\gamma_2$ extends $\lt 0$.

Moreover, the composite-up-to-homotopy of these two paths is an entire surface $\mathbb{R}^2 \to X$ in $X$, but what only matters for this surface qualifying as a compositor of $\gamma_1 and \gamma_2$ is that its $\delta_2$-segment $\{(x,y) \in \mathbb{R}^2 | 0 \leq x \leq 1, y = 0\}$ and its $\delta_0$segment $\{(x,y) \in \mathbb{R}^2 | 0 \leq y \leq 1, x = 0\}$ coincide with the corresponding segments in $\gamma_1$ and $\gamma_2$.

More on this in the following example section.

=--

#### Example: standard intervals, cubes and simplices in $Top$ and $Diff$

Let $X = $ [[Top]] or $C = $ [[Diff]] be the category of [[topological space]]s
or of [[manifold]]s.

A standard choice of interval object in $C$ is $I = [0,1] \subset \mathbb{R}$
with the obvious two boundary inclusions $0,1 : {*} \to [0,1]$.

But another possible choice is to let $I = \mathbb{R}$ be the whole
real line, but still equipped with the two maps $0,1 : {*} \to \mathbb{R}$,
that hit the $0 \in \mathbb{R}$ and $1 \in \mathbb{R}$, respectively.

Either of these two examples will do in the following discussion.
The second choice is to be thought of as obtained from the first choice
by adding "infinitely wide collars" at both boundaries of $[0,1]$. While
${*} \stackrel{0}{\to}[0,1]
\stackrel{1}{\leftarrow} {*}$ may seem like a more natural choice for a representative of the
idea of the "standard interval", the choice ${*} \stackrel{0}{\to} \mathbb{R}
\stackrel{1}{\leftarrow} {*}$ is actually more useful for many
[[category theory|abstract nonsense]] constructions.

But since it is hard to draw the full real line, in the following we depict the situation for the choice $I = [0,1]$.

Then for low $n \in \mathbb{N}$ the above construction yields this

* $n=0$ -- here $\Delta_I^0 = I^{\times 0} = {*}$ is the [[point]].

* $n=1$ -- here $\Delta_I^1 = I^{\times 1} = I$ is just the interval itself

  $$
    \array{
      (0) \to (1)

    }
  $$

  The two face maps $\delta_1 {*} \to I$ and $\delta_0 : {*} \to I$ pick
  the boundary points in the obvious way. The unique degeneracy map
  $\sigma_0 : I \to {*}$ maps all points of the interval to the single point
  of the point.

* $n=2$ -- here $\Delta_i^2 = I^{\times 2} = I \times I$
  is the **standard square**

  $$
    \array{
      (0,1) &\to& (1,1)
      \\
      \uparrow && \uparrow
      \\
      (0,0) &\to& (1,0)
    }
  $$

  But the three face maps $\delta_i : I \to I\times I$ of the cosimplicial
  object $\Delta_I$ constructed above don't regard the full square here, but
  just a triangle sitting inside it, in that pictorially they
  identify $(\Delta_I^1 = I)$-shaped boundaries in $I \times I$ as follows:

  $$
    \array{
      (0,1) &\to& (1,1)
      \\
      \uparrow &^{= \delta_1(I)}\nearrow& \uparrow^{ = \delta_0(I)}
      \\
      (0,0) &\stackrel{= \delta_2(I)}{\to}& (1,0)
    }
  $$

  (here the arrows do not depict morphisms, but the standard topological interval,
  i don't know how to typeset just lines without arrow heads in this fashion!)

* $n=3$ -- here $\Delta_i^3 = I^{\times 3} = I \times I \times I$
  is the **standard cube**

  +-- {: .un_example}
  ###### Exercise
  Insert the analog of the above discussion here
  and upload a nice graphics that shows the standard cube and how the
  cosimplicial object $\Delta_I$ picks a solid tetrahedron inside it.
  =--

As a start, we can illustrate how there are 6 3-simplices sitting inside each 3-cube.

Once you see how the 3-simplices sit inside the 3-cube, the facemaps can be illustrated as follows:

Note that these face maps are to be thought of as maps into 3-simplices sitting inside a 3-cube.

### Fundamental little 1-cubes space induced from an interval {#Dendroidal}

+-- {: .query}

[[Urs Schreiber]]: something I am thinking about...

=--

The following is supposed to give an [[(∞,1)-operad]]ic incarnation of the notion of fundamental $\infty$-groupoid induced from an interval object. It should resemble a _geometric operadic_ version of the _algebraic operadic_ version described further above.

Let $\Omega^p$ be the category of planar [[tree]]s, so that a [[presheaf]] on $\Omega^p$ is a planar [[dendroidal set]].

Given an interval object ${*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*}$ in a category $C$, assume one isomorphism
$\phi_n \in [I,I^{\vee n}]$ for each $n$ has been chosen.

Then there is a planar co-dendroidal object $\Omega^p_C : \Omega^p \to C$ in $C$ given by:

* a tree $T$ with $n$ leaves is sent to $I^{\vee n}$

  (we think of the
  $k$-th copy of $I$ here as being associated to the $k$th
  leaf of the planar tree);

* every degeneracy map
  is sent to the corresponding identity morphism

  (this corresponds to the fact that the co-dendroidal object encodes
  no nontrivial unary (co)operations, only the $(n \gt 1)$-ary
  operations encoded nontrivial infomation);

* an _outer face map_ on an $n$-ary vertex is the identity on all
  copies of $I$ corresponding to the unaffected leaves and is
  $\phi_n$ on the affected leaf;

* an _inner face map_ that contracts a $k_1$-ary vertex with
  a $k_2$-ary one is the identity on all unaffected leaves and is
  on the affected leaves the composition

  $$
    I^{\vee (k_1+k_2)}
    \stackrel{\phi_{k_1+k_2}^{-1}}{\to}
    I
    \stackrel{\phi_{k_1}}{\to}
    I^{\vee k_1}
    \stackrel{(Id, \cdots, Id,\phi_{k_2},Id, \cdots, Id )}{\to}
    I^{\vee (k_1+k_2)}
    \,.
  $$

**Example**. In [[Top]] with $I = [0,1]$ the standard interval,
and $I^{\vee n} = [0,n]$ let $\phi_n : [0,1] \to [0,n]$ be the map
given by multiplication of real numbers by $n$.

Then for the planar tree $T_1$ given by

$$
  T_1 =
  \left[
    \array{
      \searrow && \swarrow
      \\
      & \bullet
      \\
      && \searrow && \swarrow
      \\
      &&& \bullet
      \\
      &&& \downarrow
    }
  \right]
$$

the inclusion of the tree $\left[\downarrow\right]$ into $T$ given by identifying it with its root is sent to the map $f : [0,1] \to [0,3]$ that is the composite of the map $(-)\cdot 2 : [0,1] \to [0,2]$ wth the map $h : [0,2] \to [0,3]$ that is multiplication by two on $[0,1]$ and addition by 1 on $[1,2]$.

On the other hand the inner face map from

$$
  T_2 =
  \left[
    \array{
      \searrow & \downarrow & \swarrow
      \\
      & \bullet
      \\
      & \downarrow
    }
  \right]
$$

to $T_1$ corresponds to the map $[0,3] \to [0,3]$ that is the composite of the map $(-)/3 : [0,3] \to [0,1]$ with the above map $[0,1] \to [0,3]$.

Now for $X \in C$ any object, we obtain the planar [[dendroidal set]]

$$
  Paths X : T \mapsto Hom_C( \Omega^p_C[T], X )
  \,.
$$

It assigns to any tree with $n$ leaves the hom-set  $Hom(I^{\vee n},X)$. This we can think of as the set of standard parameterized paths of parameter length $n$ in $X$. The action of tree morphisms $T_1 \to T_2$ on these sets is the _reparameterization_ of these paths as encoded in the tree structure.

