Clifford algebra

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Super-Algebra and Super-Geometry
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[[!include supergeometry - contents]]
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#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Given a [[commutative ring]] $R$ and $R$-[[module]]s $M$ and $N$, an __$R$-[[quadratic function]]__ on $M$ with values in $N$ is a map $q: M \to N$ such that the following properties hold:

* (cube relation) For any $x,y,z \in M$, we have
$$q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0.$$

* (homogeneous of degree 2) For any $x \in M$ and any $r \in R$, we have
$$q(r x) = r^2q(x).$$

A **quadratic $R$-module** is an $R$-module $M$ equipped with a __[[quadratic form]]__: an $R$-quadratic function on $M$ with values in $R$.

The **Clifford algebra** $Cl(M,q)$ of a quadratic $R$-module $(M,q)$ can be defined as the [[quotient]] of the [[tensor algebra]] $T_R(M)$ by the [[ideal]] generated by the relations $x \otimes x - q(x)$ for all $x \in M$.

Equivalently, it is the [[initial object]] in the [[category]] whose [[object]]s are pairs $(A,\phi)$ where $A$ is an associative unital $R$-[[associative algebra|algebra]], and $\phi: M \to A$ is an $R$-[[linear map]] satisfying $\phi(x)^2 = q(x) 1_A$ for all $x \in M$, and whose morphisms $(A,\phi)\to (A',\phi')$ are the associative $R$-algebra maps $\chi: A\to A'$ such that $\chi\circ\phi=\phi'$.

## Examples

Examples in low [[rank]] can be calculated easily.  If $M$ is freely generated by a single element $e$, with quadratic form $q(e) = 1$, then $Cl(M,q) = R[e]/(e^2-1)$.  Note that the opposite sign convention is often used in the differential geometry literature, so one may see the assertion that the Clifford algebra of the real line with a positive definite metric is isomorphic to the complex numbers $\mathbb{R}[e]/(e^2+1)$.  Similarly, the Clifford algebra of a negative definite two dimensional real vector space is isomorphic to the (non-split) quaternions in our convention, but one may see the assertion that it is isomorphic to $M_2(\mathbb{R})$.  Complexification removes the difference between positive definite and negative definite, and the two complexified algebras are isomorphic.

Let $M = L \oplus L^\vee$ for $L$ projective of rank $d$ over $R$, and $L^\vee = Hom_R(L,R)$ the dual module.  One can define the canonical quadratic form $q(f+x) = f(x)$ for $f \in L^\vee$ and $x \in L$.  In this case, $Cl(M,q) \cong M_{2^d}(R)$.  In general, the Clifford algebra arising from a nondegenerate form is flat-locally (on $\operatorname{Spec} R$) isomorphic to a matrix algebra (when rank is even) or a direct sum of two matrix algebras (when rank is odd).

If $M$ is a projective $R$-module of rank $d$, then independently of $q$, the Clifford algebra $Cl(M,q)$ is projective of rank $2^d$, and is (noncanonically) isomorphic to $\bigwedge M$ as an $R$-module equipped with a map from $M$.  The Clifford algebra is isomorphic to the exterior algebra (as algebras equipped with $R$-module maps from $M$) if and only if $q = 0$.

If $R$ is the ring of smooth functions on a pseudo-Riemannian manifold $X$, and $M$ is the $R$-module of sections of the tangent bundle, then the metric endows $M$ with a quadratic structure, and one can form the Clifford algebra of the tangent bundle.

## Properties

### Classification and Relation to matrix algebras
 {#Classification}

#### Over the complex numbers
 {#ClassificationOverTheComplexNumbers}

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

Let $V$ be the [[vector space]] over the [[complex numbers]]
of complex [[dimension]] $d$, equipped with non-degenerate
[[bilinear form]], unique up to [[isomorphism]]. The Clifford algebra

$$
  Cl_{d}(\mathbb{C}) \coloneqq Cl(V)
$$

is  [[isomorphism|isomorphic]], as a complex [[associative algebra]]
to a  [[matrix algebra]] as follows:

$$
  Cl_d(\mathbb{C})
  \simeq
  \left\{
    \array{
       Mat_{2^{\tfrac{d}{2}}}(\mathbb{C}) & for \, d \, even
       \\
       Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) \oplus Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C})
       & for \, d \,  odd
    }
  \right.
$$

=--

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

This is one of the incarnations of [[Bott periodicity]].

=--

#### Over the real numbers
 {#ClassificationOverTheRealNumbers}

We discuss the classification of Clifford algebras over the [[real numbers]] and their relation to [[matrix algebras]] over the real numbers. A key statement is that of the mod-8 [[Bott periodicity]] of this classification (prop. \ref{RealBottPeriodicity} below).

In the following we write

*  $\Cl_{s,t}$ for the Clifford algebra of a vector space over the real numbers with

   * $s$ generators squaring to -1

   * $t$ generators squaring to +1

* $\mathbb{C}$ for the [[complex numbers]] regarded as an [[associative algebra]] over $\mathbb{R}$;

* $\mathbb{H}$ for the [[quaternions]] regarded as an [[associative algebra]] over $\mathbb{R}$;

* $\mathbb{K}[n] \coloneqq Mat_{n \times n}(\mathbb{K})$ for the [[matrix algebra]] of $n \times n$ matrices with [[coefficients]] in $\mathbb{K}$.

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

For low dimensions of real Clifford algebras, there are the following isomorphisms of [[associative algebras]] over $\mathbb{R}$

$$
  Cl_{0,1}
  \simeq
  \mathbb{R} \oplus \mathbb{R}
$$

$$
  Cl_{1,1}
   \simeq
  \mathbb{R}[2]
$$

$$
   Cl_{1,0}
    \simeq
   \mathbb{C}
$$

$$
  Cl_{2,0} \simeq \mathbb{H}
$$

=--

(e. g. [Figueroa-O'Farrill, lemma32](#FigueroaOFarrill))

+-- {: .num_prop #Smooth0TypeIsSheavesOnSmoothMfd}
###### Proposition

For $n,s,t \in \mathbb{N}$ then there are the following [[isomorphisms]] of [[associative algebras]] over $\mathbb{R}$:

$$
  Cl_{n+2,0} \simeq Cl_{0,n} \otimes_{\mathbb{R}} Cl_{2,0}
$$

$$
  Cl_{0,n+2} \simeq Cl_{n,0} \otimes_{\mathbb{R}} Cl_{0,2}
$$

$$
  Cl_{s+1.t+1} \simeq Cl_{s,t} \otimes_{\mathbb{R}} Cl_{1,1}
$$

=--

(e.g. [Lawson-Michelsohn 89, theorem 4.1](#LawsonMichelsohn89), [Figueroa-O'Farrill, lemma 2](#FigueroaOFarrill))

+-- {: .num_prop #Smooth0TypeIsSheavesOnSmoothMfd}
###### Proposition

For $n,n_1, n_2 \in \mathbb{N}$ there are the following [[isomorphisms]] of [[associative algebras]] over $\mathbb{R}$:

* $\mathbb{R}[n_1] \otimes_{\mathbb{R}} \mathbb{R}[n_2] \simeq \mathbb{R}[n_1 n_2]$;

* $\mathbb{R}[n] \otimes_{\mathbb{R}} \mathbb{K} \simeq \mathbb{K}[n]$;

* $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus \mathbb{C}$.

* $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{H} \simeq \mathbb{C}[2]$;

* $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H} \simeq \mathbb{R}[4]$

=--

(e.g. [Lawson-Michelsohn 89, proposition 4.2](#LawsonMichelsohn89))

Now the incarnation in Clifford algebras of [[Bott periodicity]] over the [[real numbers]] is the following:

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

For all $n \in \mathbb{N}$ there are [[isomorphisms]] of [[associative algebras]] over $\mathbb{R}$ as follows:

* $Cl_{n+8,0} \simeq Cl_{n,0} \otimes_{\mathbb{R}} Cl_{8,0}$;

* $Cl_{0,n+8} \simeq Cl_{0,n} \otimes_{\mathbb{R}} Cl_{0,8}$.

where

* $Cl_{8,0} \simeq Cl_{0,8} \simeq \mathbb{R}[16]$.

=--

(e.g. [Lawson-Michelsohn 89, theorem 4.3](#LawsonMichelsohn89))

### As a superalgebra

While the [[tensor algebra]] of an $R$-module $M$ has a natural [[integer]] grading, the quadratic relation collapses this to a natural $\mathbb{Z}/2\mathbb{Z}$-grading on $Cl(M,q)$.

$$
  Cl(M,q) = Cl(M,q)^{ev} \oplus Cl(M,q)^{odd}
$$

When $M$ is projective of rank $d$, each homogeneous piece is projective of rank $2^{d-1}$.  When $q$ is nondegenerate, the even part of the Clifford algebra is also flat-locally isomorphic to a matrix ring or a sum of two matrix rings.

One can view the Clifford algebra multiplication as a quantization of the commutative [[super algebra]] $\bigwedge_R M$.

\linebreak

### Relation to orthogonal Lie algebras
  {#RelationToOrthogonalLieAlgebras}

Let $\{\Gamma_{a}\}_{a = 1}^n$ be the generators of an $n$-dimensional [[Clifford algebra]] over the [[real numbers]] corresponding to an [[inner product]]/metric $g$, hence with this [[anti-commutator]]:

\[
  \label{CliffordGeneratorsAnticommutator}
  \big\{
    \Gamma_{a}
    ,\;
    \Gamma_b
  \big\}
  \;=\;
  2 g_{a b}
  \,.
\]

Then:

+-- {: .num_prop }
###### Proposition
**([[special orthogonal Lie algebra]] via [[Clifford algebra]])**

The elements of the Clifford algera (eq:CliffordGeneratorsAnticommutator) given by

\[
  \label{CliffordLieAlgebra}
  \begin{aligned}
    r_{a_1 a_2}
    & \coloneqq\;
    \tfrac{1}{4}
    \big[
      \Gamma_{a_1},
      \Gamma_{a_2}
    \big]
    \\
    & =
    \left\{
    \array{
      \tfrac{1}{2}
      \Gamma_{a_1}
      \Gamma_{a_2}
      &\vert&
      a_1 \neq a_2
      \\
      0 &\vert& otherwise
    }
    \right.
  \end{aligned}
\]

for all $a_1, a_2 \;\in\; \{1, \cdots, n\}$,

and equipped with the [[commutator bracket]] $[a,b] \coloneqq a b - b a$

[[linear span|span]] a [[Lie algebra]] which is [[isomorphism|isomorphic]] to the [[special orthogonal Lie algebra]] $\mathfrak{so}(n,g)$ with respect to $g$, in that their [[commutator]] [[Lie bracket]] is:

$$
  \big[
    r_{a_1 a_2}
    \,,
    r_{b_1 b_2}
  \big]
  \;=\;
    g_{a_2 b_1}
    r_{a_1 b_2}
    -
    g_{a_1 b_1}
    r_{a_2 b_2}
    +
    g_{a_2 b_2}
    r_{b_1 a_1}
    -
    g_{a_1 b_2}
    r_{b_1 a_2}
$$

=--

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

First observe that

\[
  \label{CliffordLieActionOnVectorsInComponents}
  \begin{aligned}
  \big[
    \tfrac{1}{2}
    \Gamma_{a_1} \Gamma_{a_2}
    ,\;
    \Gamma_b
  \big]
  &
  =
  \tfrac{1}{2}
  \Gamma_{a_1}
  \big\{
    \Gamma_{a_2},
    \;
    \Gamma_b
  \big\}
  -
  \tfrac{1}{2}
  \big\{
    \Gamma_{a_1},
    \;
    \Gamma_b
  \big\}
  \Gamma_{a_2}
  \\
  & =
  g_{a_2 b}
  \Gamma_{a_1}
  -
  g_{a_1 b}
  \Gamma_{a_2}
  \end{aligned}
\]

Here the first step may be thought of as the _graded_ [[Jacobi identity]] in the Clifford [[super Lie algebra]], but it is also immediately verified by inspection. The second step evaluates the defining anti-commutators (eq:CliffordLieActionOnVectorsInComponents).

