Poincaré duality algebra

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#### Operator algebra
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## Idea

Generally, a _Poincaré duality dg-algebra_ is a [[dg-algebra]] with structure mimicking [[Poincaré duality]] in [[ordinary cohomology]].

On the other hand a _Poincaré duality $C^\ast$-algebra_
is a [[C*-algebra]] which represents a [[space]] in [[noncommutative topology]] for which there is a sensible notion of [[Poincaré duality]] in [[K-theory]] ([[operator K-theory]]/[[K-homology]]).

## Definition

### For graded-commutative algebras
 {#ForGradedCommutativeAlgebras}

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

The structure of a Poincaré duality algebra in [[dimension]] $n$
on a graded-commutative [[graded vector space|graded]] [[associative algebra]] $A$ is a [[linear function]] $\epsilon \colon A_n \to k$ to the ground field such that all the induced [[bilinear forms]]

$$
  A_k \otimes A_{n-k} \stackrel{\otimes}{\to} A_n \stackrel{\epsilon}{\to} k
$$

are non-degenerate.

=--

e.g. ([Lambrechst-Stanley 07](#LambrechstStanley07))

### For $C^\ast$-algebras
 {#ForCStarAlgebras}

For [[C*-algebras]] hence in [[noncommutative topology]] there is the following notion of Poincaré duality, which is really Poincaré with respect not to [[ordinary cohomology]] but [[K-theory]] ([[operator K-theory]]).

We start with the definition of Poincaré _self_-duality and then generalize to Poincaré dual pairs.

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

A [[separable C*-algebra]] $A \in $ [[C*Alg]] is a **Poincaré duality algebra** (or _PD algebra_, for short ) if it is [[dualizable object]] when regarded as an object of the [[KK-theory]]-category, with [[dual object]] its [[opposite algebra]].

The element $\Delta$ in def. \ref{PDAlgebra} is called a **[[fundamental class]]** of $A$.

=--

This appears as ([BMRS 07, def. 2.1](#BMRS07), following [Connes, p. 601](#Connes)) following ([Connes](#Connes)).

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

Explicitly def. \ref{PDAlgebra} says that $A$ is a PD algebra if there exists $\Delta \in KK(A \otimes A^{op}, \mathbb{C})$ and $\Delta^\vee \in KK(\mathbb{C}, A \otimes A^{op})$ such that

$$
  \Delta^\vee \otimes_{A^{op}} \Delta = id_A \in KK(A,A)
$$

and

$$
  \Delta^\vee \otimes_A \Delta = id_{A^{op}} \in KK(A^{op}, A^{op})
  \,.
$$

=--

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

For $A$ $B$ two Poincaré duality algebras, def. \ref{PDAlgebra\}, and for $f \colon A \to B$ a [[homomorphism]] between them, regarded as a morphism $f^\ast \colon B \to A$ in [[KK-theory]], the correspondung [[dual morphism]] $f! \colon A \to B$ is the one such that postcomposition in $KK$ with this corresponds to the [[Umkehr map]]/[[push forward in generalized cohomology]] in [[KK-theory]].

=--

For more on this see below at _[Properties -- K-Orientation and Umkehr mpas](#PropertiesKOrientationAndUmkehrMaps)_.

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

For $C^\ast$-algebras which are [[groupoid convolution algebras]] $C^\ast(\mathcal{G})$ the [[opposite algebra]] is [[Morita equivalence|Morita equivlant]] (since a [[groupoid]] $\mathcal{G}$ is [[equivalence of categories|equivalent]] to its [[opposite category|opposite groupoid]] $\mathcal{G}^{op}$, the equivalence being induced by the [[functor]] which sends a morphism to its [[inverse]]). But given a [[circle 2-bundle]] $\chi \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ the corresponding [[twisted groupoid convolution algebra]] is such that passing to the opposite corresponds to passing to the inverse twist $-\chi$.

=--

Therefore it makes sense to consider more generally

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

For $A$ a [[C*-algebra]] a _Poincaré dual_ for $A$ is a [[dual object]] $B \in C^\ast Alg \to KK$ in [[KK-theory]].

=--

Below in the [Proposition-Section](#PropertiesForCStarAlgebras) is discussed how under Poincaré-duality the [[twisted K-theory|twist]] changes.

## Properties

### For dg-Algebras

### For $C^\ast$-algebras
 {#PropertiesForCStarAlgebras}

#### Duals and twists
 {#PropertiesCAstDualsAndTwists}

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

Let $X$ be a [[closed manifold]] with [[spin^c-structure]]. Then there is a Poincaré duality [[isomorphism]]

$$
  K^\bullet(X) \simeq K_\bullet(X)
  \,.
$$

=--

For instance ([Connes, chapter 2.7, prop. 5](#Connes)).

(...) The relaton between Poincaré duality on algebras of functions and [[spin^c-structure]] is discussed in ([Connes, around p. 603](#Connes)). (...)

Notice that the [[obstruction]] to [[spin^c structure]] is the third [[integral Stiefel-Whitney class]] $W_3 \colon B SO \to B^2 U(1)$. If this does not vanish on a [[manifold]], then a Poincaré dual/[[dual object]] in [[KK-theory]] still exists, but is the same manifold equipped with a [[twisted K-theory|twist]] _shifted_ by $W_3(\tau_X)$, where $\tau_X$ denotes the ([[cotangent bundle|co]])[[tangent bundle]] of $X$.

+-- {: .num_defn #CStarAlgebraOf2BundleOnManifold}
###### Definition/Notation

For $X$ a ([[compact topological space|compact]]) [[manifold]] and $c \in H^3(X,\mathbb{Z})$ the class of a [[circle 2-bundle]]/[[bundle gerbe]] $\mathcal{G}$ on $X$, write

$$
  C_c(X) \in C^\ast Alg \to KK
$$

for the corresponding [[twisted groupoid convolution algebra]], the one whose [[operator K-theory]] is the $c$-[[twisted K-theory]] of $X$:

$$
  KK_\bullet(\mathbb{C}, C_c(X)) \simeq K_{\bullet + c}(X)
  \,.
$$

=--

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

Let $X$ be a [[compact topological space|compact]] [[manifold]] with [[tangent bundle]] $\tau_X$ and let $c \in H^3(X,\mathbb{Z})$ be a [[twisted K-theory|twist]]. Then the [[C*-algebra]] $C_{c}(X)$ of def. \ref{CStarAlgebraOf2BundleOnManifold} has a [[dual object]] in the [[full subcategory]] of [[KK-theory]] on [[separable C*-algebras]], given by

$$
  (C_c(X))^\vee
  \simeq
  C_{\frac{1}{c\otimes W_3(\tau_X)}}(X)
  \,,
$$

hence by the same manifold but with twist the inverse of the third [[integral Stiefel-Whitney class]] and the original twist.

The same remains true in $G$-equivariant KK-theory, for $G$ a [[locally compact topological space|locally compact]] [[topological group]].

=--

The non-equivariant case is in ([Brodzki-Mathai-Rosenberg-Szabo 06, section 7.3](#BrodzkiMathaiRosenbergSzabo06)) and the generalization to the equivariant case in ([Tu 06, theorem 3.1](#Tu06)) (where we use remark \ref{OppositeConvolutionAlgebraAndInverseTwist} in order to interpret the opposite twisted convolution algebra up to equivalence as inducing the inverse twist).

#### K-Orientation and Umkehr maps
 {#PropertiesKOrientationAndUmkehrMaps}

We discuss [[Umkehr maps]]/[[fiber integration in generalized cohomology]] in [[K-theory]] using Poincaré duality algebras / [[dual objects]] in [[KK-theory]].

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

Every homomorphism $f \colon A \to B$ between PD $C^\ast$-algebras is [[K-orientation|K-orientable in KK-theory]]. The [[K-orientation]] is given by the corresponding [[dual morphism]], hence the element $f! \colon B \to A$ given as the composite

$$
  f!
  \coloneqq
  \Delta^\vee_A \otimes_{A^{op}} f^{op} \otimes_{B^{op}} \Delta_B
  \,.
$$

=--

([BMRS 07, 3.3](#BMRS07))

More generally we have the following.

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

Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form

$$
  C_{i^\ast \chi}(Q)
  \stackrel{i^\ast}{\longleftarrow}
  C_{\chi}(X)
  \,,
$$

with notation as in def. \ref{CStarAlgebraOf2BundleOnManifold}. By prop. \ref{DualOfCompactManifoldWithTwist} the [[dual morphism]] is of the form

$$
  C_{\frac{1}{i^\ast \chi \otimes W_3(T Q)}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{1}{\chi \otimes W_3(T X)}}(X)
  \,.
$$

If we redefine the twist on $X$ to absorb this "quantum correction" as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is

$$
  C_{i^\ast \chi\frac{W_3(i^\ast T X)}{W_3(T Q)}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\chi}(X)
  \,,
$$

Postcomposition with this map in [[KK-theory]] now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-[[twisted K-theory]] of $Q$ to the $\chi$-[[twisted K-theory]] of $X$:

$$
  i_! \colon K_{\bullet + W_3(N Q) + i^\ast \chi}(Q) \to K_{\bullet +\chi}
  \,.
$$

This is the _[[twisted Umkehr map]]_ in this context.

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a [[D-brane]] [[worldvolume]], then $\chi$ would be the class of the [[background gauge field|background]] [[B-field]] and an element

$$
  [\xi] \in K_{\bullet + W_3(N Q) + i^\ast \chi}(Q)
$$

is called (the K-class of) a _[[Chan-Paton gauge field]]_ on the D-brane satisfying the [[Freed-Witten-Kapustin anomaly cancellation]] mechanism. (The orginal [[Freed-Witten anomaly cancellation]] assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin^c structure]] on $Q$.)
Finally its [[fiber integration|push-forward]]

$$
  [i_! \xi] \in K_{\bullet- \chi}(X)
$$

is called the corresponding _[[D-brane charge]]_.

=--

See ([Nuiten 13](#Nuiten13)).

## Examples

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

For $A = C_0(X)$ the [[algebra of functions]] on a [[compact topological space|compact]] [[complex manifold]] $X$, then $A$ is a PD algebra with fundamental class $\Delta$ in [[K-homology]] given by the [[Dolbeault operator]] on $X \times X$.

=--

([BMRS 07, example 3.2](#BMRS07))

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

For $A = C_0(X)$ the [[algebra of functions]] [[vanishing at infinity]] of a [[manifold]] $X$ with [[spin^c structure]]. Take $B = C_0(T^\ast X) \simeq_{KK} A^{op} \simeq A$. Then $\Delta$ constructed from the [[Dirac operator]] on the [[Clifford algebra bundle]] over $T^\ast X$ is a fundamental class.

=--

([BMRS 07, proof of theorem 2.9](#BMRS07))

## Related concepts

* [[fundamental class]]

* [[virtual fundamental class]]

* [[Poincaré duality complex]]

## References

### For graded associative algebras

* Pascal Lambrechts, Don Stanley, _Poincaré duality and commutative differential graded algebras_ ([arXiv:math/0701309](http://arxiv.org/abs/math/0701309))
 {#LambrechstStanley07}

### For $C^\ast$-algebras

For [[C*-algebras]]/in [[noncommutative topology]]:

* [[Henri Moscovici]], _Eigenvalue inequalities and Poincaré duality in noncommutative geometry_, Commun. Math. Phys. 184 , 3 (1997) 619

Chapter 6.4 $\beta$ (starting p. 601) in

* [[Alain Connes]], _[[Noncommutative Geometry]]_
 {#Connes}

Def. 2.1 in

* Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))
 {#BMRS07}

Duality including the [[twisted K-theory]] induced by [[twisted spin^c structure]] over [[manifolds]] is discussed in section 7 of

* Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_, Commun. Math. Phys. 277:643-706,2008 ([arXiv:hep-th/0607020](http://arxiv.org/abs/hep-th/0607020))
 {#BrodzkiMathaiRosenbergSzabo06}

and generalized to equivariant [[KK-theory]] in

* [[Jean-Louis Tu]], _Twisted K-theory and Poincaré duality_ ([arXiv:math/0609556](http://arxiv.org/abs/math/0609556))
 {#Tu06}

More on dual objects in KK is in

* [[Heath Emerson]], [[Ralf Meyer]], _Bivariant K-theory via correspondences_, Adv. Math. 225 (2010), 2883-2919 ([arXiv:0812.4949](http://arxiv.org/abs/0812.4949))

* [[Heath Emerson]], [[Ralf Meyer]], _Dualities in equivariant Kasparov theory_ ([arXiv:0711.0025](http://arxiv.org/abs/0711.0025))

Discussion of the [[twisted Umkehr map]] and the [[Freed-Witten-Kapustin anomaly]] in this context is in

* [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, master thesis, August 2013
 {#Nuiten13}

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[[!redirects Poincare duality algebra]]
[[!redirects Poincare duality algebras]]

[[!redirects Poincaré duality C*-algebra]]
[[!redirects Poincaré duality C*-algebras]]

[[!redirects Poincare duality C*-algebra]]
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Poincaré duality space

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## Definition

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

A [[topological space]] for which there is $d \in \mathbb{N}$ and a  class $[X] \in H_d(X)$ such that the [[cap product]] induces [[isomorphisms]]

$$
  (-) \cap [X] \;\colon\; H^\bullet(X) \stackrel{\simeq}{\to} H_{d-\bullet}(X)
$$

between [[ordinary cohomology]] and [[ordinary homology]] groups as indicated, is called a **Poincaré duality space**.