In particular, we have the dendroidal set $Paths(I)$ of the interval object itself. This is something like the [[little cubes operad|little 1-cubes]] operad as seen by $I$.

I think for every $X$ there is an evident morphism of dendroidal sets

$$
  Paths X \to Paths I
  \,.
$$

The component over the tree $T$ sends all of $(Paths X)(T) = Hom_C(I^{\vee n},X)$ to ....

For $X$ a [[pointed object]], there is the sub-dendroidal set $\Omega X \subset Paths X$ of paths whose endpoints map to the basepoint.

## Related concepts

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

* [[fundamental category]], **fundamental $(\infty,1)$-category**

## References

The above definition, evident as it may be, does not seem to be in the literature. Articles that do discuss proposals for aspects of fundamental higher categories of directed spaces include

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

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

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

The [[simplicially enriched categories]] obtained from a directed space according to the latter have as morphisms directed paths, and as 2-morphisms _directed homotopies_  (paths of directed paths). This is in contrast to the above definition, where the 2-cells away from their boundary are unconstrained.

Accordingly it seems that the definition by Porter produces indeed not a Kan-complex enriched category (a model for an $(\infty,1)$-category) but a quasi-category-enriched category, hence actually a model for an [[(∞,2)-categories]], a candidate for the _fundamental $(\infty,2)$-category_ of a space. Generally, in the fundamental [[(∞,n)-category]] the $(k \leq n)$-cells should be directed maps, and the $k \gt n$-cells be otherwise unconstrained.


fundamental 0-groupoid

Another word for the set of [[connected components]].

## Related concepts

* [[fundamental groupoid]]

* [[fundamental infinity-groupoid]]

[[!redirects fundamental 0-groupoids]]

[[!redirects fundamental zero-groupoid]]
[[!redirects fundamental zero-groupoids]]


fundamental 2-groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a well-bahaved [[topological space]] $X$, its *fundamental 2-groupoid* is the [[2-groupoid]] whose

* [[objects]] are the [[points]] ([[elements]]) of $X$;

* [[1-morphisms]] are [[continuous function|continuous]] [[paths]] $[0,1] \to X$ (or [[thin homotopy|thin]] [[homotopy classes]] of such, to yield a [[strict 2-groupoid]]);

* [[2-morphisms]] are [[homotopies]] between such paths, fixing their endpoints;

* [[composition]] is given by [[concatenation]] of paths and homotopies.

This may equivalently be understood as:

* the [[2-groupoid|2-groupoidal]]-[[categorification]] of the notion of the [[fundamental groupoid]];

* the [[homotopy 2-category]] of the [[fundamental infinity-groupoid|fundamental $\infty$-groupoid]];

At least if $X$ admits the [[structure]] of a [[CW-complex]], its fundamental 2-groupoid captures the underlying [[truncation|2-truncated]] [[weak homotopy type]] (the [[homotopy 2-type]]).

## Related concepts

* [[path 2-groupoid]]

## References

Original discussion of the fundamental 2-groupoid of [[Hausdorff topological spaces]] as a [[strict 2-groupoid]] ([[Grpd-enriched category]]):

* [[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)]

and 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)]

[[!redirects fundamental 2-groupoid]]


* [[David Roberts]], _[[Fundamental Bigroupoids and 2-Covering Spaces]]_ ([pdf](http://ncatlab.org/davidroberts/files/DMRthesis_final.pdf))

This PhD thesis is about a special case of the notion of [[Whitehead tower in an (∞,1)-topos]] in the context of the [[(∞,1)-topos]] of [[topological ∞-groupoid]]s. Notably it introduces the notion of a **2-covering space** of a [[topological space]] as a certain topological groupoid over that space.

An exposition of central ideas can be found in the slides

* [[David Roberts]] _2-Covering spaces_ - talk at the 2009 Annual Meeting of the Australian Mathematical Society, at the University of South Australia, 30 September 2009 ([slides](http://ncatlab.org/davidroberts/files/DMRAustMS09.pdf))

For more details see

Whitehead tower -- For topological ∞-groupoids.

category: reference


fundamental category

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

#Contents#
* table of contents
{:toc}

## Idea

### For directed spaces

In generalization to how a [[topological space]] $X$ has a [[fundamental groupoid]] whose [[morphism]]s are [[homotopy]]-classes of paths in $X$ and whose composition operation is the concatenation of paths, a [[directed space]] has a **fundamental category** whose morphisms are _directed_ paths in $X$.

### For stratified spaces

A [[stratified space]] has a 'fundamental [[(infinity,n)-category with duals|n-category with duals]]', which  generalizes the [[fundamental groupoid|fundamental n-groupoid]] of a plain old space.  When a path crosses a codimension-$1$ stratum, "something interesting happens" – i.e., a [[catastrophe]].   So, we say such a path gives a noninvertible morphism.   The idea is that going along such a path and then going back is not "the same" as having stayed put. So, going back along such a path is not its inverse, just its [[dualizable object|dual]].

See Café [discussion] (http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006196) and paper it inspired, [[J. Woolf]] [Transversal homotopy theory](http://arxiv.org/abs/0910.3322).

### For simplicial sets

The [[left adjoint]] of the [[nerve]] functor $N:Cat \to SSet$, which takes a [[simplicial set]] to a category, is sometimes called the *fundamental category* functor.  One notation for it is $\tau_1$.  Explicitly, for a simplicial set $X$, $\tau_1(X)$ is the [[category]] [[free category|freely generated]] by the [[directed graph]] whose [[vertices]] are 0-[[simplices]] of $X$ and whose [[edges]] are 1-[[simplices]] (the source and target are defined by the [[face maps]]), modulo the relations $s^0(x) \sim id_x$ for $x \in X_0$ and $d^1(x) \sim d^0(x) \circ d^2(x)$ for $x \in X_2$.  Here $s^i$ and $d^i$ denote the [[degeneracy map|degeneracy]] and [[face maps]], respectively.

If $X$ is a [[quasicategory]], then its fundamental category is equivalent to its [[homotopy category of an (infinity,1)-category|homotopy category]].

$$
  \array{
     QuasiCat &&\hookrightarrow&& sSet
     \\
     & {}_{\mathllap{Ho}}\searrow && \swarrow_{\mathrlap{\tau_1}}
    \\
     && Cat
  }
  \,.
$$

## Related concepts

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

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

  * [[equivariant fundamental groupoid]]

* [[directed homotopy theory]]

## References

* [[Marco Grandis]], _Directed algebraic topology, categories and higher categories_ ([pdf](http://www.dima.unige.it/~grandis/DAT.Intro.pdf))

* [[Andre Joyal]], [[Myles Tierney]], _Notes on simplicial homotopy theory_, 2008, [citeseerx](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.187.2533)

* [[J. Woolf]], _Transversal homotopy theory_, Theory and applications of categories, Vol 24, Issue 7, pp 148-178, 2010. (arXiv:0910.3322)

* [[J. Woolf]], _The fundamental category of a stratified space_, Journal of Homotopy and Related Structures, Vol 4, Issue 1 pp 359-387, 2009. (arXiv:0811.2580)

[[!redirects fundamental categories]]


fundamental class

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

#Contents#
* table of contents
{:toc}

## Definition

### For manifolds

For $X$ a [[connected]] [[orientation|oriented]] [[boundary|closed]] [[manifold]], its integral [[homology]] group in degree the [[dimension]] of $X$ is [[isomorphic]] to the [[integer]]s

$$
  H_{dim X}(X, \mathbb{Z}) \simeq \mathbb{Z}
  \,.
$$

The generator of this corresponding to the choice of [[orientation]] is called the **fundamental class** of $X$.