With this we compute as follows, assuming, without restriction of generality, that $a_1 \neq a_2$ and $b_1 \neq b_2$:

$$
  \begin{aligned}
    \big[
      \tfrac{1}{4}
      [\Gamma_{a_1}, \Gamma_{a_2}]
      ,
      \;
      \tfrac{1}{4}
      [\Gamma_{b_1}, \Gamma_{b_2}]
    \big]
    & =
    \tfrac{1}{4}
    \Big[
      \big[
        \tfrac{1}{4}
        [\Gamma_{a_1},\Gamma_{a_2}]
        ,\,
        \Gamma_{b_1}
      \big]
      ,\,
      \Gamma_{b_2}
    \Big]
    +
    \tfrac{1}{4}
    \Big[
      \Gamma_{b_1},
      \big[
        \tfrac{1}{4}
        [\Gamma_{a_1},\Gamma_{a_2}]
        ,\,
        \Gamma_{b_2}
      \big]
    \Big]
    \\
    & =
    g_{a_2 b_1}
    \tfrac{1}{4}
    [\Gamma_{a_1},\Gamma_{b_2}]
    -
    g_{a_1 b_1}
    \tfrac{1}{4}
    [\Gamma_{a_2},\Gamma_{b_2}]
    +
    g_{a_2 b_2}
    \tfrac{1}{4}
    [\Gamma_{b_1},\Gamma_{a_1}]
    -
    g_{a_1 b_2}
    \tfrac{1}{4}
    [\Gamma_{b_1}, \Gamma_{a_2}]
  \end{aligned}
$$

Here the first step is again the graded [[Jacobi identity]] (and is again also immediely checked by inspection), while the second step uses (eq:CliffordLieActionOnVectorsInComponents).

=--

\linebreak

### Relation to $Spin$ groups

Let $M$ be a projective $R$-module of finite [[rank]], and let $q$ be nondegenerate.  Write $Cl(M,q)^\times$ for the [[group of units]] of the Clifford algebra $Cl(M,q)$.

The **Clifford group** $\Gamma_{M,q}(R)$ is the subgroup of elements $x$ for which twisted conjugation stabilizes the submodule $M \subset Cl(M,q)$.  Here, twisted conjugation is defined by $y \mapsto x y\alpha(x)^{-1}$, where $\alpha$ is the automorphism of $CL(M,q)$ induced by the $-1$ map on $M$.  Since twisted conjugation by $M$-stabilizing elements amounts to reflection $y \mapsto y - 2\frac{(x,y)}{q(x)}x$, there is a canonical map $\Gamma_{M,q}(R) \to O(M,q)$, and the Clifford group is in fact a central extension of the orthogonal group by $R^\times$.

The Clifford group is made of homogeneous elements in the $\mathbb{Z}/2\mathbb{Z}$-grading, and the subgroup of even elements is a normal subgroup of index two.  One also has a [[spinor]] norm $Q: \Gamma_{M,q}(R) \to R^\times$ on the Clifford group, defined by $Q(x) = x^t x$, where $x \mapsto x^t$ is the anti-involution of the Clifford algebra defined by opposite multiplication in the tensor algebra.

The [[Pin group]] $Pin_{M,q}(R)$ is the group elements of the Clifford group with [[spinor]] norm 1.  The [[Spin group]] $Spin_{M,q}(R)$ is the group of elements in the even subgroup of the Clifford group with [[spinor]] norm 1.

The restriction of the map $\Gamma_{M,q}(R) \to O(M,q)$ to the Pin group may not be surjective, but it is for positive definite real vector spaces.  The kernel is the group $\mu_2(R)$ of elements of $R$ that square to 1.  Similarly, the [[Spin group]] has a map to the special orthogonal group with kernel $\mu_2(R)$, but it may not be surjective in general.

One can use base change to define the groups given above as functors on commutative $R$-algebras.

### Relation to exterior algebra (quantization)
 {#AsQuantizedExteriorAlgebra}

For $V$ an [[inner product space]], the [[symbol map]] (see there) constitutes an [[isomorphism]] of the underlying [[super vector space]]s of the Clifford algebra with the [[exterior algebra]] on $V$.

One may understand the Clifford algebra as the [[quantization]] [[Grassmann algebra]] induced from the [[inner product]] regarded as an odd [[symplectic form]].

## Spinors

For nondegenerate [[quadratic forms]] on real vector spaces, [[spinors]]/[[spin representations]] are distinguished linear representations of [[Spin groups]] that are not pulled back from the corresponding special orthogonal groups.  In other words, the central element $-1$ acts nontrivially.  They can be realized as restrictions of representations of the even parts of Clifford algebras.  Since even parts of Clifford algebras are (up to complexification) the sum of one or two matrix rings, their representation theory is quite simple.

The specific nature of spinor representations possible depends on the signature of the vector space modulo 8.  This is a manifestation of Bott periodicity.  One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra.  In even dimensions, this splits into two Weyl spinors (called half-spin representations).  One may also have real representations called Majorana spinors, and these may decompose into Majorana-Weyl spinors.

There are infinite dimensional Clifford algebra constructions that appear in conformal field theory.  One may extend the above discussion to topological $R$-modules and continuous quadratic forms, and one obtains canonical central extensions of infinite dimensional groups and algebras by a relative determinant construction.  Semi-infinite wedge spaces are spinor modules for Clifford algebras of quadratic Tate $R$-modules.

## Warnings

There is a difference of sign convention between differential geometers (following Atiyah) and everyone else.

Clifford algebras are often defined using bilinear forms instead of quadratic forms (and one often sees incorrect definitions of quadratic forms in terms of bilinear forms).  Such definitions will yield wrong (or boring) objects when 2 is not invertible.

Special orthogonal groups are often defined as the kernel of the determinant map on the corresponding orthogonal groups, but in characteristic 2, the determinant is trivial, while the Clifford grading (called the Dickson invariant) is not.

The Clifford group is sometimes defined without the twist in the conjugation, and this means the map to the orthogonal group may not be a surjection, and the action on $M$ is by negative reflections.

The spinor norm is sometimes defined with the opposite sign.

Special orthogonal groups over the reals are sometimes defined to be the connected component of the identity in the orthogonal group.  In indefinite signature, this defines an index two subgroup of the special orthogonal group.

Spin groups in signature $(m,n)$ for $m,n \geq 2$ have fundamental groups of order two.  They are simply connected when $m$ or $n$ is at most one.

## Related concepts

* [[geometric algebra]]

* [[Clifford module]],

* [[Clifford bundle]]

* [[Clifford module bundle]]

* [[Fierz identity]]

* [[spin geometry]], [[spin group]]

* [[2-Clifford algebra]]

* [[Feynman slash notation]]

* [[Clifford ring]]

* [[Pythagorean ring]]

## References

The notion of Clifford algebra is due to

* [[William Clifford]], *Applications of Grassmann's extensive algebra*, American Journal of Mathematics **1** 4 (1878) 350-358 [[doi:10.2307/2369379](https://doi.org/10.2307/2369379), [jstor:2369379](https://www.jstor.org/stable/2369379)]

  > (referring to [[Hermann Grassmann]]'s *[[Ausdehnungslehre]]*)

Further early discussion:

* [[Emil Artin]], §V.4 in: *[[Geometric Algebra]]*, Wiley 1957 (1988) [[ISBN:978-1-118-16454-9](https://www.wiley.com/en-ae/Geometric+Algebra-p-9781118164549), Wikipedia entry, [ark:/13960/t4nk37034](https://archive.org/details/geometricalgebra033556mbp/page/n5/mode/2upa)]

* {#Cartan66} [[Élie Cartan]], _Theory of Spinors_, Dover,  (1966)

Brief introductions:

* [[Jean Gallier]], _Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions_ ([arXiv:0805.0311](https://arxiv.org/abs/0805.0311))

* Marc Lachieze-Rey, _Spin and Clifford algebras, an introduction_ ([arXiv:1007.2481](https://arxiv.org/abs/1007.2481))

Standard textbook accounts:

* {#LawsonMichelsohn89} [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989) ISBN 0-691-08542-0

* {#Lounesto} [[Pertti Lounesto]], _Clifford  Algebras and Spinors_, London Mathematical Society No. 286. Cambridge University Press, second edition, 2001 ([doi:10.1017/CBO9780511526022](https://doi.org/10.1017/CBO9780511526022))

See also

* [[Eckhard Meinrenken]], _Clifford algebras and Lie groups_ ([doi:10.1007/978-3-642-36216-3](https://link.springer.com/book/10.1007/978-3-642-36216-3)

For a program that promotes the use of Clifford algebra as a good expositional tool in introductory [[mechanics]] see _[[Geometric Algebra]]_.

See also the discussion of [[Majorana spinors]]

* {#FigueroaOFarrill} [[José Figueroa-O'Farrill]], _Majorana spinors_ ([pdf](http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/Majorana.pdf))

On [[subgroups]] of Clifford algebras reminiscent of [[gauge groups]] of relevance in the [[standard model of particle physics]] and [[grand unified theories]] thereof:

* {#Wilson20} [[Robert A. Wilson]], *A group-theorist's perspective on symmetry groups in physics* ([arXiv:2009.14613](https://arxiv.org/abs/2009.14613))

* {#Wilson21} [[Robert A. Wilson]], *On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics*, Adv. Appl. Clifford Algebras 31, 59 (2021) ([doi:10.1007/s00006-021-01160-5]( https://doi.org/10.1007/s00006-021-01160-5))

(For similar investigations, see also [here](Albert+algebra#ReferencesRelationStandardModel) at *[[Albert algebra]]*.)

[[!redirects Clifford algebras]]


Clifford bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Clifford bundle_ is a [[fiber bundle]] of [[Clifford algebras]].

## Examples

Given a [[spin bundle]] $P$, then the [[associated bundle]] $P \underset{Spin(n)}{\times} Cl_n$ is a Clifford bundle.

## Related concepts

* [[Clifford module bundle]]

* [[twisted K-theory]]

* [[Karoubi K-theory]]

* [[fiber integration in K-theory]]

## References

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter II, section 3 _[[Spin geometry]]_, Princeton University Press (1989)

[[!redirects Clifford bundles]]

[[!redirects Clifford algebra bundle]]
[[!redirects Clifford algebra bundles]]


Clifford Johnson

* [webpage](http://physics.usc.edu/~johnson1/)

## Selected writings

On [[K3]] [[orientifolds]]:

* [[Eric Gimon]], [[Clifford Johnson]], _K3 Orientifolds_, Nucl. Phys. B477: 715-745, 1996 ([arXiv:hep-th/9604129](https://arxiv.org/abs/hep-th/9604129))

On [[D-branes]] and [[duality in string theory]]:

* {#Johnson98} [[Clifford Johnson]], _Études on D-Branes_, in: [[Mike Duff]] et. al. (eds.) _Nonperturbative aspects of strings, branes and supersymmetry_, Proceedings, Trieste, Italy, March 23-April 3, 1998 ([arXiv:hep-th/9812196](https://arxiv.org/abs/hep-th/9812196), [spire:481393](https://inspirehep.net/literature/481393))

## Related entries

* [[D-brane]]

category: people


Clifford module

#Contents#
* table of contents
{:toc}

## Idea

A [[module]] for a [[Clifford algebra]].

## Related pages

* [[Atiyah-Bott-Shapiro isomorphism]]

* [[Clifford module bundle]]

## References

* [[Michael Atiyah]], [[Raoul Bott]], [[Arnold Shapiro]],  (1964), _Clifford Modules_, Topology 3 ((Suppl. 1)): 3–38, doi:10.1016/0040-9383(64)90003-5 ([pdf](http://www.ma.utexas.edu/users/dafr/Index/ABS.pdf))

[[!redirects Clifford modules]]


Clifford module bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[fiber bundle]] of [[Clifford modules]].

For instance a [[spinor bundle]].

## Related pages

* [[Clifford bundle]]

* [[Dirac operator]]

* [[Karoubi K-theory]]

## References

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], chapter II, section 3 _[[Spin geometry]]_, Princeton University Press (1989)

[[!redirects Clifford module bundles]]


Clifford ring

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Group theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A $\mathbb{Z}$-[[Clifford algebra]].

## Definition

Given a [[quadratic abelian group]] $G$ with a [[quadratic function]] $q:G \to \mathbb{Z}$, the **Clifford ring** $\mathrm{Cl}(G, q)$ is the [[quotient ring]] of the [[tensor ring]] $T(G)$ by the [[ideal]] generated by the relations $g \cdot g - q(g)$ for all $g \in G$.