=--

If $X$ is moreover a [[CW-complex]] then this it is sometimes called a _[[Poincaré complex]]_ or even _Poincaré manifold_.

See at _[[Poincaré duality]]_ for more.

## References

* [[James Munkres]], *Duality in Manifolds*, Chapter 8 in: *Elements of Algebraic Topology*, Addison-Wesley (1984) [[pdf](https://people.dm.unipi.it/benedett/MUNKRES-ETA.pdf)]

In the general concext of [[spectral geometry]] ([[spectral triples]]):

* {#Connes95} [[Alain Connes]], page 10 of _Noncommutative geometry and reality_, J. Math. Phys. 36 (11), 1995 ([pdf](http://www.alainconnes.org/docs/reality.pdf))

[[!redirects Poincaré duality spaces]]

[[!redirects Poincare duality space]]
[[!redirects Poincare duality spaces]]


Poincaré group

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## Definition

The **Poincaré group** $ISO(d,1)$ is the [[isometry group]] of [[Minkowski spacetime]] of [[dimension]] $d+1$. The classical case is $ISO(3, 1)$, the [[group]]  of affine transformations on $\mathbb{R}^4$ which preserve the [[pseudo-Riemannian metric|Minkowski "metric"]], i.e., the group of maps $f: \mathbb{R}^4 \to \mathbb{R}^4$ of the form $f(\vec{x}) = M \vec{x} + \vec{b}$ such that

$$Q(f(\vec{x}) - f(\vec{y})) = Q(\vec{x} - \vec{y})$$

where the quadratic form $Q(t, x, y, z) = t^2 - x^2 - y^2 - z^2$ is often called the Minkowski "norm". The group elements are multiplied by composing maps.

The Poincaré group $G$ may also be described as a [[semidirect product]]

$$(\mathbb{R} \oplus \mathbb{R}^3) \rtimes O(1, 3)$$

where $O(1, 3)$, the **Lorentz group**, consists of all _linear_ transformations $L: \mathbb{R}^4 \to \mathbb{R}^4$ that preserve the Minkowski inner product of signature $(1, 3)$.

This is a linear [[algebraic group]] (e.g. [arXiv:0905.1217, p. 6-7](http://arxiv.org/abs/0905.1217)).

## Properties

### Basic structure of the Lorentz group

The [[Lorentz group]] is a 6-dimensional [[Lie group]]. It has four connected components; the [[connected space|connected component]] of the identity is called the **special Lorentz group** and is denoted $SO^+(1, 3)$.

* The superscript of $SO^+(1, 3)$ indicates that the group elements take the **forward light cone**, i.e., the set of vectors
$$\{v = (t, x, y, z): Q(v) \gt 0, t \gt 0\},$$
to itself, and the $S$ indicates of course group elements which have determinant 1 as $4 \times 4$ matrices. It easily follows that elements of $SO^+(1, 3)$ preserve orientation of the spatial component $\mathbb{R}^3$.

Passage between the four components of the full Lorentz group is effected by composing with a **time-reversal** operator

$$(t, x, y, z) \mapsto (-t, x, y, z)$$

and with a spatial inversion (or **parity-reversal**) operator

$$(t, x, y, z) \mapsto (t, -x, -y, -z)$$

The special Lorentz group (also called the **proper, orthochronous** Lorentz group: "orthochronous" here means the forward light cone is mapped to itself, and "proper" means orientation-preserving) may be analyzed further. The subgroup of $SO^+(1, 3)$ that fixes the unit time-like vector

$$u_t := (1, 0, 0, 0)$$

may be identified with the group of rotations $SO(3)$, since the restriction of the Minkowski norm to the spatial component $\mathbb{R}^3$ is minus the usual Euclidean norm, $-x^2 - y^2 - z^2$. This subgroup is of course 3-dimensional.

A general element $g \in SO^+(1, 3)$ may be decomposed uniquely in the form

$$g = \beta_v \circ \rho$$

where $\rho$ is a rotation and $\beta_v$ is a **boost** in the direction $v$ ($v \in S^2$ a unit spatial vector), mapping

$$u_t \mapsto \cosh(\beta)u_t + \sinh(\beta)v$$

$$v \mapsto \sinh(\beta)u_t + \cosh(\beta)v$$

for some parameter $\beta$ (called the **rapidity**), and acting as the identity on the spatial plane orthogonal to $v$. Thus a boost is described by a pair $(v, \beta)$, involving 3 parameters. (Warning: boosts do not compose to form a subgroup.) A boost can be thought of as a relativistic coordinate change from a "laboratory" frame of reference to the frame of reference of an observer moving inertially in the direction $v$ with speed $\tanh(\beta)$ (relative to the speed of light $c = 1$), as measured in the laboratory frame.

### Universal spin covering
 {#UniversalSpinCovering}

We discuss aspects of the [[Poincaré spinor group]].

The universal cover of $SO^+(1, 3)$ is a double cover (the **spin double cover**)

$$SL_2(\mathbb{C}) \to SO^+(1, 3)$$

constructed as follows: to each $x = (x_0, x_1, x_2, x_3)$ one associates a [[Hermitian matrix]]

$$X = \left(
\array{
x_0 + x_1 & x_2 + i x_3 \\
x_2 - i x_3 & x_0 - x_1
}
\right)
= x_0 \left(
\array{
1 & 0 \\
0 & 1
}
\right)
+ x_1 \left(
\array{
1 & 0 \\
0 & -1
}
\right)
+ x_2 \left(
\array{0 & 1 \\
1 & 0
}
\right)
+ x_3 \left(
\array{
0 & i \\
-i & 0
}
\right)
$$

whose determinant is the Minkowski norm of $x$. We thus identify $\mathbb{R}^4$ with the space $H$ of Hermitian matrices, and define an action of $SL_2(\mathbb{C})$ on $H$:

$$A \cdot X = A X A^*, A \in SL_2(\mathbb{C}), X \in H$$

Observe that $A X A^*$ belongs to $H$. Also, $det(A \cdot X) = det(X)$ since $A$ has determinant 1, so the action preserves the Minkowski norm. Therefore the action

$$SL_2(\mathbb{C}) \to GL(\mathbb{R}^4)$$

factors through the inclusion $O(1, 3) \hookrightarrow GL(\mathbb{R}^4)$. Furthermore, since $SL_2(\mathbb{C})$ is connected, the action $SL_2(\mathbb{C}) \to O(1, 3)$ factors through the connected component $SO^+(1, 3)$ of $O(1, 3)$. It is not hard to check that the kernel of the action is $\{I, -I\}$; therefore the map

$$SL_2(\mathbb{C}) \to SO^+(1, 3)$$

is an open homomorphism between connected Lie groups of the same dimension, and is therefore surjective. In this way, we have produced an explicit identification

$$SL_2(\mathbb{C})/\{\pm I\} \cong SO^+(1, 3)$$

which exhibits $SL_2(\mathbb{C})$ as a double cover of $SO^+(1, 3)$; another way to say it is that $SO^+(1, 3)$ is identified with the group $PSL_2(\mathbb{C})$ of complex [[Moebius transformation]]s. Finally, there is morphism of covering spaces

$$\array{
SU(2) & \hookrightarrow & SL_2(\mathbb{C}) \\
\pi \downarrow & & \downarrow \pi \\
SO(3) & \hookrightarrow & SO^+(1, 3)
}$$

Here the inclusions are homotopy equivalences and the left projection is a universal covering map (as $SU(2) \cong S^3$ is simply connected), therefore $SL_2(\mathbb{C})$ is also simply connected and the projection on the right is a universal covering map. This is the _spin double cover_; it is crucial for getting a correct mathematical description of [[fermion]]s in particle theory.

With regard to the inclusion maps above being homotopy equivalences, we remark in passing that the [[homogeneous space]]

$$SO^+(1, 3)/SO(3)$$

is identified with the space of boost maps; concretely, each coset of $SO^+(1, 3)/SO(3)$ is of the form $\beta_v SO(3)$ for a unique boost map $\beta_v$. Topologically, the space of boost maps is $\mathbb{R}^3$ or a $3$-ball, hence contractible, and from the long exact homotopy sequence applied to the fibration

$$SO(3) \to SO^+(1, 3) \to SO^+(1, 3)/SO(3)$$

we deduce that the inclusion $i: SO(3) \to SO^+(1, 3)$ is a homotopy equivalence.

Similarly the lift to the double covers $SU(2) \to SL_2(\mathbb{C})$ is a homotopy equivalence. Since $SU(2) \cong S^3$ is identified with the space of unit quaternions, $SU(2)$ and $SL_2(\mathbb{C})$ are simply connected and hence the respective universal covering spaces of $SO(3)$ and $SO^+(1, 3)$.

* **Remark:** Geometrically the space of boosts carries hyperbolic structure as well, in other words the space of boosts carries a structure of hyperbolic 3-space $H^3$.

### Lie algebra presentations

As for any Lie group, there are various mechanisms for describing the [[Lie algebra]]s of the Lorentz group and of the Poincaré group: by left-invariant [[vector field]]s, or by studying "infinitesimal generators" of 1-parameter subgroups, etc. We begin with the Lorentz group. See also [[Poincaré Lie algebra]].

In the vector field picture, one often chooses a basis of the [[Lorentz algebra]] consisting of six elements: the first three

$$M_{12} = x\partial_y - y\partial_x \qquad M_{23} = y\partial_z - z\partial_y \qquad M_{13} = z\partial_x - x\partial_z$$

describe rotational flows (around the $z$-, $x$-, and $y$-axes respectively), and the last three

$$M_{01} = x\partial_t + t\partial_x \qquad M_{02} = y\partial_t + t\partial_y \qquad M_{03} = z\partial_t + t\partial_z$$

describe hyperbolic or boost flows, (where the boosts are in the directions of the $x$-, $y$-, and $z$-axes, respectively). One may easily compute the commutators by hand and reproduce the standard gobbledygook formula given in physics texts:

$$[M_{\mu\nu}, M_{\rho\sigma}] = \eta_{\mu\rho} M_{\nu\sigma} - \eta_{\mu\sigma} M_{\nu\rho} - \eta_{\nu\rho} M_{\mu\sigma} + \eta_{\nu\sigma} M_{\mu\rho}$$

where lower-case Greek letters range over $0, 1, 2, 3$ and $\eta$ is the $4 \times 4$ matrix representing the Minkowski quadratic form.

_(I could be off by a sign here. It depends on whether $\eta$ has one plus and three minuses, or one minus and three pluses.)_

By integrating the vector fields, we obtain in each of these cases flows or 1-parameter subgroups, e.g.,

$$s \mapsto \exp(s M_{12}) = \left(
\array{
1 & 0 & 0 & 0 \\
0 & \cos(s) & -\sin(s) & 0 \\
0 & \sin(s) & \cos(s) & 0 \\
0 & 0 & 0 & 1
}
\right)$$

$\,$

$$s \mapsto \exp(s M_{01}) = \left(
\array{
\cosh(s) & \sinh(s) & 0 & 0 \\
\sinh(s) & \cosh(s) & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
}
\right)$$

Differentiating these maps at $s = 0$ gives matrix representations for the Lie algebra elements $M_{\mu\nu}$.