### For $(n-1)$-connected spaces

If a [[topological space]] $X$ is $(n-1)$-[[n-connected space|connected]] for $n\geq2,$ then by the [[Hurewicz theorem]] there is an isomorphism $h\colon\pi_n(X)\to H_n(X)$. By the [[universal coefficient theorem]], we have $H^n(X;\pi_n(X))=\hom(H_n(X),\pi_n(X))$. Hence $h^{-1}$ represents an element of $H^n(X;\pi_n(X))$ called the fundamental class of $X$. In particular, the [[Eilenberg-MacLane space]] $K(G,n)$ has a fundamental class $\iota$ which represents the identity map $1\in [K(G,n),K(G,n)]\cong H^n(K(G,n);G).$ This is the universal cohomology class, in the sense that all cohomology classes are pullbacks of this one by classifying maps. ref Mosher and Tangora.

### Virtual fundamental class

(...)

* [[virtual fundamental class]].

## Related concepts

* [[Poincaré duality complex]]

* [[Poincaré duality algebra]]

* [[intersection theory]]

* [[wrapped brane]]

[[!redirects fundamental classes]]


fundamental frequency

\tableofcontents

\section{Idea}

In [[acoustics]] and [[signal processing]], the **fundamental frequency** or the **first frequency** is the lowest [[frequency]] of a [[periodic function|periodic]] [[waveform]]

\section{See also}

* [[frequency]]

* [[waveform]]

* [[periodic function]]

* [[harmonic]]

* [[Fourier analysis]]

\section{References}

See also:

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

[[!redirects fundamental frequencies]]
[[!redirects first frequency]]
[[!redirects first frequencies]]


fundamental group

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

# Contents
* table of contents
{: toc}

## Idea

The _fundamental group_ $\pi_1(X,x)$ of a [[pointed object|pointed]] [[topological space]] $(X,x)$ is the group of based [[homotopy classes]] of [[loops]] at $x$, with multiplication defined by concatenation (following one path by another).

This is also called _the first [[homotopy group]]_ of $X$.

The notion of _fundamental group_ $\pi_1(X,x)$ generalises in one direction to the [[fundamental groupoid]] $\Pi_1(X)$, or in another direction to the [[homotopy groups]] $\pi_n(X,a)$ for $n \in \mathbb{N}$. Both of this is contained within the [[fundamental ∞-groupoid]] $\Pi(X)$.

## Definition

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

For $X$ a [[topological space]] and $x : * \to X$ a [[point]]. A [[loop space|loop]] in $X$ based at $x$ is a [[continuous function]]

$$
  \gamma : \Delta^1  \to X
$$

from the topological 1-[[simplex]], such that $\gamma(0) = \gamma(1) = x$.

A _based [[homotopy]]_ between two loops is a [[homotopy]]

$$
  \array{
    \Delta^1
    \\
    \downarrow^{\mathrlap{(id,\delta_0)}} & \searrow^{\mathrlap{f}}
    \\
    \Delta^1 \times \Delta^1 &\stackrel{\eta}{\to}& X
    \\
    \uparrow^{\mathrlap{(id,\delta_1)}} & \nearrow_{\mathrlap{g}}
    \\
    \Delta^1
  }
$$

such that $\eta(0,-) = \eta(1,-) = x$.

=--

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

This notion of based homotopy is an [[equivalence relation]].

=--

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

This is directly checked. It is also a special case of the general discussion at _[[homotopy]]_.

=--

+-- {: .num_definition #Concatenation}
###### Definition

Given two loops $\gamma_1, \gamma_2 : \Delta^1 \to X$, define their
**concatenation** to be the loop

$$
  \gamma_2 \cdot \gamma_1 : t \mapsto
  \left\{
    \array{
      \gamma_1(2 t) & ( 0 \leq t \leq 1/2 )
      \\
      \gamma_2(2 (t-1/2)) & (1/2 \leq t \leq 1)
    }
  \right.
  \,.
$$

=--

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

Concatenation of loops respects based homotopy classes
where it becomes an [[associativity|associative]], [[unital]]
binary pairing with [[inverses]], hence the product in a [[group]].

=--

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

See also at _[[path groupoid]]_ for similar constructions.

=--

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

For $X$ a topological space and $x \in X$ a point, the set of based homotopy equivalence classes of based loops in $X$ equipped with the group structure from prop. \ref{GroupStructure} is the **fundamental group** or **first [[homotopy group]]** of $(X,x)$, denoted

$$
  \pi_1(X,x) \in Grp
  \,.
$$

=--

Hence if we write $[\gamma] \in p_1(X,x)$ for the based homotopy class of a loop $p$, then then group operation is

$$
  [\gamma_1] \cdot [\gamma_2] \coloneqq [\gamma_1 \cdot \gamma_2]
  \,.
$$

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

A [[topological space]] whose fundamental group is trivial is called a
_[[simply connected topological space]]_.

=--

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

Conversely, a topological space whose only non-trivial [[homotopy group]] is the fundamental group is called an [[Eilenberg-MacLane space]] denoted $K(\pi_1(X), 1)$.

=--

## Properties

### Naturality

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

If a topological space $X$ is path-[[connected space]], then all of the fundamental groups $\pi_1(X,x)$ are [[isomorphism|isomorphic]], for all choices of base points $x \in X$.

=--

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

For $x_0, x_1 \in X$ any two basepoints, there is by assumption a path connecting them, hence a [[continuous function]]

$$
  p : \Delta^1 \to X
$$

such that $p(0) = x_0$ and $p(1) = x_1$.

Write $\bar p \coloneqq (p(1-(-)))$ for the same path with the orientation reversed. Then for $\gamma_0$ any loop based at $x_0$, the concatenation
$ [ p \cdot (\gamma_0 \cdot \bar p) ]$, def. \ref{Concatenation} yield a loop based at $x_1$ (obtained from $[\gamma_0]$ by [[conjugation]] with $[p]$).

It is immediate to check that this induces an [[isomorphism]]

$$
  Ad_{[p]} : \pi_1(X,x_0) \to \pi_1(X,x_1)
  \,.
$$

=--

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

Therefore one sometimes loosely speaks of 'the' fundamental group of a connected space. But beware that the isomorphism in the above construction is not unique. Therefore forming fundamental groups is not a [[functor]] on connected spaces.
=--

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

It is, however, a functor on [[pointed object|pointed topological spaces]]:
$\pi_1(-) :$ [[Top]] ${}^{*/} \to$ [[Grp]].

=--

### Relation to singular homology
 {#RelationToSingularHomology}

The fundamental group is in general non-abelian (i.e. is not an [[abelian group]]), the  [Examples](#Examples) below. For a connected topological space $X$, its _[[abelianization]]_ is equivalent to the first [[singular homology]] group

$$
  \pi_1(X,x)^{ab} \simeq H_1(X)
  \,.
$$

See at _[singular homology -- Relation to homotopy groups](singular%20homology#RelationToHomotopyGroups)_ for more on this.

### Relation to universal covers and Galois groups

There is a relation to [[universal cover|universal covers]]: Under suitable conditions the group of cover automorphisms of a universal cover is isomorphic to the fundamental group of the covered space. This is the topic of the _[[étale fundamental group]]_, also referred to at _[[Chevalley fundamental group]]_, see there for more.

In particular in [[algebraic geometry]] and [[arithmetic geometry]] this essentially identifies the concept of fundamental group with that of _[[Galois groups]]_. For this reason one also speaks of the _[[algebraic fundamental group]]_ in this context. See at _[[Galois theory]]_ for more on this.

See also at _[[link between Galois theory and fundamental groups]]_.