### Universal property

Given a [[quadratic abelian group]] $G$ with a [[quadratic function]] $q:G \to \mathbb{Z}$, the **Clifford ring** is a [[ring]] $\mathrm{Cl}(G, q)$ with canonical ring homomorphism $j:\mathbb{Z} \to \mathrm{Cl}(G, q)$ with a abelian group [[homomorphism]] $g:G \to \mathrm{Cl}(G, q)$ such that

* for every element $a:G$, $g(a) \cdot g(a) = j(q(g(a)))$

* for every other ring $R$ with abelian group homomorphism $h:G \to R$ where for every element $a:G$, $h(a) \cdot h(a) = j_R(q(h(a)))$, there is a unique ring homomorphism $i:\mathrm{Cl}(G, q) \to R$ such that $i \circ g = h$.

## See also

* [[ring]]

* [[tensor ring]]

* [[super ring]]

* [[exterior ring]]

* [[Clifford algebra]]

* [[quadratic abelian group]]

[[!redirects Clifford rings]]


Clifford Taubes

* [Wikipedia entry](http://en.wikipedia.org/wiki/Clifford_Taubes)

## Selected writings

On the [[rigidity theorem for elliptic genera]]:

* [[Clifford Taubes]], _$S^1$-actions and elliptic genera_, Comm. Math. Phys. Volume 122, Number 3 (1989), 455-526 ([euclid:cmp/1104178471](https://projecteuclid.org/euclid.cmp/1104178471))

* [[Raoul Bott]], [[Clifford Taubes]], _On the Rigidity Theorems of Witten_, Journal of the American Mathematical Society
Vol. 2, No. 1 (Jan., 1989), pp. 137-186 ([doi:10.2307/1990915](https://doi.org/10.2307/1990915))

category: people


Clifford Truesdell

* [Wikipedia entry](http://en.wikipedia.org/wiki/Clifford_Truesdell)

Truesdell championed an [[axiom|axiomatic]] approach to [[continuum physics]]/[[irreversible thermodynamics]] which goes by the name "[[rational thermodynamics]]". Exposure to this axiomatic approach partly motivated the development of [[synthetic differential geometry]] and [[cohesion]] by his student [[William Lawvere]] (see there for more) as a [[foundations]] for [[continuum mechanics]].

## Selected writings

On [[classical field theory]]:

* [[Clifford Truesdell]], [[Richard A. Toupin]], *The Classical Field Theories*, in: *Principles of Classical Mechanics and Field Theory*, Encyclopedia of Physics, volume III/1, Springer (1960) 226-858 [[doi:10.1007/978-3-642-45943-6_2](https://doi.org/10.1007/978-3-642-45943-6_2)]

* _Rational thermodynamics_, Springer (1972)

* _A first course in rational continuum mechanics_. Pure and Applied Mathematics, Academic Press, New York, San Francisco, London, 1977, xxiii + 280 pp., $23.00 ([review](http://projecteuclid.org/euclid.bams/1183548435))

category: people


Clifford-Klein space form

[[!redirects Clifford-Klein form]]

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##Idea

A **Clifford–Klein space form** is a [[double coset space]] $\Gamma \backslash G/H$, where $G$ is a [[Lie group]], $H$ a [[closed subspace|closed]] [[subgroup]] of $G$, and $\Gamma$ a [[discrete group|discrete]] [[subgroup]] of $G$ that [[action|acts]] [[properly discontinuous action|properly discontinuously]] and [[free action|freely]] on the [[homogeneous space]] $G/H$.

The classical space forms are the [[cosets]] of the [[n-sphere]] $S^n$ or the [[Cartesian space]] $\mathbb{R}^n$ or the [[hyperbolic space]] $\mathbb{H}^n$ by [[discrete group|discrete]] [[subgroups]] of their [[isometry group]] [[action|acting]] [[properly discontinuous action|properly discontinuously]] ([Carmo 92, chapt 8](#Carmo92)).

The [[properly discontinuous action|properly discontinuous]] [[free action|free]] [[quotient space|quotients]] of [[n-spheres]] by [[discrete groups]] of [[isometries]] are also called _[[spherical space forms]]_. The classification of these was raised as an open problem by [Killing 1891](#Killing1891) and a complete solution was finally compiled by ([Wolf 74](#Wolf74)). For more on this see at _[[group actions on n-spheres]]_.

## Related entries

* [[locally Klein geometry]]

* [[Bertrand Russell]], _[[An Essay on the Foundations of Geometry]]_

| [[synthetic geometry]]  |
|-------------------------|
| [[Euclidean geometry]]  |
| [[hyperbolic geometry]] |
| [[elliptic geometry]]   |

## References

* {#Killing1891} [[Wilhelm Killing]], _Ueber die Clifford-Klein’schen Raumformen_, Math. Ann. 39 (1891), 257–278 ([doi:10.1007/BF01206655](https://doi.org/10.1007/BF01206655))

* [[Heinz Hopf]], *Zum Clifford-Kleinschen Raumproblem*,  Math. Ann. 95, 313–339 (1926).  ([doi:10.1007/BF01206614](https://doi.org/10.1007/BF01206614), [pdf](https://link.springer.com/content/pdf/10.1007/BF01206614.pdf))

* Toshiyki Koyabashi, _Discontinuous Groups and Clifford—Klein Forms of Pseudo-Riemannian Homogeneous Manifolds_, ([pdf](http://www.ms.u-tokyo.ac.jp/~toshi/texpdf/33-sum94lec-preprint.pdf))

* {#Wolf74} [[Joseph Wolf]], _Spaces of constant curvature_, Third ed.: Publish or Perish, Boston, 1974, Sixth edition: AMS Chelsea Publishing 2011 ([doi:10.1090/chel/372](https://doi.org/10.1090/chel/372))

* {#Carmo92} M. P. do Carmo, _Riemannian geometry_, Mathematics: Theory & Applications, Birkhaeuser Boston Inc., Boston, MA, 1992, Translated from the second Portuguese edition by Francis Flaherty.

* Ian Hambleton, _Topological spherical space forms_, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston ([arXiv:1412.8187](https://arxiv.org/abs/1412.8187))

[[!redirects Clifford-Klein forms]]

[[!redirects Clifford-Klein space form]]
[[!redirects Clifford-Klein space forms]]

[[!redirects space form]]
[[!redirects space forms]]


clique

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## Definition

A __clique__ of a category $C$ is a [[functor]] $T \to C$ from a [[(-2)-groupoid]] $T$, or equivalently an [[anafunctor]] to $C$ from the [[trivial category]].

So this is a pair of a category $T$ which is weakly equivalent to $1$ (i.e., $T$ is the [[indiscrete category]] on an [[inhabited set|inhabited collection]] of [[object]]s) and a [[functor]] $A\colon T \rightarrow C$.

A clique is also sometimes called an **anaobject**, since an [[object]] of $C$ is a [[functor]] (not anafunctor) to $C$ from the trivial category.

We can form a category __$Clique(C)$__ whose objects are cliques of $C$, and whose morphisms and compositions are given as follows: Given two such cliques $(T_0, A_0)$ and $(T_1, A_1)$ in $C$, say that a morphism between them is a natural transformation from $T_0 \times T_1 \stackrel{\pi}{\to} T_0 \stackrel{A_0}{\to} C$ to $T_0 \times T_1 \stackrel{\pi}{\to} T_1 \stackrel{A_1}{\to} C$, where the $\pi$ are the appropriate projections. Given such morphisms $m : (T_0, A_0) \rightarrow (T_1, A_1)$ and $n : (T_1, A_1) \rightarrow (T_2, A_2)$, and $(t_0, t_2) \in Ob(T_0 \times T_2)$, note that the composite $n_{(t_1, t_2)} m_{(t_0, t_1)}$ of corresponding components has the same value no matter what the choice of $t_1 \in Ob(T_1)$, and there is at least one such choice. Accordingly, we can take this to give a well-defined component $(n m)_{(t_0, t_2)}$, thus defining binary composition of morphisms of cliques. Similarly, we can take the identity on a clique $(T, A)$ to be the natural transformation whose component on $(t, t') \in Ob(T \times T)$ is the value of $A$ on the unique morphism from $t$ to $t'$ in $T$.

## Applications

### Objects with universal properties

Many [[universal properties]] that are commonly considered as defining "an object" actually define a clique.  For example, given two objects $a$ and $b$ of a category $C$, their [[cartesian product]] can be considered as the clique $T\to C$, where $T$ is the indiscrete category whose objects are product diagrams $a \overset{\leftarrow}{p} c \overset{\to}{q} b$, and where the functor $T\to C$ sends each such diagram to the object $c$ and each morphism to the unique comparison isomorphism between two cartesian products.  Note that unlike "the product" of $a$ and $b$ considered as a single object, this clique is defined without making any arbitrary choices.  This of course is the same philosophy which leads to [[anafunctors]], and so cliques are closely related to anafunctors.

### Cliques and anafunctors

There is an obvious [[anafunctor]] from $Clique(C)$ into $C$, through which every other anafunctor into $C$ factors in an essentially unique way into a genuine functor. If we ignore size issues (like that $Clique(C)$ will be large even for small $C$), this induces for $Clique(-)$ the structure of a (2-)monad on $Str Cat$ (the (2-)category of "genuine" functors between categories), such that the [[Kleisli category]] for this monad will be $Cat_{ana}$ (the (2-)category of anafunctors between categories).  This monad can also be described more explicitly; in particular the unit (a "genuine" functor) $C\to Clique(C)$ takes each object $c\in C$ to the corresponding clique $c\colon 1\to C$ defined on the domain $1$.  Note that this functor is a [[weak equivalence]], i.e. fully faithful and essentially surjective on objects, but not a strong equivalence unless one assumes the [[axiom of choice]].

In particular, we can use cliques to *define* anafunctors, taking an anafunctor from $C$ into $D$ to simply be a genuine functor from $C$ into $Clique(D)$. (With composition of these defined in a straightforward way, and natural transformations between these being simply natural transformations of the corresponding genuine functors into $Clique(D)$). Accordingly, $Clique(-)$ is itself the same as $Cat_{ana}(1, -)$, and this can also be taken as a definition of a clique (hence the alternate name *anaobject*).

### Monoidal strictifications

Unsurprisingly, cliques provide a useful technical device for describing strictifications of [[monoidal category|monoidal categories]].

It is relevant first to recall the original form of Mac Lane's [[coherence theorems|coherence theorem]]: the free monoidal category on one generator, $F[1]$, is monoidally equivalent to the discrete monoidal category $(\mathbb{N}, +, 0)$. Thus each connected component $C_n$ of $F[1]$ is an indiscrete category whose objects are the possible $n$-fold tensor products of the generator, possibly with instances of the unit object folded in; the indiscreteness says that "all diagrams built from associativity and unit constraints commute".

One canonical way to strictify a monoidal category $M$ is by considering cliques in $M$ where the domains are the $C_n$ and the functors model associativity and unit constraints, in the following precise sense:

1. We may form a monoidal category $Oper(M)$ whose objects are functors
$$F: M^j \to M$$
for natural numbers $j$, and whose morphisms are natural transformations between such functors. The tensor product of $F: M^j \to M$ and $G: M^k \to M$ in $Oper(M)$ is the composite
$$M^{j+k} \cong M^j \times M^k \stackrel{F \times G}{\to} M \times M \stackrel{\otimes}{\to} M$$
and the rest of the monoidal structure on $Oper(M)$ is inherited from the monoidal structure on $M$.

1. By freeness of $F[1]$, we have a (strict) monoidal functor
$$\kappa: F[1] \to Oper(M)$$
uniquely determined as the one which sends the generator $1$ of $F[1]$ to $Id_M$. On each connected component $C_n$ of $F[1]$, this restricts to a functor
$$C_n \stackrel{\kappa|}{\to} Cat(M^n, M)$$

1. Then, for each $n$-tuple of objects $(x_1, \ldots, x_n)$ of objects of $M$, there is an associated clique $\kappa_{x_1, \ldots, x_n}$ in $M$:
$$C_n \stackrel{\kappa|}{\to} Cat(M^n, M) \stackrel{eval_{(x_1, \ldots, x_n)}}{\to} M$$

1. Finally, the objects of the strictification $M^{st}$ are $n$-tuples $(x_1, \ldots, x_n)$ of objects of $M$. A morphism
$$(x_1, \ldots, x_m) \to (y_1, \ldots, y_n)$$
is by definition a clique morphism $\kappa_{x_1, \ldots, x_m} \to \kappa_{y_1, \ldots, y_n}$. There is an evident strict monoidal category structure on $M^{st}$ which at the object level is just concatenation of tuples.