For the 10-dimensional Poincaré algebra, we need to give, in addition to infinitesimal generators for rotations and boosts, four more elements which generate translations. In the vector field picture, these Lie algebra elements are represented by

$$P_0 = \partial_t, P_1 = \partial_x, P_2 = \partial_y, P_3 = \partial_z$$

and these of course commute; the brackets with the $M_{\mu\nu}$ are

$$[M_{\mu\nu}, P_\rho] = \eta_{\mu\rho} P_\nu - \eta_{\nu\rho} P_\mu$$

Integration of the vector fields $P_\rho$ leads to the expected translations, e.g.,

$$\exp(s P_0)(t, x, y, z) = (t + s, x, y, z)$$

## The Poincare group in physics

The Poincaré group is basic to relativistic physics, since the fundamental principle of relativity is that physical laws are required to be invariant with respect to the action of the Poincaré group on spacetime. (There is some fine print here: in some cases, e.g., where physical laws are not invariant under space or time inversions, one must restrict to the action of the group $\mathbb{R}^4 \rtimes SO^+(1, 3)$. If one is dealing with fermions, one considers invariance with respect to an action of the universal cover $\mathbb{R}^4 \rtimes SU_2(\mathbb{C})$.)

Such physical laws may be classical or quantum, according to the description of physical states and observables in the theory. For example, in classical mechanics, pure states correspond to points in a symplectic phase space such as the cotangent bundle of Minkowski 4-space; in quantum mechanics, pure states are described by unit vectors in a suitable Hilbert space such as $L^2(\mathbb{R}^4)$. In either case, a relativistic theory will involve an action or representation of the Poincaré group, together with a structure which governs the dynamics of the theory, e.g., a Hamiltonian.

In the quantum case, a fundamental relativistic condition is that probability amplitudes $\langle \psi|\phi \rangle$, obtained by pairing an initial state $\phi$ with a final state $\psi$, are invariant under the action of the Poincaré group. This condition says

$$\langle \psi|\phi \rangle = \langle g \cdot \psi|g \cdot \phi \rangle$$

for every $g$ in the Poincaré group. Thus the representation of the Poincaré group on Hilbert space is required to be _unitary_. Due to the noncompactness of the Poincaré group, unitary representations on finite-dimensional Hilbert spaces are scarce; one must really pass to unitary representations on infinite-dimensional Hilbert spaces to get anything interesting.

In particular, an **elementary particle** in quantum physics is sometimes defined to be an [[unitary irreps of the Poincare group|irreducible unitary representation]] of the Poincaré group on $L^2(\mathbb{R}^4)$.

#### Unitary representations

* [[unitary irrep of the Poincare group]]

## Related entries

* [[Lorentzian geometry]]

* [[orthogonal group]], [[affine group]]

* [[Euclidean group]]

  * [[super Euclidean group]]

* [[super Poincaré group]]

* [[Poincaré Lie algebra]]

  * [[super Poincaré Lie algebra]]

[[!include table of orthogonal groups and related]]

[[!redirects Poincare group]]

[[!redirects Poincaré symmetry]]
[[!redirects Poincare symmetry]]

[[!redirects Poincaré symmetries]]
[[!redirects Poincare symmetries]]

[[!redirects Poincaré symmetry transformation]]
[[!redirects Poincare symmetry transformation]]

[[!redirects Poincaré symmetry transformations]]
[[!redirects Poincare symmetry transformations]]


Poincaré homology sphere

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{:toc}

## Idea

Poincaré's homology sphere is a [[closed manifold|closed]] [[3-manifold]] with the same [[homology]] as the [[3-sphere]] but with a [[fundamental group]] which is non-trivial.

## Realizations

...the [[coset space]] $SU(2)/2I$ of the [[special unitary group]] in [[dimension]] 2 by the [[binary icosahedral group]].

## References

* [[Manifold Atlas]], _[Poincaré's homology sphere](http://www.map.mpim-bonn.mpg.de/Poincaré's_homology_sphere)_

[[!redirects Poincare homology sphere]]


Poincaré lemma

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## Idea

The **Poincaré Lemma** in [[differential geometry]] and [[complex analytic geometry]] asserts that "every [[differential form]] $\omega$ which is closed, $d_{dR}\omega = 0$, is _locally_ exact, $\omega|_U = d_{dR}\kappa$."

In more detail: if $X$ is [[contractible topological space|contractible]] then for every closed [[differential form]] $\omega \in \Omega^k_{cl}(X)$ with $k \geq 1$ there exists a differential form $\lambda \in \Omega^{k-1}(X)$ such that

$$
  \omega
  \,=\,
  d_{dR} \lambda
  \,.
$$

Moreover, for $\omega$ a smooth family of closed forms, there is a smooth family of $\lambda$s satisfying this condition.

This statement has several more abstract incarnations. One is that it says that on a [[Cartesian space]] (or a complex [[polydisc]]) the [[de Rham cohomology]] (the [[holomorphic de Rham complex|holomorphic de Rham cohomology]]) vanishes in positive degree.

Still more abstractly this says that the canonical morphisms of [[sheaves of chain complexes]]

$$
  \mathbb{R} \to \Omega^\bullet_{dR}
$$

$$
  \mathbb{C} \to \Omega^\bullet_{hol}
$$

from the [[locally constant sheaf]] on the [[real numbers]] (the [[complex numbers]]) to the [[de Rham complex]] ([[holomorphic de Rham complex]]) is a [[stalk]]-wise [[quasi-isomorphism]] -- hence an [[equivalence]] in the [[derived category]] and hence induce an equivalence in [[hypercohomology|hyper]]-[[abelian sheaf cohomology]]. (The latter statement _fails_ in general in complex [[algebraic geometry]], see ([Illusie 12, 1.](#Illusie12)) and see also at _[[GAGA]]_.) (A variant of such [[resolutions]] of constant sheaves for the case over [[Klein geometries]] are [[BGG resolutions]].)

The Poincaré lemma is a special case of the more general statement that the pullbacks of differential forms along [[homotopy|homotopic]] [[smooth function]] are related by a [[chain homotopy]].

## Statement
 {#Statement}

### Over contractible domains

\begin{proposition}
\label{HomotopyToChainHomotopy}
Let

* $f_0, f_1 \,\colon\, X \to Y$ be a [[pair]] of [[smooth functions]] between [[smooth manifolds]]

* $\Psi \colon [0,1] \times X \to Y$ a [[smooth homotopy]] between them.

Then there is a [[chain homotopy]] between the induced operations of [[pullback of differential forms]]:

$$
  f_0^*, f_1^*
  \;\colon\;
  \Omega^\bullet(Y) \to \Omega^\bullet(X)
$$

on the [[de Rham complexes]] of $X$ and $Y$, an explicit formula for which is given by the following "[[homotopy operator]]":

$$
  \psi
   \;\colon\;
  \omega \mapsto
  \Big(
    x
     \mapsto
    \textstyle{\int}_{[0,1]}
    \big(
      \iota_{\partial_t} \Psi^*\omega)(x)
    \big)
    d t
  \Big)
  \,.
$$

In particular, $f_0^*$ and $f_1^*$ coincide on [[de Rham cohomology]]:

$$
  H_{dR}^\bullet(f_0^*)
    \,\simeq\,
  H_{dR}^\bullet(f_1^*)
  \,.
$$
\end{proposition}

Here $\iota_{\partial t}$ denotes [[tensor contraction|contraction]] (cf. *[[Cartan calculus]]*) with the canonical [[vector field]] tangent to $[0,1]$, and the [[integration]] is that of functions with values in the [[vector space]] of differential forms.

\begin{proof}
\label{ProofOfHomotopyToChainHomotopy}

We compute as follows:

$$
  \begin{array}{l}
    \mathrm{d}_{X} \psi(\omega)
    +
    \psi( \mathrm{d}_Y \omega )
    \\
    \;=\;
    \textstyle{\int}_{[0,1]}
      \mathrm{d}_{X}
      \iota_{\partial_t} \Psi^*(\omega)
    d t
    +
    \textstyle{\int}_{[0,1]}
      \iota_{\partial_t} \Psi^*(\mathrm{d}_Y \omega)
    d t
    \\
    \;=\;
    \textstyle{\int}_{[0,1]}
      \mathrm{d}_{X}
      \iota_{\partial_t} \Psi^*(\omega)
    d t
    +
    \textstyle{\int}_{[0,1]}
      \iota_{\partial_t}
      \big(\mathrm{d}_X + \mathrm{d}_{[0,1]}\big)
      \Psi^*( \omega)
    d t
    \\
    \;=\;
    \textstyle{\int}_{[0,1]}
      \iota_{\partial_t}
      \mathrm{d}_{[0,1]}
      \Psi^*(\omega)
    d t
    \\
    \;=\;
    \textstyle{\int}_{[0,1]}
      \mathrm{d}_{[0,1]}
      \Psi^*(\omega)
    \\
    \;=\;
    \Psi_1^\ast(\omega)
    -
    \Psi_0^\ast(\omega)
    \\
    \;=\;
    f_1^\ast \omega - f_0^\ast \omega
    \,,
  \end{array}
$$

where in the second step we used that [[exterior differential]] commutes with [[pullback of differential forms]] ([this Prop.](pullback+of+a+differential+form#CompatibilityWithDeRhamDifferential)), and in the last step the [[Stokes theorem]].
\end{proof}

The **Poincaré lemma** proper is the special case of this statement for the case that $f_2 = const_y$ is a function constant on a point $y \in Y$:

\begin{corollary}
If a [[smooth manifold]] $X$ admits a smooth contraction

$$
  \array{
    X
    \\
    \downarrow^{\mathrlap{(id,0)}} & \searrow^{\mathrlap{id}}
    \\
    X \times [0,1] & \stackrel{\Psi}{\to} & X
    \\
    \uparrow^{\mathrlap{(id,1)}} & \nearrow_{\mathrlap{const_x}}
    \\
    X
  }
$$

then the [[de Rham cohomology]] of $X$ is concentrated on the
ground field in degree 0. Moreover, for $\omega$ any closed
form on $X$ in positive degree, an explicit formula for a form
$\lambda$ with $d \lambda = \omega$ is given by

$$
  \lambda
  \,=\,
  - \textstyle{\int}_{[0,1]}
  \iota_{\partial_t}\Psi^*(\omega)
  d t
  \,.
$$
\end{corollary}

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

This is the special case of Prop. \ref{HomotopyToChainHomotopy} where $f_0^\ast = id$ and $f_1^* = 0$ in positive degree.

=--

### Over $n$-connected domains
 {#OverNConnectedDomains}

More generally, the conclusion of the Poincaré lemma for differential forms of bounded degree $\leq n$ follows already on [[n-connected topological spaces|$n$-connected spaces]] (for instance by combining the [[Hurewicz theorem]] first with the [[universal coefficient theorem#UniversalCoefficientTheoremInRationalCohomology|universal coefficient theorem]] and then with the [[de Rham theorem]]).

Explicitly:

\begin{proposition}
\label{ForOneFormsOnSimplyConnectedDomain}
  On a [[simply-connected topological space|simply-connected]] (i.e.: 1-connected) [[smooth manifold]], a closed [[differential 1-form]] $\omega$ is exact, with potential function given at $x \in X$ by the [[integration of differential forms|integral]] of $\omega$ from any fixed base point along any smooth path to $x$.
\end{proposition}

This follows locally for instance by the fiberwise Stokes theorem ([here](Stokes+theorem#StokesTheoremForFiberIntegration)) and then globally due to the independence of the choice of path, by the assumption of simple-connectivity and the plain [[Stokes theorem]].

Textbook accounts which make this explicit include [do Carmo 1994, Prop. 3 (p. 24) in §3](#Carmo94). Exposition is also in [Armstrong 2017](https://unapologetic.wordpress.com/2011/12/17/simply-connected-spaces-and-cohomology/).