In [[Grothendieck's Galois theory]], the role of the basepoint is replaced by considering a 'fibre functor' $F:\mathcal{C}\to Sets$ or to $FinSets$, where $\mathcal{C}$ is the category of coverings of the given space. This theory extends to other situations and the term [[algebraic fundamental group]] is used in particular for the case of [[scheme]]s (of a suitable type); see (SGA1).

## Generalizations

### Non-locally 'nice' spaces and 'generalised' spaces

The definition of fundamental group in terms of homotopy classes of loops at a base point does not work well for the spaces that occur in [[algebraic geometry]], nor for many spaces considered in analysis as there may be very few loops.  For instance, for a [[scheme]] there are in general very few paths, and [[Grothendieck]] gave a definition of a fundamental group in SGA1 which is closely related to the [[Galois group|Galois groups]] of number theory, but in cases where both the path-based group and this [[algebraic fundamental group]] make sense, the algebraic form tends to be related to the [[profinite completion of a group|profinite completion]] of the topological fundamental group; see the example in that entry.

A similar type of construction gives the [[fundamental group of a topos]]. Other related forms include a Čech version of the fundamental group used in shape theory, and linked to Čech homology groups of a compact space.

The notion of fundamental group generalizes to that of [[fundamental groupoid]] in both the loop based theory and in [[Grothendieck's Galois theory]] as described in [[SGA1]]. In this form it has been used to give generalisations for simplicial profinite spaces in work by Quick and to [[pro-spaces]] in work of Isaksen.

### Proper fundamental groups

In the context of [[proper homotopy theory]] there are two related fundamental groups for single ended spaces.

## Examples
 {#Examples}

+-- {: .num_example #EuclideanSpaceFundamentalGroup}
###### Example
**(Euclidean space is [[simply connected topological space|simply connected]])**

For $n \in \mathbb{N}$, let $\mathbb{R}^n$ be the $n$-dimensional
[[Euclidean space]] with its [[metric topology]]. Then for every
point $x \in \mathbb{R}^n$ the fundamental group is trivial:

$$
  \pi_1(\mathbb{R}^n, x) = 1
  \,.
$$

=--

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

Let

$$
  \gamma \;\colon\; [0,1] \longrightarrow \mathbb{R}^n
$$

be [[loop]] at $x$, hence a [[continuous function]] with $\gamma(0) = x$ and $\gamma(1) = x$. Using the [[real vector space]] structure on $\mathbb{R}^n$, we may define the function

$$
  \array{
    [0,1] \times [0,1] &\overset{\eta}{\longrightarrow}& \mathbb{R}^n
    \\
    (t,s) &\mapsto& x + s (\gamma(t) - x)
  }
  \,.
$$

This is a [[continuous function]], since it is the composite of the continuous function $(id_{[0,1]} \times \gamma) \;\colon\; (t,s) \mapsto (\gamma(t),s)$ (which is continuous as the [[product topological space|product]] of two continuous functions) and the function $(v,s) \mapsto x + s ( v - x )$ (which is continuous since [[polynomials are continuous]]).

Moreover, by construction we have

$$
  \eta(-,1) = \gamma(-)
   \phantom{AAAA}
  \eta(-,0) = const_x
  \,.
$$

Therefore this is a [[homotopy]] from $\gamma$ to the constant loop at $x$.

=--

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

By definition \ref{SimplyConnectedSpace}, the fundamental group of every [[simply connected topological space]] is trivial.

=--

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

The [[fundamental group of the circle is the integers]]:

$$
  \pi_1(S^1) \simeq \mathbb{Z}
  \,.
$$

=--

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

An instructive formalization of this basic statement in [[homotopy type theory]] is in ([Shulman](#Shulman)).

=--

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

By definition \ref{EMSpace}, the fundamental group of any [[Eilenberg-MacLane space]] $K(G,1)$ is $G$: $\pi_1(K(G,1)) = G$.

=--

## Related concept

* [[fundamental theorem of covering spaces]]

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

* [[winding number]]

* [[algebraic fundamental group]]

* [[anabelian geometry]]

* [[Chevalley fundamental group]]

* [[etale homotopy]]

* [[Grothendieck's Galois theory]]

## References

Historical origins:

* [[Henri Poincaré]], *[[Analysis Situs]]*, Journal de l'École Polytechnique. (2). 1: 1–123  (1895) (Orig.: [gallica:12148/bpt6k4337198/f7](https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7), Engl.: [[John Stillwell]]: [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf), [[Stillwell_AnalysisSitus.pdf:file]])

Textbook accounts:

* {#AGP02} [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], Section 2.5 of: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

Review and Exposition:

* {#Moller11} [[Jesper Møller]], _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]]]

* Alberto Santini, *Topological groupoids* (2011) [[pdf](http://web.math.ku.dk/~moller/students/alberto.pdf), [[Santini-Groupoids.pdf:file]]]

  > (about [[groupoids]] in [[topology]], notably [[fundamental groupoids]] -- not about [[topological groupoids]])

* *[[Introduction to Topology -- 2]]*

Discussion from the point of view of [[Galois theory]] is in

* [[Luis Javier Hernández-Paricio]], _Fundamental pro-groupoids and covering projections_([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf))

See also:

* [[D. C. Isaksen]], _A model structure on the category of pro-simplicial sets_,  Trans. Amer. Math. Soc., 353, (2001), 2805–2841

* [[G. Quick]], _Profinite homotopy theory_, Documenta Mathematica, 13, (2008), 585–612.

Proof in [[homotopy type theory]] that the [[fundamental group of the circle is the integers]]:

* [[Daniel Licata]], [[Michael Shulman]], _Calculating the Fundamental Group of the Circle in Homotopy Type Theory_,  ([arXiv:1301.3443](https://arxiv.org/abs/1301.3443))

* [[UF-IAS-2012|Univalent Foundations Project]], section 8.1 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

the HoTT-[[Coq]]-code is at

* {#Shulman} [[Mike Shulman]], _[P1S1.v](https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Pi1S1.v)_

[[!redirects fundamental groups]]


fundamental group of a topos

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

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

## Idea

By the logic of [[space]], one may think of a [[topos]] as being (the [[category of sheaves]] on) some generalization of a [[topological space]]. The construction of the first [[fundamental group]] of a topological space happens to generalize to an arbitrary topos, even without a predefined notion of path or [[interval object]]. Instead, by the logic of [[Grothendieck's Galois theory]], the homotopy groups can be detected from the behaviour of [[locally constant sheaf|locally constant sheaves]].

### More details

> (At present the following is just lifted/adapted from another article, (see references below) and needs some editing and possibly correcting. [[Tim Porter|T.P.]])

Let $\mathbb{E}$ be a [[Grothendieck topos]] (think of $\mathbb{E}$ as the
category, $Sh(X)$, of set valued [[sheaf|sheaves]] on a space $X$).  Within
$\mathbb{E}$, we can pick out a [[subcategory]], $\mathbb{C}$, of locally finite,
[[locally constant sheaf|locally constant object]]s in $\mathbb{E}$.  (If $X$ is a space with
$\mathbb{E}= Sh(X)$, $\mathbb{C}$ corresponds to those sheaves whose
'_[[etale space|espace étalé]]_' is a finite
[[covering space]] of $X$.)  Picking a base point in $X$ generalises to picking a
'[[fiber functor]]' $F :\mathbb{C} \to \mathbf{Sets_{fin}}$, a functor satisfying
  various conditions implying that it is [[pro-representable functor|pro-representable]]. (If $x_0 \in X$
  is a base point $\{x_0\}\to X$ induces a 'fibre functor' $ Sh(X)\to
  Sh\{x_0\} \cong \mathbf{Sets}$, by pullback.)