It is straightforward to check that the natural inclusion

$$i: M \to M^{st},$$

which interprets each object as a 1-tuple and each morphism as an evident clique morphism, is a monoidal equivalence. The essential idea is that there is a canonical clique isomorphism

$$(x_1, x_2, \ldots, x_n) \to i(Bracketing(x_1 \otimes \ldots \otimes x_n))$$

for every choice of bracketing the tensor product on the right in $M$ (possibly with units thrown in).

## In graph theory

There is a notion of clique in an undirected simple [[graph]] familiar to graph-theorists: a _clique_ in a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that any two distinct vertices $x,y \in C$ are connected by an edge. The cliques of cardinal $1$ are exactly the vertices and the cliques of cardinal $2$ are exactly the edges. Thus, if we know all the cliques of a graph, we know automatically this graph. If we know a graph, we can compute the cliques of this graph. There is thus a bijection between the undirected simple graphs and the sets of all cliques of an undirected simple graph. But it would be better to have an intrinsic definition of what is the set of all cliques of an undirected simple graph without reference to graphs in order to state this bijection. The set of all cliques of an undirected simple graph can be seen as a [[coherence space]] and we then have a bijection between undirected simple graphs and coherence spaces. Note that although it is a bijection, it requires some work to compute the set of all cliques of an undirected graph for a human or a computer. Thus, a coherence space can be seen as a richer description of an undirected simple graph.

This definition is specialized to simple graphs, however, and a more general definition that works for arbitrary undirected graphs (possibly containing loops and multiple edges) takes a clique (of size $n$) in $G$ to be a graph homomorphism $C : K_n \to G$ from the [[complete graph]] on $n$ vertices.  Indeed, this latter definition could also be taken as a reasonable notion of clique in any undirected graph/[[quiver]].  Equivalently, a clique in this sense is a subgraph $C$ of $G$ which is _indiscrete_: there is exactly one edge in $C$ from $x$ to $y$ for any vertices $x$, $y$ of $C$.

The categorical notion of clique is one step removed from that: a clique in a category $C$ is a functor $i: K \to C$ where the underlying graph of $K$ is indiscrete. The generic "picture" of a clique in a category is reminiscent of (and no doubt the etymology derives from) the graph-theoretic notion, even if the notions are technically distinct.

[[!redirects clique]]
[[!redirects cliques]]


clone

# Clones

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## Idea

An __abstract clone__ is a structure that describes a single-sorted [[algebraic theory]] in a presentation-invariant way.

An abstract clone is equivalent to a [[cartesian operad]], that is, a [[cartesian multicategory]] with one object. However, clones are presented differently, and quite economically, with projections rather than with symmetries, contraction and weakening.

Abstract clones are equivalent to [[Lawvere theories]], and also to [[finitary monads]]. These are all different ways of giving presentation-invariant descriptions of single-sorted algebraic theories.

## Definition

A set of algebraic operations on a fixed set $S$ is a __concrete clone__
on $S$ if it contains all (component) projections $S^{n}\to S$ and is closed under composition ("superposition").

An **abstract clone** consists of an abstract set of "$n$-ary operations" for every $n$ together with projection and composition operations.  This is the notion that's equivalent to a cartesian operad or a Lawvere theory.

More precisely, the usual presentation of abstract clone comprises

* a set $T(n)$ for each natural number $n$;

* functions $\eta_n : n \to T(n)$;

* functions $c_{m,n} : T(m) \times T(n)^m\to T(n)$

such that

* $\eta(i) \rhd j. t=t$;

* $t \rhd i.\eta(i) =t$;

* $t \rhd i.(u(i) \rhd j.v(j)) = (t \rhd i.u(i)) \rhd j. v(j)$

where we omit subscripts and write $(t \rhd i.u(i))$ for $c(t,(u_i)_i)$.

This resembles the [[monad (in computer science)#RelationToMonadsInCategoryTheory|Kleisli triple presentation]] of a [[monad]], except that $n$ is a natural number rather than an arbitrary set; in this sense it is the Kleisli triple form of a [[relative monad]] for the embedding $\mathbf{FinSet}\to\mathbf{Set}$.

## Examples

### Concrete clones as abstract clones

Suppose a concrete clone is given on a set $S$, i.e. a set of operations $S^n\to S$ of different arities, closed under composition and containing all the projections. We regard this as an abstract clone by putting $T(n)$ as the set of admitted operations $S^n\to S$. The functions $\eta_n$ pick out the projections, and $c_{m,n}$ is multi-ary function composition.

### Abstract clones from algebraic signatures

An [[signature (in logic)|algebraic signature]] comprises, for each number $n$, a set $\Sigma_n$ of formal $n$-ary operations. For example, the theory of groups has $\Sigma_0=\{1\}$; $\Sigma_1=\{(-)^{-1}\}$; $\Sigma_2=\{\cdot\}$.

We then build for all $n$ the terms in $n$ variables, written $x_1\dots x_n\vdash x_i$, by the following inductive definition:

* $x_1\dots x_n\vdash x_i$ is a term if $1\leq i\leq n$;

* if $n\vdash t_1\ \ \dots\ \  n\vdash t_m$ are terms and $\sigma\in \Sigma_m$ then $n\vdash \sigma(t_1,\dots, t_m)$ is a term. Here we write $n =(x_1\dots x_n)$.

Now we form an abstract clone by putting $T_\Sigma(n)=\{t\ \ |\ \ (n\vdash t\ \ \text{ is a term})\}$.
The functions $\eta_n$ picks out the variables, and the function $c_{m,n}$ amount to substitutions of terms for variables, which produces well-formed terms.

The inhabitants of $T_\Sigma$ are sometimes called "derived operations", since although they are not in the signature they have an interpretation in any model.

### Abstract clones from presentations of algebraic theories

An algebraic theory over a signature is given by a collection of axioms of the form $\Gamma\vdash t=u$, where $\Gamma \vdash t$ and $\Gamma \vdash u$. We then form  equivalence relations $\sim_n$ of terms by closing under substitution instances, congruence, and reflexivity, symmetry and transitivity. These are the equations that are true in all models of the theory.

We can now form an abstract clone for the theory, derived from the abstract clone for the signature. Put $T(n)=T_\Sigma(n)/_{\sim_n}$. The $\eta$ and $c$ respect equivalence relations because we closed under substitution instances and congruence.

For example, starting from the theory of groups, $T(n)$ will be the free group on $n$ generators.

### Algebraic theories from abstract clones

Conversely, any abstract clone can be regarded as a presentation of an algebraic theory. The signature has $\Sigma_n=T(n)$, and we include the axioms

* $\eta_n(i) = x_i$

* $s(t_1\dots t_n) = (s \rhd i. t(i))$.

An abstract clone is a presentation of a "saturated" algebraic theory: all derived operations are equivalent to basic operations that are already in the signature.

### Lawvere theories and abstract clones

If $T$ is an abstract clone, we can form a category whose objects are natural numbers and where a morphism $m\to n$ is a tuple $T(m)^n$. The identity morphisms are $\eta_n$. Composition is $(g\circ f)(i)=(g(i)\rhd j.f(j))$. This category is a [[Lawvere theory]].

(This is the [[opposite category]] of the [[Kleisli category]] of $T$ regarded as a [[relative monad]].)

Conversely, if $L$ is a Lawvere theory then we can build a clone $T(n)=L(n,1)$. This gives an equivalence between abstract clones and Lawvere theories.

## Algebras of abstract clones and connection with finitary monads

We can define an __algebra of an abstract clone $T$__ to be a set $X$ together with $n$-ary function $t \rhd : X^n\to X$ for each $t\in T(n)$, such that

* $\eta_i \rhd i.x(i)= x(i)$;

* $(t\rhd i.u(i))\rhd j.x(j)= t\rhd i.(u(i)\rhd j.x(j))$.

Here we are writing $\rhd$ for both the clone composition and the algebra structure, and again writing $t \rhd i.x(i)$ for $t\rhd(x_i)_i$.

For example, an algebra for the abstract clone of groups is a group.

Now the category of $T$-algebras has a forgetful functor

$$ T\mathbf{-Alg} \to \mathbf{Set}$$

One can show that this functor is [[monadic]] by showing it preserves limits, satisfies the solution set condition, reflects isomorphisms, and has reflexive coequalizers. Thus every abstract clone gives rise to a monad.

Another way to define this monad is to note that $T$ can be regarded as a functor $\mathbf{FinSet}\to \mathbf{Set}$, with $T(f):T(m)\to T(n)$ given by $T(f)(t)=(t\rhd i.\eta(f(i)))$.
Then we extend this to a functor $M:\mathbf{Set}\to\mathbf{Set}$ by a [[coend]] or by [[Kan extension]]:

$$ M(X) = \int^n T(n)\times X^n = (\mathrm{Lan}\,T)(X)$$

In other words, an inhabitant of $M(X)$ is a derived operation $t\in T(n)$ formally applied to an $n$-tuple in $X^n$, modulo change of variables.
This functor $M$ can be given the structure of a monad on $\mathbf{Set}$, and its [[Eilenberg-Moore algebras]] are the $T$-algebras.

Conversely, suppose that $M$ is a [[monad]] on the category of sets. We can define an abstract clone by regarding a natural number $n$ as a finite set and putting $T(n)=M(n)$. Then $\eta$ is the unit of the monad and $c$ is the Kleisli composition (called "monadic bind").

This gives an equivalence between [[finitary monads]] and abstract clones. They are also the [[monad with arities|monads with arities]] in $\mathbf{FinSet}$.

## Monadicity of the category of abstract clones

The definition of abstract clones given here is itself a presentation of algebraic structure. Indeed we can consider an obvious notion of homomorphism of clones, and then the forgetful functor

$$ \mathbf{AbstractClones} \to [\mathbf{FinSet},\mathbf{Set}]$$

is [[monadic]].

In fact this adjunction is [[enriched]] in $[\mathbf{FinSet},\mathbf{Set}]$. Moreover, we can consider a more general notion of abstract clone enriched in a [[locally presentable category]], where the arities $n$ are replaced by finitely presentable objects. Then the plain abstract clones themselves are algebras for an abstract clone enriched in $[\mathbf{FinSet},\mathbf{Set}]$. One way to present this formally is in terms of the substitution algebras of Fiore, Plotkin and Turi.

Another view of this adjunction starts from viewing $[\mathbf{FinSet},\mathbf{Set}]$ as a [[monoidal category]]. To get this, we can understand functors $\mathbf{FinSet}\to \mathbf{Set}$ as [[filtered colimit]] preserving endofunctors $\mathbf{Set}\to\mathbf{Set}$, and then composition forms a monoidal structure on a category of endofunctors. In detail, the unit is $\mathbf{FinSet}(1,-)$ and tensor product is

$$(F \otimes G) (n) = \int^m F(m) \times G(n)^m = ((\mathrm{Lan}\,F)\circ G)(n)$$

Then the category of abstract clones is (equivalent to) the category of [[monoids]] for this monoidal structure. Since a finitary monad is a filtered-colimit-preserving "monoid in the category of endofunctors", this also gives an abstract view of the connection between abstract clones and finitary monads on $\mathbf{Set}$.

## References

* Ágnes Szendrei, _Clones in universal algebra_, Séminaire de mathématiques supérieures 99, Les presses de l'université de Montreal, 1986. — 166 p.

A rather general framework is discussed in

* Zhaohua Luo, _Clone theory, its syntax and semantics, applications to universal algebra, lambda calculus and algebraic logic_, [arxiv/0810.3162](https://arxiv.org/abs/0810.3162)
* Dietlinde Lau, _Function algebras on finite sets: Basic course on many-valued logic and clone theory_, Springer Monographs in Mathematics

A common generalization of a clone and of an operad is proposed, using a new notion of a verbal category, in

* S. Tronin, _Abstract clones and operads_, Siberian Mathematical Journal 43, No.4, 746–755, 2002 [link](http://rd.springer.com/content/pdf/10.1023%2FA%3A1016340806503.pdf)

Another unification of clones and operads is via the formalism in

* Pierre-Louis Curien, _Operads, clones, and distributive laws_, [arxiv/1205.3050](https://arxiv.org/abs/1205.3050)

See also the thesis

* Miles Gould, _Coherence for operadic theories_, Glasgow 2009 [pdf](http://theses.gla.ac.uk/689/1/2009gouldphd.pdf)

Algebraic theories are treated as abstract clones in the enriched setting (but abstractly), and monadicity is dealt with, in

* [[Max Kelly]], [[John Power]], _Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads_. JPAA 1993. [doi:10.1016/0022-4049(93)90092-8](https://doi.org/10.1016/0022-4049%2893%2990092-8).