## Related concepts

* **Poincaré lemma**

  * [[Kähler potential]]

* [[Stokes theorem]]

* [[de Rham theorem]]

* [[Hodge theorem]]

* [[variational sequence]]

## References

Textbook account:

* {#Carmo94} [[Manfredo P. do Carmo]], §4.3 of *Differential Forms and Applications*, Springer (1994) [[doi:10.1007/978-3-642-57951-6](https://doi.org/10.1007/978-3-642-57951-6)]

Course notes:

* [[Daniel Litt]], _The Poincaré lemma and de Rham cohomology_ ([pdf](http://math.stanford.edu/~dlitt/exposnotes/poincare_lemma.pdf))

Discussion in [[complex analytic geometry]]:

* {#Illusie12} [[Luc Illusie]], _Around the Poincaré lemma, after Beilinson_, talk notes 2012 ([pdf](https://www.imo.universite-paris-saclay.fr/~luc.illusie/derived-deRham3a1.pdf))

following

* {#Beilinson12} [[Alexander Beilinson]], _$p$-adic periods and de Rham cohomology_, J. of the AMS 25 (2012), 715-738

[[!redirects Poincaré lemma]]
[[!redirects Poincare lemma]]


Poincaré Lie algebra

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## Idea

The _Poincaré Lie algebra_ $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ is the [[Lie algebra]] of the [[isometry group]] of [[Minkowski spacetime]]: the [[Poincaré group]]. This happens to be the [[semidirect product]] of the [[special orthogonal Lie algebra]] $\mathfrak{so}(d-1,1)$ with the the abelian translation Lie algebra $\mathbb{R}^{d-1,1}$.

## Definition

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

For $d \in \mathbb{N}$, write $\mathbb{R}^{d-1,1}$ for [[Minkowski spacetime]], regarded as the [[inner product space]] whose underlying [[vector space]] is $\mathbb{R}^d$ and equipped with the [[bilinear form]] given in the canonical [[linear basis]] of $\mathbb{R}^d$ by

$$
  \eta \coloneqq diag(-1,+1,+1, \cdots, +1)
  \,.
$$

The [[Poincaré group]] $Iso(\mathbb{R}^{d-1,1})$ is the [[isometry group]] of this inner product space. The _Poincaré Lie algebra_ $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ is the [[Lie algebra]] of this [[Lie group]] (its [[Lie differentiation]])

$$
  \mathfrak{iso}(\mathbb{R}^{d-1,1})
  \coloneqq
  Lie(Iso(\mathbb{R}^{d-1,1}))
  \,.
$$

=--

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

The [[Poincaré group]] is the [[semidirect product group]]

$$
  Iso(\mathbb{R}^{d-1,1})
    \simeq
  \mathbb{R}^{d-1,1} \rtimes O(d-1,1)
$$

of the [[Lorentz group]] $O(d-1,1)$ (the group of [[linear map|linear]] [[isometries]] of [[Minkowski spacetime]]) with the $\mathbb{R}^d$ regarded as the [[translation group]] along itself, via the defining [[action]].

Accordingly, the Poincaré Lie algebra is the [[semidirect product Lie algebra]]

$$
  \mathfrak{iso}(\mathbb{R}^{d-1,1})
  \simeq
  \mathbb{R}^{d-1,1}
    \rtimes
  \mathfrak{so}^+(d-1,1)
$$

of the abelian Lie algebra on $\mathbb{R}^d$ with the (orthochronous) [[special orthogonal Lie algebra]] $\mathfrak{so}(d-1,1)$.

=--

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

For $\{P_a\}$ the canonical [[linear basis]] of $\mathbb{R}^d$, and for $\{L_{a b} = - L_{b a}\}$ the corresponding canonical basis of $\mathfrak{so}(d-1,1)$, then the [[Lie bracket]] in $\mathfrak{iso}(\mathbb{R}^{d-1,1})$ is given as follows:

$$
  \begin{aligned}
    [P_a, P_b] & = 0
    \\
    [L_{a b}, L_{c d}]
     & =
      \eta_{d a} L_{b c}
      -\eta_{b c} L_{a d}
      +\eta_{a c} L_{b d}
      -\eta_{d b} L_{a c}
     \\
    [L_{a b}, P_c] & = \eta_{a c} P_b -\eta_{bc} P_a
  \end{aligned}
$$

=--

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

Since [[Lie differentiation]] sees only the [[connected component]] of a [[Lie group]], and does not distinguish betwee a Lie group and any of its discrete [[covering spaces]], we may equivalently consider the Lie algebra of the [[spin group]] $Spin(d-1,1) \to SO^+(d-1,1)$ (the double cover of the [[proper orthochronous Lorentz group]]) and its [[action]] on $\mathbb{R}^{d-1,1}$.

By the discussion at _[[spin group]]_, the Lie algebra of $Spin(d-1,1)$ is the Lie algebra spanned by the [[Clifford algebra]] [[bivectors]]

$$
  L_{a b} \leftrightarrow \Gamma_a \Gamma_b
$$

and its [[action]] on itself as well as on the vectors, identified with single Clifford generators

$$
  P_a \leftrightarrow \Gamma_a
$$

is given by forming [[commutators]] in the [[Clifford algebra]]:

$$
  [L_{a b}, P_c]
    \leftrightarrow
  \tfrac{1}{2}[\Gamma_{a b}, \Gamma_c  ]
$$

$$
  [L_{a b}, L_{c d}]
  \leftrightarrow
  \tfrac{1}{2}[\Gamma_{a b}, \Gamma_{c d}  ]
  \,.
$$

Via the Clifford relation

$$
  \Gamma_a \Gamma_b + \Gamma_b \Gamma_a = -2 \eta_{a b}
$$

this yields the claim.

=--

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

Dually, the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{iso}(\mathbb{R}^{d-1})$ is generated from $\mathbb{R}^{d,1}$ and $\wedge^2 \mathbb{R}^{d,1}$. For $\{t_a\}$ the standard basis of $\mathbb{R}^{d-1,1}$ we write $\{\omega^{a b}\}$ and $\{e^a\}$ for these generators. With $(\eta_{a b})$ the components of the [[Minkowski metric]] we write

$$
  \omega^{a}{}_b \coloneqq \omega^{a c}\eta_{c b}
  \,.
$$

In terms of this the CE-differential that defines the Lie algebra structure is

$$
  d_{CE} \colon \omega^{a b} = \omega^a{}_c \wedge \omega^{c b}
$$

$$
  d_{CE} \colon e^a \mapsto \omega^{a}{}_b  \wedge t^b
$$

=--

## Properties

### Cohomology

We discuss some elements in the [[Lie algebra cohomology]] of $\mathfrak{iso}(d-1,1)$.

The canonical degree-3 $\mathfrak{so}(d-1,1)$-cocycle is

$$
  \omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a
  \in
  CE(\mathfrak{iso}(d-1,1))
  \,.
$$

The _volume cocycle_ is the [[volume form]]

$$
  vol = \epsilon_{a_1 \cdots a_{d}} e^{a_1} \wedge \cdots \wedge e^{a_d} \in CE(\mathfrak{iso}(d-1,1))
  \,.
$$
{#VolumeCocycle}

### Invariant polynomials and Chern-Simons elements

With the basis elements $(e^a, \omega^{a b})$ as above, denote the shifted generators of the [[Weil algebra]] $W(\mathfrak{iso}(d-1,1))$ by $\theta^a$ and $r^{a b}$, respectively.

We have the [[Bianchi identity]]

$$
  d_W : r^{a b}
   \mapsto
   \omega^{a c} \wedge R_c{}^d
   -
   R^{a c} \wedge \omega_c{}^b
$$

and

$$
  d_W : \theta^a
   \mapsto
    \omega^a{}_b \theta^b
    -
    R^{a}{}_b e^b
   \,.
$$

The element $\eta_{a b} \theta^a \wedge \theta^b \in W(\mathfrak{iso}(d-1,1))$ is an [[invariant polynomial]]. A [[Chern-Simons element]] for it is $cs = \eta_{a b} e^a \wedge \theta^b$. So this transgresses to the trivial cocycle.

Another invariant polynomial is $r^{a b} \wedge r_{a b}$. This is the [[Killing form]] of $\mathfrak{so}(d-1,1)$. Accordingly, it transgresses to a multiple of $\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a$.

This is the first in an infinite series of Pontryagin invariant polynomials

$$
  P_n
  :=
  r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3}
  \wedge \cdots \wedge r^{a_n}{}_{a_1}
  \,.
$$

There is also an infinite series of mixed invariant polynomials

$$
  C_{2n + 2}
  :=
  \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3}
  \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge \theta^{a_n}
  \,.
$$

[[Chern-Simons element]]s for these are

$$
  B_{2n + 1}
  :=
  \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3}
  \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n}
  \,.
$$

### Lie algebra valued forms

A [[Lie algebra-valued form]] with values in $\mathfrak{iso}(d-1,1)$

$$
  \Omega^\bullet(X) \leftarrow W(\mathfrak{iso}(d-1,1)) : (E,\Omega)
$$

is

* a [[vielbein]] $E$ on $X$;

* a "[[spin connection]]" $\Omega$ on $X$.

The [[curvature]] 2-form $(T, R)$ consists of

* the [[torsion of a metric connection|torsion]] $T = d E + [\Omega \wedge E]$;

* the [[Riemannian curvature]] $R = d \Omega + [\Omega \wedge \Omega]$.

If the torsion vanishes, then $\Omega$ is a [[Levi-Civita connection]] for the [[metric]] $E^a \otimes E^b \eta_{a b}$ defined by $E$.

The [[volume form]] is the image of the [volume cocycle](#VolumeCocycle)

$$
  \Omega^\bullet(X)
  \stackrel{(E,\Omega)}{\leftarrow}
  W(\mathfrak{iso}(d-1,1))
  \stackrel{vol}{\leftarrow}
  W(b^{d-1} \mathbb{R})
  :
  vol(E)
  \,.
$$

We have

$$
  vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d}
  \,.
$$

If the torsion vanishes, this is indeed a closed form.

## Related structures

* [[special orthogonal Lie algebra]]

* [[super Poincaré Lie algebra]]

[[!redirects Poincaré Lie algebras]]

[[!redirects Poincare Lie algebra]]
[[!redirects Poincaré Lie algebra]]


Poincaré line bundle

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## Idea

The canonical [[line bundle]] on the [[product]] of an [[abelian group]] ([[abelian variety]]) and its dual [[Picard scheme]]. Alternatively its [[sheaf]] of [[sections]]: the _Poincaré sheaf_.

## References

The idea is briefly summarized in

* [[Akhil Mathew]], section 10 of _[Duality for abelian varieties](http://amathew.wordpress.com/2013/03/21/duality-for-abelian-varieties/)_

Lecture notes include

* John Meth, Kelly McKinnie, _The Poincaré bundle_ ([pdf](https://www.ma.utexas.edu/users/gfarkas/teaching/abvar/projectmeth.pdf))

Textbook accounts include

* {#Polsihchuk03} [[Alexander Polishchuk]], section 16.3 of _Abelian varieties, Theta functions and the Fourier transform_, Cambridge University Press (2003) ([pdf](http://math1.unice.fr/~beauvill/pubs/poli.pdf))

[[!redirects Poincaré line bundles]]

[[!redirects Poincaré bundle]]
[[!redirects Poincaré bundles]]

[[!redirects Poincaré sheaf]]
[[!redirects Poincaré sheafs]]

[[!redirects Poincare line bundle]]
[[!redirects Poincare line bundles]]

[[!redirects Poincare bundle]]
[[!redirects Poincare bundles]]

[[!redirects Poincare sheaf]]
[[!redirects Poincare sheafs]]


Poincaré sphere

#Contents#
* table of contents
{:toc}

## Idea

The **Poincaré sphere** is a "fake" 3-dimensional sphere: it is a smooth 3-dimensional [[manifold]] which has the same [[homology]] as the usual [[3-sphere]], but which is not [[homeomorphism|homeomorphic]] to it.

Specifically, it is obtained as $SO(3)/I$ where $I$ is the [[icosahedral group]] of rotations (the subgroup of rotations which [[stabilizer subgroup|stabilize]] an [[icosahedron]]).