+--{: .query}
[[Mike Shulman|Mike]]: I presume that more generally any "point" of $\mathbb{E}$, meaning a [[geometric morphism]] $Set\to \mathbb{E}$, supplies a fibre functor (its inverse image)?  Of course, in general $\mathbb{E}$ might not have a point.  Are there other examples of fibre functors when $\mathbb{E}=Sh(X)$?
=--

If $F$ is 'pro-representable' by $P$, then $\pi_1(\mathbb{E},F)$ is defined to
be $Aut(P)$, which is a [[profinite group]].  (Usually we will simply write
$\pi_1(\mathbb{E})$, for this.)  Grothendieck proves there is an [[equivalence
of categories]]

$$\mathbb{C} \simeq \pi_1(\mathbb{E})-\mathbf{Sets_{fin}},$$

the
category of finite $\pi_1(\mathbb{E})$-sets. (This is what is called [[Grothendieck's Galois theory]].)

If $X$ is a locally nicely
behaved space such as a [[CW complex]] and $\mathbb{E} = Sh(X)$, then
$\pi_1(\mathbb{E})$ is the [[profinite completion of a group|profinite completion]] of $\pi_1(X)$.  This profinite
completion occurs only because Grothendieck considers locally [[finite object]]s.
Without this restriction, a covering space $Y$ of $X$ would correspond to a
$\pi_1(X)$-set, $Y^\prime$, but if $Y$ is a finite covering of $X$ then the
homomorphism from $\pi_1(X)$ to the finite group of transformations of $Y$
factors through the profinite completion of $\pi_1(X)$.

### Remarks

 This idea of using covering spaces or their analogue in
$\mathbb{E}$ raises several important points:

* these are *homotopy theoretic results, but no paths (no [[interval object]]) are used*.  The argument
involving [[sheaf and topos theory|sheaf theory]], the theory of [[pro-representable functor]]s, etc., is of
a purely categorical nature.  This means it is applicable to spaces where the
use of paths, and other homotopies is impossible because of bad (or unknown)
local properties.  Such spaces have been studied within [[shape theory]] and
[[strong shape theory]], although not by using exactly Grothendieck's fundamental group,
nor using sheaf theory. (See below for more on this connection and such
sources as _Lisica and Mardešić_, _Edwards and
Hastings_, _Cordier and Porter_, _Mardešić and Segal_  for more
information on Shape and Strong Shape).

*  As no paths are used, these methods can also be applied to 'non-spaces',
e.g. [[locale|locales]] and possibly to their
non-commutative analogues, [[quantale|quantales]].

For instance, classically one could
consider a [[field]] $k$ and an algebraic closure $K$ of $k$ and then choose
$\mathbb{C}$ to be a category of étale algebras over $k$, in such a way that
$\pi_1(\mathbb{E}) \cong Gal(K/k)$, the Galois group of $k$.  A beautiful
treatment of this can be found in Douady and Douady, (see below), and
the link with locales (which is very strong) is explored in Joyal and Tierney.  It, in fact, leads to a classification theorem for
[[Grothendieck topos]]es.  From this viewpoint, low dimensional [[homotopy theory]] is
seen as being part of [[Galois theory]], or _vice versa_. Of course, the really interesting question is how to fit higher dimensional homotopy theory into a higher dimensional Galois theory, and, again, -vice versa-.  See A. Grothendieck, (1975?), [[Pursuing Stacks|Letter to L. Breen]].  *NOT to Quillen as is sometimes claimed.* More on that at [[homotopy group of an ∞-stack]].

* This underlines the fact that $\pi_1(X)$ classifies [[covering space]]s -- but
for $i \gt 1$, $\pi_i(X)$ does not seem to classify anything other than maps
from $S^i$ into $X$!

+--{: .query}
[[Mike Shulman|Mike]]: From a higher-categorical perspective, the reason $\pi_1(X)$ classifies covering spaces is that covering spaces are fibrations with discrete fibers, and so are classified by functors $\Pi_\infty(X)\to Set$.  But since $Set$ is a 1-category, any such functors factors through the 1-categorical reflection of $\Pi_\infty(X)$, which is the ordinary [[fundamental groupoid]] $\Pi_1(X)$.  Thus, to ask what higher homotopy groups classify, one should consider not $\pi_i(X)$ but $\Pi_i(X)$, which one might expect to classify fibrations over $X$ whose fibers are homotopy $(i-1)$-types.

Does a topos have a fundamental groupoid?  A fundamental $i$-groupoid?  A fundamental $\infty$-groupoid?

[[Tim Porter|Tim]]: I am not an expert on the 'not enough points' case, but do know that a long time ago people generalised to a fundamental groupoid. (I think it was a preprint from Montpellier in which I saw this, but I know it was also taken up by others in (?) the 1970s. More recently Marta Bunge and Eduardo Dubuc have published on this and look at at Eduardo's:

http://arxiv.org/abs/0706.1771

[[David Corfield|David]]: The paper [Higher Monodromy](http://lanl.arxiv.org/abs/math/0407507) shows what the fundamental 2-groupoid classifies.

[[Mike Shulman|Mike]]: Just looking at the abstract of _Higher Monodromy_, what I was saying above (for $i=2$) looks like the special case of their theory for locally constant stacks with values in the 2-category $Gpd$ of 1-types.  They don't seem to treat the topos case, though.

=--

## Higher categorical generalization

One may consider the [[vertical categorification]] of this situation from the notion of [[topos]] to that of [[(∞,1)-topos]]. The discussion of the construction of homotopy groups of objects in such a higher topos is at

* [[homotopy group of an ∞-stack]].

## References

The above was taken from an article:

* [[Tim Porter]], Abstract Homotopy Theory, the interaction of category theory and homotopy theory , Cubo Matematica Educacional, 5, (2003), 115--165.

Fundamental groups of toposes are already on [[SGA4]], Expose IV _Exercice 2.7.5. and its relations with [[etale topos]] are on Expose VIII Proposition 2.1_

The general construction of the first geometric homotopy group of objects in a [[Grothendieck topos]] is in section 8.4 of

* [[Peter Johnstone]], _Topos theory_

A discussion of the fundamental groupoid for a general topos is in

* [[Eduardo Dubuc]], _The fundamental progroupoid of a general topos_ ([arXiv](http://arxiv.org/abs/0706.1771))

A discussion for "locally simply connected" toposes is in

* [[Michael Barr]], [[Radu Diaconescu]], _On locally simply connected toposes and their fundamental groups_ ([NUMDAM](http://www.numdam.org/item?id=CTGDC_1981__22_3_301_0))
{#BarrDiaconescu}

Other references are

* D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.

* J.T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glasnik Mat. 19(39) (1984) 335-399.

* J.-M. Cordier and [[Tim Porter]], (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).
* S. Mardešić and J. Segal, (1982) _Shape Theory_, North Holland.
* S. Mardešić, _Strong Shape and Homology_, Springer monographs in mathematics, Springer-Verlag.

* On categorical Galois theory

  * F. Borceux and G. Janelidze, (2001), Galois Theories, Cambridge Studies in Advanced Mathematics, 72, Cambridge University Press.

* and for a more traditional approach:

  * A. Douady and R. Douady, (1977) Algébre et Théories Galoisiennes, Cedic / F. Nathan.

*  For the link with [[locale]] theory

   * [[Andre Joyal]] and M. Tierney, (1984), An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309.

* When we have no paths, in internal case, one may find the article of Pataraia useful (beware many typoses, but the article is dense with content):

   * D. Pataraia, Internal categories in the left exact cosimplicial category. Georgian Math. J. 4 (1997), No. 6, 533-556. (link)

* [[Luis Javier Hernández-Paricio]], _Fundamental pro-groupoids and covering projections_([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf))


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

#Contents#
* table of contents
{:toc}

## Statement

A basic statement in [[homotopy theory]]:

+-- {: .num_prop}
###### Proposition
**(fundamental group of the circle is the integers)**

The [[fundamental group]] $\pi_1$ of the [[circle]] $S^1$ is the additive group of [[integers]]:

$$
  \pi_1(S^1) \overset{\simeq}{\longrightarrow} \mathbb{Z}
$$

and the isomorphism is given by assigning [[winding number]].