Substitution algebras as a theory over $[\mathbf{FinSet},\mathbf{Set}]$ are presented in:

* [[Marcelo Fiore]], [[Gordon Plotkin]] and Daniele Turi. Abstract syntax and variable binding. Proc LICS 1999. [preprint](https://www.dcs.ed.ac.uk/home/dt/FiorePlotkinTuri.pdf).

One description of enriched abstract clones is in Section 5 of

* [[Sam Staton]], An algebraic presentation of predicate logic. FOSSACS 2013. [preprint](http://www.cs.ox.ac.uk/people/samuel.staton/papers/fossacs13.pdf).

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[[!redirects abstract clone]]
[[!redirects abstract clones]]
[[!redirects cartesian operad]]
[[!redirects cartesian operads]]


clopen set

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## Definition

A [[subset]] of a [[topological space]] is called **clopen** if it is both [[open set|open]] and [[closed set|closed]].  Equivalently, a clopen set is a [[complement|complemented element]] of the [[frame]] of open subsets, a definition which makes as good sense for [[locales]] as for spaces.

## Properties

The set of clopen sets in any space forms a [[Boolean algebra]].  If the space is a [[Stone space]], then it can be reconstructed from its Boolean algebra of clopens.  On the other hand, a space is [[connected space|connected]] just when the only clopens are the [[empty set]] and the whole space.

## Related concepts

* [[open subset]], [[closed subset]]

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[[!redirects clopen subset]]
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closed bicategory

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## Definition

A **closed bicategory** is a [[bicategory]] $B$ admitting all [[Kan extension|right extensions]] and [[Kan lift|right lifts]], equivalently a bicategory whose composition functor

$${\circ}_{x, y, z} \colon B(y,z) \times B(x,y) \to B(x,z)$$

participates in a [[two-variable adjunction]]. Closed bicategories were introduced by Lawvere in unpublished lecture notes *Closed categories and biclosed bicategories* (1971).

## Remarks

A closed bicategory is a [[horizontal categorification]] of a [[closed monoidal category]].  It is not to be confused with a closed [[monoidal bicategory]], which is a [[vertical categorification]] of the same concept.

Dually, a bicategory admitting all *left* extensions and lifts is called a **coclosed bicategory**, and is analogously the horizontal categorification of a [[coclosed monoidal category]]. A bicategory admitting all (right and left) extensions and lifts is a **biclosed bicategory**.

## Examples

- [[Prof]]
- The bicategory $Span(E)$ is closed if and only if the category $E$ is [[locally cartesian closed]]; see ([Day 1974, Proposition 4.1](#Day74)).

## Related entries

* [[symmetric bicategory]]

* An [[closed category|extension system]] is to a closed bicategory what a [[closed category]] is to a [[monoidal category]].

* [[locally internal category]]

## References

* {#Day74} [[Brian Day]], _Limit spaces and closed span categories_,  Lecture Notes in Mathematics, 420, 1974 ([doi:10.1007/BFb0063100](https://doi.org/10.1007/BFb0063100))

* {#BettiWalters1985} [[Renato Betti]], [[Robert F. C. Walters]], *Closed bicategories and variable category theory*, Universita degli Studi di Milano (1985), reprinted in: Reprints in Theory and Applications of Categories, **26** (2020) 1-27 $[$[tac:tr26](http://www.tac.mta.ca/tac/reprints/articles/26/tr26abs.html)$]$

[[!redirects closed bicategories]]
[[!redirects coclosed bicategory]]
[[!redirects coclosed bicategories]]
[[!redirects biclosed bicategory]]
[[!redirects biclosed bicategories]]


closed category

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### Context
#### Category theory
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[[!include category theory - contents]]
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# Contents
* table of contents
{: toc}

## Idea

A [[category]] $C$ is _closed_ if for any pair $a, b$ of [[objects]] the collection of [[morphisms]] from $a$ to $b$ can be regarded as forming itself an object of $C$.

This object is often denoted $hom(a,b)$ or $[a,b]$ or similar and often addressed as the _internal [[hom-object]]_ or simply the [[internal hom]].

A familiar kind of closed categories are [[closed monoidal category|closed monoidal categories]]. However, there is also a definition of _closed category_ that does not require the category to already be monoidal.  A monoidal structure $\otimes$, if it exists, can then be universally characterized as an (enriched) _left_ adjoint to the internal-hom, dual to the above characterization of internal-homs as right adjoints to $\otimes$; see below.

Although it may seem that in general the closed structure is less intuitive to work with than the monoidal structure, in some cases it is in fact more obvious what the correct internal-homs are than what the correct tensor product is, so the latter was originally defined as an adjoint to the former.  This is the case for the [[Gray tensor product]] and the [[projective tensor product]], for example, and was probably the case for [[abelian groups]] (the original notion of [[tensor product]]) as well.

## Definition

### Basic definition

A __closed category__ is a [[category]] $C$ together with the following data:

* A [[functor]] $[-,-] : C^{op} \times C \to C$, called the [[internal hom]]-functor.

* An [[object]] $I\in C$ called the *[[unit object]]*.

* A [[natural isomorphism]] $i\colon Id_C \cong [I,-]$.

* A transformation $j_X\colon I \to [X,X]$, [[extranatural]] in $X$.

* A transformation $L^X_{Y Z} \colon  [Y,Z] \to [ [X,Y],[X,Z]]$, natural in $Y$ and $Z$ and extranatural in $X$.

which is required to satisfy the following axioms.

* The following [[diagram]] commutes for any $X,Y$.

  $$\array{I & \overset{j_Y}{\to} & [Y,Y]\\
  & _{\mathllap{j_{[X,Y]}}}\searrow & \downarrow^{L^X_{Y Y}}\\
  & & [ [X,Y],[X,Y]]}$$

* The following diagram commutes for any $X,Y$.

  $$\array{[X,Y] & \overset{L^X_{X Y}}{\to} & [ [X,X],[X,Y]]\\
  & _{\mathllap{i_{[X,Y]}}}\searrow & \downarrow^{[j_X,1]}\\
  & & [I,[X,Y]]}$$

* The following diagram commutes for any $Y,Z$.

  $$\array{[Y,Z] & \overset{L^I_{Y Z}}{\to} & [ [I,Y],[I,Z]]\\
  & _{\mathllap{[1,i_Z]}}\searrow & \downarrow^{[i_Y,1]}\\
  & & [Y,[I,Z]]}$$

* The following diagram commutes for any $X,Y,U,V$.

  $$\array{[U,V] & \overset{L^Y_{U V}}{\to} & [ [Y,U],[Y,V]]\\
  ^{L^X_{U V}}\downarrow && \\
  [ [X,U],[X,V]] & &  \downarrow^{[1,L^X_{Y V}]} \\
  ^{L^{[X,Y]}_{[X,U],[X,V]}}\downarrow && \\
  [ [ [X,Y],[X,U]],[ [X,Y],[X,V]]] & \underset{[L^X_{Y U},1]}{\to} & [ [Y,U],[ [X,Y],[X,V]]]}$$

* Finally, the map $\gamma\colon C(X,Y) \to C(I,[X,Y])$ defined by $f \mapsto [1,f](j_X)$ is a [[bijection]].

Note that both a left-closed (non-symmetric) monoidal category and a right-closed (non-symmetric) monoidal category give rise to a closed category in this sense. See the discussion [on the nForum](https://nforum.ncatlab.org/discussion/4632/closed-category/?Focus=66659#Comment_66659).

### The Eilenberg-Kelly definition

The above definition is the one used in [LaPlaza](#LaPlaza77) and [Manzyuk](#Manzyuk09).  It differs slightly from the original definition in [Eilenberg & Kelly 1965](#EK65), which omitted the axiom involving $\gamma$ but assumed an "underlying-set-functor" $U\colon C \to Set$ as part of the structure, with an axiom asserting that $U([X,Y]) = C(X,Y)$ (naturally, i.e. as a strict equality of functors $C^{op}\times C\to Set$) and that the resulting isomorphism
$$ C(X,X) = U([X,X]) \overset{U i_{[X,X]}}{\to} U([I,[X,X]]) = C(I,[X,X]) $$
sends $1_X$ to $j_X$.  (This is then essentially a definition of $j_X$, so that it could be omitted from the definition.)

The two are essentially equivalent, and the one given here is perhaps a little simpler, as well as obeying the [[principle of equivalence]].  To be precise, every closed category in the sense of Eilenberg-Kelly is also one in the above sense, and conversely every closed category in the above sense is *isomorphic* to one in the sense of [Eilenberg & Kelly 1965](#EK65).  The latter has the same objects and internal-homs, but new hom-sets $\hat{C}(X,Y) \coloneqq C(I,[X,Y])$, and underlying-set functor $U(X) \coloneqq C(I,X)$ (note that this is different from $\hat{C}(I,X) = C(I,[I,X])$).  See [Manzyuk](#Manzyuk09) for the proof.

### Omitting units: prounital closed categories

The fact (discussed below) that closed categories, in the above sense, are equivalent to closed [[multicategories]] with unit, suggests that there ought to be a version of closed categories that *don't* necessarily have a unit object $I$.  In fact in this case the definition becomes simpler because we can omit all the $i$'s and $j$'s, but there is an additional wrinkle that we have to include "left evaluation" as a new operation.  Furthermore, it seems that we have to use the Eilenberg-Kelly version of the definition, not the LaPlaza one; the "underlying set" functor $U:C\to Set$ corresponds to the nullary homsets $C(;-)$ of a multicategory.

A **prounital closed category** is a [[category]] $C$ together with the following data:

* A [[functor]] $[-,-] : C^{op} \times C \to C$, called the [[internal hom]]-functor.

* A functor $U : C \to Set$.

* A transformation $L^X_{Y Z} \colon  [Y,Z] \to [ [X,Y],[X,Z]]$, natural in $Y$ and $Z$ and extranatural in $X$.

* A transformation $ev : U(X) \to C([X,Y], Y)$, natural in $X$ and extranatural in $Y$.

which is required to satisfy the following axioms.

* $U([X,Y]) = C(X,Y)$ strictly and naturally, i.e. $U \circ [-,-] = C(-,-)$ as functors $C^{op}\times C\to Set$.  (Note that this can be phrased in accord with the [[principle of equivalence]] by considering $U : C \to Set$ as a [[displayed category]] over $Set$.)

* For any $x\in U(X)$ and $f\in C(X,Y) = U([X,Y])$ we have $U(ev_x)(f) = U(f)(x)$ in $U(Y)$.

* For any $X,Y\in C$ we have $U(L^X_{Y Y})(1_Y) = 1_{[X,Y]}$.  (Note that $L^X_{Y Y} : [X,X] \to [ [X,Y],[X,Y]]$, so $U(L^X_{Y Y})(1_Y) : C(X,X) \to C([X,Y],[X,Y])$.)

* For any $X,Y\in C$ the composite $[X,Y] \xrightarrow{L^X_{X Y}} [ [X,X],[X,Y]] \xrightarrow{ev_{1_X}} [X,Y]$ is the identity.

* The following diagram commutes for any $X,Y,U,V$.

  $$\array{[U,V] & \overset{L^Y_{U V}}{\to} & [ [Y,U],[Y,V]]\\
  ^{L^X_{U V}}\downarrow && \\
  [ [X,U],[X,V]] & &  \downarrow^{[1,L^X_{Y V}]} \\
  ^{L^{[X,Y]}_{[X,U],[X,V]}}\downarrow && \\
  [ [ [X,Y],[X,U]],[ [X,Y],[X,V]]] & \underset{[L^X_{Y U},1]}{\to} & [ [Y,U],[ [X,Y],[X,V]]]}$$

* Some axiom(s) relating $L$ to $ev$.  For instance, for any $x\in U(X)$ the following diagram should commute:

  $$\array{ [Y,Z] & \overset{L^X_{Y Z}}{\to} & [ [X,Y],[X,Z]]\\
  & _{[ev_x,Z]}\searrow & \downarrow^{[ [X,Y],ev_x]} \\
  && [ [X,Y],Z]} $$

The third and fourth axioms, plus naturality of $ev$, should imply a more general form of the third axiom: for any $f:X\to Y$, the composite $[Y,Z] \xrightarrow{L^X_{Y Z}} [ [X,Y],[X,Z]] \xrightarrow{ev_f} [X,Z]$ equals the functorial action $[f,Z]$.