The [[universal cover]] of the Poincaré sphere $S$ is the standard 3-sphere. This is exhibited in the following diagram:

$$\array{
B & \hookrightarrow & SU(2) \cong S^3 & \stackrel{p}{\to} & S \\
\downarrow & & \downarrow \pi & & \downarrow \cong \\
I & \hookrightarrow & SO(3) & \to & SO(3)/I
}$$

where the double cover $\pi$ is the universal cover of $SO(3)$, and the double cover $B$ of $I$ is called the _binary icosahedral group_. It follows that $\pi_1(S) \cong B$, and since $B$ is a [[perfect group]], its abelianization is trivial, and therefore $p$ induces an isomorphism in first homology. It induces an isomorphism in homotopy groups $\pi_i$ for $i \geq 2$ by an easy computation involving the long exact homotopy sequence applied to the top row.

## Related concepts

* [[ADE classification]]

## References

* _[The Poincare sphere](http://sites.psu.edu/surgeryforamateurs/2013/09/03/the-poincare-sphere/)_

[[!redirects Poincare sphere]]


Poincaré spinor group

## Idea

The [[spin group]]-extension of the [[Poincaré group]].

For the moment see at _[Poincare group -- Universal spin covering](http://ncatlab.org/nlab/show/Poincare+group#UniversalSpinCovering)_.

## Related concepts

[[!include table of orthogonal groups and related]]

[[!redirects Poincare spinor group]]

[[!redirects Poincaré spin group]]
[[!redirects Poincare spin group]]


Poincaré–Hopf theorem

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## Idea

The Poincaré–Hopf theorem says that for any [[vector field]] $v \in \Gamma(T X)$ with a [[finite set]] of isolated vanishing points $\{x_i\}$ on an [[orientation|orientable]] [[compact topological space|compact]] [[differential manifold]] $X$, the [[sum]] over the $x_i \in X$ of the [[degree of a continuous function|degrees]] of the vector in the vicinity of these points, regarded as [[cohomotopy]] classes

$$
  v/{\vert v\vert}_{\vert x_i}
  \;\colon\;
  \partial D_{x_i} \longrightarrow S(T_{x_i} X)
$$

and called the _Poincaré–Hopf [[index]]_ of $f$ at $x_i$

$$
  ind_{x_i}(v)
  \;\coloneqq\;
  deg\big(  v/{\vert v\vert}_{\vert {x_i} } \big)
$$

is given by the [[Euler characteristic]], hence by the value of the [[Euler class]] on the [[tangent bundle]]:

$$
  \chi(X)
  \coloneqq
  \chi[T X]
  \;=\;
  \underset{
    {\text{isolated zero}}
    \atop
    { x_i }
  }{\sum}
  ind_{x_i}(v)
  \,.
$$

\begin{center}
\begin{imagefromfile}
  "file_name": "PHIndices.jpg",
  "width": 570
\end{imagefromfile}
\end{center}

In particular, the existence of a nowhere vanishing [[vector field]] (for which the above [[sum]] is empty) implies that the [[Euler characteristic]] vanishes.

The converse also holds: If the Euler characteristic of a closed manifold vanishes, then a nowhere vanishing vector field exists (e.g. [MO:a/47553/58526](https://math.stackexchange.com/a/47553/58526)).

Also: Given any connected closed manifold, the [[complement]] of any [[positive number]] of [[disjoint union|disjoint]] [[open balls]] carries a nowhere vanishing vector field (namely: choose any vector field with isolated zeros on the original manifold and then deform continuously to push all these zeros into inside the open balls). Smoothly scaling this vector field to zero towards the centers of these open balls gives a vector field on the original manifold which may be used in the PH-theorem.

## Related concepts

* [[Hopf degree theorem]], [[equivariant Hopf degree theorem]]

* [[Pontrjagin theorem]], [[equivariant Pontrjagin theorem]]

* [[Gauss-Bonnet theorem]]

* [[twisted cohomotopy]]

## References

Named after [[Henri Poincaré]] and [[Heinz Hopf]].

Textbook accounts:

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], Chapter 11 of  _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

* {#DubrovinNovikovFomenko85} B. A. Dubrovin, [[S. P. Novikov]], A. T. Fomenko, section 15.2 of _Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds_, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985 ([doi:10.1007/978-1-4612-1100-6](https://link.springer.com/book/10.1007/978-1-4612-1100-6))

* [[John Milnor]], Chapter 6 of: _Topology from the differential viewpoint_, Princeton University Press, 1997. ([ISBN:9780691048338](https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf))

* {#Walschap04} Gerard Walschap, chapter 6.7 of _Metric Structures in Differential Geometry_, Graduate Texts in Mathematics, Springer 2004

Review:

* Alex Wright, Kael Dixon, _The Poincaré–Hopf theorem_ ([pdf](https://pdfs.semanticscholar.org/cd21/dbffb8cbc3a3636c40a58cb921789b0eaac9.pdf))

* Ariel Hafftka, _Differential topology and the Poincaré–Hopf theorem_ ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Hafftka.pdf))

See also:

* Wikipedia, _[Poincaré–Hopf theorem](https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem)_

Discussion in a broader perspective of [[K-theory]] and [[index theorems]]:

* Omar Mohsen, _Poincaré–Hopf theorem and groupoids_ ([pdf](https://webusers.imj-prg.fr/uploads//omar.mohsen/Poincare-Hopf.pdf), [[MohsenPHIndex.pdf:file]])

A comment on the version for complex vector fields is in

* Howard Jacobowitz, _Non-vanishing complex vector fields and the Euler characteristic_ ([arXiv:0901.0893](https://arxiv.org/abs/0901.0893))

Generalization to [[orbifolds]]:

* Christopher Seaton, _Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary_, Differential Geometry and its Applications
Volume 26, Issue 1, February 2008, Pages 42-51 ([doi:10.1016/j.difgeo.2007.11.002](https://doi.org/10.1016/j.difgeo.2007.11.002))

[[!redirects Poincaré-Hopf theorem]]
[[!redirects Poincare–Hopf theorem]]
[[!redirects Poincare-Hopf theorem]]

[[!redirects Poincaré–Hopf index theorem]]
[[!redirects Poincaré-Hopf index theorem]]
[[!redirects Poincare–Hopf index theorem]]
[[!redirects Poincare-Hopf index theorem]]


point

# Points
* table of contents
{: toc}

## The abstract point

The **point** is what is shown here:

$$\bullet$$

As a [[category]], we can interpret this as a category with a single [[object]] $\bullet$ and a single [[morphism]] (the [[identity morphism]] on the object, which is not shown in the picture since it is automatic).  This can be generalised recursively to [[higher category theory|higher categories]]; as an $(n+1)$-[[n-category|category]], the point consists of a single object $\bullet$ whose endomorphism $n$-category is the point (now understood as an $n$-category).  Of course, to make this work, the point must be a [[symmetric monoidal category|symmetric monoidal]] $n$-category at each stage, but it is (in a unique way).  In the limit, the point can even be understood as an $\infty$-[[infinity-category|category]], with a unique $j$-morphism for each $j \geq 0$ (each of which is an identity for $j \gt 0$).

In the other direction, the point is a [[singleton]] [[set]] with a unique element $\bullet$.  It can also be seen as a [[truth value]] that is [[true]].  It can even be understood as the $(-2)$-[[(-2)-category|category]].

As a [[topological space]], the _[[point space]]_ is the usual point from [[geometry]]: that which has no part.  In more modern language, we might say that it has no *structure* ---except that something exists.  (So it is *not* [[empty space|empty]]!)  This is consistent with the preceding paragraphs using the interpretation of a topological space as an $\infty$-[[infinity-groupoid|groupoid]].  (But up to [[homotopy equivalence]], any [[contractible space]] qualifies as a point.)

In all of the above, the point can be seen as a [[terminal object]] in an appropriate category (or $\infty$-category).  However, you can also see it as a [[zero object|null object]] in a category of [[pointed objects]].  (Of course, it\'s always true that a terminal object $1$ in $C$ becomes a null object in $1/C$, but the dual argument also holds, so the question is which is the primary picture.)

But perhaps the point is best seen as the unique object in itself:
$$ \bullet = \{\bullet\} ,$$
an equation that makes sense as a definition in the theory of ill-founded [[pure sets]].  Another possible definition (this time well-founded) in pure set theory is that the point is $\{\emptyset\}$, but this doesn\'t capture the picture that we get from higher category theory: the $(-1)$-[[(-1)-category|category]] ([[truth value]]) of the $(-2)$-category (the point) is [[true]] (which is also the point), the $0$-[[0-category|category]] (set) of the true truth value is [[generalized the|the]] [[singleton]] (which is also the point), the $1$-[[1-category|category]] (category) of the singleton (and *all* of its [[endofunctions]]!) is the [[terminal category]] (which is also the point), and so on.  That is:
$$ \bullet \in \bullet \in \bullet \in \bullet \in \cdots .$$

## Concrete points

The term 'point' is often used for a [[global element]]; that is the meaning, for example, in the sense of a [[point of a topos]] or a [[point of a locale]].  The connection is that a global element of $X$ is a map from the point to $X$.  So one may describe the point above as the *abstract* point, while a global element is a *concrete* point.

## Related concepts

### As homotopy types

[[!include homotopy n-types - table]]

### As directed homotopy types

* [[terminal category]]

### As geometric objects

* [[geometric point]]

* [[generalized element]]

* [[infinitesimally thickened point]]

* [[superpoint]]

[[!redirects point]]
[[!redirects points]]

[[!redirects abstract point]]
[[!redirects abstract points]]

[[!redirects concrete point]]
[[!redirects concrete points]]


point of a locale

A __point__ of a [[locale]] $X$ is a [[continuous map]] (in the sense of locales) to $X$ from the abstract [[point]] (seen as a locale whose corresponding [[frame]] is the frame of [[truth values]]).

If $X^*$ is the [[frame]] that corresponds to $X$, then a point of $X$ is the same as a frame [[homomorphism]] from $X^*$ to the frame of truth values.  This is the same as a [[completely prime filter]] in $X^*$.

A point of $X$ is the same as a point (in the usual sense) of the [[topological space]] $pt(X)$; that is, the [[underlying set]] of $pt(X)$ is the set of points (in the sense above) of $X$.  (Thus, we call $pt(X)$ the _space of points_ of $X$.)  Conversely, if $S$ is a [[topological space]], then every point of $S$ determines a point of the locale $op(S)$ of opens of $S$.  This map $S \to pt(op(S))$ (which is a [[continuous map]] of topological spaces) is [[injection|injective]] iff $S$ is $T_0$ (see [[separation axioms]]); it is a [[homeomorphism]] iff $S$ is [[sober space|sober]].

[[!redirects point of a locale]]
[[!redirects points of a locale]]
[[!redirects points of locales]]
[[!redirects point of the locale]]
[[!redirects points of the locale]]
[[!redirects points of the locales]]


point of a topos

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

# The points of a topos #
* table of contents
{: toc}

## Definition ##

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

A **point $x$ of a [[topos]] $E$** is a [[geometric morphism]]

$$
  x : Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} E
$$

from the base topos [[Set]] to $E$.

For $A \in E$ an object, its [[inverse image]] $x^* A \in Set$ under such a point is called the [[stalk]] of $A$ at $x$.

If $x$ is given by an [[essential geometric morphism]] we say that it is an **essential point** of $E$.

=--

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

Since [[Set]] is the [[terminal object]] in the category [[Topos|GrothendieckTopos]] of [[Grothendieck topos]]es, for $E$ a [[sheaf topos]] this is a [[global element]] of the topos.

Since $Set = Sh(*)$ is the [[category of sheaves]] on the one-point [[locale]], the notion of point of a topos is indeed the direct analog of a point of a [[locale]] (under [[localic reflection]]).

=--

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

A [[topos]] is said to have **enough points** if isomorphy can be tested [[stalk]]wise, i.e. if the
[[inverse image]] functors from all of its points are jointly [[conservative functor|conservative]].

More explicity: $E$ has enough points if for any morphism $f : A \to B$, we have that if for every point $p$ of $E$, the morphism of [[stalks]] $p^* f : p^* A \to p^* B$ is an [[isomorphism]], then $f$ itself is an isomorphism.

=--

## Properties

### In presheaf toposes

+-- {: .num_prop}

###### Proposition

For $C$ a [[small category]], the points of the [[presheaf topos]] $[C^{op}, Set]$ are the [[flat functor]]s
$C \to Set$:

there is an [[equivalence of categories]]

$$
  Topos(Set, [C^{op}, Set])
  \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}}
  FlatFunc(C,Set)
  \,.
$$

=--

This appears for instance as ([MacLaneMoerdijk, theorem VII.5.2](#MacLaneMoerdijk)).