=--

Here in the context of [[topological homotopy theory]] the [[circle]] $S^1$ is the [[topological subspace]] $S^1 = \{x \in \mathbb{R}^2 \,\vert\, x_1^2 + x_2^2 = 1 \} \subset \mathbb{R}^2$ of the [[Euclidean plane]] with its [[metric topology]], or any [[topological space]] of the same [[homotopy type]]. More generally, the circle in question is, as a [[homotopy type]], the [[homotopy pushout]]

$$
  S^1 \simeq \ast \underset{\ast \sqcup \ast}{\coprod} \ast
  \,,
$$

hence the [[homotopy type]] with the [[universal property]] that it makes a homotopy commuting diagram of the form

$$
  \array{
    \ast \sqcup \ast &\longrightarrow& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \ast &\longrightarrow& S^1
  }
  \,.
$$

## Proofs

### In topological point-set homotopy theory
 {#ProofInTopologicalHomotopyTheory}

We discuss the classical proof in [[point-set topology|point-set]] [[topological homotopy theory]].

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

The [[universal covering space]] $\widehat{S^1}$ of $S^1$ is the [[real line]] (by [this example](universal+covering+space#UniversalCoveringOfCircleRealLine)):

$$
  p \coloneqq (cos(2 \pi(-)), \sin(2 \pi(-)))
   \;\colon\; \mathbb{R}^1 \longrightarrow S^1
  \,.
$$

Since the [[circle]] is [[locally path-connected topological space|locally path-connected]] ([this example](locally+path-connected+space#LocallyPathConnectedCircle)) and [[semi-locally simply connected topological space|semi-locally simply connected]] ([this example](semi-locally+simply+connected+topological+space#LocallySimplyConnectedCircle)) the [[fundamental theorem of covering spaces]] applies and gives that the [[automorphism group]] of $\mathbb{R}^1$ over $S^1$ equals the automorphism group of its [[monodromy]] [[permutation representation]]:

$$
  Aut_{Cov(S^1)}(\mathbb{R}^1)
    \;\simeq\;
  Aut_{\pi_1(S^1) Set}(Fib_{S^1})
  \,.
$$

Moreover, as a corollary of the [[fundamental theorem of covering spaces]] we have that the [[monodromy]] representation of a [[universal covering space]] is given by the [[action]] of the [[fundamental group]] $\pi_1(S)$ on itself ([this prop.](universal+covering+space#ReconstructCoveringForFreeAndTransitiveMonodromyRepresentation)).

But the [[automorphism group]] of any group regarded as an [[action]] on itself by left multiplication is canonically isomorphic to that group itself (by [this example](permutation+representation#AutomorphismsOfGAsGTorsor)), hence we have

$$
  Aut_{{\pi_1(S^1)} Set}(Fib_{S^1})
   \;\simeq\;
  Aut_{{\pi_1(S^1)} Set}( \pi_1(S^1) )
   \;\simeq\;
  \pi_1(S^1)
  \,.
$$

Therefore to conclude the proof it is now sufficient to show that

$$
  \mathrm{Aut}_{Cov(S^1)}(\mathbb{R}^1) \simeq \mathbb{Z}
  \,.
$$
#
To that end, consider a [[homeomorphism]] of the form

$$
  \array{
    \mathbb{R}^1 && \underoverset{\simeq}{f}{\longrightarrow} && \mathbb{R}^1
    \\
    & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{p}}
    \\
    && S^1
  }
  \,.
$$

Let $s \in S^1$ be any point, and consider the restriction of $f$ to the fibers over the [[complement]]:

$$
  \array{
    p^{-1}(S^1 \setminus \{s\}) && \underoverset{\simeq}{f}{\longrightarrow} && p^{-1}(S^1 \setminus \{s\})
    \\
    & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{p}}
    \\
    && S^1 \setminus \{s\}
  }
  \,.
$$

By the [[covering space]] property we have (via [this example](universal+covering+space#UniversalCoveringOfCircleRealLine)) a [[homeomorphism]]

$$
  p^{-1}(S^1 \setminus \{s\})
   \simeq
  (0,1) \times Disc(\mathbb{Z})
  \,.
$$

Therefore, up to homeomorphism,  the restricted function is of the form

$$
  \array{
    (0,1)\times Disc(\mathbb{Z})
      &&
      \underoverset{\simeq}{f}{\longrightarrow}
      &&
    (0,1) \times Disc(\mathbb{Z})
    \\
    & {}_{pr_1}\searrow && \swarrow_{pr_1}
    \\
    && (0,1)
  }
  \,.
$$

By the [[universal property]] of the [[product topological space]] this means that $f$ is equivalently given by its two components

$$
  (0,1) \times Disc(\mathbb{Z})
    \overset{pr_1 \circ f}{\longrightarrow}
  (0,1)
  \phantom{AAAA}
  (0,1) \times Disc(\mathbb{Z})
    \overset{pr_2 \circ f}{\longrightarrow}
  Disc(\mathbb{Z})
  \,.
$$

By the [[commuting diagram|commutativity]] of the above [[diagram]], the first component is fixed to be $pr_1$. Moreover, by the fact that $Disc(\mathbb{Z})$ is a [[discrete space]] it follows that the second component is a [[locally constant function]] (by [this example](locally+constant+function#LocallyConstantFunctionIntoDiscreteSpace)). Therefore, since the [[product space]] with a [[discrete space]] is a [[disjoint union space]]  (via [this example](product+topological+space#ProductWithDiscreteFiniteTopologicalSpace))

$$
  (0,1) \times Disc(\mathbb{Z}) \simeq \underset{n \in \mathbb{Z}}{\sqcup}(0,1)
$$

and since the disjoint summands $(0,1)$ are [[connected topological spaces]] ([this example](connected+space#ConnectedSubspacesOfRealLineAreTheIntervals)), it follows that the second component is a [[constant function]] on each  of these summands (by [this example](connected+space#LocallyConstantFunctionsOnConnectedSpaces)).

Finally, since every function out of a [[discrete topological space]] is continuous, it follows in conclusion that the restriction of $f$ to the fibers over $S^1 \setminus \{s\}$ is entirely encoded in an [[endofunction]] of the set of [[integers]]

$$
  \phi \;\colon\; \mathbb{Z} \to \mathbb{Z}
$$

by

$$
  \array{
    S^1 \setminus \{s\} \times Disc(\mathbb{Z})
      &\overset{f}{\longrightarrow}&
    S^1 \setminus \{s\} \times Disc(\mathbb{Z})
    \\
    (t,k) &\mapsto& (t, \phi(k))
  }
  \,.
$$

Now let $s' \in S^1$ be another point, distinct from $s$. The same analysis as above applies now to the restriction of $f$ to $S^1 \setminus \{s'\}$ and yields a function

$$
  \phi' \;\colon\; \mathbb{Z} \longrightarrow \mathbb{Z}
  \,.
$$

Since

$$
  \left\{
    p^{-1}(S^1 \setminus \{s\})
    \subset \mathbb{R}^1
     \,,\,
    p^{-1}(S^1 \setminus \{s'\})
    \subset \mathbb{R}^1
  \right\}
$$

is an [[open cover]] of $\mathbb{R}^1$, it follows that $f$ is unqiuely fixed by its restrictions to these two subsets.

Now unwinding the definition of $p$ shows that the condition that the two restrictions coincide on the intersection $S^1 \setminus \{s,s'\}$ implies that there is $n \in \mathbb{Z}$ such that $\phi(k) = k+ n$ and $\phi'(k) = k+n$.