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I haven't worked out exactly what axioms are required here.
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### Symmetric and cartesian versions

A **symmetric closed category** is additionally equipped with natural isomorphisms $C^{X Y}_Z : [X,[Y,Z]] \cong [Y,[X,Z]]$ satisfying appropriate axioms; see [deSchipper75](#deSchipper75) and [DayLaPlaza78](#DayLaPlaza78).  These should correspond to [[symmetric multicategories]].

Similarly, a "cartesian closed category" (there is an unfortunate terminological clash with [[cartesian closed category]]) should furthermore have morphisms $K^X_Y : Y\to [X,Y]$ and $T^X_Y : [X,[X,Y]]\to [X,Y]$ satisfying appropriate axioms.  They should correspond to [[cartesian multicategories]].  The relation to [[combinatory logic]] suggests that it should also be possible to axiomatize the cartesian case by replacing $L$, $C$, and $T$ with a single family of morphisms $S^X_{Y Z} : [X,[Y,Z]]\to [ [X,Y],[X,Z]]$.

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I don't know of anywhere that the details are worked out for the cartesian case.
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## Examples

* Any [[closed monoidal category]] gives a closed category, by simply forgetting the tensor product and remembering only the internal-hom.  Most examples seem to be of this sort (in fact, as discussed [below](#EmbedIntoCloseMon), *every* closed category arises as the full [[subcategory]] of a closed monoidal category).  However, as remarked above it, is often the case that the closed structure is "primary" and the tensor product is defined as a [[left adjoint]] to it (see below).  For instance:

  * In the category [[Ab]] of [[abelian groups]] the internal-hom $[X,Y]$ is the set of group homomorphisms $X\to Y$ with the pointwise addition, giving a closed category.

  * More generally, for any [[commutative theory]] $T$, the category of $T$-algebras has an internal-hom $[X,Y]$ that is the set of $T$-algebra maps with the pointwise operations (this is again a $T$-algebra by commutativity).

  * In the category [[2Cat]] of [[strict 2-categories]], the internal-hom $[X,Y]$ can be taken to be the category of strict 2-functors $X\to Y$ and [[lax natural transformation|pseudo, lax, or oplax natural transformations]] between them, giving three closed categories.

  * In the category $Unif$ of [[uniform spaces]] the internal-hom $[X,Y]$ can be taken to be the set of uniformly continuous functions $X\to Y$ with either the uniformity of uniform convergence or of pointwise convergence, giving two closed categories.

  In all these examples there is a corresponding tensor product, but its construction is much less intuitive, being essentially defined by a sort of [[adjoint functor theorem]] or [[initial lift]] that gives little information about what it actually looks like.

* Any [[multicategory]] which has a unit, i.e. an object $I$ such that $C(;Y) \cong C(I;Y)$ naturally, and is closed in the sense that for any $Y,Z$ there is an object $[Y,Z]$ with natural isomorphisms $C(X_1,\dots,X_n,Y;Z) \cong C(X_1,\dots,X_n; [Y,Z])$, gives rise to a closed category.  Conversely, from any closed category we can construct a multicategory of this sort, by defining the multimaps as $C(X_1,\dots,X_n; Z) = C(I, [X_1,\dots,[X_n,Z]])$.  Thus closed categories are essentially equivalent to closed unital multicategories.

## Properties

### The closed enrichment context

The closed structure on a category $\mathcal{V}_0$ is an enrichment context $\mathcal{V}$ relative to which we can define a [[category of V-enriched categories]]. From this point of view, a closed structure is more natural than a monoidal structure since most (if not all) of this structure is forced if one insists for an enrichment context $\mathcal{V}$ for which $\mathcal{V}_0$ is self-enriched, that is, isomorphic to the underlying category $\mathcal{V}^e_0$ of some $\mathcal{V}$-enriched category $\mathcal{V}^e$. See [[category of V-enriched categories]] for details.

### Reconstruction of monoidal structure

Suppose $C$ is a closed category such that the functor $[A,[B,-]] : C \to C$ is [[representable functor|representable]] as a $C$-enriched functor.  This means we have an object $A\otimes B$ and a $C$-natural isomorphism

$$ [A,[B,X]] \cong [A\otimes B,X]. $$

Then $C$ is a (closed) monoidal category.  See [Eilenberg--Kelly (1965)](#EK65) for details.

Note that although this is analogous to the fact that a monoidal category with right adjoints to tensor product is a closed category, the requisite hypothesis is stronger.  When $C$ is given as monoidal, we need only $Set$-natural isomorphisms of hom-sets $C(A\otimes B,X) \cong C(A,[B,X])$ to make it closed, whereas when $C$ is closed we need the above $C$-natural isomorphism of *internal* hom-objects to make it monoidal.

However, in the case when $C$ is a *symmetric* closed category, this stronger condition is unnecessary: to make $C$ (symmetric) monoidal it suffices to have external natural isomorphisms of hom-sets

$$ C(A, [B,X]) \cong C(A\otimes B, X). $$

Put differently, any symmetric closed category has an underlying (symmetric) [[promonoidal category]], whereas for a non-symmetric closed category the associativity of the promonoidal structure is an extra condition.  See [Day-Laplaza, Proposition 2.3](#DayLaPlaza78).

### Embedding into closed monoidal categories {#EmbedIntoCloseMon}

By a result due to Miguel LaPlaza [(1977)](#LaPlaza77), every closed category embeds [[full and faithful functor|fully and faithfully]] into a [[closed monoidal category]] by a strong [[closed functor]], i.e., one respecting closed structure up to suitably coherent isomorphism, and this closed functor is also strong monoidal if the original closed category is closed monoidal.  In particular, this functor is defined as the composition $q = y \circ p$ of the functor

$$p : C \to E^\op$$

sending an object $X \in C$ to the object (endofunctor) $[X,-] \in E = [C,C]$, together with the [[Yoneda embedding]] $y : E^\op \to [E, Set]$.  The closed monoidal structure on the presheaf category $[E,Set]$ is given by [[Day convolution]], using the monoidal structure of the category of endofunctors $E$ (see [[closed monoidal structure on presheaves#presheaves_over_a_monoidal_category|closed monoidal structure on presheaves over a monoidal category]]).

An alternative embedding was also sketched by Day [(1974)](#Day74), and later studied by Day and LaPlaza [(1978)](#DayLaPlaza78) in the case of symmetric closed categories.  This embedding is based on the fact that the construction of a closed monoidal category of presheaves does not actually require the base category to be monoidal: it suffices for it to be a [[promonoidal category#notes|promonoidal category]], and any closed category gives rise to such a promonoidal structure (perhaps a "lax" one with noninvertible associativity).  So, instead of using the category of copresheaves on the monoidal category of endofunctors $[ [C,C],Set]$ as in (LaPlaza 1977), the embedding in (Day 1974) and (Day and LaPlaza 1978) uses $[C,Set]$ itself.  One can then restrict to a small subcategory $A$ of $[C,Set]$ and compose further with a restricted version of its Yoneda embedding in order to obtain an embedding of $C$ into a closed monoidal category that also preserves certain colimits in $C$.

### Monadicity and 2-categories

Since the notion of closed category involves a contravariant functor and extranatural transformations, it cannot be expected to be [[2-monad|2-monadic]] over the [[2-category]] [[Cat]].  It is, however, 2-monadic over the 2-category $Cat_g$ of categories, functors, and natural isomorphisms, the [[core]] of [[Cat]].  In this way we obtain a 2-category $ClCat$ of closed categories, strong [[closed functors]], and [[closed natural transformations]].  One can also define a notion of non-strong, or "lax," closed functor; although these do not seemingly arise from the 2-monad in question, they generalize lax monoidal functors between closed monoidal categories.

## Extension systems

There is a notion that is related to a [[bicategory]] in the same way that a closed category is related to a [[monoidal category]], i.e. a [[horizontal categorification]] of a closed category.  It has a set of "objects", for any two objects $A,B$ a category $\mathcal{B}(A,B)$, and internal-hom functors like those in a [[closed bicategory]].  It was defined by [Street](#Street74) under the name **extension system**, although our page [[extension system]] is about an unrelated concept with the same name.

## Related concepts

* [[closed functor]]

* [[dual object in a closed category]]

* [[dualizing object in a closed category]]

* [[Kelly–Mac Lane graph]]

* [[locally cartesian closed category]]

## References

Closed categories were first defined in:

* {#EK65} [[Samuel Eilenberg]], [[G. Max Kelly]], *Closed Categories*, in:  [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Their [[coherence theorem]] was considered in terms of [[Kelly-Mac Lane graphs]] in

* {#KM71} [[Max Kelly]], [[Saunders MacLane]], _Coherence in closed categories_, JPAA 1 (1971), 97-140 ([web](http://www.sciencedirect.com/science/article/pii/0022404971900132))

The relationship to [[monoidal categories]] was studied more deeply in:

* Jan Pavelka, _A note on closed categories_, Commentationes Mathematicae Universitatis Carolinae 17.2 (1976) 261-272. [[10338.dmlcz/105692](https://dml.cz/handle/10338.dmlcz/105692), [pdf](https://dml.cz/bitstream/handle/10338.dmlcz/105692/CommentatMathUnivCarol_017-1976-2_5.pdf)]

They were shown to be equivalent to closed unital [[multicategories]] here:

* Oleksandr Manzyuk, _Closed categories vs. closed multicategories_, [arXiv](http://arxiv.org/abs/0904.3137), 2009
 {#Manzyuk09}, [TAC 26, 2012] (http://www.tac.mta.ca/tac/volumes/26/5/26-05abs.html)

You can get some of the idea from a [post by Owen Biesel at the $n$-Café](http://golem.ph.utexas.edu/category/2009/02/monoidal_closed_categories_and.html#c022133).

LaPlaza's theorem on embedding closed categories in closed monoidal categories is given in

* {#LaPlaza77} Miguel LaPlaza, [_Embedding of Closed Categories Into Monoidal Closed Categories_](http://www.jstor.org/pss/1997823), Trans. AMS, Vol. 233 (1977).

The alternative embedding of closed categories based on promonoidal Day convolution is discussed in

* {#Day74} [[Brian Day]], An embedding theorem for closed categories, _Lecture Notes in Mathematics_ 420 (1974), 55-64.

* {#DayLaPlaza78} B. J. Day and M. L. Laplaza, On Embedding Closed Categories, _Bull. Austral. Math. Soc._ 18 (1978), 357-371.

The latter uses the notion of symmetric closed category, which was first defined in

* {#deSchipper75} W.J. de Schipper, _Symmetric  closed categories_ (Mathematical Centre Tracts,  64.  Mathematisch Centrum, Amsterdam, 1975)

The generalization to a bicategory-like notion ("extension systems") appears in

* {#Street74} [[Ross Street]], *Elementary cosmoi I*, Category Seminar (Proc. Sem., Sydney, 1972/1973), 134–180, Lecture Notes in Math., vol. 420, Springer, Berlin, 1974.

[[!redirects closed categories]]
[[!redirects symmetric closed category]]
[[!redirects symmetric closed categories]]
[[!redirects semiclosed category]]
[[!redirects semiclosed categories]]
[[!redirects semi-closed category]]
[[!redirects semi-closed categories]]
[[!redirects prounital closed category]]
[[!redirects prounital closed categories]]


closed cover

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#### Topology
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[[!include topology - contents]]
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# Contents
* table of contents
{:toc}

## Definition

A **closed cover** of a [[topological space]] $X$ is a collection $\{U_i \subset X\}$ of [[closed subsets]] of $X$ whose [[union]] equals $X$: $\cup_i U_i = X$.

Often
it is also required that every point $x \in X$ is in the [[interior]] of one of the $U_i$ (e.g  in [Floyd 1957](#Floyd57), but e.g. not in [Karoubi & Weibel 2016](#KaroubiWeibel16)).

## Properties
 {#Properties}

Closed covers can be obtained from [[open covers]] by forming the [[closed subset|closure]] of each of the [[open subsets]]. The result clearly satisfies the clause that every point is in the interior of one of the closed subsets.