### In localic sheaf toposes

For the special case that $E = Sh(X)$ is the [[category of sheaves]] on a [[category of open subsets]] $Op(X)$ of a [[sober topological space|sober]] [[topological space]] $X$ the notion of topos points comes from the ordinary notion of points of $X$.

For notice that

* $Set = Sh(*)$ is simply the [[topos]] of [[sheaf|sheaves]] on a one-[[point]] [[topological space|space]].

* [[geometric morphisms]] $f : Sh(Y) \to Sh(X)$ between [[category of sheaves|sheaf topoi]] are in a bijection with continuous functions of topological spaces $f : Y \to X$ (denoted by the same letter, by convenient abuse of notation); for this to hold $X$ needs to be sober.

It follows that for $E = Sh(X)$ points of $E$ in the sense of points of topoi are in bijection with the ordinary points of $X$.

The action of the [[direct image]] $x_* : Set \to Sh(X)$ and the [[inverse image]] $x^* : Sh(X) \to Set$ of a point $x : Set \to Sh(X)$ of a sheaf topos have special interpretation and relevance:

* The [[direct image]] of a set $S$ under the point $x : {*} \to X$ is, by definition of [[direct image]] the sheaf
  $$
    x_*(S) : (U \subset X) \mapsto S(x^{-1}(U)) =
    \left\{
      \array{
        S & \text{ if } x \in U
        \\
        {*} & \text{otherwise}
      }
    \right.
  $$
 This is the [[skyscraper sheaf]] $skysc_x(S)$ with value $S$ supported at $x$. (In the first line on the right in the above we identify the set $S$ with the unique sheaf on the point it defines. Notice that $S(\emptyset) = pt$).

* The [[inverse image]] of a sheaf $A$ under the point $x : {*} \to X$ is by definition of [[inverse image]] (see the [[Kan extension]] formula discussed there), the set
  $$
    \begin{aligned}
       x^*(A) & =
       colim_{{*} \to x^{-1}(V)} A(V)
       \\
       &=
       colim_{V\subset X| x \in V} A(V)
    \end{aligned}
    \,.
  $$
  This is the [[stalk]] of $A$ at the point $x$,
  $$
    x^*(-) = stalk_x(-)
    \,.
  $$

By definition of [[geometric morphisms]], taking the stalk at $x$ is [[left adjoint]] to forming the skyscraper sheaf at $x$:

for all $S \in Set$ and $A \in Sh(X)$ we have
$$
  Hom_{Set}(stalk_x(A), S) \simeq
  Hom_{Sh(X)}(A, skysc_x(S))
  \,.
$$

Note that the observation that the points of $Sh(X)$ are in bijection with the points of $X$ actually factors over an intermediate concept, namely that of [[point of a locale|points of a locale]]. Firstly, any topological space gives rise to a locale; if the space is sober, its points are in bijection with the locale-theoretic points of the induced locale. Secondly, for any locale ([[topological locale|spatial]] or not), its locale-theoretic points correspond to the points of its induced sheaf topos.

### In sheaf toposes

The following characterization of points in [[sheaf toposes]] a special case of the general statements at [[morphism of sites]].

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

For $C$ a [[site]], there is an [[equivalence of categories]]

$$
  Topos(Set, Sh(C))
  \simeq
  Site(C,Set)
  \,.
$$

(Morphisms of sites $C\to Set$ are precisely the [[cover-preserving functor|continuous]] [[flat functors]].)
=--

This appears for instance as ([MacLaneMoerdijk, corollary VII.5.4](#MacLaneMoerdijk)).

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

If $E$ is a [[Grothendieck topos]] with enough points (def. \ref{EnoughPoints}), there is a  *[[small set]]* of points of $E$ which are jointly conservative, and therefore a [[geometric morphism]] $Set/X \to E$, for some set $X$, which is [[surjective geometric morphism|surjective]].

=--

This appears as ([Johnstone, Lemma C2.2.11, C2.2.12](#Johnstone)).

(In general, of course, a topos can have a proper class of non-isomorphic points.)

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

A Grothendieck topos has enough points (def. \ref{EnoughPoints}) precisely when it underlies a bounded [[ionad]].

=--

### In classifying toposes

From the above it follows that if $E$ is the [[classifying topos]] of a [[geometric theory]] $T$, then a point of $E$ is the same as a [[model]] of $T$ in [[Set]].

### Of toposes with enough points

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

If a [[sheaf topos]] $E$ has _enough points_ (def. \ref{EnoughPoints}) then

* there exists a [[topological space]] $X$ whose [[cohomology]] and [[homotopy theory]] is the [[cohomology|intrinsic cohomology]] and [[homotopy groups in an (infinity,1)-topos|intrinsic homotopy theory]] of the topos;

* such that $E$ is the category of [[equivariant cohomology|equivariant]] objects in the [[sheaf topos]] $Sh(X)$ with respect to some groupoid action on $X$.

=--

This is due to ([Butz](#Butz)) and ([Moerdijk](#Moerdijk)).

## Examples

### General

* For $X$ any [[topological space]], the  [[category of sheaves|topos of sheaves]] on (the [[category of open subsets]] of) $X$ has enough points (def. \ref{EnoughPoints}): a morphism of sheaves is a mono-/epi-/isomorphism precisely if it is so on every [[stalk]].

* Points of [[over-toposes]] are discussed at over topos -- points.

### Rings and algebraic theories
 {#Rings}

The category of points of the presheaf topos over $Ring_{fp}^{op}$, the dual of the category of finitely presented [[rings]], is the category of all rings (without a size or presentation restriction). In fact this holds for any [[algebraic theory]], not only for the theory of commutative rings. See also at  _[[Gabriel-Ulmer duality]]_, _[[flat functors]]_.

### Of a local topos

A [[local topos]] $(\Delta \dashv \Gamma \dashv coDisc) : E \to Set$ has a canonical point, $(\Gamma \dashv coDisc) : Set \to E$. Morover, this point is an [[initial object]] in the category of all points of $E$ (see _[Equivalent characterizations](local+geometric+morphism#EquivalentCharacterizations)_ at _[[local topos]]_.)

### Over $\infty$-cohesive sites

* Let [[Diff]] be a [[small category]] version of the category of smooth manifolds (for instance take it to be the category of manifolds embedded in $\mathbb{R}^\infty$). Then the sheaf topos $Sh(Diff)$ has precisely one point $p_n$ per natural number $n \in \mathbb{N}$ , corresponding to the $n$-ball: the [[stalk]] of a sheaf on $Diff$ at that point is the colimit over the result of evaluating the sheaf on all $n$-dimensional smooth balls.

  This is discussed for instance in ([Dugger, p. 36](#Dugger)) in the context of the [[model structure on simplicial presheaves]].

### Toposes with enough points

The following classes of topos have enough points (def. \ref{EnoughPoints}).

* every [[presheaf topos]];

* every [[coherent topos]] (due to the [[Deligne completeness theorem]]);

* every [[Galois topos]] (see ([Zoghaib](#Zoghaib))).

### Toposes without points

In contrast with point-set topology where non empty spaces tautologically have points there are non trivial toposes lacking points just like in pointfree topology there are nontrivial [[locale|locales]] without points. From a logical perspective these toposes correspond to consistent [[geometric theory|geometric theories]] lacking models in Set. Classical examples are provided by toposes of sheaves on complete atomless Boolean algebras (see Barr ([1974](#Barr74)) or for an example due to [[Pierre Deligne|Deligne]] [[SGA4]], p.412).

## Related concepts

* [[topological locale]]

* [[Barr's theorem]]

## References ##

Textbook references are section 7.5 of

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
  {#MacLaneMoerdijk}

as well as section C2.2 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

In

* Carsten Butz, _Logical and cohomological aspects of the space of points of a topos_ ([web](http://www.academia.edu/9139781/Logical_and_Cohomological_Aspects_of_the_Space_of_Points_of_a_Topos))
 {#Butz}

is a discussion of how for every topos with enough points there is a [[topological space]] whose [[cohomology]] and [[homotopy theory]] is related to the [[cohomology|intrinsic cohomology]] and [[homotopy groups in an (infinity,1)-topos|intrinsic homotopy theory]] of the topos.

More on this is in

* [[Ieke Moerdijk]], _Classifying toposes for toposes with enough points_ , Milan Journal of Mathematics Volume 66, Number 1, 377-389
 {#Moerdijk}

See also

* Sam Zoghaib, _A few points in topos theory_ ([pdf](http://www.normalesup.org/~zoghaib/math/points-tt.pdf))
 {#Zoghaib}

Points of the sheaf topos over the category of [[manifold]]s are discussed in

* [[Dan Dugger]], _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))
 {#Dugger}

Toposes without points are discused in the homonymous paper

* {#Barr74} [[Michael Barr]], *Toposes without points*, J. Pure Appl. Algebra **5** (1974) 265-280 [[pdf](https://www.math.mcgill.ca/barr/papers/top.no.pt.pdf), [[Barr-ToposesWithoutPoints.pdf:file]]]

[[!redirects points of a topos]]

[[!redirects points of toposes]]
[[!redirects points of topoi]]

[[!redirects point of topos]]

[[!redirects enough points]]
[[!redirects topos with enough points]]
[[!redirects topos without points]]
[[!redirects pointless topos]]

[[!redirects topos point]]
[[!redirects topos points]]


point space

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

#Contents#
* table of contents
{:toc}

## Definition

The _point space_ $\ast$ is the [[topological space]] whose underlying set is [[generalized the|the]] [[singleton]], and equipped with the unique topology that this set carries.

## Properties

### General

The point space is the [[terminal object]] in the [[category]] [[Top]] of topological spaces.

For $X$ any [[topological space]], then for every element of its underlying set there is a [[continuous function]] from the point space

$$
  \ast \longrightarrow X
$$

whose image is that point, and every such continuous function arises this way

### Relation to irreducible closed subspaces

For the following we write the point space explicitly as

$$
  \ast =
  \left\{
     \{1\}, \, \tau_\ast = \left\{ \emptyset, \{1\} \right\}
  \right\}
$$

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

For $(X,\tau)$ a [[topological space]], then there is a [[bijection]] between
the [[irreducible closed subspaces]] of $(X,\tau)$ and the
[[frame]] [[homomorphisms]] from $\tau_X$ to $\tau_\ast$ from the [[frame of opens]] of $X$ to that of the point space. Moreover, this is  given by

$$
  \array{
    Hom_{Frame}(\tau_X, \tau_\ast)
      &\underoverset{\simeq}{}{\longrightarrow}&
    IrrClSub(X)
    \\
    \phi
      &\mapsto&
    X \backslash U_\emptyset(\phi)
  }
$$

where $U_\emptyset(\phi)$ is the [[union]] of all elements $U \in \tau_x$ such that $\phi(U) = \emptyset$:

$$
  U_{\emptyset}(\phi)
    \coloneqq
  \underset{{U \in \tau_X} \atop {\phi(U) = \emptyset} }{\cup}
   U
  \,.
$$

=--

See also ([[Stone Spaces|Johnstone 82, II 1.3]]).

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

First we need to show that the function is well defined in that given
a frame homomorphism $\phi \colon \tau_X \to \tau_\ast$ then  $X \backslash U_\emptyset(\phi)$
is indeed an irreducible closed subspace.

To that end observe that:

$(\ast)$ _If there are two elements $U_1, U_2 \in \tau_X$ with $U_1 \cap U_2 \subset U_{\emptyset}(\phi)$
then $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$._

This is because

$$
  \begin{aligned}
    \phi(U_1) \cap \phi(U_2)
    & =
    \phi(U_1 \cap U_2)
    \\
    & \subset \phi(U_{\emptyset}(\phi))
    \\
    & =
    \emptyset
  \end{aligned}
  \,,
$$

where the first equality holds because $\phi$ preserves finite intersections by def. \ref{HomomorphismOfFramesOfOpens}, the inclusion holds because $\phi$ respects
inclusions by remark \ref{PreservationOfInclusionsByFrameHomomorphism}, and the second equality holds because $\phi$ preserves arbitrary unions
by def. \ref{HomomorphismOfFramesOfOpens}.
But in $\tau_\ast = \{\emptyset, \{1\}\}$ the intersection of two open subsets is empty precisely if at least one of them is empty,
hence $\phi(U_1) = \emptyset$ or $\phi(U_2) = \emptyset$. But this means that $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$, as claimed.