This shows that $Aut_{Cov(S^1)}(\mathbb{R}^1) \simeq \mathbb{Z}$.

=--

### In homotopy type theory
 {#ProofInHomotopyTypeTheory}

There is also a purely synthetic a proof in [[homotopy type theory]] ([Licata-Shulman 13](#LicataShulman13), [UF, corollary 8.1.10](#UF)).

## Consequences
 {#Consequences}

+-- {: .num_example #CoveringOfCircleAndConjugacyClassesOfSymmetricGroup}
###### Example
**([[isomorphism classes]] of [[covering spaces|coverings]] of the circle are [[conjugacy classes]] in the [[symmetric group]])**

The [[monodromy]] construction assigns to an [[isomorphism class]] of covering spaces
over the [[circle]] $S^1$
with [[fibers]] consisting of $n$ elements [[conjugacy classes]]
of elements the [[symmetric group]] $\Sigma(n)$:

$$
  \left \{
    \array{
      \text{isomorphism classes of}
      \\
      \text{finite covering spaces }
      \\
      \text{over the circle}
    }
  \right\}
   \;\simeq\;
  \left\{
    \array{
      \text{conjugacy classes of}
      \\
      \text{elements of a symmetric group}
    }
  \right\}
$$

To see this,
we may without restriction (via [this prop.](groupoid#representation#GroupoidRepresentationsAreProductsOfGroupRepresentations)) choose a basepoint $x \in S^1$ so that
a monodromy representation is equivalently a groupoid morphism of the form

$$
  \rho
    \;\colon\;
  B \mathbb{Z}
    \overset{\simeq}{\longrightarrow}
  B \pi_1(S^1,x)
    \overset{\rho}{\longrightarrow}
  Core(Set)
  \,.
$$

Since $\mathbb{Z}$ is the [[free abelian group]] on a single generator, such as morphism
is uniquely determined by the image of $1 \in \mathbb{Z}$. This is taken to some
isomorphism of the set $p^{-1}(x)$. If we choose any identification $\phi \colon p^{-1}(x) \overset{\simeq}{\to} \{1, \cdots, n\}$,
then this defines an element $\sigma \in \Sigma(n)$ in the [[symmetric group]]:

$$
  \array{
    x &\mapsto&  p^{-1}(x) &\underoverset{\simeq}{\phi}{\longrightarrow}& \{1, \cdots, n\}
    \\
    {}^{\mathllap{1}}\downarrow && {}^{\mathllap{\rho(1)}}\downarrow && \downarrow^{\sigma}
    \\
    x &\mapsto& p^{-1}(x) &\underoverset{\phi}{\simeq}{\longrightarrow}& \{1, \cdots, n\}
  }
  \,.
$$

Now if

$$
  f \;\colon\; E_1 \overset{\simeq}{\longrightarrow} E_2
$$

is an isomorphism of covering spaces,
then by the [[fundamental theorem of covering spaces]] this corresponds bijectively to a homomorphism of representations

$$
  Fib(f) \;\colon\; Fib_{E_1} \overset{\simeq}{\longrightarrow} Fib_{E_2}
$$

which in turn is by definition a homotopy (natural isomorphism) between the monodromy functors
$Fib_{E_i} \;\colon\; B \mathbb{Z} \to Core(Set)$.

The combination of the naturality square of this natural isomorphism with the above identification
yields the following diagram

$$
  \array{
    \{1,\cdots, n\}
      &\overset{\phi_1^{-1}}{\longrightarrow}&
    p_1^{-1}(x)
      &\overset{f\vert_{\{x\}}}{\longrightarrow}&
    p_2^{-1}(x)
      &\overset{\phi_2}{\longrightarrow}&
    \{1, \cdots, n\}
    \\
    {}^{\mathllap{\sigma_1}}
    \downarrow
      &&
    {}^{\mathllap{ Fib_{E_1}(1) }}\downarrow
      &&
    \downarrow^{\mathrlap{ Fib_{E_2}(1) }}
      &&
    \downarrow^{\mathrlap{ \sigma_2 }}
    \\
    \{1,\cdots, n\}
      &\underset{\phi_1^{-1}}{\longrightarrow}&
    p_1^{-1}(x)
      &\underset{f\vert_{\{x\}}}{\longrightarrow}&
    p_2^{-1}(x)
      &\underset{\phi_2}{\longrightarrow}&
    \{1, \cdots, n\}
  }
  \,.
$$

The commutativity of the total rectangle says that the permutations $\sigma_1$ and $\sigma_2$ are related by conjugation
with the element $\phi_2 \circ f\vert_{\{x\}}\circ \phi_1^{-1}$.

=--

+-- {: .num_example }
###### Example
**(three-sheeted covers of the circle)**

Consider the three-sheeted [[covering spaces]] of the [[circle]].

By example \ref{CoveringOfCircleAndConjugacyClassesOfSymmetricGroup}
these are, up to isomorphism, given by the [[conjugacy classes]] of the elements
of the [[symmetric group]] $\Sigma(3)$ on three elements. These in turn are labeled by the
[[cycle]] structure of the elements ([this prop.](symmetric+group#ConjugacycClassesOfSymmetricGroupCorrespondToCycleSet)).

For the symmetric group on
three elements there are three such classes

$$
  \array{
    (1\; 2\; 3)
    \\
    (1 \; 2) (3)
    \\
    (1) (2) (3)

  }
$$

The corresponding covering spaces of the circle are shown in the graphics.

> graphics grabbed from [Hatcher](homotopy+equivalence#Hatcher)

=--

## References

### In topological homotopy theory

Lecture notes on the classical discussion include

* {#Waldhausen} [[Friedhelm Waldhausen]], p. 63-77 of  _Topologie_ ([pdf](https://www.math.uni-bielefeld.de/~fw/ein.pdf))

* {#Moller11} [[Jesper Møller]], theorem 3.1 in _The fundamental group and covering spaces_ (2011) ([pdf](http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf))

### In homotopy type theory

The proof in [[homotopy type theory]] is discussed in

* {#LicataShulman13} [[Daniel Licata]], [[Michael Shulman]], _Calculating the Fundamental Group of the Circle in Homotopy Type Theory_,  ([arXiv:1301.3443](https://arxiv.org/abs/1301.3443))

* {#UF} [[UF-IAS-2012|Univalent Foundations Project]], section 8.1 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

the HoTT-[[Coq]]-code is at

* {#Shulman} [[Mike Shulman]], _[P1S1.v](https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Pi1S1.v)_


fundamental groupoid

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

# Contents
* toc
{: toc}

## Idea

The **fundamental groupoid** of a space $X$ is a [[groupoid]] whose objects are the points of $X$ and whose morphisms are [[paths]] in $X$, identified up to endpoint-preserving [[homotopy]].

In parts of the literature the fundamental groupoid, and more generally the [[fundamental ∞-groupoid]], is called the **Poincaré groupoid**.

## Definition

The **fundamental groupoid** $\Pi_1(X)$ of a topological space $X$ is the [[groupoid]] whose set of objects is $X$ and whose morphisms from $x$ to $y$ are the equivalence classes of homotopy of homotopy relative to $\partial I$ $[\gamma]$ of continuous maps $\gamma : [0,1] \to X$ whose endpoints map to $x$ and $y$ (which the homotopies are required to fix). Composition is by concatenation (and reparametrization) of representative maps. Under the [[homotopy]]-[[equivalence relation]] this becomes an associative and unital composition with respect to which every morphism has an inverse; hence $\Pi_1(X)$ is a groupoid.

The use of the fundamental groupoid of a manifold for describing the monodromy principle on the extension of local morphisms is discussed in the paper by Brown/Mucuk listed below.