## Related concepts

* [[open cover]]

* [[schedule]]

## References

### General

* Dragan Janković, Chariklia Konstadilaki, _On covering properties by regular closed sets_, Mathematica Pannonica, 7/1 (1996) 97-111 [[pdf](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/MP/index_elemei/mp07-1/mp07-1-097-111.pdf)]

* {#KaroubiWeibel16} [[Max Karoubi]], [[Charles Weibel]], *On the covering type of a space*, L’Enseignement Mathématique, **62** 3/4 (2016) 457-474 [[arXiv:1612.00532](https://arxiv.org/abs/1612.00532), [doi:10.4171/LEM/62-3/4-4](https://www.ems-ph.org/journals/show_abstract.php?issn=0013-8584&vol=62&iss=3&rank=4)]

Applications of closed covers in [[Čech homology]]:

* {#Floyd57} [[Edwin E. Floyd]], *Closed coverings in Čech homology theory*, Trans. Amer. Math. Soc. **84** (1957) 319-337  [[doi:10.1090/S0002-9947-1957-0087100-2](https://doi.org/10.1090/S0002-9947-1957-0087100-2), [pdf](http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0087100-2/S0002-9947-1957-0087100-2.pdf)]

Related discussion is also in [this MO thread](http://mathoverflow.net/questions/24103/ech-cohomology-of-compact-spaces-via-closed-covers)

### Examples

In [[analytic geometry]], [[affinoid domains]] have [[closed sets]] as [[analytic spectra]] and hence the [[topological space]] underlying a [[Berkovich analytic spaces]] is equipped by a closed cover by affioid domains

* {#Berkovich09} [[Vladimir Berkovich]], _Non-archimedean analytic spaces_, lectures at the _Advanced School on $p$-adic Analysis and Applications_, ICTP, Trieste, 31 August - 11 September 2009 ([pdf](http://www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf))

[[!redirects closed covers]]

[[!redirects closed covering]]
[[!redirects closed coverings]]


closed differential form

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#Contents#
* table of contents
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## Idea

A [[differential form]] $\omega \in \Omega^p(X)$ is called _closed_ if the [[de Rham differential]] $d \colon \Omega^{p}(X) \to \Omega^{p+1}(X)$ sends it to zero: $d \omega = 0$, hence if it is in the [[kernel]] of the de Rham.

A differential form $\omega \in \Omega^{p+1}(X)$ is called _exact_ if it is in the [[image]] of the de Rham differential: $\omega = d \alpha$, for some $\alpha \in \Omega^{p}(X)$.

The [[quotient]] of the vector space of closed differential forms by the exact differential forms of degree $p$ is the _[[de Rham cohomology]]_ of $X$ in degree $p$.

Formalization of closed and co-exact differential forms in [[cohesive homotopy theory]] is discussed at _[[differential cohomology hexagon]]_.

## Properties

* [[Stokes theorem]]

* [[kernel of integration is the exact differential forms]]

[[!redirects closed differential forms]]

[[!redirects exact differential form]]
[[!redirects exact differential forms]]


closed form

A [[differential form]] $\omega \in \Omega^n(X)$ is _closed_ if $d_{dR} \omega = 0$.


closed functor

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#### Monoidal categories
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#Contents#
* table of contents
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## Idea

Just as a [[functor]] is a [[homomorphism]] between [[categories]], so a _closed functor_ is a homomorphism between [[closed categories]].  Like [[monoidal functors]], closed functors come in varying levels of strictness and strength.

## Definition

A __strict closed functor__ is a [[functor]] $F : C \to D$ between [[closed categories]] that preserves all the structure on the nose.  In particular, it must preserve the [[internal homs]] and the unit object:

$$ F(I_C) = I_D ,$$

$$ F([X,Y]) = [F(X),F(Y)].$$

but it should also preserve all of the additional structure transformations. Strict closed functors are not common "in nature" (since their definition violates the [[principle of equivalence]]), but they are sometimes useful to consider in [[syntax|syntactic]] studies of closed categories, as for example in [[coherence theorem|coherence problems]].

More commonly occurring in nature are functors that preserve the closed structure merely up to a transformation, which may perhaps be invertible.  Just as [[lax monoidal functors]] are often called simply "monoidal functors," the "lax" sort of closed functor for which the transformation is not invertible are often called simply *closed functors*, the ones where it is invertible being called *strong*.

To be precise: a **(lax) closed functor** between closed categories $C$ and $D$ is a functor $F\colon C\to D$ together with:

* A [[natural transformation|transformation]] $\hat{F}\colon F([X,Y]_C) \to [F(X),F(Y)]_D$, natural in $X$ and $Y$.

* A morphism $F^0\colon I_D \to F(I_C)$.

which satisfy the following axioms.

* The following [[diagram]] commutes for any $X$.
  $$\array{F(I_C) & \overset{F(j)}{\to} & F([X,X])\\
  ^{\mathllap{F^0}}\uparrow && \downarrow^{\mathrlap{\hat{F}}}\\
  I_D & \underset{j}{\to} & [F(X),F(X)]}$$

* The following diagram commutes for any $X$.
  $$\array{F([I,X]) & \overset{\hat{F}}{\to} & [F(I),F(X)]\\
  ^{\mathllap{F(i^{-1})}}\downarrow && \downarrow^{\mathrlap{[F^0 ,1]}}\\
  F(X) & \underset{i}{\to} & [I,F(X)]}$$

* The following diagram commutes for any $X,Y,Z$.
  $$\array{F([Y,Z]) & \overset{F(L)}{\to} & F([[X,Y],[X,Z]]) & \overset{\hat{F}}{\to} & [F([X,Y]),F([X,Z])]\\
  ^{\mathllap{\hat{F}}}\downarrow &&&& \downarrow^{\mathrlap{[1,\hat{F}]}}\\
  [F(Y),F(Z)] & \underset{L}{\to} & [[F(X),F(Y)],[F(X),F(Z)]] & \underset{[\hat{F},1]}{\to} & [F([X,Y]),[F(X),F(Z)]]}$$

A **strong closed functor** is a closed functor such that $\hat{F}$ and $F^0$ are [[isomorphisms]].

Together with [[closed natural transformations]], closed categories and closed functors form a [[2-category]].

## Examples

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

Suppose that $C$ and $D$ are [[closed monoidal categories]].  Then any [[lax monoidal functor]] $F\colon C\to D$ gives rise to a lax closed functor by defining $F^0$ to be the unit constraint of $F$, and $\hat{F}\colon F([X,Y]) \to [F(X),F(Y)]$ to be the [[adjunct]] under the [[internal-hom]] adjunction of the composite
  $$ F([X,Y]) \otimes F(X) \to F([X,Y] \otimes X) \to F(Y). $$
  Conversely, from a lax closed functor between closed monoidal categories we can recover a [[lax monoidal functor]], with multiplication constraint $F(X)\otimes F(Y) \to F(X\otimes Y)$ being adjunct to the composite
  $$ F(X) \to F([Y,X\otimes Y]) \to [F(Y), F(X\otimes Y)]$$
  where the map $X\to [Y,X\otimes Y]$ is adjunct to the identity of $X\otimes Y$.  In this way, lax monoidal and lax closed functors between closed monoidal categories are in [[bijective]] correspondence.

  Note, however, that for such a functor to be *[[strong monoidal functor|strong monoidal]]* or *strong closed* are generally independent conditions.  Since any lax (or strong) monoidal functor is automatically a lax closed functor, the term **closed monoidal functor** is usually used to mean one which is *strong* closed (or at least for which $\hat{F}$ is an isomorphism).

=--

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

The same idea works more generally for closed unital [[multicategories]], since arbitrary "multifunctors" between multicategories correspond to lax monoidal functors.

=--

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

However, there does not seem to be a natural notion of "colax" functor between closed categories.  One could, of course, simply ask for transformations in the other direction, but such things do not seem to arise much in practice, and would not correspond to colax monoidal functors in the same way.

=--

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

If $C$ and $D$ are [[cartesian closed categories]], then any functor $F\colon C\to D$ is automatically colax monoidal, and it is strong (hence also lax) monoidal iff it preserves [[products]].  By the above argument, any product-preserving functor between cartesian closed categories is automatically a lax closed functor.  If it is moreover a *strong* closed functor, we call it a **[[cartesian closed functor]]**. (See also _[[locally cartesian closed functor]]_.)

=--

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

The [[inverse image]] [[base change]] in an [[etale geometric morphism]] between [[toposes]] is a [[cartesian closed functor]] (see there).

=--

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

An [[adjoint triple]] of functors between [[symmetric monoidal category|symmetric]] [[closed monoidal categories]] is a "[[Wirthmüller context]] of [[six operations]]" when the middle functor is closed monoidal.

=--

+-- {: .num_example}
###### Example
A class of examples of strong closed functors that are not strong monoidal comes from the inclusions of [[reflective subcategory|reflective]] [[exponential ideals]].  [[Day's reflection theorem]] implies that if $C$ is a closed monoidal category and $E\subseteq C$ is a reflective subcategory that is an exponential ideal in the monoidal sense (i.e. $x\in C$ and $y\in E$ imply $[x,y]\in E$), then $E$ is a closed monoidal category with the induced internal-hom and a *reflected* tensor product $x\otimes_E y = R(x\otimes_C y)$ where $R$ is the reflector.  Then the inclusion functor $E\hookrightarrow C$ preserves internal-homs but is only lax monoidal (in contrast to its left adjoint, the reflection, which is strong monoidal).  It doesn't in general preserve the unit strongly, but there are cases when it does.

For instance, let $C = P A$ be a [[presheaf category]] with a [[Day convolution]] monoidal structure induced by a monoidal structure on a small category $A$, and let $E$ be the subcategory of $\Phi$-[[continuous functor|continuous]] presheaves for some set $\Phi$ of colimits in $A$ that are preserved on both sides by the tensor product (e.g. if $A$ is itself closed).  Then $E$ is a reflective exponential ideal, and contains the unit object since the latter is a [[representable presheaf]] (at the [[unit object]] of $A$) hence preserves all colimits.
=--

## Related concepts

* [[indexed closed monoidal category]]

* [[six operations]], [[dependent linear type theory]]

## References

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §I.3 of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

* Geoffrey Lewis, ["Coherence for a Closed Functor"](http://handle.unsw.edu.au/1959.4/63475) UNSW Ph.D. thesis 1974

[[!redirects closed functor]]
[[!redirects closed functors]]
[[!redirects closed monoidal functor]]
[[!redirects closed monoidal functors]]
[[!redirects lax closed functor]]
[[!redirects lax closed functors]]
[[!redirects strong closed functor]]
[[!redirects strong closed functors]]


closed graph theorem

#Contents#
* tic
{:toc}

The closed graph theorem is a fundamental theorem of [[functional analysis]], see

* Wikipedia on the [closed graph theorem](http://en.wikipedia.org/wiki/Closed_graph_theorem)

[[!redirects Closed Graph Theorem]]
[[!redirects Closed Graph Theorems]]
[[!redirects closed graph theorems]]


closed immersion of schemes

#Contents#
* table of contents
{:toc}

## Definition

A [[homomorphism]] $f:X\to Y$ of [[schemes]] is a __closed immersion__ if it is tracked by a [[homeomorphism]] of the underlying [[topological space]] $X$ onto a closed subspace of $Y$, and the [[comorphism]] $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is an [[epimorphism]] of sheaves on $Y$.

More generally, let us consider some category of spaces, i.e. [[sheaves]] of sets on $C = Aff$ equipped with a [[subcanonical Grothendieck topology]]. Then a morphism $F\to G$ of spaces is said to be __closed immersion__ if it is [[representable morphism|representable]] by a [[strict monomorphism]].

## Related concepts

* [[open immersion of schemes]]

## References

* Stacks project [29.2 Closed immersions](https://stacks.math.columbia.edu/tag/01QN) (tag/01QN)

Discussion in the context of [[higher geometry]]/[[higher algebra]] is in

* [[Jacob Lurie]], _[[Closed Immersions]]_

[[!redirects closed immersion]]
[[!redirects closed immersions]]

[[!redirects closed immersions of schemes]]


Closed Immersions

This page collects links related to the article

* [[Jacob Lurie]]

  _Closed Immersions_

  ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-IX.pdf))

on the refinement of the notions of _[[closed immersion of schemes]]_ and of _[[complex analytic space]]s_ from [[geometry]]/[[algebra]] to [[higher geometry]]/[[higher algebra]], specifically to [[E-∞ geometry]]

* [[derived complex analytic space]]

category: reference


closed injections are embeddings

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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #OpenClosedContinuousInjectionsAreEmbeddings}
###### Proposition
**(open/closed continuous injections are embeddings)**

A [[continuous function]] $f \colon (X, \tau_X) \to (Y,\tau_Y)$
which is

1. an [[injective function]]

1. an [[open map]] or a [[closed map]]

is a [[topological embedding]].