Now according to prop. \ref{OpenSubsetVersionOfClosedIrreducible}
the condition $(\ast)$ identifies the [[complement]]
$X \backslash U_{\emptyset}(\phi)$ as an [[irreducible closed subspace]] of $(X,\tau)$.

Conversely, given an irreducible closed subset $X \backslash U_0$, define $\phi$ by

$$
  \phi
    \;\colon\;
  U
    \mapsto
  \left\{
    \array{
      \emptyset & \vert \, \text{if} \, U \subset U_0
      \\
      \{1\} & \vert \, \text{otherwise}
    }
  \right.
  \,.
$$

This does preserve

1. arbitrary unions

   because $\phi(\underset{i}{\cup} U_i) = \{\emptyset\}$ precisely if $\underset{i}{\cup}U_i \subset U_0$ which is the
   case precisely if all $U_i \subset U_0$, which means that all $\phi(U_i) = \emptyset$ and because $\underset{i}{\cup}\emptyset = \emptyset$;

   while $\phi(\underset{i}{\cup}U_1) = \{1\}$ as soon as one of the $U_i$ is not contained in $U_0$, which means that
   one of the $\phi(U_i) = \{1\}$ which means that $\underset{i}{\cup} \phi(U_i) = \{1\}$;

1. finite intersections

   because if $U_1 \cap U_2 \subset U_0$, then by $(\ast)$ $U_1 \in U_0$ or $U_2 \in U_0$, whence $\phi(U_1) = \emptyset$
   or $\phi(U_2) = \emptyset$, whence with $\phi(U_1 \cap U_2) = \emptyset$ also $\phi(U_1) \cap \phi(U_2) = \emptyset$;

   while if $U_1 \cap U_2$ is not contained in $U_0$ then neither $U_1$ nor $U_2$ is contained in $U_0$ and hence with
   $\phi(U_1 \cap U_2) = \{1\}$ also $\phi(U_1) \cap \phi(U_2) = \{1\} \cap \{1\} = \{1\}$.

Hence this is indeed a frame homomorphism $\tau_X \to \tau_\ast$.

Finally, it is clear that these two operations are inverse to each other.

=--

## Related concepts

[[!include universal constructions of topological spaces -- table]]

* [[Sierpinski space]]

[[!redirects point spaces]]

[[!redirects point topological space]]
[[!redirects point topological spaces]]


point symmetry

In the context of the [[variational bicomplex]], a symmetry ([[evolutionary vector field]]) on the [[jet bundle]] of a [[field bundle]] is called a _point symmetry_ if it comes from a vector field on the [[field bundle]] itself.

[[!redirects point symmetries]]


point-free topology

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

# Contents
* table of contents
{: toc}

## Idea

__Point-free topology__ refers to various formulations of [[topology]] that are not based on the notion of [[topological space]] as a [[set]] of [[points]] equipped with [[extra structure]].  What they generally have in common is that instead the points are described as [[models]] of a [[geometric theory]]. This change has some important consequences.

First, it conveniently describes the points not only in one's favourite [[category of sets]], but also in arbitrary [[Grothendieck topos|Grothendieck toposes]].

Second, it opens up the possibility of using predicate geometric theories, to obtain generalized spaces (the [[classifying toposes]]) in the sense of Grothendieck.
This article will mostly concern itself with the ungeneralized point-free spaces, corresponding to propositional geometric theories.

The phrase "point-free topology" is often taken as synonymous with _pointless topology_ [Johnstone 83](#Johnstone83). However, let us reserve "pointless" for approaches that avoid mentioning points and refer directly to the geometric theory presentation and structures deriving from it.

## Examples and Definitions

### Locale theory

In [[locale theory]], one studies the set of [[open subspaces]] (with the extra structure of a [[frame]]) as the fundamental notion. If $A$ is the frame, then the [[geometric theory]] is that of completely prime filters of $A$. It has a propositional symbol $\phi_a$ for each element of $A$, and axioms -

  - $\phi_a \vdash \phi_b$ (if $a \leq b$)

  - $\top \vdash \phi_1$

  - $\phi_a \wedge \phi_b \vdash \phi_{a\wedge b}$

  - $\phi_{\bigvee_i a_i} \vdash \bigvee_i \phi_{a_i}$

This simply makes a direct correspondence between the algebra of $A$ and the geometric logic. Of course it also corresponds to the finite intersections and arbitrary unions of open sets, which shows how geometric logic is matched to topology.

The classifying topos is the topos of sheaves over $A$.

The frame is a presentation-independent representation of the theory. It can be recovered from the theory as the Lindenbaum algebra of formulae modulo equivalence.

### Propositional geometric theories

A presentation of a frame by generators and relations corresponds more directly to a general propositional [[geometric theory]]. The generators are the signature (propositional symbols) and the relations can be straightforwardly converted to axioms.

### Formal topology

In [[formal topology]], one studies a set $B$ of *[[base for a topology|basic]]* open subspaces, with a _cover relation_ $a\triangleleft U$ between elements and subsets of $B$, to show when one basic open is covered by a family of others. There are various flavours of this. $\triangleleft$ may be the full cover relation or a generating part, and there may sometime be included the extra structure of a [[positivity predicate|positivity]]. In each case the formal topology has the structure of a [[posite]] and so gives rise to a geometric theory of flat continuous functors. Its propositional  symbols are $\phi_a$ for each $a\in B$. The axioms are -

  - $\top \vdash \bigvee_{a\in B} \phi_a$

  - $\phi_a \wedge \phi_b \vdash \bigvee\{\phi_c\mid c\triangleleft\{a\} \wedge c\triangleleft\{b\}\}$

  - $\phi_a \vdash \bigvee_{b\in U}\phi_{b}$ (if $a\triangleleft U$)

The models are known as _formal points_.

### General definition

It is probably impossible to give a real definition of "a notion of point-free topology".  However, one way to capture the notion of "point-free-ness" is by a version of [[well-pointed category|well-pointedness]].

Specifically, suppose that one defines a [[category]] (or $(\infty,1)$-[[(infinity,1)-category|category]]) $S$, whose [[objects]] one thinks of as [[spaces]], and that one intends $Hom(X,Y)$ to be the [[set]] (or $\infty$-[[infinity-groupoid|groupoid]]) of *all* continuous maps from $X$ to $Y$ (whereas $Ob(S)$ need not be the class of all spaces).  Also suppose that $S$ has a [[terminal object]] $pt$, which one interprets as the [[point]].  Then we may call this a __pointwise__ formulation of topology if $pt$ is a [[generator]] in $S$, and __pointless__ if this is not provable (or, for a stronger variation, provably false).

## Pointwise reasoning

The traditional way of doing topology using points may be called __pointwise topology__. Obviously that is natural in point-set topology, but for point-free there is an apparent problem: there may not be enough points to support, semantically, all the syntactic distinctions between formulae in the geometric logic. In other words, geometric logic is not necessarily complete.

It is possible to circumvent this by reasoning in an entirely _pointless_ style, without reference to the points.
For instance with locales, the maps are defined to be the frame homomorphisms, backwards.

Surprisingly, however, it is also possible to reason pointwise: a map from $X$ to $Y$ can be defined by saying how points of $X$ transform to points of $Y$. What makes this work is that there are enough points (models of the geometric theory) if you allow for those that live outside your favourite category of sets. In essence, if the construction is defined for the generic point of $X$ in the topos of sheaves over $X$, then the universal property of classifying toposes gives a point-free map (geometric morphism). The catch is that, to allow the construction to be transported from one topos to another, it has to be _geometric_ - preserved by inverse image functors. Thus continuity becomes a logical issue, of geometricity.

Further details and references are in Vickers [2007](#Vickers07) and [2014](#Vickers14).

## Bundles

A decisive advantage of point-free topology is that it gives a fibrewise topology of **bundles**, understood broadly as maps into a base space. (Note - for generalized spaces, i.e. toposes, bundles must be understood as _bounded_ geometric morphisms.)

It is shown in [Joyal, Tierney](#JoyalTierney84) that, for any elementary topos $\mathcal{E}$, there is an equivalence between localic bundles over $\mathcal{E}$ (i.e. localic geometric morphisms into it) and internal _point-free spaces_ in $\mathcal{E}$. The result does not work point-set.

Joyal and Tierney proved their result for internal frames. However, in some ways it works better for presentations of frames by generators and relations, if there is some geometric theory $\mathbb{T}_0$ of which the systems of generators and relations are models. This is not the case for frames - the frame structure is not geometric.  [Vickers 2007](#Vickers07) works it out for the general case of "GRD-systems", systems of generators, relations and "disjuncts".
Each model $P$ of the geometric theory $\mathbb{T}_0$ itself presents another geometric theory (for its point-free space), and this can be used to make a theory $\mathbb{T}_1$, extending $\mathbb{T}_0$, that classifies "pointed" presentations.
There is a forgetful map (forget the point) $p\colon \mathcal{S}[\mathbb{T}_1] \to \mathcal{S}[\mathbb{T}_0]$ between the classifying toposes.
Then a presentation in $\mathcal{E}$ is a map $P\colon\mathcal{E}\to\mathcal{S}[\mathbb{T}_0]$, and the localic bundle constructed by Joyal and Tierney is a bipullback of $p$ along $P$.

What this account reveals is that the bundle is - using $P$ - a continuous space-valued map on $\mathcal{E}$. For each point $x$ of $\mathcal{E}$, the fibre is $P(x)$ - because fibres are just pullbacks.

## Set of points

Depending on the foundational setting, a point-free space may or may not have a set of points, a discrete coreflection. This can be done in topos theory, but relies on an impredicative use of power_sets_. (Formal topology arose out of a desire to work predicatively.)

However, even when it exists, the set of points does not generally represent the space well.

This is best understood in terms of bundles over $X$. The "set of points" of a bundle $p\colon Y\to X$ is a sheaf over $X$, hence a local homeomorphism $q\colon Z\to X$. As a discrete coreflection it must also have a bundle map $f$ from $q$ to $p$. However, local homeomorphisms have an _opfibrational_ property that limits their ability to approximate general bundles. Suppose $x \sqsubseteq x'$ are two points of $X$ that are related by the specialization order. Then there is a corresponding fibre map $q^{-1}(x) \to q^{-1}(x')$.

As an example, take $X$ to be the Sierpinski space $\mathbb{S}$. Its points are subsets of a singleton $\{\ast\}$ and include $\bot=\emptyset\sqsubseteq\top=\{\ast\}$.
Its sheaves are functions - the fibre maps $Z_{\bot}\to Z_{\top}$.

Now consider the bundle $p\colon 1\to\mathbb{S}$ that picks out the closed point $\bot$. Its fibres over $\bot$ and $\top$ are singleton and empty. Let $q\colon Z \to \mathbb{S}$ be its discrete coreflection. Because there is a bundle map from $q$ to $p$, the fibre $Z_\top$ must be empty. But then because $q$ is a local homeomorphism, $Z_\bot$ must also be empty. $Z$, the "set of points" of $p$, is the empty space over $\mathbb{S}$.

We see that a non-trivial space can have have an empty set of points for purely topological reasons, nothing to do with logic or a pathological nature of point-free spaces.

## Applications

* [[Banach-Tarski paradox]]

## Related concepts

* [[localic homotopy theory]]

## References

An introduction to [[locale theory]] is

* {#Johnstone83} [[Peter Johnstone]] (1983); _The point of pointless topology_; Bull. Amer. Math. Soc. (N.S.) Volume 8, Number 1, 41--53; [Euclid](http://projecteuclid.org/euclid.bams/1183550014).

This is, in its own words, to be read as the trailer for Johnstone's book _[[Stone Spaces]]_, which see.

For a constructive treatment more appropriate to the ideas on this page see

* {#JoyalTierney84}[[Andre Joyal|André Joyal]], [[Myles Tierney]], _An extension of the Galois theory of Grothendieck_, Memoirs of the American Mathematical Society **51** (1984), no. 309

For [[formal topology]], see

*  [[Giovanni Sambin]] (2001); _Some points in formal topology_; [pdf](http://www.math.unipd.it/~sambin/txt/SP.pdf).