## Remarks

### Relationship to fundamental group

For any $x$ in $X$ the first homotopy group $\pi_1(X,x)$ of $X$ based at $x$ arises as the [[automorphism group]] of $x$ in $\Pi_1(X)$:
$$
  \pi_1(X,x) = Aut_{\Pi_1(X)}(x)
  \,.
$$
So the fundamental groupoid gets rid of the choice of basepoint for the fundamental group, and this is valuable for some applications. The set of connected components of $\Pi_1(X)$ is precisely the set $\Pi_0(X)$ of path-components of $X$.  (This is not to be confused with the set of connected components of $X$, sometimes denoted by the same symbol.  Of course they are the same when $X$ is locally path-connected.)

### Topologizing the fundamental groupoid

The fundamental groupoid $\Pi_1(X)$ can be made into a [[topological groupoid]] (i.e. a [[internal groupoid|groupoid internal]] to [[Top]]) when $X$ is [[path-connected space|path-connected]], [[locally path-connected space|locally path-connected]] and [[semi-locally simply connected space|semi-locally simply connected]]. This is a special case of ([Brown 06, 10.5.8](#Browno6)).
This construction is closely linked with the construction of a [[universal covering space]] for a path-connected pointed space. The object space of this groupoid is just the space $X$.

When $X$ is not semi-locally simply connected, the set of arrows of the fundamental groupoid inherits the [[quotient space|quotient topology]] from the path space such that the fibres of $(s,t):Mor(\Pi_1(X)) \to X\times X$ are not discrete, which is an obstruction to the above-mentioned source fibre\'s being a covering space. However the composition is no longer continuous. When $X$ is not locally path-connected, $\Pi_0(X)$ also inherits a non-discrete topology (the quotient topology of $X$ by the relation of path connections).

In circumstances like these more sophisticated methods are appropriate, such as [[shape theory]].  This is also related to the [[fundamental group of a topos]], which is in general a [[progroup]] or a [[localic group]] rather than an ordinary group.

### $\Pi_1(X)$ with a chosen set of basepoints

An improvement on the fundamental group and the total fundamental groupoid  relevant to the [[van Kampen theorem]] for computing the fundamental group or groupoid is to use $\Pi_1(X,A)$, defined for a set $A$ to be the full subgroupoid of $\Pi_1(X)$ on the set $A\cap X$, thus giving a set  of base points which can be  chosen according to the geometry at hand. Thus if $X$ is the union of two open sets $U,V$ with intersection $W$ then we can take $A$ large enough to meet each path-component of $U,V,W$; note that by the above definition we can write $\Pi_1(U,A)$, etc.  If $X$ has an action of a group $G$ then $G$ acts on $\Pi_1(X,A)$ if $A$ is a union of orbits of the action. Thus $\Pi_1(X,A)$ can represent some symmetry of a given situation.

 The notion of $\Pi_1(X,A)$ was introduced in 1967 by [[Ronnie Brown]]  to give a version of the Seifert-van Kampen Theorem which allowed the determination  of the fundamental group of a connected space which is the union of connected subspaces with nonconnected intersection, such as the  circle, a space  which is, after all, THE basic example in topology.

Grothendieck writes in his 1984 [[Esquisse d'un Programme]] (English translation):

" ..,people still obstinately persist,  when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under  the symmetries of the situation, which thus get lost on the way.  In certain situations (such as descent theorems for fundamental groups \`a la van  Kampen) it is much more elegant,  even indispensable for understanding  something, to work with fundamental groupoids with respect to a suitable    packet of base points,..".

Notice that $\Pi_1(X,X)$ recovers the full fundamental groupoid, while $\Pi_1(X,\{x\})$ is simply  the [[fundamental group]] $\pi_1(X,x)$.

 Basically, $\Pi_1(X,A)$  allows for the _computation of homotopy 1-types_; the theory was developed in  _Elements of Modern Topology_ (1968), now available as _Topology and Groupoids_ (2006). These accounts show  the use of the algebra of  groupoids in 1-dimensional  [[homotopy theory]], for example for [[covering space]]s,  and, in the later edition,  for [[orbit spaces]]. spring

### In higher category theory

See [[fundamental ∞-groupoid]].

### Simplicial version

See [[simplicial fundamental groupoid]].

## Related concepts

* [[fundamental group]],

* [[fundamental 2-groupoid]], [[fundamental ∞-groupoid]]

* [[fundamental theorem of covering spaces]]

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

* [[simplicial fundamental groupoid]]

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

* [[equivariant fundamental groupoid]]

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

## References

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.

Monograph:

* [[Philip J. Higgins]], §6 of: *Categories and Groupoids*, Mathematical  Studies **32**, van Nostrand New  York (1971), Reprints in Theory and Applications of Categories **7** (2005) 1-195 [[tac:tr7] (http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/7/tr7-2a4.pdf)]

Review and Exposition:

* {#Moller11} [[Jesper Møller]], _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]]]

* Alberto Santini, *Topological groupoids* (2011) [[pdf](http://web.math.ku.dk/~moller/students/alberto.pdf), [[Santini-Groupoids.pdf:file]]]

  > (about [[groupoids]] in [[topology]], notably [[fundamental groupoids]] -- not about [[topological groupoids]])

* *[[Introduction to Topology -- 2]]*

See also:

* R. Brown, Groupoids and Van Kampen's theorem, _Proc. London
Math. Soc_. (3)   17 (1967) 385-401.

* R. Brown and G. Danesh-Naruie,  The fundamental groupoid as a
topological groupoid, _Proc. Edinburgh Math. Soc._ 19 (1975)
237-244.

* R. Brown and O. Mucuk,  The monodromy groupoid of a Lie
groupoid, _Cah. Top. G\'eom. Diff. Cat_. 36 (1995) 345-369.

* {#Brown06} [[Ronnie Brown]], _Topology and Groupoids_, Booksurge (2006). (See particularly 10.5.8, using lifted topologies to topologise $(\pi_1 X)/N$ where $N$ is a normal, totally disconnected subgroupoid of $\pi_1 X$, and $X$ admits a universal cover). ([more info] (http://pages.bangor.ac.uk/~mas010/topgpds.html))

Discussion from the point of view of [[Galois theory]] is in

* [[Luis Javier Hernández-Paricio]], _Fundamental pro-groupoids and covering projections_, Fund. Math. (1998), ([pdf](http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf))

* The use of many base points is  discussed at  this ([mathoverflow page] (http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one)).

Discussion of the fundamental groupoid (for good [[topological spaces]] and for [[noetherian schemes]]) as the [[costack]] (via the [[Seifert-van Kampen theorem]]) characterized as being 2-[[terminal object|terminal]] is in

* {#Pirashvili14a} [[Ilia Pirashvili]], _The fundamental groupoid as a terminal costack_ ([arXiv:1406.4419](https://arxiv.org/abs/1406.4419))

* {#Pirashvili14b} [[Ilia Pirashvili]], _The Étale Fundamental Groupoid as a Terminal Costack_ ([arXiv:1412.5473](https://arxiv.org/abs/1412.5473))

Discussion in the context of [[dynamical systems]]:

* Emmanuel Paul, Jean-Pierre Ramis, *Dynamics on Wild Character Varieties*, _SIGMA_ **11** (2015) 068 [[arXiv:1508.03122](https://arxiv.org/abs/1508.03122), [doi:10.3842/SIGMA.2015.068](https://doi.org/10.3842/SIGMA.2015.068)]

[[!redirects fundamental groupoid]]
[[!redirects fundamental groupoids]]

[[!redirects Poincaré groupoid]]
[[!redirects Poincaré groupoids]]
[[!redirects Poincaré-groupoid]]
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[[!redirects Poincare groupoid]]
[[!redirects Poincare groupoids]]
[[!redirects Poincare-groupoid]]
[[!redirects Poincare-groupoids]]