=--

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

If $f$ is injective, then the map onto its [[image]] $X \to f(X) \subset Y$ is a [[bijection]]. Moreover, it is still continuous with respect to the [[subspace topology]] on $f(X)$. Now a bijective continuous function is a homeomorphism precisely if it is an [[open map]] or a [[closed map]] (by [this prop.](homeomorphism#HomeoContinuousOpenBijection)). But the image projection of $f$ has this property, respectively, if $f$ does (by [this prop.](open+map#OpenClosedImageProjection)).

=--

## Related concepts

* [[proper maps to locally compact spaces are closed]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[CW-complexes are Hausdorff spaces]]

* [[Hausdorff spaces are sober]], [[schemes are sober]]

* [[continuous image of a compact space is compact]]

* [[closed subsets of compact spaces are compact]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff]]

* [[paracompact Hausdorff spaces are normal]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

[[!redirects open injections are embeddings]]
[[!redirects open/closed injections are embeddings]]
[[!redirects closed/open injections are embeddings]]


closed intervals are compact topological spaces

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

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

For any $a \lt b \in \mathbb{R}$ the [[closed interval]]

$$
  [a,b] \subset \mathbb{R}
$$

regarded with its [[topological subspace|subspace topology]]
of [[Euclidean space]] with its [[metric topology]] is
a [[compact topological space]].
=--

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

Since all the closed intervals are [[homeomorphism|homeomorphic]]
it is sufficient to show the statement for $[0,1]$. Hence let $\{U_i \subset [0,1]\}_{i \in I}$ be an [[open cover]]. We need to show that it has an [open subcover](/nlab/show/compact+space#OpenCover).

Say that an element $x \in [0,1]$ is _admissible_ if the closed sub-interval $[0,x]$ is covered by
finitely many of the $U_i$. In this terminology, what we need to show is that $1$ is admissible.

Observe from the definition that

1. 0 is admissible,

1. if $y \lt x \in [0,1]$ and $x$ is admissible, then also $y$ is admissible.

This means that the set of admissible $x$ forms either

1. an [[open interval]] $[0,g)$

1. or a [[closed interval]] $[0,g]$,

for some $g \in [0,1]$. We need to show that the latter is true, and for $g = 1$. We do so by observing that the alternatives lead to [[contradictions]]:

1. Assume that the set of admissible values were an open interval $[0,g)$. Pick an $i_0 \in I$ such that $g \in U_{i_0}$ (this exists because of the covering property). Since such $U_{i_0}$ is an open neighbourhood of $g$, there is a positive real number $\epsilon$ such that the open ball $B^\circ_g(\epsilon) \subset U_{i_0}$ is still contained in the patch. It follows that there is an element $x \in B^\circ_g(\epsilon) \cap [0,g) \subset U_{i_0} \cap [0,g)$ and such that there is a finite subset $J \subset I$ with $\{U_i \subset [0,1]\}_{i \in J \subset I}$ a finite open cover of $[0,x)$. It follows that $\{U_i \subset [0,1]\}_{i \in J \subset I} \sqcup \{U_{i_0}\}$ were a finite open cover of $[0,g]$, hence that $g$ itself were still admissible, in contradiction to the assumption.

1. Assume that the set of admissible values were a closed interval $[0,g]$ for $g \lt 1$.
   By assumption there would then be a finite set $J \subset I$ such that $\{U_i \subset [0,1]\}_{i \in J \subset I}$
   were a finite cover of $[0,g]$. Hence there would be an index $i_g \in J$ such that $g \in U_{i_g}$.
   But then by the nature of open subsets in the Euclidean space $\mathbb{R}$, this $U_{i_g}$ would also
   contain an open ball $B^\circ_g(\epsilon) = (g-\epsilon, g + \epsilon)$. This would mean that
   the set of admissible values includes the open interval $[0,g+ \epsilon)$, contradicting the assumption.

This gives a [[proof by contradiction]].

=--

[[!redirects closed intervals are compact]]


closed manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[manifold]] is called **closed** if it is

1. [[compact topological space|compact]];

1. without [[boundary]].

In contrast, a manifold without [[boundary]] all whose [[connected components]] are non-compact is called an *[[open manifold]]*.

## Related concepts

* [[manifold with boundary]]

* [[compact hyperkähler manifold]]

* [[cobordism]]

## Literature

See also

* Wikipedia, *[Closed manifold](https://en.wikipedia.org/wiki/Closed_manifold)*

[[!redirects closed manifolds]]


closed map

> This page is about the concept in [[topology]]. For the more general concept see at _[[closed morphism]]_.

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

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #OpenMap}
###### Definition
**([[open maps]] and [[closed maps]])**

A [[continuous function]] $f \colon (X,\tau_X) \to (Y, \tau_Y)$ is called

*  an _[[open map]]_ if the [[image]] under $f$ of an [[open subset]] of $X$ is an open subset of $Y$;

* a _[[closed map]]_ if the [[image]] under $f$ of a [[closed subset]] of $X$ (def. \ref{ClosedSubset}) is a closed subset of $Y$.

=--

## Examples

+-- {: .num_example}
###### Example
**([[image]] projections of [[open map|open]]/[[closed maps]] are themselves open/closed)**

If a [[continuous function]] $f \colon (X,\tau_X) \to (Y,\tau_Y)$ is an [[open map]] or [[closed map]] (def. \ref{OpenMap})
then so its its [[image]] projection $X \to f(X) \subset Y$, respectively, for $f(X) \subset Y$ regarded with its
[[subspace topology]].

=--

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

If $f$ is an open map, and $O \subset X$ is an open subset, so that $f(O) \subset Y$ is also open in $Y$, then, since $f(O) = f(O) \cap f(X)$, it is also still open in the subspace topology, hence $X \to f(X)$ is an open map.

If $f$ is a closed map, and $C \subset X$ is a closed subset so that also $f(C) \subset Y$ is a closed subset, then the [[complement]] $Y \backslash f(C)$ is open in $Y$ and  hence $(Y \backslash f(C)) \cap f(X) = f(X) \backslash f(C)$ is open in the subspace topology, which means that $f(C)$ is closed in the subspace topology.

=--

+-- {: .num_prop #MapsFromCompactSpacesToHausdorffSpacesAreClosed}
###### Proposition
**([[maps from compact spaces to Hausdorff spaces are closed and proper]])**

Let $f \colon (X, \tau_X) \longrightarrow (Y, \tau_Y)$ be a [[continuous function]] between [[topological spaces]] such that

1. $(X,\tau_X)$ is a [[compact topological space]];

1. $(Y,\tau_Y)$ is a [[Hausdorff topological space]].

Then $f$ is

1. a [[closed map]] (def. \ref{OpenMap});

1. a [[proper map]].

=--

+-- {: .num_prop}
###### Proposition
**([[proper maps to locally compact spaces are closed]])**

Let

1. $(X,\tau_X)$ be a [[topological space]],

1. $(Y,\tau_Y)$ a [[locally compact topological space]] according to def. \ref{LocallyCompactSpace},

1. $f \colon X \to Y$ a [[continuous function]].

Then:

If $f$ is a [[proper map]], then it is a [[closed map]].

=--

## Related concepts

* [[closed morphism]]

* [[embedding of topological spaces]]

* [[open map]], [[proper map]]

[[!redirects closed maps]]
[[!redirects closed mapping]]
[[!redirects closed mappints]]


closed map > history

< [[closed map]]


closed midpoint algebra

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

#Contents#
* table of contents
{:toc}

## Idea

The idea of a closed midpoint algebra comes from [[Peter Freyd]].

## Definition

A __closed midpoint algebra__ is a [[cancellative midpoint algebra]] $(M,\vert)$ equipped with two chosen elements $\bot:M$ and $\top:M$.

## Examples

The [[unit interval]] with $a \vert b \coloneqq \frac{a + b}{2}$, $\bot = 0$, and $\top = 1$ is an example of a closed midpoint algebra.

The set of truth values in Girard's [[linear logic]] is a closed midpoint algebra.

## Properties

Every closed midpoint algebra with $\bot = \top$ is [[trivial object|trivial]].

### Partial order

Every closed midpoint algebra has a natural [[partial order]] defined as $a \leq b$ for elements $a$ and $b$ in $M$ if there exists $c$ in $M$ such that $a \vert \top = b \vert c$. Dually, $a \leq b$ for elements $a$ and $b$ in $M$ if there exists $c$ in $M$ such that $c \vert a = \bot \vert b$.

Every closed midpoint algebra [[homomorphism]] is a [[monotone]].

### Function algebras

For any set $S$ and any closed midpoint algebra $M$, the set of functions from $S$ to $M$ is a closed midpoint algebra $(S\to M,\vert_{S\to M},\bot_{S\to M},\top_{S\to M})$, defined by

* $(f\vert_{S\to M} g)(x) = f(x)\vert_M g(x)$ for all $f,g:S\to M$ and $x:M$

* $(\bot_{S\to M})(x) = \bot_M$ for all $x:M$

* $(\top_{S\to M})(x) = \top_M$ for all $x:M$

### Definite integration

Let $M$ and $N$ be [[continuous space|continuous]] closed midpoint algebras and let $M\to_C N$ be the space of [[continuous functions]] from $M$ to $N$. Then there exists an [[operator]] $\int (-)(x) \mathrm{d}x: (M\to_C N)\to (M\to_C N)$ called __[[definite integration]]__ such that

$$\int \top_{M\to N}(x) \mathrm{d}x = \top_N$$

$$\int \bot_{M\to N}(x) \mathrm{d}x = \bot_N$$

$$\int f(x) \vert_N g(x) \mathrm{d}x = \int f(x) \mathrm{d}x \vert_N \int g(x) \mathrm{d}x$$

$$\int f(x) \mathrm{d}x = \int f(\bot_M \vert_M x) \mathrm{d}x \vert_N \int f(x \vert_M \top_M) \mathrm{d}x$$

for all $x$ in $M$.

When $M$ is the [[unit interval]] on the [[real numbers]] $[0,1]$, and $N$ is an interval $[a,b]$ on the real numbers, written in conventional notation the axioms become:

$$\int_{0}^{1} a(x) \mathrm{d}x = a$$

$$\int_{0}^{1} b(x) \mathrm{d}x = b$$

$$\int_{0}^{1} \frac{f(x) + g(x)}{2} \mathrm{d}x = \frac{\int_{0}^{1} f(x) \mathrm{d}x + \int_{0}^{1} g(x) \mathrm{d}x}{2}$$

$$\int_{0}^{1} f(x) \mathrm{d}x = \frac{\int_{0}^{1} f(\frac{0 + x}{2}) \mathrm{d}x + \int_{0}^{1} f(\frac{x + 1}{2}) \mathrm{d}x}{2}$$

## Related concepts

* [[definite integration]]

* [[cancellative midpoint algebra]]

* [[symmetric closed midpoint algebra]]

## References

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))


closed monoidal (infinity,1)-category

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

#Contents#
* table of contents
{:toc}

## Definition

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

A [[symmetric monoidal (∞,1)-category]] $(C,\otimes)$ is **closed** if for each [[object]] $X \in C$ the [[(∞,1)-functor]]

$$
  X \otimes (-) : C \to C
$$

given by forming the [[tensor product]] with $C$ has a [[right adjoint|right]] [[adjoint (∞,1)-functor]]

$$
  (X \otimes(-)\dashv [X,-] ) : C \stackrel{\overset{X \otimes (-)}{\leftarrow}}{\underset{[X,-]}{\to}} C
  \,.
$$

(In cases where the monoidal structure is not assumed symmetric, the property of possessing a right adjoint to tensoring on the left (resp. right) is called *left* (resp. *right*) *closed*, while *closed* is used for the properties jointly (Def. Definition 4.1.1.15 of [[Higher Algebra]]).)
=--

## Examples

* Every [[(∞,1)-topos]] with its structure of a [[cartesian monoidal (∞,1)-category]] is closed. See there for details.

* The [[(∞,1)-category of (∞,1)-modules]] over an [[E-∞ ring]] is closed.

## Related concepts

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

* [[symmetric monoidal category]], [[symmetric monoidal (∞,1)-category]]

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

* [[trace-class map]]

[[!redirects closed monoidal (∞,1)-category]]

[[!redirects closed monoidal (infinity,1)-categories]]
[[!redirects closed monoidal (∞,1)-categories]]