* {#Palmgren05} [[Erik Palmgren]], _From Intuitionistic to Point-Free Topology: On the Foundation of Homotopy Theory_, Logicism, Intuitionism, and Formalism Volume 341 of the series Synthese Library pp 237-253, 2005 ([pdf](http://www2.math.uu.se/~palmgren/homotopy_rev2.pdf))

The pointwise reasoning is explained in the next references (though it was surely understood before them):

* {#Vickers04} [[Steve Vickers]], _The double powerlocale and exponentiation: a case study in geometric logic_, Theory and Applications of Categories **12** (2004) pp. 372-422 ([pdf](http://www.tac.mta.ca/tac/volumes/12/13/12-13.pdf))

* {#Vickers07} [[Steve Vickers]], _Locales and toposes as spaces_, Chapter 8 in _Handbook of Spatial Logics_ (ed. Aiello, Pratt-Hartman, van Bentham), Springer, 2007, pp. 429-496.

* {#Vickers14} [[Steve Vickers]], _Continuity and geometric logic_, Journal of Applied Logic **12 (1)** (2014), pages 14-27 ([pdf](http://www.cs.bham.ac.uk/~sjv/GeoAspects.pdf))

* [[Steven Vickers]], *Generalized point-free spaces, pointwise* [[arXiv:2206.01113](https://arxiv.org/abs/2206.01113)]

[[!redirects pointless topology]]

[[!redirects pointwise topology]]
[[!redirects point-wise topology]]

[[!redirects pointfree topology]]
[[!redirects point-free topology]]


point-supported distribution

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## Idea

A _point-supported distribution_ is a [[distribution]] whose [[support of a distribution]] is a single point. These turn out to be precisely the sums of multiples of the [[delta distribution]] and its [[derivative of a distribution|derivatives]] at that point (prop. \ref{PointSupportedDistributionsAreSumsOfDerivativesOfDeltaDistibutions} below).

## Definition

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

A [[distribution]] $u \in \mathcal{D}'(X)$ is _point-supported_ if its [[support of a distribution]] is a [[singleton]] set:

$$
  supp(u) = \{p\}
$$

for some $p \in X$.

=--

## Properties

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

Every point-supported distribution $u$ (def. \ref{PointSupportedDistribution}) with $supp(u) = \{p\}$ is a [[finite set|finite]] [[sum]] of multiplies of [[derivative of a distribution|derivatives]] of the [[delta distribution]] at that point:

$$
  u =
  \underset{ {\alpha \in \mathbb{N}^n} \atop { {\vert \alpha\vert} \leq k } }{\sum}  c^\alpha \partial_\alpha \delta(p)
$$

where $\{c^\alpha \in \mathbb{R}\}_\alpha$,
and for $k \in \mathbb{N}$ the [[order of a distribution|order]] of $u$.

=--

(e.g. [Hörmander 90, theorem 2.3.4](#Hoermander90))

Clearly a point-supported distribution is in particular a [[compactly supported distribution]].

## Related concepts

* [[extension of distributions]]

## References

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

[[!redirects point-supported distributions]]

[[!redirects point supported distribution]]
[[!redirects point supported distributions]]

[[!redirects distribution supported at a point]]
[[!redirects distributions supported at points]]

[[!redirects distribution with point support]]
[[!redirects distributions with point support]]


pointed

* [[pointed object]]

* [[pointed category]]


pointed (infinity,1)-category

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## Definition

Under a _pointed $(\infty,1)$-category_ is often understood an [[(∞,1)-category]] that admits a [[zero object]], i.e. an [[object]] which is both [[initial object in an (∞,1)-category|initial]] and [[terminal object in an (∞,1)-category|terminal]] (this then is unique up to [[equivalence in an (∞,1)-category|equivalence]]).

The terminology "pointed $(\infty,1)$-category", in this sense, is commonly used, for instance, when speaking about [[stable (∞,1)-categories]], which are such pointed $(\infty,1)$-categories with further properties.

If a pointed $(\infty,1)$-category in this sense happens to be just a [[1-category]], then it is a _[[pointed category]]_.

The same terminological caveat applies as applies to "[[pointed categories]]":

More generally, a _pointed (∞,1)-category_ could be taken to be a [[pointed object]] in [[(∞,1)Categories]], i.e. an [[(∞,1)-category]] with _any_ of its [[objects]] singled out, and with [[(∞,1)-functors]] between such pointed $(\infty,1)$-categories required to preserved these chosen objects.

## See also

* [[pointed category]], [[pointed model category]]

* [[stable (∞,1)-category]]

* [[zero morphism]], [[null homotopy]]

* [[based loop space object]], [[reduced suspension object]]

* [[Toda bracket]]

## References

* [[Jacob Lurie]], chapter 1 of: _[[Higher Algebra]]_

[[!redirects pointed (infinity,1)-categories]]

[[!redirects pointed (∞,1)-category]]
[[!redirects pointed (∞,1)-categories]]


pointed abelian group

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\section{Definition}

\begin{definition}\label{PointedAbelianGroups}
A **pointed abelian group** is an [[abelian group]] $(A, +, 0, -)$ equipped with the [[structure|choice]] of an [[element]] $1 \in A$. A [[homomorphism]] of pointed abelian groups is a homomorphism of [[underlying]] groups which preserves the choice of the extra points.
\end{definition}

\begin{remark}
Pointed abelian groups are "pointed" in the sense of *[[pointed objects in a monoidal category]]*, in that the choice of an element $1 \in A$ is equivalently the choice of a [[homomorphism]] $\mathbb{Z} \longrightarrow A$ out of the additive group of [[integers]] (which is fixed by its [[image]] of $1 \in \mathbb{Z}$).

Therefore the [[category]] of pointed abelian groups is equivalently the [[coslice category]] $Ab^{\mathbb{Z}/}$ of [[Ab]] under $\mathbb{Z}$.
\end{remark}

\begin{remark}\label{AsCosliceObjects}
The additive [[neutral element]] $0 \in A$ is *not* actually needed in the definition of apointed abelian groups (Def. \ref{PointedAbelianGroups}): Pointed abelian groups could equally be defined as [[pointed set|pointed]] [[commutative semigroup|commutative]] [[invertible semigroups]], hence as [[commutative semigroups]] $(A, +)$ equipped with an element $1 \in A$ and a [[function]] $- \colon A \to A$ such that for all [[elements]] $a \in A$ and $b \in A$, $a + b + (-b) = a$.

From [[commutativity]] and [[associativity]] one may derive the other three invertibility properties for an [[invertible semigroup]]: $a + (-b) + b = a$, $b + (-b) + a = a$, and $(-b) + b + a = a$. The element $1 + (-1)$ is both left [[unit|unital]] and right unital with respect to the binary operation $+$, so by defining $0 \coloneqq 1 + (-1)$, $A$ becomes an [[abelian group]] $(A, +, 0, -)$ with an additional point $1 \in A$; hence a pointed abelian group.
\end{remark}

\section{Examples}

\begin{example}\label{TrivialPointing}
Every [[abelian group]] becomes a pointed abelian group (Def. \ref{PointedAbelianGroups}) by taking the point to be the [[neutral element]] $0$. Since the neutral element $0$ is necessarily preserved by an [[group homomorphism]], this constitutes a [[full subcategory]]-inclusion
$$
  Ab \hookrightarrow Ab^{\mathbb{Z}/}
  \,.
$$
\end{example}

On the other hand:
\begin{example}
The notation $1 \in A$ is motivated from the case of [[rings]] $(R,0,+,1,\cdot)$, [[underlying]] which is the pointed abelian group $(R,0,+,1)$ with the "point" being the ring's [[unit]] element $1 \in R$.
\end{example}

\section{Properties}

\begin{remark}
Despite the inclusion Exp. \ref{TrivialPointing}, the [[category]] of pointed abelian groups (Def. \ref{PointedAbelianGroups}) is not [[equivalence of categories|equivalent]] to the [[category of abelian groups]] ($\mathrm{Ab}$):

In [[Ab]], the [[initial object]] and [[terminal objects]] are the same, both are given by the [[trivial group]]. However, in the category of pointed abelian groups, while the [[terminal object]] is still the [[trivial group]], the [[initial object]] is the additive group [[integers]] $\mathbb{Z}$ (manifestly so from Rem. \ref{AsCosliceObjects}). In this, the category of pointed abelian groups has more in common with the categories [[Ring]] and [[CRing]] of [[rings]] and of [[commutative rings]], respectively.
\end{remark}

\section{Related concepts}

* [[pointed module]]

* [[pointed object in a monoidal category]]

* [[commutative invertible semigroup]]

* [[abelian group]]

* [[Tarski group]]

* [[lattice]]

## References

* W. Edwin Clark, Xiang-dong Hou, *Galkin Quandles, Pointed Abelian Groups, and Sequence A000712* [[arXiv:1108.2215](https://arxiv.org/abs/1108.2215)]

* A. M. Nurakunov, *Quasivariety Lattices of Pointed Abelian Groups*, Algebra and Logic **53** (2014) 238–257 [[doi:10.1007/s10469-014-9286-5](https://doi.org/10.1007/s10469-014-9286-5)]

[[!redirects pointed abelian group]]
[[!redirects pointed abelian groups]]

[[!redirects category of pointed abelian groups]]

[[!redirects pointed commutative invertible semigroup]]
[[!redirects pointed commutative invertible semigroups]]

[[!redirects category of pointed commutative invertible semigroups]]


pointed category

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# Pointed categories
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## Definition

A [[category]] is sometimes called __pointed__ if it has a [[zero object]] (e.g. [Quillen 67, II.2 def. 4](#Quillen67)), i.e. if it has an [[initial object]] and a [[terminal object]] and they are [[isomorphism|isomorphic]].

(If the morphism from the [[initial object]] to the [[terminal object]] is not necessarily an [[isomorphism]] but just a [[monomorphism]] then one speaks also of a [[quasi-pointed category]].)

Every category of [[pointed objects]] is a pointed category in this sense, and this is probably the motivation for the terminology.

Beware that the concept of _[[pointed objects]] in [[Cat]]_, which may also be called "pointed categories" is more general and more restricted: these are categories with any one [[object]] singled out, and with [[functors]] between them required to preserve these objects.

Beware also that the concept of _[[well-pointed topos]]_ is unrelated.

## Properties

### Canonical enrichment in pointed sets

Every [[locally small category|locally small]] pointed category is naturally  [[enriched category|enriched]] over the category of [[pointed sets]] using the [[smash product]] as [[tensor product]]; the converse is true if and only if the enriched category has an initial object.

## Related concepts

* [[pointed model category]]

  * [[linear model category]]

  * [[stable model category]]

* [[pointed (infinity,1)-category]]

* [[zero object]]

* [[semiadditive category]]

* [[biproducts]]

## References

* {#Quillen67} [[Daniel Quillen]], chapter I, section 2, def. 4 of _Homotopical algebra_, Lecture Notes in Mathematics __43__, Springer-Verlag 1967, iv+156 pp.

[[!redirects pointed category]]
[[!redirects pointed categories]]
[[!redirects punctuated category]]


pointed connected groupoid

[[!redirects pointed connected groupoids]]

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## Definition

A **pointed connected groupoid** is a [[groupoid]] that is [[pointed object|pointed]] and [[connected]].

Under the [[looping and delooping]]-[[equivalence of (infinity,1)-categories|equivalence]], this is equivalently the *[[delooping groupoid]]* of a [[group]].

### In homotopy type theory

In [[homotopy type theory]], a __pointed connected groupoid__ consists of

* A type $G$
* A basepoint $e:G$
* A 0-connector
$$\kappa_1:\prod_{f:G \to \mathbb{1}} \prod_{a:\mathbb{1}} \mathrm{isContr}(\left[\mathrm{fiber}(f, a)\right]_{0})$$
* A 1-truncator:
$$\tau_2:\mathrm{isGroupoid}(G)$$

## See also ##

* [[delooping groupoid]]

* [[group]]

* [[groupoid]]

* [[infinity-group]]

## References ##

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], and [[Daniel R. Grayson]], [Symmetry book](https://unimath.github.io/SymmetryBook/book.pdf) (2021)